Béla Valek
Modern Physics Handbook I.
General Relativity
Béla Valek Modern Physics Handbook I. General Relativity
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Introduction
Introduction This series presents modern physics through examples and derivations. The organization of topics does not follow the traditional historical approach, it was determined by practical considerations instead. We build up the mathematical framework of the models first, and then completely derive the most important consequences and compare them to recent experimental results. The volumes can be used in the specialized fields as reference materials, and are suitable for self-study in each topic. They may also serve the needs of university lectures as well. The first volume deals with the general theory of relativity. This description has only historical relevance by now, since during the last century, much experimental evidence was found for the consequences of Einstein's theory. It is a classical field, where centuries-old scientific and philosophical ideas got mathematical formulation, and the experimental confirmation. It is important to point out, that the traditional mechanical worldview, that is often considered to be easier to grasp, is in fact an incomplete intellectual achievement. The basic assumptions of general relativity draw from everyday experiences, and the recognition of the curved nature of spacetime follows naturally. It is comparable to the understanding of Earth's spherical shape, and if we are familiar with the mathematical foundations, it does not even require too much imagination. The Reader is assumed to have some basic knowledge in higher mathematics and among the more traditional subjects in physics, but there is only as much mathematical depth in this book, as absolutely necessary. We use the traditional symbols of differential calculus and index notation. The SI system of measurement is used in all physical derivations. Béla Valek
4
Introduction
Contents Introduction......................................................................................................................................4 Figures.............................................................................................................................................8 Observations....................................................................................................................................9 Notation and constants...................................................................................................................10 1. Foundations................................................................................................................................12 1.1 Coordinate systems.............................................................................................................12 1.2 Tensors................................................................................................................................15 1.3 Straight lines.......................................................................................................................17 1.4 Parallel displacement..........................................................................................................20 1.5 Connection and metric tensor..............................................................................................23 1.6 Derivation............................................................................................................................25 1.7 Invariant derivative.............................................................................................................26 1.8 Derivative along a curve.....................................................................................................29 1.9 Curvature.............................................................................................................................32 1.10 Geodesic deviation............................................................................................................36 1.11 Integration.........................................................................................................................38 1.12 Variation and action principle...........................................................................................39 1.13 Runge-Kutta approximation method.................................................................................41 2. Examples....................................................................................................................................43 2.1 Curvature on a two dimensional surface.............................................................................43 2.2 Plain....................................................................................................................................45 2.3 Cylinder...............................................................................................................................46 2.4 Cone....................................................................................................................................47 2.5 Sphere..................................................................................................................................48 2.6 Paraboloid...........................................................................................................................51 2.7 Hyperboloid........................................................................................................................52 2.8 Bolyai surface.....................................................................................................................54 2.9 Catenoid..............................................................................................................................56 2.10 Helicoid.............................................................................................................................58 2.11 Hyperbolic paraboloid.......................................................................................................59 2.12 Torus..................................................................................................................................61 3. Flat and general spacetime.........................................................................................................64 3.1 Proper time..........................................................................................................................64 3.2 Lorentz-transformation.......................................................................................................65 3.3 Addition of velocity and acceleration.................................................................................71 3.4 Aberration of light...............................................................................................................74 3.5 Doppler-effect.....................................................................................................................75 3.6 Sequence of events..............................................................................................................76 3.7 Energy and momentum.......................................................................................................78 3.8 Relativistic rocket...............................................................................................................83 3.9 Faster than light particles....................................................................................................85 3.10 Circular motion and Thomas precession...........................................................................87 3.11 Gravitational redshift.........................................................................................................89 4. Spherically symmetric spacetime..............................................................................................90 4.1 Spherically symmetric coordinate system...........................................................................90 5
Contents 4.2 Schwarzschild coordinates..................................................................................................91 4.3 Geodesic equations...........................................................................................................100 4.4 Gravitational redshift........................................................................................................101 4.5 Wormhole..........................................................................................................................102 4.6 Newtonian approximation.................................................................................................104 4.7 Circular orbit.....................................................................................................................110 4.8 Surface acceleration and hovering....................................................................................113 4.9 Geodesic precession..........................................................................................................116 4.10 Stability of circular orbits................................................................................................119 4.11 Perihelion precession.......................................................................................................121 4.12 Bending of light..............................................................................................................124 4.13 Tides................................................................................................................................129 4.14 Falling orbit.....................................................................................................................134 4.15 Isotropic coordinates.......................................................................................................138 4.16 Gaussian polar coordinates.............................................................................................143 4.17 Rotating Schwarzschild coordinates...............................................................................145 4.18 Kruskal-Szekeres coordinates.........................................................................................149 4.19 Kruskal-Szekeres spacetime............................................................................................156 5. Spacetime of the rotating black hole........................................................................................158 5.1 Axially symmetric spacetime............................................................................................158 5.2 Ernst equation...................................................................................................................162 5.3 The derivation of the Kerr solution...................................................................................167 5.4 Coordinate singularities....................................................................................................174 5.5 Redshift.............................................................................................................................175 5.6 Frame dragging.................................................................................................................175 5.7 Equatorial circular orbit....................................................................................................177 5.8 Kerr-Schild metrics...........................................................................................................179 5.9 Tomimatsu-Sato spacetimes..............................................................................................181 6. Spacetime with matter.............................................................................................................185 6.1 Energy-momentum tensor.................................................................................................185 6.2 Einstein equation with matter...........................................................................................186 6.3 Perfect fluid.......................................................................................................................188 6.4 Spherically symmetric celestial body................................................................................189 6.5 Sphere with constant density.............................................................................................195 6.6 Fall into the centre.............................................................................................................201 6.7 Relativistic dust.................................................................................................................206 6.8 Collapsing spherical dust cloud........................................................................................207 6.9 Electromagnetic interaction..............................................................................................212 6.10 Electromagnetic waves....................................................................................................216 6.11 Unification of interactions...............................................................................................218 6.12 Klein-Gordon equation...................................................................................................224 6.13 Proca equation.................................................................................................................225 6.14 Dirac equation.................................................................................................................226 6.15 Weyl equation..................................................................................................................228 7. Gravitational waves.................................................................................................................229 7.1 Splitting the metric tensor.................................................................................................229 7.2 Examining the metric........................................................................................................231 7.3 Plane wave solutions.........................................................................................................234 6
Contents 7.4 Second order approximation.............................................................................................236 7.5 Examples...........................................................................................................................239 8. Spacetime of the Universe.......................................................................................................243 8.1 Assumptions......................................................................................................................243 8.2 Positive curvature..............................................................................................................244 8.3 Negative curvature............................................................................................................245 8.4 Zero curvature...................................................................................................................246 8.5 Cosmological redshift.......................................................................................................246 8.6 Hubble law........................................................................................................................248 8.7 Flat geometry....................................................................................................................249 8.8 General Friedmann equations...........................................................................................251 8.9 World models....................................................................................................................255 Summary......................................................................................................................................259 Bibliography................................................................................................................................260 Index............................................................................................................................................263
7
Contents
Figures Galilei, Clausius, Maxwell, Einstein, Kaluza.......................................................................................1 plain with rectangular coordinates......................................................................................................45 plain with polar coordinates...............................................................................................................46 cylinder...............................................................................................................................................46 cone with polar coordinates................................................................................................................47 cone with rectangular coordinates......................................................................................................47 sphere with polar coordinates.............................................................................................................48 sphere with rectangular coordinates...................................................................................................49 paraboloid with polar coordinates......................................................................................................51 paraboloid with rectangular coordinates............................................................................................51 single-sheet hyperboloid.....................................................................................................................53 double-sheet hyperboloid with polar coordinates...............................................................................54 double-sheet hyperboloid with rectangular coordinates.....................................................................54 tractroid...............................................................................................................................................56 catenoid...............................................................................................................................................57 helicoid...............................................................................................................................................58 hyperbolic paraboloid.........................................................................................................................59 torus....................................................................................................................................................62 lightcone.............................................................................................................................................65 wormhole with polar coordinates.....................................................................................................104 wormhole with rectangular coordinates...........................................................................................104 acceleration of hovering body..........................................................................................................115 geometric potential...........................................................................................................................121 gravitational lens..............................................................................................................................129 tides caused by the Sun on Earth......................................................................................................131 tides caused by a black hole on a test body......................................................................................131 path of a falling body in proper time................................................................................................136 path of a falling body in coordinate time..........................................................................................138 Kruskal-Szekeres coordinates..........................................................................................................152 complete wormhole in polar coordinates.........................................................................................153 complete wormhole in rectangular coordinates................................................................................153 Kruskal-Szekeres spacetime.............................................................................................................157 longitudinal section of a rotating black hole....................................................................................175 fall inside a planetary body...............................................................................................................205 acceleration inside a planetary body.................................................................................................206 collapsing dust..................................................................................................................................212
8
Observations
Observations Maybe our senses mislead us, but we do not notice that while we sit calmly in a room, Earth is actually speeding around the Sun at nearly 30 km/s. In fact we cannot even tell if we are standing still in the dock or moving on open water, while we are inside an ocean liner. The fact is, that not only us, but (in the ideal case) not even our instruments are able to tell the difference. They may be inaccurate, but it is also possible that there is a principal reason keeping us from telling our absolute velocity. This is the principle of relativity by Galileo Galilei. Our journey in time in one direction from the past to the future, and the ordering between cause and effect, is such a self-evident experience, that it is surprising that we have to explicitly state it as a condition. Serious logical flaws and paradoxes would arise, if it were not so. But from a practical point of view, we cannot tell anything except that we have not yet observed the contrary. This idea gained importance, when Rudolf Clausius recognized entropy, the change of which shows the direction of time. We must keep in mind however, that nothing has been said on how fast time passes, or if it passes in a constant manner. Light is travelling imperceptibly fast to our notions. However already a thousand years ago the Arabic scientist Ibn al-Haytham suggested, that since light is a propagating phenomenon, it may have a finite propagating velocity. We recognize it only at astronomical distances, or with the help of our instruments, having much better reaction time than our naked eyes. It is an important fact that its speed in vacuum is always the same and constant, for all observers, regardless of their motion. The theoretical foundation for this comes from James Clerk Maxwell's equations. We trust this observation so much, that this forms the basis of the definition of meter in the SI metric system. If it were not so, a very fast observer could outrun light, and measure a different value for its velocity, thus measure his/her absolute speed, something impossible, as we believe. Astronaut candidates in an aeroplane on a paraboloidal path (the “vomit comet”) can experience weightlessness for a short period of time. Seats of the visitors in fun park simulators are tipped back to simulate gravity, they feel only their own weights, but they are led to believe that they accelerate. If the simulator would actually move away from its position and not only tilt, the person sitting inside would not be able to tell the difference. There are two indistinguishable phenomena again, let us declare that they are the same, this is Albert Einstein's principle of equivalence. Bodies accelerating under electromagnetic influence behave as if gravitation would act on them, a similar empirical law describes their motion. However, this force depends on whether they have a net charge, moreover, it is not only attracting, but also repelling. Nevertheless, the trajectory of a particle moving in a general electromagnetic and gravitational field may be described by purely geometric means, as shown by Theodor Kaluza. Our statement essentially means, that an instrument does not measure any difference between acceleration under gravitational or electric influence, or the state of weightlessness. The general theory of relativity is based on these observations, and describes the behavior of spacetime, and its interaction with the matter it contains. Thus it creates a framework where all the other physical models can be described.
9
Notation and constants
Notation and constants We use index notation in the entire book. Indices are always single letters, and the following table summarizes how and where they are used: spaces
free indices
summation indices
3D or general space (1…N)
i, j, k, l, m, n
a, b, c, d, e, f
4D spacetime (0…3)
η, κ, μ, ν, ξ, σ
α, β, γ, δ, ε, ζ
5D spacetime (0…4)
P, Q, R, S, T, U
A, B, C, D, E, F
Both sides of the equations must have the same number of free indices, since in fact, we have as many equations as the number of dimensions, multiplied with the number of free indices: i
i
v =a⋅u b
i
1
→
1
1
v =a⋅u b
, v 2 =a⋅u 2b 2 , v 3=a⋅u 3b3
The terms that contain summation indices are summed, as many times as the number of dimensions: N
s=v a⋅u a=∑ v a⋅u a =v 1⋅u 1v 2⋅u 2v 3⋅u3 a=1
The Kronecker delta:
{
i j = 1, i= j 0, i≠ j
The coordinate systems where the quantities are written down, are indicated with a lower-left index. Quantities in different points are always denoted with a different letter, wherever possible. The determinant of a matrix with two indices can be calculated using the following recursive formula: ∣M ij∣=−1a 1⋅M 1 a⋅∣M i≠1
∣
j≠a
Calculating the components of the twice contravariant metric tensor from the twice covariant metric tensor: g kl =
−1k l⋅∣g i≠k ∣g ij∣
∣
j≠l
The derivative and integral of the Lambert function: dW 1 = W x dx e ⋅W x1
∫ W x ⋅dx= x⋅ W x W 1 x −1
Natural constants:
10
C
Notation and constants speed of light:
c=2.99792458⋅108
gravitational constant:
=6.67428⋅10−11
m s
m3 kg⋅s 2
Units of time and distance: 7
Julian year:
1 a=365.25 days=3.15576⋅10 s
astronomical unit:
1 AU =1.49597870691⋅1011 m
lightyear:
1 ly=9.4607304725808⋅10 m
parsec:
1 Pc=3.08567758128⋅10 m 5 =2.06264806245⋅10 AU =3.26156377695 ly
15
16
11
1. Foundations
1. Foundations In this chapter we introduce the mathematical language used by the theory of general relativity. Our goal is to find measurable quantities that we can use to describe multi-dimensional surfaces. This problem is dealth with by differential geometry, and the methods used are similar to those found in geodesy. The word “geometry” itself means “measuring Earth” by which our topic is related to an ancient science that has been cultivated in the antique world with great expertise. While the geodeses survey the Earth's curved surface, we will explore the curved spacetime, but our goals are exactly the same: to orientate, to measure distances, or to find the shortest path between two points, and so on. We are going to see that starting with naïve, down to earth assumptions about space, we can build up a geometry with various properties, where we can identify all the necessary geometric quantities. From this we can draw the lesson, that when we think about “simple” flat space, we apply several unspoken assumptions and limitations, that do not follow automatically from our starting conditions.
1.1 Coordinate systems Let us imagine an arbitrary space, where we identify the points with individual sequences of numbers called vectors, in other words we set up a coordinate system. By definition, we denote each coordinate with an index in the upper-right corner. The i.-th coordinate of point x: xi
(1.1.1)
These sequences of numbers are not exclusive of course, the points in space can be arbitrarily renumbered. If we distribute the numbers according to a different logic, we are setting up another coordinate system, denoted by 2 in the lower left corner, where the coordinates depend on the coordinates in the first coordinate system: xi = 2xi(1xj)
(1.1.2)
2
Let us now assign a number, or otherwise known as a scalar, to every point in space. A scalar field is defined as a function of the points in space, and the value of this so called scalar function is of course independent of the choice of coordinates: i
i
i
(1.1.3)
2 x = 1 x
x
Let us take a look at two different points in a coordinate system: xi
yi
(1.1.4)
The difference between the values of the scalar field in those points is independent from our choice of the coordinate system:
12
1.1 Coordinate systems i
i
i
i
(1.1.5)
1 x − 1 y = 2 x − 2 y
This is also true if we reduce the distance between the points infinitesimally, which is the total derivative: ∂ ∂ ⋅ dx a= ⋅ dx a a 1 a 2 1∂ x 2∂ x
(1.1.6)
However the change in the scalar field along a coordinate depends on the choice of the coordinate system. In fact, we are examining how much the change along a coordinate looks along a coordinate of a different coordinate system. To make the equations less crowded, we omit the 1-index most of the time: Transforming the partial derivative of the scalar field:
∂ ∂ ∂ x a = a⋅ i i ∂ x 2∂ x 2∂ x
(1.1.7)
Transforming the coordinate differential:
i i 2∂ x dx = ⋅dx a 2 a ∂x
(1.1.8)
Their scalar multiple is coordinate system independent. This demonstrates that the transformation formulas are correct, since they behave as expected: a ∂ ∂ ∂ x b 2∂ x ∂ a ⋅ dx = ⋅ ⋅ ⋅dx c = b⋅dx b a 2 b a c ∂ x 2∂ x ∂ x ∂x 2∂ x
(1.1.9)
In the case of infinitesimal displacement, coordinates change in both coordinate systems. The ratios of these changes between the coordinate systems form a square matrix, called the transformation matrix: i ∂ xi i 1∂ x j= ≠ j = j j 1∂ x 2∂ x i
2
(1.1.10)
The quantities that are transformed like the partial derivatives of the scalar field are called covariant vectors, which are also numbering the points in space, just by a different logic. By definition, the index is placed into the lower right corner: ∂ ∂ = a⋅ i a i ∂x 2∂ x
vi =
∂ i ∂x
Covariant vector: a v quantity that transforms like the partial derivatives of the scalar field between coordinate systems: a
∂x =v a⋅i a i 2∂ x
2v i =v a⋅
(1.1.11)
13
1.1 Coordinate systems The quantities that are transformed like coordinate differentials are called contravariant vectors. The position of the index remains unchanged: ∂ xi ⋅dx a 2dx = a ∂x i
2
i
v =dx
i
Contravariant vector: a v quantity that transforms like the coordinate differentials between coordinate systems: ∂ xi a ⋅v =i a⋅v a 2v = a ∂x i
2
(1.1.12)
If we swap the coordinate systems, the reverse transformation formulas are: ∂ xi v= ⋅ v a = ai⋅2v a a 2 2∂ x i
2
v i =2v a⋅
∂ xa
∂ xi
=2v a⋅ ai
(1.1.13)
The reciprocal of the differential transforms like a covariant vector: 1 1 1 ∂ xa 1 1 = ⋅ = ⋅ a = i a⋅ a i i a i a dx 2∂ x dx dx 2dx
(1.1.14)
It follows from above, that the scalar multiple of the covariant and contravariant vectors is also coordinate system independent, and the result is a scalar: a
b
a
c
v ⋅ u =v b⋅a ⋅ c⋅u =v b⋅u
b
(1.1.15)
2 a 2
We perform a double transformation, we write down a vector in a different coordinate system, and then we return to the original: ∂ xi v= ⋅ v a = ai⋅2v a a 2 2∂ x i
i
i
a
b
i
v =b ⋅v ⋅ a= a⋅v
∂ xi =v a⋅i a 2v =v ⋅ a ∂x i
a 2
a
The scalar multiple of the transformation matrices is the Kronecker delta: a ∂ xi 2∂ x a ⋅ j= = ⋅ j a ∂x 2∂ x i
a
i j
(1.1.16)
14
1.2 Tensors
1.2 Tensors The results of vector products belong to the family of tensors. These are special kinds of tensors, because they depend only on n∙m numbers, where m is the number of vectors and n is the number of coordinates: v i⋅u j⋅w k⋅⋅pl⋅q m⋅=T ijk lm
(1.2.1)
On the other hand, a matrix representing such a tensor might have nm independent components, and this is also true about the general tensor. This suggests the following transformation rule: Tensor: a T quantity that transforms in the following way between coordinate systems: 2
T
ijk
i
lm
j
k
= a⋅ b⋅ c⋅⋅T
abc de
d
e
⋅l ⋅m⋅
(1.2.2)
This formula clearly shows, that if all components of the tensor are zero, then this remains so under any coordinate transformation. The rank of a tensor is the number of indices, a scalar is a tensor of zero rank, the vector is a first rank tensor, and so on. The product of arbitrary tensors is also a tensor, for example: Aijk⋅Blm=T ijk lm
(1.2.3)
The sum of tensors is interpreted only if they have the same rank, for example: ij
2
ij
i
j
ab
ab
c
(1.2.4)
A k 2 B k = a⋅ b⋅ A c B c ⋅ k
Let us multiply an arbitrary tensor with vectors in such a way, that we perform a summation for every index. According to the definition above, the result has to be an invariant scalar: T abc de⋅v a⋅u b⋅w c p d⋅qe =s
(1.2.5)
If we do not know the nature of a quantity having multiple indices, the formula above can determine if it is a tensor. Because if we write it down in a different coordinate system, then the transformation matrices will cancel out only, if the quantity with multiple indices deploys the same number of transformation matrices, as there are for transforming the vectors. The value of the Kronecker delta is one in the case of corresponding indices, and zero if the indices are different. According to the formula above its a special kind of tensor, whose components are separately invariant: ab⋅v a⋅ub =v b⋅ub=s
(1.2.6)
Covariant vectors can also be used to number the points in space, these sequences of numbers are made of covariant coordinates: xi
(1.2.7)
15
1.2 Tensors In fact, this is just another coordinate system, one of many possibilities, therefore the contravariant coordinates of a point certainly can be transformed into covariant ones. Therefore let the other coordinate system be the covariant one, thus the mutual ratios of the coordinate changes form a symmetric gij quantity: i
∂ xi
∂xj
∂x j 1∂ x
→
g ij =
i i a 2∂ x v =v ⋅ 2 ∂ xa
→
∂x v i =v a⋅ ai =v a⋅g ia ∂x
→
g ij =
→
∂x v i =v a⋅ =v a⋅g ai ∂ xi
2
∂x
j
=
∂ xi
=g ji
(1.2.8)
(1.2.9)
In the reverse direction: ∂ xi j 2∂ x 1
∂ xa i 2∂ x
∂ xi ∂xj = =g ji ∂xj ∂ xi
(1.2.10)
a
2v i =v a⋅
(1.2.11)
The scalar product of the contravariant and covariant representation of the vector is a scalar – the length of the vector – therefore our new quantity is a tensor: v a⋅v a =v a⋅v b⋅g ab=v b⋅v a⋅g ba
(1.2.12)
It is called the metric tensor, and it can raise and lower indices: d
e
ai
bj
ck
i
j
k
v a⋅u b⋅w c⋅⋅p ⋅q ⋅⋅g ⋅g ⋅g ⋅⋅g dl⋅g em⋅=v ⋅u ⋅w ⋅⋅pl⋅q m⋅ de
ai
bj
ck
T abc ⋅g ⋅g ⋅g ⋅⋅g dl⋅g em⋅=T
ijk lm
(1.2.13)
We perform a double transformation, we write down a contravariant vector in covariant form, then we transform it back into contravariant form: i
a
bi
a
i
v =v ⋅g ab⋅g =v ⋅a
The scalar multiple of the metric tensors is the Kronecker delta: g ia⋅g aj =ij =
∂ xi ∂ x a ⋅ ∂xa ∂ x j
(1.2.14)
If we perform a summation on all indices, the result is the number of dimensions: ab
b
(1.2.15)
g ab⋅g =b =N
16
1.2 Tensors An arbitrary tensor can always be split into the sum of a symmetric and an antisymmetric tensor. In the case of two indices, it is possible to create a symmetric tensor with the averaging of the opposing tensor components: sz
1 1 T ij = ⋅T ij T ji = ⋅T ji T ij =sz T ji 2 2
(1.2.16)
Subtracting it from the general tensor, the result is an antisymmetric tensor: an
1 1 1 T ij =T ij − sz T ij =T ij − ⋅T ij T ji = ⋅T ij −T ji =− ⋅T ji −T ij =−anT 2 2 2
The diagonal elements are zeroes:
an
1 T ii = ⋅T ii −T ii =0 2
ji
(1.2.17) (1.2.18)
Their sum recreates the original tensor:
sz
1 1 T ij anT ij = ⋅T ij T ji ⋅T ij −T ji =T ij 2 2
(1.2.19)
Reversing the indices of a general tensor: T ij =T ji 2⋅anT ij
(1.2.20)
1.3 Straight lines In an arbitrary space, a curve with a constant tangent vector is the closest thing to a straight line. The change of coordinates with respect to an invariant quantity is the following: ∂ xi u= ∂ i
(1.3.1)
Let it be the tangent vector of the straight line. We write it down in another coordinate system: ∂ x i 2∂ x i ∂ x a = ⋅ ∂ ∂ x a ∂
2
/
∂ ∂
(1.3.2)
The formula which describes that it is not changing, that the derivative is zero, is the equation of the straight line: 2 i i ∂2 xi ∂ x a ∂ x b 2∂ x ∂ 2 x a 2∂ x = ⋅ ⋅ ⋅ =0 ∂2 ∂ x a⋅∂ x b ∂ ∂ ∂ x a ∂2
2
i c : 17
∂xj i 2∂ x
/⋅
1.3 Straight lines 2 c c ∂ x j 2∂ x ∂ x a ∂ x b ∂ x j 2∂ x ∂ 2 x a ⋅ ⋅ ⋅ ⋅ ⋅ 2 =0 c a b c a ∂ ∂ ∂ x ∂ x ⋅∂ x ∂ x ∂ x ∂ 2 2 2
c
We introduce the connection in the first term:
j ∂ x ∂x ab= ⋅ 2a c b 2∂ x ∂ x ⋅∂ x
The index changes in the second term:
j 2 a 2 a 2 j ∂ x 2∂ x ∂ x ∂ x j ∂ x ⋅ ⋅ = ⋅ = a c a 2 2 2 ∂ x ∂ ∂ ∂ 2∂ x
j
(1.3.3)
c
The geodesic equation: ∂2 x j ∂ xa ∂ xb j ⋅ ⋅ =0 ab ∂ ∂ ∂ 2
(1.3.4)
We write down the geodesic equation in two different coordinate systems: 2
i
a
a b ∂2 x i i 2∂ x 2∂ x 2 ab⋅ ⋅ =0 ∂ ∂ ∂ 2
b
∂ x ∂ x ∂x i ab⋅ ⋅ =0 2 ∂ ∂ ∂
2
(1.3.5)
The transformation of the first and second derivative with respect to the invariant: ∂ x i 2∂ x i ∂ x a = ⋅ ∂ ∂ x a ∂
2
2 i i a b 2 a ∂2 xi ∂ x ∂ x 2∂ x ∂ x 2∂ x = ⋅ ⋅ ⋅ ∂2 ∂ x a⋅∂ x b ∂ ∂ ∂ x a ∂2
2
(1.3.6)
Insert them into the geodesic equation: a c
bd 2
i
i
a
b
c d 2 c c d ∂ x ∂ x 2∂ x ∂ x ∂x i 2∂ x 2∂ x ∂ x ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ =0 2 ab c d c 2 c d ∂ x ⋅∂ x ∂ ∂ ∂ x ∂ ∂ x ∂ ∂x ∂ 2
∂ x
Rearrange the formula and multiply with a new factor:
2 i a b ∂ x i ∂2 x c ∂ xc ∂ xd i 2∂ x 2∂ x 2∂ x ⋅ ⋅ ⋅ ⋅ ⋅ =0 2 ab c 2 c d c d ∂ ∂ ∂ x ∂ ∂ x ⋅∂ x ∂ x ∂x
2
i e
Write down the change in the first term in detail: 18
∂xj i 2∂ x
/⋅
(1.3.7)
1.3 Straight lines 2 c ∂ x e ∂ x j ∂2 x c ∂2 x j j ∂ x ⋅ ⋅ = ⋅ = c ∂ x c 2∂ x e ∂ 2 ∂ 2 ∂ 2
2
Let us reinsert it:
2 e a b ∂2 x j ∂xj ∂ x j ∂ xc ∂ xd e 2∂ x 2∂ x 2∂ x ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ =0 2 ab ∂ 2 ∂ x c⋅∂ x d 2∂ x e ∂ x c ∂ x d 2∂ x e ∂ ∂
(1.3.8)
Our result is in the form of an equation of a straight line. We can recognize the connection, from which we identify the transformation rule: c i
d k
a b j j ∂x ∂x e 2∂ x 2∂ x ik = i ⋅ 2 ab⋅ i ⋅ k ⋅ e k e ∂ x ⋅∂ x 2∂ x ∂ x ∂ x 2∂ x j
2
∂2 x e
(1.3.9)
The connection is therefore not a tensor-like quantity. The symmetric part of the general connection: 1 C k ij = ⋅ kij k ji 2
(1.3.10)
The antisymmetric part of the general connection is the torsion: 1 S k ij = ⋅ kij − k ji 2
k
k
ij = ji 2⋅S
k ij
(1.3.11)
We insert the transformation law of the connection:
a b 2 e b a ∂2 x e ∂ x k 1 ∂xk ∂ xk ∂ xk e 2∂ x 2∂ x 2∂ x e 2∂ x 2∂ x S k ij = ⋅ 2 i ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅ 2 ba 2 ∂ x ⋅∂ x j 2∂ x e 2 ab ∂ x i ∂ x j 2∂ x e ∂ x j⋅∂ x i 2∂ x e ∂ x j ∂ x i 2∂ x e
Simplify, the transformation of the torsion: ∂ xa ∂ xb ∂ xk 1 S k ij = ⋅ 2 e ab−2 e ba ⋅2 i ⋅2 j ⋅ 2 ∂ x ∂ x 2 ∂ xe
(1.3.12)
Judging from the transformation law of the torsion, it is a tensor with three indices: S k ij =2 S kij⋅ai⋅b j⋅ e k
(1.3.13)
The variations of the connection transform like tensors:
19
1.3 Straight lines j
j
j
first coordinate system:
ik = ik x − ik x x
second coordinate system:
2
j
j
j
ik =2 ik 2 x −2 ik 2 x 2 x
(1.3.14)
Insert into the first equation the transformation of the connection: j
ik = 2 e a b 2 e a b ∂xj ∂xj ∂xj ∂xj 2∂ x e 2∂ x 2∂ x 2∂ x e 2∂ x 2∂ x ⋅ x ⋅ ⋅ ⋅ − ⋅ − x x ⋅ ⋅ ⋅ 2 ab 2 ab ∂ x i⋅∂ x k 2∂ x e ∂ x i ∂ x k 2∂ x e ∂ x i⋅∂ x k 2∂ x e ∂ x i ∂ x k 2∂ x e a b ∂ x a 2∂ x b ∂ x j ∂xj e 2∂ x 2∂ x ik =2 ab x ⋅ i ⋅ k ⋅ e −2 ab x x⋅ i ⋅ k ⋅ ∂ x ∂ x 2∂ x ∂ x ∂ x 2∂ x e j
e
2
(1.3.15)
The variation of the connection is a tensor-like quantity: a
b
∂ x 2∂ x ∂ x j ik =2 ik⋅ i ⋅ k ⋅ e ∂ x ∂ x 2∂ x j
j
2
(1.3.16)
1.4 Parallel displacement Let us take a look at two infinitesimally close points in space, where we define vectors in each of them, and describe it with two different coordinate systems at the same time. In order to avoid confusion, we summarize the notation here: First coordinate system
Second coordinate system
First point
xi
2
Second point
yi
2
Vectors in the first point
vi, wi
2
Vectors in the second point
ui, qi
2
xi yi vi ui
We parallel transfer a vector from the first point to the second, and we define the first coordinate system in such a way, that every components of the new vector are identical to the old one: u i=vi
(1.4.1)
The change in the second coordinate system is more general: i
2
i
u = 2v 2dv
i
(1.4.2)
We transform the coordinates of the vectors in the following way:
20
1.4 Parallel displacement i
∂x v= ⋅ va a 2 2∂ x i
i
∂y u= ⋅ ua a 2 2∂ y i
(1.4.3)
Let us insert it into the formula written down in the first coordinate system: ∂ xi ∂ yi a ⋅ v = ⋅ ua a 2 a 2 2∂ x 2∂ y
(1.4.4)
The difference of the two points is the total derivative, where we now differentiate with respect to the coordinates of the second coordinate system: y i= xi
∂ xi ⋅ dx a a 2 2∂ x
/
∂ j 2∂ x
∂ yi ∂ xi ∂2 x i = ⋅ dx a j j j a 2 2∂ y 2∂ x 2∂ x ⋅2∂ x
(1.4.5)
Substitute it, and then expand the parentheses:
∂ xi ∂ xi ∂2 x i a ⋅ v = ⋅ dx a ⋅ 2v b2 dv b a 2 b b a 2 2∂ x 2∂ x 2∂ x ⋅2∂ x
(1.4.6)
∂ xi ∂ xi ∂ xi ∂2 x i ∂2 x i a b b a b ⋅v = ⋅v ⋅ dv ⋅ dx ⋅2 v ⋅ dx a⋅2dv b a 2 b 2 b 2 b a 2 b a 2 2∂ x 2∂ x 2∂ x 2∂ x ⋅2∂ x 2∂ x ⋅2∂ x We simplify and then ignore the last term, where the infinitesimally small quantities are on a higher power: 0=
∂ xi ∂2 x i b ⋅ dv ⋅ dx a⋅2v b b 2 b a 2 2∂ x 2∂ x ⋅2∂ x
∂xj /⋅2 i ∂x
i c j ∂ x j ∂ xc ∂2 x c b 2∂ x ⋅ ⋅ dv =− ⋅ ⋅ dx a⋅2v b c b 2 c b a 2 ∂ x 2∂ x ∂ x 2∂ x ⋅2∂ x
2
(1.4.7)
On the left side of the equation, we substitute and apply the Kronecker delta, on the right side we substitute the connection: j
2 c ∂x ∂ x = ⋅ 2 ba c b a ∂ x 2∂ x ⋅2∂ x j
2
j
2
j
a
dv =−2 ba⋅2 dx ⋅2 v
(1.4.8)
b
21
1.4 Parallel displacement We reinsert this into the parallel transfer formula of the contravariant vector, without the identification numbers of the coordinate systems. It is possible to swap the lower indices, we do so to have the same index convention that is used in a later definition of the connection: i
i
i
a
u =v − ba⋅v ⋅dx
b
(1.4.9)
We take the scalar product of two vectors in the first point. If we parallel transfer them into the second point, the resulting scalar does not change: a
a
(1.4.10)
v ⋅w a =u ⋅qa
The parallel displacement of the contravariant vector: i
i
i
i
i
a
u =v dv =v − ba⋅v ⋅dx
b
(1.4.11)
The parallel transfer of the covariant vector: q i=widw i
(1.4.12)
Substitute them into the scalar product: v a⋅w a = v a − adc⋅v c⋅dx d ⋅ wa dw a
(1.4.13)
v a⋅w a =v a⋅w av a⋅dwa − adc⋅v c⋅dx d⋅w a− adc⋅v c⋅dx d⋅dwa Simplify and omit the last term, where the infinitesimal quantities are on a higher power: a
b
dw i= bi⋅dx ⋅w a
We reinsert this into the parallel displacement of the covariant vector: q i=wi abi⋅w a⋅dx b
(1.4.14)
Based on these, we can determine the parallel transfer formula of any tensor. We are already not following the notation introduced in the table earlier: v i⋅u j⋅w k⋅⋅pl⋅q m⋅=T ijk lm We substitute the parallel transfer formulas, and neglect the terms containing higher powers of infinitesimal quantities. Therefore only the products of vectors remain, and those terms, that contain the connection only once:
v i − i ba⋅v a⋅dx b ⋅ u j− jba⋅ua⋅dx b ⋅ w k− kba⋅wa⋅dx b ⋅⋅ p l abl⋅p a⋅dx b ⋅ q m abm⋅qa⋅dx b ⋅= T ijk lm− i ba⋅T ajk lm− jba⋅T iak lm− k ba⋅T ijalm− abl⋅T ijk am abm⋅T ijk la⋅dx b (1.4.15) 22
1.4 Parallel displacement
1.5 Connection and metric tensor Like in the previous chapter, we take the scalar product of two vectors in the first point. If we parallel transfer them into the second point, the resulting scalar does not change: a
b
a
g ab x⋅v ⋅w = g ab y ⋅u ⋅q
b
(1.5.1)
We wrote on the left side of the equation the scalar product in the first point, and on the right side the scalar product in the second point. We approach a metric tensor from another with a Taylor series: ∂ g ij x a 1 ∂ 2 g ij x g ij y =g ij x ⋅dx ⋅ a ⋅dx a⋅dx b a b 2 ∂ x ⋅∂ x ∂x
(1.5.2)
We substitute the parallel displacement formula for vectors and the Taylor series of the metric tensor, neglecting the terms containing products of infinitesimal quantities:
g ab x⋅v a⋅wb= g ab x
∂ g ab x c ⋅dx ⋅ v a− adc⋅v c⋅dx d ⋅ wb− bdc⋅wc⋅dx d c ∂x
(1.5.3)
Let us simplify and neglect the higher order terms of infinitesimal quantities again, and then step by step rewrite the summation indices into free indices. We must take care about what indices belongs to what factors: 0=
0=
0=
∂ g ab
1 ⋅dx c⋅v a⋅wb− g ab⋅v a⋅ bdc⋅w c⋅dx d − g ab⋅ adc⋅v c⋅dx d⋅w b /⋅ k ∂x dx c
∂ g ab ∂x
k
⋅v a⋅w b−g ab⋅v a⋅ bkc⋅w c −g ab⋅ akc⋅v c⋅wb
1 /⋅ i v
∂ g ib b 1 ⋅w −g ib⋅ bkc⋅wc − g ab⋅ aki⋅w b /⋅ j k ∂x w
g ia⋅ akjg aj⋅ aki =
∂ g ij
(1.5.4)
∂ xk
Using this result it can be shown, that the parallel displacement of the metric tensor transforms it into the local metric tensor of the destination, thus we are recovering the Taylor series: g ij y =g ij x abi⋅g aj x abj⋅g ia x ⋅dx b =g ij x
23
∂ g ij x b ⋅dx ∂ xb
(1.5.5)
1.5 Connection and metric tensor Permute the indices: ∂ g ij
(1)
g ia⋅ akjg aj⋅ aki =
(2)
g ja⋅ aik g ak⋅ aij =
(3)
g ka⋅ a ji g ai⋅ a jk =
∂ xk ∂ g jk ∂ xi ∂ g ki ∂x
(1.5.6)
j
Summarize the equations in the following way: (1) + (2) – (3): g ia⋅ akjg aj⋅ aki g ja⋅ aik g ak⋅ aij − g ka⋅ a ji −g ai⋅ a jk =
∂ g ij ∂x
k
∂ g jk ∂x
−
i
∂ g ki ∂xj
(1.5.7)
Reorder the connections according to their lower indices: g ia⋅ akj − a jk g aj⋅ aki aik g ak⋅ aij − a ji =
∂ g ij ∂x
k
∂ g jk ∂x
i
−
∂ g ki ∂x
j
1 /⋅ 2
Arrange the antisymmetric terms to the right side:
1 ∂ g ij ∂ g jk ∂ g ki g aj⋅C aki= ⋅ − − g ia⋅S akj −g ak⋅S aij k i j 2 ∂x ∂x ∂x
/⋅g jb
(1.5.8)
Apply the Kronecker-delta on the left side:
∂ g ib ∂ g bk ∂ g ki 1 C jki = ⋅g jb⋅ − b −g jb⋅g ia⋅S akb− g jb⋅g ak⋅S aib /S jki k i 2 ∂x ∂x ∂x
(1.5.9)
The general connection is the sum of the symmetric and antisymmetric parts:
∂ g ib ∂ g bk ∂ g ki 1 jki = ⋅g jb⋅ − −g jb⋅g ia⋅S akb− g jb⋅g ak⋅S aib S jki k i b 2 ∂x ∂x ∂x
(1.5.10)
If we use a symmetric connection since the beginning, our formula gives the relationship between the connection and the metric tensor:
∂ g ia ∂ g ak ∂ g ki 1 jki = ⋅g ja⋅ − 2 ∂ xk ∂ xi ∂ xa
(1.5.11)
The infinitesimal surrounding of every point can be approximated with a flat space, where it is possible to set up a coordinate system, where the partial derivative of the metric tensor is zero. In 24
1.5 Connection and metric tensor other words, we approach the point with a Taylor series, where the first derivative is zero, but others are not. In this case the symmetric connection is zero, its partial derivative however is not: ∂2 g ij ≠0 ∂ x l⋅∂ x k
∂ g ij ∂x
=0 , k
i jk =0 ,
→
∂ i jk ≠0 ∂ xl
(1.5.12)
1.6 Derivation First in order to deduce the transformation rule of the second partial derivative, we differentiate both sides of the transformation formula of the partial derivative of the scalar function: ∂ ∂ ∂ x a = a⋅ i i ∂ x 2∂ x 2∂ x 2
/ 2
∂ j 2∂ x a
2
a
∂ ∂ ∂x ∂ ∂ x = ⋅ a⋅ j i j a i j i ∂ x 2∂ x ⋅2∂ x 2∂ x ⋅2∂ x 2∂ x ⋅∂ x 2∂ x
(1.6.1)
In the first term of the right side, we transform one of the denominator differential from the second to the first coordinate system: 2
2
b
a
2
a
∂ ∂ ∂x ∂x ∂ ∂ x = b ⋅ ⋅ a⋅ j i a j i j i ∂ x ⋅∂ x 2∂ x 2 ∂ x ∂ x 2∂ x ⋅2∂ x 2∂ x ⋅2∂ x
(1.6.2)
The second partial derivative of the scalar field, that is the partial derivative of the covariant vector, does not transform like a tensor. Substitute it to the formula above: vi =
∂ ∂ xi
2
∂ vi
2
∂ vi
∂v a ∂ x b ∂ x a ∂2 x a = b⋅ ⋅ v a⋅ j j i j i ∂ x 2∂ x 2∂ x 2∂ x 2∂ x ⋅2∂ x
2
∂x
(1.6.3)
2 a ∂ va ∂ xb ∂ xa ∂ x = ⋅ ⋅ i j i b j ∂ x ⋅ ∂ x ∂ x ∂ x 2 2 2 2∂ x
−v a⋅ j
(1.6.4)
In the left side, we transform the vector in the second term, so it would be written in the same coordinate system as the partial derivative in the first term. The transformation rule of the derivative of the covariant vector in the general case: b ∂ vi ∂ va ∂ xb ∂ xa ∂2 x a 2∂ x − v ⋅ ⋅ = ⋅ ⋅ i 2 b j a j i b j ∂ x ∂ x ∂ x ⋅ ∂ x ∂ x ∂ x 2 2 2 2 2∂ x 2
25
(1.6.5)
1.6 Derivation This quantity does not transform like a tensor. But if we switch the indices in it and subtract it from the original expression, we get a tensor-like quantity called rotation. The right side of the equation:
∂ va ∂ xb ∂ xa ∂ vb ∂ x a ∂ xb ∂ va ∂ vb ∂ xb ∂ xa ⋅ ⋅ − ⋅ ⋅ = − a ⋅ ⋅ =T ab⋅ jb⋅i a b j i a i j b j i ∂ x 2∂ x 2∂ x ∂ x 2 ∂ x 2 ∂ x ∂ x ∂ x 2∂ x 2∂ x The left side of the equation:
∂v i
2
∂ xb
b ∂v ∂v ∂2 x a ∂2 x a 2∂ v j 2∂ x − v ⋅ ⋅ − − v ⋅ ⋅ = 2 ij − 2 ij =2T ij 2 b 2 b j a j i i a i j ∂ x 2∂ x ⋅2∂ x ∂ x 2∂ x ⋅2∂ x 2∂ x 2∂ x 2∂ x 2∂ x 2
The transformation rule of the rotation corresponds to tensors of second rank: b
a
(1.6.6)
T ab⋅ j ⋅i =2T ij
This tensor is antisymmetric: T ij =−T ji =
∂v i ∂x
j
−
∂v j ∂x
i
=−
∂v j ∂x
i
−
∂ vi ∂ xj
(1.6.7)
T ii =
Its diagonal elements are always zero:
∂ vi ∂x
i
−
∂ vi ∂ xi
(1.6.8)
=0
If we cyclic permute the indices of the partial derivative of the antisymmetric tensor and add them together, the result is zero, because the second derivatives of the vectors cancel out: ∂T ij ∂x
k
∂ ∂ vi ∂ v j ∂ ∂ v j ∂v k ∂ ∂ vk ∂ vi − i i − j − = k j k j ∂x ∂x ∂x ∂ x ∂x ∂x ∂x ∂x ∂ x ∂ xi ∂ x k ∂2 v i ∂2 v j ∂2 v j ∂2 v k ∂2 v k ∂2 v i − − − =0 ∂ x k⋅∂ x j ∂ x k⋅∂ x i ∂ x i⋅∂ x k ∂ x i⋅∂ x j ∂ x j⋅∂ x i ∂ x j⋅∂ x k
∂T jk i
∂T ki j
=
(1.6.9)
1.7 Invariant derivative Let us assume, that a covariant vector field is constant in a given coordinate system, its partial derivative is zero everywhere: ∂ vi ∂x
j
(1.7.1)
=0
Rewrite this formula in another coordinate system, utilizing the results of the previous chapter:
26
1.7 Invariant derivative b
2 a ∂ v a ∂ x b ∂ x a 2∂ v i ∂ x 2∂ x ⋅ ⋅ = − v ⋅ ⋅ =0 2 b b j i j a j i ∂ x 2∂ x 2∂ x 2 ∂ x ∂ x 2∂ x ⋅2∂ x
(1.7.2)
Insert the connection in the second term: b ∂2 x a b 2∂ x = ⋅ 2 ji ∂ x a 2∂ x j⋅2∂ x i
(1.7.3)
This formula describes in an arbitrary coordinate system, that the partial derivative of the vector field vanishes in the original coordinate system: 2 2
∂ vi
∂x
j
− 2v b⋅2 b ji =0
(1.7.4)
Let us define the invariant derivative of the covariant vector: ∇ j vi =
∂ vi ∂x
j
−v b⋅ b ji
(1.7.5)
Connection: a Γ quantity, that makes sure, that in the case of a coordinate transformation, the invariant derivative transforms like a tensor: ∇ j vi =
∂ vi ∂x
j
−v a⋅ a ji
2
∇ j 2 v i=∇ b v a⋅ jb⋅ i a
(1.7.6)
Transformation of the invariant derivative, substitute the transformation rules of the partial derivative of the covariant vector and the connection: 2
∂ vi
j 2∂ x
− 2v d⋅2 d ji =
d d 2 a c 2 e b a ∂ va ∂ xb ∂ xa ∂ x ∂x ∂ x ∂ x ∂ x 2∂ x e 2∂ x ⋅ ⋅ v a⋅ −v c⋅ ⋅ ⋅ e ba⋅ ⋅ ⋅ e = j i d j i j i ∂ x b 2∂ x j 2∂ x i ∂x ∂x 2∂ x ⋅2∂ x 2∂ x 2∂ x ⋅2∂ x 2∂ x 2∂ x d
d
2 a c 2 e c b a ∂ va ∂ xb ∂ xa ∂ x ∂x ∂ x ∂x ∂ x ∂ x 2∂ x 2∂ x e ⋅ ⋅ v ⋅ −v ⋅ ⋅ ⋅ −v ⋅ ⋅ ⋅ ⋅ ⋅ e= a c c ba b j i j i d j i e d j i ∂ x 2∂ x 2∂ x ∂ x ⋅ ∂ x ∂ x ∂ x ⋅ ∂ x ∂ x ∂ x ∂ x ∂ x ∂x 2 2 2 2 2 2 2 2
We recognize the Kronecker-delta in the third and fourth terms: ∂ va ∂ xb ∂ xa ∂2 x a ∂2 x e ∂ xb ∂ xa c c e ⋅ ⋅ v ⋅ −v ⋅ ⋅ −v ⋅ ⋅ ⋅ ⋅ i= a c e c e ba j i j i j ∂ x b 2∂ x j 2∂ x i 2∂ x ⋅2∂ x 2∂ x ⋅2∂ x 2∂ x 2∂ x The second and the third terms cancel out, we pull out the transformation matrices from the first and 27
1.7 Invariant derivative fourth terms:
∂ va ∂ xb ∂ xa ∂ va ∂ xb ∂ xa ∂ xb ∂ xa c c ⋅ ⋅ −v ⋅ ⋅ ⋅ = −v ⋅ ⋅ ⋅ i c ba c ba j i j ∂ x b 2∂ x j 2∂ x i ∂ xb 2∂ x 2∂ x 2∂ x 2∂ x 2 2
∂ vi
∂x
− 2v d⋅2 d ji = j
∂ va ∂x
∂ xb ∂ xa ⋅ j i 2∂ x 2∂ x
−v c⋅ c ba ⋅ b
(1.7.7)
The partial derivative of a scalar field is a tensor, therefore it coincides with the invariant derivative: ∇ i =
∂ ∂ xi
(1.7.9)
The scalar product of the covariant and the contravariant vector is a scalar: a
∇ i u ⋅v a =
∂u a⋅v a ∂ xi
(1.7.10)
a ∂v ∂u ∇ i u ⋅v au ⋅∇ i v a = i ⋅v aua⋅ ai ∂x ∂x a
a
∂v ∂v ∂u a ∇ i u a⋅v au a⋅ ai −u a⋅v b⋅ bia= i ⋅v aua⋅ ai ∂x ∂x ∂x ∇ i u a⋅v a=
∂ ua 1 ⋅v a u a⋅v b⋅ bia /⋅ i v ∂x a
The invariant derivative of the contravariant vector: ∇ i u j=
∂uj u a⋅ jia i ∂x
(1.7.11)
Now we can determine the invariant derivative of arbitrary tensors, using the products of contravariant and covariant vectors: ∇n T
ijk
i
∇n T
ijk lm
j
k
=∇ n v ⋅u ⋅w ⋅⋅p l⋅qm⋅
lm
i
j
k
i
j
k
=∇ n v ⋅u ⋅w ⋅⋅pl⋅q m⋅v ⋅∇ n u ⋅w ⋅⋅p l⋅q m⋅
∇ n T ijk lm=
∂v i j k ⋅u ⋅w ⋅⋅p l⋅qm⋅v a⋅ i na⋅u j⋅w k⋅⋅pl⋅q m⋅ ∂ xn
Group together the vectors on the right side:
28
1.7 Invariant derivative ∇ n T ijk lm= ∂T ijk lm i na⋅T ajk lm jna⋅T iak lm k na⋅T ijalm− anl⋅T ijk am − anm⋅T ijk la − n ∂x (1.7.12) It is possible for certain quantities to serve as the connection, that cannot be formulated like the previously defined version, therefore we have to spend some time with the properties of the general case. The rotation, using the invariant derivative: ∇ j v i −∇ i v j=
∂ vi ∂x
j
−
∂vj ∂x
i
−v b⋅ b ji − bij =
∂v i ∂x
j
−
∂v j ∂ xi
(1.7.13)
This equation is satisfied only if the connection is symmetric: k ij = k ji
(1.7.14)
The invariant derivative of the Kronecker-delta: ∇ k ij =
∂ij iak⋅ aj− ajk⋅ia =0 ijk − ijk =0 k ∂x
(1.7.15)
The formula for the invariant derivative of the metric tensor is the same, that we permuted three times during the derivation of the formula for the relationship between the connection and the metric tensor, and its zero: ∇ k g ij =
∂ g ij ∂x
k
−g ia⋅ akj −g aj⋅ aki =0
(1.7.16)
The invariant derivative of the twice contravariant metric tensor: ia
i
∇ k g ⋅g aj =∇ k j=0 ia
ia
ia
ia
∇ k g ⋅g aj g ⋅∇ n g aj =0 ∇ k g ⋅g aj g ⋅0=0
Since the twice covariant metric tensor is not zero in the general case, the invariant derivative of the twice contravariant metric tensor has to be zero: ij
(1.7.17)
∇ k g =0
1.8 Derivative along a curve
29
1.8 Derivative along a curve Let us push a vector of a vector field a bit along a curve. The curve is parametrized by an invariant quantity: The vector field at a certain point of the curve:
v i
The vector field at an infinitesimally nearby point of the curve:
v i
The parallel translated vector along the curve into the second point: i
i
i
a
w =v − ba⋅v ⋅dx
b
The difference between two infinitesimally close points of a vector field is the total differential: i
i
i
(1.8.1)
dv =v −v
The differential along a curve is the difference between the parallel translated vector and the local vector of the vector field in that point: Dv i=v i −w i
(1.8.2)
Dv i=v i − v i − i ba⋅v a ⋅dx b i
i
i
a
Dv =dv ba⋅v ⋅dx
b
1 /⋅ d dx i =ui d
Substitute the tangent vector:
When the derivative of a contravariant vector field along a curve is zero, it has the same form as the geodesic equation with tangent vectors: Đvi =
dv i i ba⋅v a⋅u b d
(1.8.3)
We proceed the same way with covariant vector fields: Dv i=v i −w i
(1.8.4)
Dv i=v i − v i abi⋅v a ⋅dx b a
Dv i=dv i − bi⋅v a ⋅dx
b
1 /⋅ d
The derivative along a curve of a covariant vector field:
30
1.8 Derivative along a curve Đvi =
dv i − abi⋅v a⋅u b d
(1.8.5)
Continue to example the two formulas. Since they have the same structure, we perform the following changes on both of them simultaneously. Let us expand the first term and reorder it: Đvi =
dx b dv i dx b dv i a b ⋅ − ⋅v ⋅u = ⋅ b − abi⋅v a⋅ub bi a b d d dx dx
dx b dv i dx a dv i i a b Đv = b⋅ ba⋅v ⋅u = ⋅ a i ba⋅v a⋅u b d dx dx d i
(1.8.6)
Substitute the tangent vector:
dv dv Đvi =u b⋅ bi − abi⋅v a⋅u b=ub⋅ bi − abi⋅v a dx dx
dv i dv i i a b b Đv =u ⋅ b ba⋅v ⋅u =u ⋅ b i ba⋅v a dx dx i
b
(1.8.7)
We identify the invariant derivative inside the parentheses: ∇ j vi =
∂ vi ∂x
−v a⋅ a ji j
∇ i u j=
∂uj u a⋅ jia i ∂x
the relationship between the derivative along a curve and the invariant derivative: b
i
Đvi =u ⋅∇ b v i
b
Đv =u ⋅∇ b v
j
(1.8.8)
Now we can write down the derivative along a curve of arbitrary tensors: ĐT
ijk lm
a
=u ⋅∇ a T
ijk
a
i
j
k
=u ⋅∇ a v ⋅u ⋅w ⋅⋅pl⋅q m⋅
lm
(1.8.9)
The derivative along a curve of the tangent vector of the curve is zero, because if it is displaced along the curve, it will coincide with the local tangent vector at the destination. Therefore the derivative of the tangent vector along a curve is the geodesic equation: Đui=
du i ∂2 x j ∂ x a ∂ xb j i ba⋅u a⋅u b= ⋅ ⋅ =0 ab d ∂ ∂ ∂2
(1.8.10)
This condition is obviously satisfied only if the connection is symmetric. The derivative along a curve of the covariant tangent vector of the curve:
31
1.8 Derivative along a curve Đui=
du i − abi⋅ua⋅u b=0 d
We rewrite the covariant vector to contravariant in the second term, using the metric tensor: Đu i=
du i −g ac⋅ abi⋅uc⋅u b=0 d
Đu i=
du i 1 − ⋅ g ⋅ a g ab⋅ aci ⋅u c⋅ub =0 d 2 ac bi
(1.8.11)
If the connection is symmetric, then in the second term we can identify the invariant derivative of the metric tensor inside the parentheses: ∇ k g ij = ∂ g ij ∂x
k
∂ g ij ∂x
k
−g ia⋅ akj− g aj⋅ aki =0
=g ia⋅ akj g aj⋅ aki
Substitute it: Đui=
du i 1 ∂ g bc c b − ⋅ ⋅u ⋅u =0 d 2 ∂ xi
(1.8.12)
If the partial derivative of the twice covariant metric tensor is zero, the corresponding covariant tangent vector of the geodesics does not change: ∂ g ij ∂x
k
=0
du k =0 d
→
(1.8.13)
1.9 Curvature We are going to examine the global properties of surfaces, that are by their nature independent of the coordinate systems. Our requirement is to be able to collect as many information about the structure of the surfaces as possible, using internally measurable quantities. The practical significance is, that we have to examine the shape of the spacetime using physical events and processes that happen inside, we do not have the option to observe them from somewhere outside. The commutator of the invariant derivative of the contravariant vector: i
∇ k ∇ j v −∇ j ∇ k v
i
(1.9.1)
The invariant derivative is a tensor-like quantity by definition: 32
1.9 Curvature
∇ j vi =
∂ vi v a⋅i ja =T i j j ∂x
(1.9.2)
Expand the repeated invariant derivation: ∂T i j ∇ k ∇ j v =∇ k T j = i kb⋅T b j− bkj⋅T i b k ∂x i
i
(1.9.3)
And substitute the vector's invariant derivative: ∇k
i
i
b
i
∂v ∂ ∂v ∂v ∂v v a⋅ i ja = k v a⋅i ja i kb⋅ v a⋅ b ja − bkj⋅ v a⋅ i ba j j j b ∂x ∂ x ∂x ∂x ∂x
Opening the parentheses: i
∇ k ∇ j v i=
2 i a b i ∂ v ∂v ∂v ∂v i a ∂ ja i i a b b ⋅ v ⋅ ⋅ ⋅v ⋅ − ⋅ − bkj⋅v a⋅ i ba ja kb kb ja kj k j k k j b ∂ x ⋅∂ x ∂ x ∂x ∂x ∂x
Doing the same with the opposite index order: ∇ j ∇ k v i=
i ∂2 v i ∂ va i ∂ vb ∂ vi a ∂ ka i i a b b ⋅ v ⋅ ⋅ ⋅v ⋅ − ⋅ − b jk⋅v a⋅ i ba ka jb jb ka jk j k j j k b ∂ x ⋅∂ x ∂ x ∂x ∂x ∂x
Subtract one of the other:
∇ k ∇ j v i−∇ j ∇ k v i=
∂ i ja ∂x
k
−
∂ i ka
∂ vi i b i b b i a b ⋅ − ⋅ 2⋅S ⋅ ⋅ v 2⋅S ⋅ kb ja jb ka jk ba jk j b ∂x ∂x (1.9.4)
Where we have substituted the antisymmetric expression with the torsion tensor: 1 S b jk = ⋅ b jk − bkj 2
(1.9.5) i
Substitute the curvature tensor:
Ri jak =
i
∂ ja ∂ ka − i kb⋅ b ja − i jb⋅ bka k j ∂x ∂x
∇ k ∇ j v i−∇ j ∇ k v i=Ri jak⋅v a2⋅S b jk⋅
(1.9.6)
∂v i i ba⋅v a =Ri jak⋅v a 2⋅S b jk⋅∇ b v i b ∂x
If the connection is symmetric, then the commutator of the invariant derivative of the contravariant vector is the curvature tensor, where we can recognize the tensor property from the form of the expression: 33
1.9 Curvature
i
i
∇ k ∇ j v −∇ j ∇ k v =R
i
⋅v
a
(1.9.7)
jak
We determine the commutator of the invariant derivative of the covariant vector using the same steps: ∇ k ∇ j v i−∇ j ∇ k v i
(1.9.8)
The invariant derivative is a tensor-like quantity: ∇ j vi =
∂ vi ∂x
j
−v a⋅ a ji =T ij
(1.9.9)
The repeated invariant derivation: ∂ T ij
∇ k T ij =
∂x
k
− akj⋅T ia − aki⋅T aj
(1.9.10)
Substitute the invariant derivative of the vector: ∇k
∂ vi ∂x
j
−v a⋅ a ji =
∂ vi ∂v b ∂ ∂ vi −v a⋅ a ji − bkj⋅ −v a⋅ abi − bki⋅ −v a⋅ a jb k j b j ∂x ∂ x ∂x ∂x
Open the parentheses: 2
a
∂ v ∂v ∂ ji ∂v ∂v ∇ k ∇ j v i= k i j − ak⋅ a ji −v a⋅ − bkj⋅ ib bkj⋅v a⋅ abi − bki⋅ bj bki⋅v a⋅ a jb k ∂ x ⋅∂ x ∂ x ∂x ∂x ∂x Doing the same with the opposite index order: ∇ j ∇ k v i=
∂2 v i ∂ va a ∂ aki ∂v ∂v − ⋅ −v ⋅ − b jk⋅ bi b jk⋅v a⋅ abi − b ji⋅ bk b ji⋅v a⋅ akb ki a j k j j ∂ x ⋅∂ x ∂ x ∂x ∂x ∂x
Subtract one of the other:
∇ k ∇ j v i−∇ j ∇ k v i=
∂ aki ∂x
j
−
∂ a ji
∂v bki⋅ a jb −b ji⋅ akb −2⋅S b jk⋅ abi ⋅v a 2⋅S b jk⋅ ib ∂x ∂x (1.9.11) k
Where we have substituted again the antisymmetric expression with the torsion tensor: S
b jk
1 b b = ⋅ jk − kj 2
(1.9.12)
34
1.9 Curvature a
a
∂ ki ∂ ji R kij = − bki⋅ a jb −b ji⋅ akb j k ∂x ∂x a
Substitute the curvature tensor:
∇ k ∇ j v i−∇ j ∇ k v i=R akij⋅v a 2⋅S b jk⋅
∂ vi ∂x
b
(1.9.13)
− abi⋅v a =R akij⋅v a 2⋅S b jk⋅∇ b v i
If the connection is symmetric, then the commutator of the invariant derivative of the covariant vector is also the curvature tensor: a
(1.9.14)
∇ k ∇ j v i−∇ j ∇ k v i=R kij⋅v a
If we set up a coordinate system in the immediate surrounding of a point, where the partial derivatives of the metric tensor and the connection are zeroes, the curvature tensor will not necessarily vanish, since the partial derivatives of the connection are not necessarily zeroes: Ri jkl =
∂ i jk ∂ xl
−
∂ i lk
(1.9.15)
∂xj
The invariant derivative: ∇ m Ri jkl =∇ m
∂ i jk ∂x
l
−
∂ i lk ∂x
j
(1.9.16)
Permute the first and the third lower indices and the index of the invariant derivation: i
i
(1)
∇ m Ri jkl =∇ m
∂ jk ∂ lk −∇ m l j ∂x ∂x
(2)
∇ j Ri lkm=∇ j
∂ i lk ∂ i mk −∇ j ∂ xm ∂ xl
(3)
∇ l Ri mkj =∇ l
∂ i mk ∂ i jk −∇ l ∂xj ∂ xm
(1.9.17)
Add the three equations: ∇ m Ri jkl ∇ j Ri lkm∇ l R i mkj= ∂ i jk ∂ i lk ∂ i lk ∂ i mk ∂ i mk ∂ i jk ∇m −∇ ∇ −∇ ∇ −∇ m j j l l ∂ xl ∂xj ∂ xm ∂ xl ∂xj ∂ xm
(1.9.18)
Since the connection coefficients are zeroes in this coordinate system, only the partial derivatives remain from the invariant derivatives, and they eventually cancel out:
35
1.9 Curvature ∇ m Ri jkl ∇ j Ri lkm∇ l R i mkj= ∂2 i jk ∂ 2 i lk ∂2 i lk ∂ 2 i mk ∂2 i mk ∂2 i jk − − − ∂ x l⋅∂ x m ∂ x m⋅∂ x j ∂ x j⋅∂ x m ∂ x j⋅∂ x l ∂ x l⋅∂ x j ∂ x l⋅∂ x m The following Bianchi identity is valid in every coordinate system: ∇ m Ri jkl ∇ j Ri lkm∇ l R i mkj=0
(1.9.19)
The curvature tensor is antisymmetric in its first and third lower indices: ∂ l ki ∂ l ji R kij = − bki⋅ l jb −b ji⋅ l kb j k ∂x ∂x
∂ l ji ∂ l ki R jik = − b ji⋅ l kb− bki⋅ l jb k j ∂x ∂x
l
R
l kij
=−R
l
l
(1.9.20)
jik
If the connection is symmetric, the sum of the cyclic permutations of the lower indices of the curvature tensor is zero: ∂ l ij ∂ l kj R ijk = − bij⋅l kb − bkj⋅ l ib k i ∂x ∂x
∂ l jk ∂ l ik R jki= − b jk⋅ l ib − bik⋅ l jb i j ∂x ∂x
l
R
l
l
kij
R ijk R
l jki
l
(1.9.21)
=0
1.10 Geodesic deviation In the immediate surrounding of any y point it is possible to set up a rectangular coordinate system, where the connection is zero. Therefore the equation of the geodesics crossing the point simplifies: ∂2 y i =0 ∂2
(1.10.1)
This is however true for the point only. In the immediate surrounding, the general equation of geodesics continues to apply: ∂2 x i ∂ xa ∂ xb i ⋅ ⋅ =0 ab ∂ ∂ ∂ 2
(1.10.2)
We approach the connection in the neighbouring x points with a Taylor series of the connection of the original point:
36
1.10 Geodesic deviation i
2
i
∂ jk y a 1 ∂ jk y a b jk x = jk y ⋅dx ⋅ a ⋅dx ⋅dx a 2 ∂ x ⋅∂ x b ∂x i
i
(1.10.3)
Substitute it into the equation of the neighbouring geodesics. Since the connection is zero in the centre, only the first derivative appears in the equation, the higher order derivatives are neglected: 2
i
i
a
b
∂ x ∂ ab c ∂ x ∂ x ⋅dx ⋅ ⋅ =0 2 c ∂ ∂ ∂ ∂x
(1.10.4)
Keep one of the components of the distance vector from the centre zero, thus we move only in a subspace around the original geodesic. Therefore the partial derivative of the connection according to this coordinate will also be zero: i
i
dx =0 dx
1
N
∂ jk =0 0 ∂x
→
dx
(1.10.5)
The tangent vector is perpendicular to this subspace, thus the centre-crossing geodesic pierces this subspace perpendicularly:
∂ xi ∂ x0 = ∂ ∂
(1.10.6)
0 0
The equation of the surrounding geodesics simplifies further: i
∂ 2 x i ∂ 00 c ∂ x 0 ∂ x 0 ⋅dx ⋅ ⋅ =0 ∂ ∂ ∂ 2 ∂ xc
(1.10.7)
Expand the connection derivative with a term that is known to be zero, and with them we produce the curvature tensor:
0 0 ∂ i 00 ∂ i c0 ∂2 x i c ∂x ∂x − ⋅ dx ⋅ ⋅ =0 2 c 0 ∂ ∂ ∂ ∂x ∂x 0 ∂2 x i ∂ x0 i c ∂x R ⋅dx ⋅ ⋅ =0 00c ∂ ∂ ∂ 2
Ri 00l =
∂ i 00 ∂ i l0 − ∂ xl ∂ x0 (1.10.8)
This formula is valid in general, we have no reason to restrict it to the indices denoted with numbers. Therefore the deviation of geodesics in the immediate surrounding of a geodesic is: 2
i
a
b
∂ x ∂x ∂x Ri abc⋅dx c⋅ ⋅ =0 2 ∂ ∂ ∂
(1.10.9)
37
1.11 Integration
1.11 Integration We integrate with respect to all coordinate-variables in space, therefore we introduce a shorthand notation for the product of the differentials: N
dx N =∏ dx n
(1.11.1)
n =1
The simple multi-variable integral of the scalar function is not invariant in the general case, since the product of the differentials depends on the coordinate system:
∫ A⋅dx N ≠∫ A⋅2 dx N
(1.11.2)
When we switch coordinate systems, the product of the differentials transforms with the determinant of the transformation matrix:
∣ ∣
dx N =
∂ xi ⋅ dx N =∣ ji∣⋅2dx N j 2 2∂ x
(1.11.3)
We insert this into the integral:
∣ ∣ i
∫ A⋅dx N =∫ A⋅ ∂∂ xx j ⋅2 dx N =∫ A⋅∣ ji∣⋅2 dx N
(1.11.4)
2
The following expression is invariant, if we build in the determinant of the transformation matrix into the expression under the integral:
∫ A⋅dx N =∫ 2 A⋅2dx N
(1.11.5)
The quantities that transform the following way are called scalar densities:
∣ ∣
∂ xi ⋅∣ ji∣ A = A⋅ =A 2 j 2∂ x
(1.11.6)
Let us examine the transformation rule of the metric tensor: a
2
b
(1.11.7)
g ij =i ⋅ j ⋅g ab
Since by matrix multiplication the determinants are also multiplied: 2
g =∣i a∣⋅∣ jb∣⋅g
(1.11.8)
The determinant of the metric tensor transforms like a scalar quantity, this also shows, that the signature of this determinant is independent from the choice of coordinates: 38
1.11 Integration
2 g =∣ ji∣⋅ g
(1.11.9)
Therefore the following integral is invariant:
∫ A⋅ g⋅dx N =∫ A⋅ 2 g⋅2 dx N
(1.11.10)
1.12 Variation and action principle Integral of a scalar function that vanishes at the boundaries, and its integral has an extremity, thus by slightly changing the input parameters the value of the integral does not change: S=∫ s x i ⋅ g⋅dx N
S =0
(1.12.1)
In order to apply the action principle, we need to choose a scalar that represents the space. For this we start with the curvature tensor and contract it to create the Ricci-tensor: Rki =R akia=
∂ aki ∂ aai − bki⋅ aab− bai⋅ akb a k ∂x ∂x
(1.12.2)
The trace of the Ricci-tensor is the curvature scalar, the simplest invariant scalar in the space: ab
(1.12.3)
R= g ⋅Rab
We use this as the scalar function:
S=∫ R⋅ g⋅dx N
(1.12.4)
Its variation: S = ∫ R⋅ g⋅dx N =∫ R⋅ g R⋅ g ⋅dx N =0
(1.12.5)
∫ g ab⋅Rab ⋅ g R⋅ g ⋅dx N =∫ g ab⋅Rab g ab⋅ Rab ⋅ gR⋅ g ⋅dx N =0 Examine the second term of the inner parentheses in a coordinate system, where the connection is zero: g ab⋅ R ab=g ab⋅
∂ cab ∂ ccb ∂ ∂v c ab ab c cb a −g ⋅ = g ⋅ − g ⋅ = ab ab ∂ xc ∂ xa ∂ xc ∂ xc
where: v i =g ab⋅ i ab−g ib⋅ aab
39
(1.12.6)
1.12 Variation and action principle Thus the expression inside the parentheses transforms like a vector. We rewrite the partial derivative into an invariant one, thus it will become valid in every coordinate system:
∫ ∇ c v c⋅ g⋅dx N =∮ v i⋅ g⋅dx N −1=0
(1.12.7)
Rewrite the integral into a surface integral using the divergence theorem, however our starting condition was that our scalar is zero on the boundaries, thus we succeeded in making this term disappear:
∫ g ab⋅Rab⋅ g R⋅ g ⋅dx N =0
(1.12.8)
Using differentiation rules we rewrite the variation of the square root of the determinant of the metric tensor (where M is the algebraic minor): g⋅ ij=M k ≠a , l≠i⋅g aj dg =M
k ≠a , l≠i
⋅dg ai
g=g⋅g ab⋅ g ab 1 g=− ⋅ g⋅g ab⋅ g ab 2
(1.12.9)
Substitute it:
∫ ∫
1 g ab⋅Rab⋅ g−R⋅ ⋅ g⋅g ab⋅ g ab ⋅dx N =0 2
1 Rab− ⋅R⋅g ab ⋅ g ab⋅ g⋅dx N =0 2
(1.12.10)
This expression is zero only if the expression inside the parentheses is zero, it is the Einstein equation: 1 Rij − ⋅R⋅g ij =0 2
(1.12.11)
The invariant derivative of the Einstein equation:
1 1 ∇ k Rij − ⋅R⋅g ij =∇ k Rij − ⋅R⋅∇ k g ij =∇ k Rij 2 2
(1.12.12)
We already know, that the invariant derivative of the metric tensor is zero. We can determine the invariant derivative of the Ricci-tensor using the Bianchi identity:
40
1.12 Variation and action principle 1 ∇ k Rij = ⋅∇ k Raija ∇ i Raajk ∇ a Rakji =0 3
(1.12.13)
Thus our result is:
1 ∇ k Rij − ⋅R⋅g ij =0 2
(1.12.14)
1.13 Runge-Kutta approximation method On surfaces with known geometry, we can examine arbitrary geodesics using numerical methods. The geodesic is uniquely identified by the coordinates of a single point it crosses, and its tangent vector in that point. With this information it is possible to recover with small steps the coordinates of the other points the geodesic is crossing, the smaller the steps are, the greater the accuracy becomes. Using the Runge-Kutta approximation method we can determine trajectories with high accuracy and by doing significantly less iteration steps, if we know the following variables at the n.th step: coordinates:
n
xi
coordinate-velocities:
n
v
connection in a given point:
change of the invariant parameter:
d
i
i
i
jk
x
(1.13.1)
It is important that the invariant parameter increases or decreases monotonically, because only in this case will it lead along the entire geodesic. The change in the parameter determines the the step size, that can be a conveniently chosen number. We approach the trajectory of the moving body with four straight sections, determine the coordinate- and coordinate-velocity-changes along the sections, and then average them: 1
a i=− i ab n x i ⋅nv a⋅n vb
1
v i=1 ai⋅d
i i i 1 2a =− ab n x
i
i
2 v =2 a ⋅d
i
1
i
(1.13.2)
x = nv ⋅d
xi va vb ⋅ n v a 1 ⋅ nv b 1 2 2 2
i i 1 2 x = n v
41
i
v ⋅d 2
(1.13.3)
1.13 Runge-Kutta approximation method
i
i i i 2 3a =− ab n x
a
x v v ⋅ n v a 2 ⋅ nv b 2 2 2 2
vi ⋅d 3 x = n v 2
i i 3 v = 3a ⋅d
i
4
ai =− i ab n x i 3 x i ⋅ n v a3 v a ⋅ n v b3 v b
4
v =4 a ⋅d
i
b
i
4
i
2
x i = nv i 3 v i ⋅d
(1.13.4)
(1.13.5)
We write down the weighted sum of the resulting coordinate- and coordinate-velocity-changes, thus we get the variables that determine the trajectory at the next step. The results calculated with this method deviate from the actual value only in the fifth order: i i 1 n1 x = n x
x i 2 xi 3 x i 4 x i 6 3 3 6 i
i
i
i
v 2 v 3 v 4 v n1v = nv 6 3 3 6 i
i
1
42
(1.13.6)
2. Examples
2. Examples In this chapter we are going to visualize the quantities derived in the previous chapter with some easy examples. We examine various two dimensional surfaces, we determine the metric tensor, its derivatives, the connection and derivatives, the Ricci-tensor and the Ricci-scalar. By doing so we demonstrate their geometric meaning and we prepare ourselves to apply them in the real four dimensional spacetime.
2.1 Curvature on a two dimensional surface The curvature of an arbitrary two dimensional surface in a given point is uniquely characterized by the reciprocal of the product of the radii of two, mutually perpendicular circles attached to the surface: K=
1 r⋅q
(2.1.1)
We approach the point with a surface, where the parametric equation is written with rectangular coordinates, utilizing the curvature radii: 2
z=
2
2
x y 2⋅r 2⋅q
z=
2
x y 2⋅r 2⋅q
(2.1.2)
Substitute this into the three dimensional element of arc length squared: ds 2=dx 2dy 2dz 2
(2.1.3)
The arc length squared on the surface:
x2 y2 2⋅x⋅y 2 ds = 1 2 ⋅dx 1 2 ⋅dy 2 ⋅dx⋅dy r⋅q r q 2
(2.1.4)
On this we can identify the metric tensor components:
x2 r2 x⋅y r⋅q
1
g ij =
x⋅y r⋅q y2 1 2 q
(2.1.5)
The metric tensor determinant:
43
2.1 Curvature on a two dimensional surface r 2⋅y 2q 2⋅x 2r 2⋅q2 ∣g ij∣=g 11⋅g 22−g 12⋅g 21 = r 2⋅q 2
(2.1.6)
The twice contravariant metric tensor: g ij =
1 r 2⋅y 2r 2⋅q 2 −r⋅q⋅x⋅y ⋅ 2 2 2 2 r 2⋅y 2q2⋅x 2r 2⋅q 2 −r⋅q⋅x⋅y q ⋅x r ⋅q
(2.1.7)
The metric tensor partial derivatives: ∂ g xx 2⋅x = 2 ∂x r
∂ g xy ∂ g yx y = = ∂x ∂ x r⋅q
∂ g xy ∂ g yx x = = ∂y ∂ y r⋅q
∂ g yy 2⋅y = 2 ∂y q
(2.1.8)
Calculating the connection:
∂ g ia ∂ g ak ∂ g ki 1 jki = ⋅g ja⋅ − a k i 2 ∂x ∂x ∂x x xx=
q 2⋅x r 2⋅y 2q 2⋅x 2r 2⋅q 2
y yy=
r 2⋅y 2 2 2 2 2 2 r ⋅y q ⋅x r ⋅q
x yy=
r⋅q⋅x 2 2 2 2 r ⋅y q ⋅x r ⋅q 2
2
(2.1.9)
The partial derivatives of the connection, we put the common factor in a separate variable: 1 =r 4⋅y 4q4⋅x 4r 4⋅q4 2⋅r 4⋅q 2⋅y 22⋅r 2⋅q 4⋅x 2 2⋅r 2⋅q 2⋅x 2⋅y 2 a ∂ x xx =a⋅q 2⋅r 2⋅y 2−q 2⋅x 2r 2⋅q 2 ∂x
∂ x yy =a⋅r⋅q⋅r 2⋅y 2−q 2⋅x 2 r 2⋅q 2 ∂x
∂ y yy ∂ x xx = =−a⋅2⋅r 2⋅q2⋅x⋅y ∂x ∂y
∂ x yy =−a⋅2⋅r 3⋅q⋅x⋅y ∂y
y
∂ yy =−a⋅r 2⋅ r 2⋅y 2 −q 2⋅x 2−r 2⋅q2 ∂y
(2.1.10)
The Ricci-tensor:
44
2.1 Curvature on a two dimensional surface a
Rij =R
a ija
a
∂ ij ∂ aj = − bij⋅ aab−baj⋅ aib a i ∂x ∂x
Rij =a⋅r⋅q⋅
q 2⋅x 2 r 2⋅q 2 r⋅q⋅x⋅y 2 r⋅q⋅x⋅y r ⋅y 2r 2⋅q 2
(2.1.11)
The Ricci-scalar: ab
3
R= g ⋅Rab=a⋅2⋅r ⋅q
3
3
R=
3
2⋅r ⋅q 4 4 4 4 4 4 4 2 2 2 4 2 2 2 2 2 r ⋅y q ⋅x r ⋅q 2⋅r ⋅q ⋅y 2⋅r ⋅q ⋅x 2⋅r ⋅q ⋅x ⋅y
(2.1.12)
The centre of the coordinate system is in the point we are discussing, where the coordinates are zeroes: x=y=0 3
R=
3
2⋅r ⋅q 2 = =2⋅K 4 4 r⋅q r ⋅q
(2.1.13)
2.2 Plain The element arc length squared and the metric tensor on the plain: ds 2=dx 2dy 2
g ij = g ij =
1 0 0 1
(2.2.1)
There are no variables in the metric tensor, therefore all its derivatives and derivable quantities are zeroes.
45
2.2 Plain The element arc length squared in polar coordinates and the other quantities: 2
2
2
2
ds =dr r ⋅d
1 0 g ij = 0 r2 ∂ g =r ∂r
r
=−
r 2
ij
g =
1 0
0 1 r2
∂ g 2 =− 3 ∂r r (2.2.2) r = r=
∂ r 1 =− ∂r 2
r 2
∂ r ∂ r 1 = = ∂r ∂r 2 (2.2.3)
The Ricci-tensor and therefore the Ricci-scalar are zeroes.
2.3 Cylinder We map the surface with a rectangular coordinate system, and the arc length squared depends on the constant radius of the cylinder: ds 2=2c⋅d 2dz 2
2 0 g ij = c 0 1
1 g = 2c 0 ij
0 1
(2.3.1) The metric contains no variables again, therefore every derivative and derivable quantity is zero. It is worth mentioning that it is possible to conceive a two dimensional surface with zero curvature, that cannot be embedded into three dimensional euclidean space. Let us imagine a cylinder where we are deforming the space it is embedded into. As the coordinate system on the cylinder differs from the plain case because one of the coordinates is made cyclic, it is possible to do so in three dimensions as well. While the x and y coordinates extend into infinity, the z coordinate returns into itself, its length is the circumference of a circle out in the fourth dimension. Since the curvature radius along the other coordinates is infinite, its easy to see that our space has 46
2.3 Cylinder zero curvature. If we define our cylinder on this manifold with the same orientation, and assume that the height of the cylinder is the same as the length of the z coordinate, we will notice that the top and bottom circles of the cylinder are touching. The newly formed surface is finite and homogeneous with both coordinates cyclic, cannot be embedded into common three dimensional space, still it has zero curvature (although it is not isotropic, circumnavigating the surface in various directions, the path taken would differ): 2
2
2
2
ds =c⋅d c⋅dz
2 c
g ij = 0
0 2c
2
ij
g =
1 2c
0
0
1 2c
(2.3.2)
2.4 Cone The cone is also a surface of zero curvature, because it can be unfolded to a plain. In rectangular and polar coordinates:
Some possible parametric equations: x=r⋅sin ⋅cos
y=r⋅sin ⋅sin
z =r⋅¿sin
x2 y2 z2 − =0 a2 b2 c2 x=
h−u ⋅r⋅cos h
y=
h−u ⋅r⋅sin h
z =u
Using the last equations, the arc length squared and the other quantities: 47
(2.4.1)
2.4 Cone
r 2c
hc −u2 ds = 1 2 ⋅du r ⋅ ⋅d 2 2 hc hc 2
g ij =
2
1
r 2c
2 c
0
2
hc
ij
h c −u2 r⋅ h2c 2 c
0
g =
h 2c 2
0
2
h c r c
h2c
0
r 2c⋅h c −u2
2⋅h2c ∂g = 2 3 ∂u r c⋅ hc −u
∂ g h −u =−2⋅r 2c⋅ c 2 ∂u hc
(2.4.2)
2
r h −u u = c2⋅ c 2 hc r 1 c2 hc u
∂ =− ∂u
u
=
u
=−
1 h c −u
2
rc
h 2c⋅ 1
∂ u ∂ u 1 = =− 2 ∂u ∂u h c −u
2 c 2 c
r h
(2.4.3)
The Ricci-tensor, and so the Ricci-scalar are also zeroes, the cone can be unfolded to a plain.
2.5 Sphere We introduce geographic latitudinal and longitudinal coordinates on a sphere, and calculate the geometric quantities from the arc length squared to the curvature: 2
2
2
2
2
ds =r c⋅d r c⋅sin ⋅d
2
r 2c 0 g ij = 2 0 r c⋅sin 2
ij
g =
1 r 2c
0
0
1 r ⋅sin 2 2 c
48
2.5 Sphere ∂ g =2⋅r 2c⋅cos ⋅sin ∂
2⋅cos ∂ g =− 2 ∂ r c⋅sin 3
=−cos ⋅sin
= =cot
∂ =sin2 −cos2 ∂
∂ ∂ cos2 = =− 2 −1 ∂ ∂ sin
Rij =
1 0 0 sin2
(2.5.1)
R=2⋅K =
(2.5.2)
2 r 2c
(2.5.3)
We calculate the surface of the sphere by integrating the infinitesimal surface element: dA=r c⋅d ⋅r c⋅sin ⋅d 2⋅
A=∫ r c⋅d ⋅∫ r c⋅sin ⋅d =r 2c⋅4⋅ 0
(2.5.4)
0
The sphere can be covered with rectangular coordinates also. Using this method we can map only one half of the sphere, thus it cannot be used to map the entire surface, just as polar coordinates cannot map the poles. The parametric equation: z =± r 2c − x 2− y 2 dz =−
x⋅dx y⋅dy r2c − x 2− y 2
(2.5.5)
The characterizing quantities of the surface, from the arc length squared to the curvature: ds 2=
1 ⋅ r 2c − y 2 ⋅dx 2 r 2c − x 2⋅dy 22⋅x⋅y⋅dx⋅dy 2 2 r −x −y 2 c
r 2c − y 2 x⋅y 1 g ij = 2 2 2⋅ r c− x − y x⋅y r 2c −x 2 ∣g ij∣=
2 2 1 r c − x −x⋅y g = 2⋅ r c −x⋅y r 2c − y 2 ij
r 2c r 2c − x 2− y 2
49
2.5 Sphere 2
2
2
∂ g 2⋅x⋅ r c − y = ∂ r 2c −x 2 − y 22
2
2
∂ g r −x 2⋅x = 2 2 ⋅ 2 c 2 −1 2 ∂ r c −x − y r c −x − y 2
∂ g ∂ g x 2⋅y 2 = = 2 2 ⋅ 1 ∂ ∂ r c −x − y 2 r 2c −x 2 − y 2 ∂ g 2⋅x =− 2 ∂ rc
=
∂ g ∂ g x = =− 2 ∂ ∂ rc
=
2
2
x ⋅y 2 2 2 2 r c⋅r c −x − y
4
2
2
2
2
2
2
2
2
2
2
2
x⋅y 2 2 2 2 r c⋅r c −x − y
4
2
4
2
2
2
2
2
2
2
2
∂ 2⋅x⋅y 3 = 2 2 2 ∂ r c⋅r c −x − y 22
∂ 2⋅x 3⋅y = 2 2 2 22 ∂ r c⋅r c −x − y
2
2
4
2
∂ ∂ x ⋅r c −x y = = 2 2 2 22 ∂ ∂ r c⋅ r c − x − y
2
2
∂ y 3⋅x ⋅y −2⋅r c⋅y −r c⋅x r c = 2 2 2 2 2 ∂ r c⋅ r c − x − y
∂ ∂ y ⋅r c x − y = = 2 2 2 22 ∂ ∂ r c⋅ r c − x − y
r −y 1 Rij =−g ij =− 2 2 ⋅ c 2 r c −x − y x⋅y
2
∂ ∂ ∂ 2⋅x⋅y⋅ r c − x = = = 2 2 2 22 ∂ ∂ ∂ r c⋅r c − x − y
∂ x 3⋅x ⋅y −2⋅r c⋅x −r c⋅y r c = 2 2 2 2 2 ∂ r c⋅ r c − x − y
2
∂ x 4−x 2⋅y 2−2⋅r 2c⋅x 2r 2c⋅y 2 r 4c = ∂ r 2c⋅r 2c −x 2− y 2 2
∂ ∂ ∂ 2⋅x⋅y⋅r c − y = = = 2 2 2 2 2 ∂ ∂ ∂ r c⋅r c −x − y
∂ g 2⋅y =− 2 ∂ rc (2.5.6)
y⋅r −x = 2 2 c 2 2 r c⋅ r c − x − y
= =
2
y⋅r 2c − y 2 r 2c⋅r 2c −x 2− y 2 2
x⋅r −x = 2 2 c 2 2 r c⋅ r c − x − y
∂ y 4− x 2⋅y 22⋅r 2c⋅y 2r 2c⋅x 2r 4c = ∂ r 2c⋅r 2c −x 2− y 2 2
2
∂ g ∂ g y 2⋅x 2 = = 2 2 ⋅ 1 ∂ ∂ r c −x − y 2 r 2c −x 2 − y 2
x⋅r 2c − y 2 r 2c⋅r 2c −x 2− y 2
= =
2
∂ g r −y 2⋅y = 2 2 2⋅ 2 c 2 2 −1 ∂ rc− x − y rc− x − y
∂ g ∂ g y = =− 2 ∂ ∂ rc
2
2
∂ g 2⋅y⋅r c −x = ∂ r 2c − x 2− y 2 2
x⋅y 2 r c −x 2
R=2⋅K =
2 r 2c
Our result is of course the same as the result from calculating with polar coordinates. 50
(2.5.7)
(2.5.8)
2.5 Sphere
2.6 Paraboloid In polar and rectangular coordinates:
The equation: z =b⋅ x 2 y 2 The parametric equations:
x=a c⋅
u ⋅cos hc
y=ac⋅
u ⋅sin hc
z =u
(2.6.1)
The characterizing geometric quantities of the surface, from the arc length squared to the curvature:
a 2c a 2c⋅u 2 2 ds = 1 ⋅du ⋅d 4⋅h c⋅u hc 2
g ij =
1
a 2c 4⋅h c⋅u
0 a2c⋅u hc
0
2 c
∂ g uu a =− ∂u 4⋅hc⋅u 2
ij
g =
4⋅hc⋅u
∂ g uu = ∂u
51
0
4⋅h c⋅ua 2c
hc a 2c⋅u
0
2
ac
a 2c 4⋅h c⋅u ⋅ 1 4⋅hc⋅u 2
2
2.6 Paraboloid h ∂g =− 2 c 2 ∂u a c⋅u
2
∂ g ac = ∂u hc
u uu
=−
a 2c
a 2c 4⋅hc⋅u ⋅ 1 4⋅h c⋅u 2
∂ uuu = ∂u
2
ac 4⋅hc⋅u ⋅ 1 4⋅h c⋅u 2
u
a 2c
∂ =−2⋅ ∂u
1 1 Ruu= − 2 4⋅u u
2
ac
4⋅hc⋅u⋅ 1
2 c
a 4⋅hc⋅u
a 2c 4⋅h c⋅u⋅ 1 4⋅hc⋅u
=−
1 ⋅ − u
a 2c
a 2c 2⋅h c⋅ 1 4⋅hc⋅u
a2c
2
ac 2⋅h c⋅u ⋅ 1 4⋅h c⋅u
2
u = u =
1 2⋅u
2
∂ u ∂ u 1 = =− ∂u ∂u 2⋅u 2
2
ac
R=2⋅K =
u
(2.6.2)
R =
ac
a 2c 4⋅h c⋅u⋅ 1 4⋅hc⋅u
⋅ 1−
(2.6.3)
ac
a 2c 4⋅hc⋅u⋅ 1 4⋅hc⋅u
4⋅h c
(2.6.4)
2 c
a 4⋅hc⋅u
Far away from the tip of the paraboloid, the curvature of the surface approaches zero: lim
u∞
4⋅hc 2 c
a 4⋅h c⋅u
=0
(2.6.5)
2.7 Hyperboloid The parametric equation of a hyperboloid of one sheet: x=a c⋅ u21⋅cos
y=ac⋅ u 21⋅sin
x 2 y 2 z 2 − 2 =1 a 2c bc
z =bc⋅u (2.7.1)
52
2.7 Hyperboloid From the arc length squared to the curvature: 2
ds =
a 2c⋅u 2 2
u 1
b 2c ⋅du 2a 2c⋅u 21⋅d 2
a c⋅u
u 21 a2c b2c ⋅u 2b 2c
0
0
1 a ⋅u 21
2
2
g ij = u 1 0
ij
g =
2
2
0
b c
2
2
a c⋅u 1
2 c
∂ g uu 2⋅a 2c⋅u u = 2 ⋅ 1− 2 ∂u u 1 u 1
∂ g 2 =2⋅a c⋅u ∂u
2⋅a 2c⋅u ∂ g uu =− 2 2 2 ∂u u ⋅a c bc b2c 2 uuu=
u
∂g 2⋅u =− 2 2 2 ∂u a c⋅ u 1
a 2c⋅u a 2c⋅u 2b 2c⋅ u2 1⋅ u21
=
u
=
u
=−
(2.7.2)
a 2c⋅u⋅u 21 a 2c⋅u 2b 2c⋅u 21
u u 1 2
∂ uuu a 2⋅a 2⋅u 2⋅3⋅u2 1b2⋅3⋅u 42⋅u2 −1 =− c c 2 2 2 2 c ∂u a c⋅u bc⋅u 1⋅u 212 ∂ u a 2c⋅a 2c⋅u2⋅u 2−1b 2c⋅ u2 12 =− ∂u a 2c⋅u2 b2c⋅u 212 ∂ u ∂ u u 2−1 = =− 2 ∂u ∂u u 12
Rij =−
2 c 2 c
b
⋅ a 2c⋅u 2b ⋅ u2 1
(2.7.3)
1 u 1
0
2
0
a 2c⋅u 21 2
2
2
2
a c⋅u bc⋅u 1
53
2.7 Hyperboloid
R=2⋅K =−
2⋅b2c a 2c⋅u2 b2c⋅u 212
(2.7.4)
2.8 Bolyai surface The Bolyai surface is an infinite surface with constant negative curvature, it cannot be embedded into a three dimensional surface with a positive arc length squared. We start with the hyperboloid of two sheets. With polar and rectangular coordinates:
The equation of the surface: z = a 2c x 2 y 2
(2.8.1)
We set up a coordinate system in the three dimensional space, based on the hyperboloid: x=r⋅sinh ⋅cos
y=r⋅sinh ⋅sin
z =r⋅cosh
(2.8.2)
Substitute this into the equation of the surface, this shows that the hyperboloid is a coordinate surface in this coordinate system:
r⋅cosh = a 2c r⋅sinh ⋅cos 2 r⋅sinh ⋅sin 2 r =a c
(2.8.3)
The arc length squared of the pseudo-euclidean three dimensional space, where we are going to embed the hyperboloid: ds 2=dx 2dy 2−dz 2 54
2.8 Bolyai surface Changes of coordinates on the embedded hyperboloid: dr =0 dx=r⋅cosh ⋅cos ⋅d −r⋅sinh ⋅sin⋅d dy=r⋅cosh ⋅sin ⋅d −r⋅sinh ⋅cos ⋅d dz =r⋅sinh⋅d
(2.8.4)
The arc length squared on the embedded hyperboloid and the other geometric quantities: ds 2=r 2c⋅d 2r 2c⋅sinh 2 ⋅d 2
2 c
r g ij = 0
0 2 r c⋅sinh 2
ij
g =
=−cosh ⋅sinh
0
1 r ⋅sinh 2 2 c
=
−1 0 0 −sinh2
(2.8.5)
=coth
∂ ∂ cosh 2 = =1− ∂ ∂ sinh2
∂ =−sinh 2 −cosh 2 ∂ Rij =
0
2⋅cosh ∂ g =− 2 ∂ r c⋅sinh 3
∂ g =2⋅r 2c⋅cosh ⋅sinh ∂
1 r 2c
R=2⋅K =−
2 2 rc
(2.8.6)
(2.8.7)
The Bolyai surface has infinite extension. There is a two dimensional surface, that although lacks the symmetry features of the previous Bolyai surface, but it has the same constant negative curvature, and it can be embedded into three dimensional space: the tractroid. This surface is created by revolving a tractrix about its asymptote:
a a a 2−2 z =a⋅arcosh − a 2− 2=a⋅ln − a 2 −2
55
(2.8.8)
2.8 Bolyai surface From the arc length to the curvature: ds 2=d 22⋅d 2 dz 2 2
ds 2=
ac d 2 2⋅d 2 2
a 2c
g ij = 0
2
0
ij
g =
2
∂ g 2⋅a 2c =− 3 ∂
2 a 2c
0
0
1 2
∂ g 2⋅ = 2 ∂ ac
∂ g =2⋅ ∂
∂ g 2 =− 3 ∂ (2.8.9)
=−
1
=−
3 2 ac
∂ 3⋅2 =− 2 ∂ ac
∂ 1 = 2 ∂
Rij =
1 2
=
0
R=2⋅K =−
2
0
=
1
∂ ∂ 1 = =− 2 ∂ ∂
−
− 2 ac
2 a 2c
(2.8.10)
(2.8.11)
2.9 Catenoid Although the curvature of the catenoid is non-zero, but it is a minimal surface, its average curvature is zero, just like the plain. The parametric equation: x=a c⋅cosh
v ⋅cos u ac
x=a c⋅cosh
56
v ⋅sin u ac
z =v
(2.9.1)
2.9 Catenoid From the arc length squared to the curvature: ds 2=cosh 2
v v ⋅dv 2cosh 2 ⋅du 2 ac ac
g ij =
cosh
v ac
0
cosh
v ac
1
g ij =
0
0
v cosh ac
1
0
cosh
∂ g vv ∂ g uu = = ∂v ∂v
2⋅cosh
v ac
v v ⋅sinh ac ac
vv
∂g ∂g = =− ∂v ∂v
ac
u
u
Rij =
−
a 2c⋅cosh 2
2
0
−
v ac
1
v a ⋅cosh ac 2 c
(2.9.2)
3
v
0
v a ⋅cosh ac 2 c
v a c⋅cosh ac
∂ uu =− ∂v
1
1
v ac
v
1 v v uu=− ⋅coth ac ac
1 v v vv= uvu= uuv = ⋅coth ac ac ∂ vv ∂ vu ∂ uv = = = ∂v ∂v ∂v
2⋅sinh
uu
2
1 a 2c⋅cosh 2
R=2⋅K =−
v ac
(2.9.3)
2 a 2c⋅cosh 4
v ac
(2.9.4) Far away from the signature funnel of the catenoid, the curvature of the surface approaches zero:
57
2.9 Catenoid
2
lim − v ∞
2 c
a ⋅cosh
4
v ac
=0
(2.9.5)
2.10 Helicoid The helicoid is also a minimal surface, and it is isometric to the catenoid, thus it is possible to deform it into a catenoid without distortion and vice versa, just as the cylinder can also be flattened into a plain. The parametric equations: x=v⋅cos y=x⋅tan
z =a c⋅v
y=v⋅sin
z ac
(2.10.1)
From the arc length squared to the curvature: ds 2=dv 2a 2c v 2⋅du2 g ij =
ij
g =
1 0 2 0 a c v 2
1
0 1 2 a c v 2
0
∂ g uu =2⋅v ∂v
∂ g uu 2⋅v =− 2 2 2 ∂v ac v
v uu=−v
uvu = uuv =
∂ v uu =−1 ∂v
∂ uvu ∂ uuv a 2c −v 2 = = 2 22 ∂v ∂v a c v
(2.10.3)
(2.10.4)
Rij =
−
a 2c a 2c v 22 0
0
v2 −1 2 2 a c v
(2.10.2)
v a v 2 2 c
R=2⋅K =−
58
2⋅a 2c a 2c v 2 2
2.10 Helicoid
2.11 Hyperbolic paraboloid The parametric equation of the surface with negative curvature in a possible coordinate system: x=u
y=v
z =x⋅y
(2.11.1)
From the arc length squared to the curvature: ds 2=1v 2 ⋅du 21u 2⋅dv 22⋅u⋅v⋅du⋅dv g ij =
1v 2 u⋅v u⋅v 1u 2 2
∣g ij∣=1u v
g ij =
2
∂ g uv ∂ g vu = =v ∂u ∂u
∂ g uv ∂ g vu = =u ∂v ∂v
∂ g uu 2⋅u⋅v 2 = ∂ u 1u 2v 22 uv
1 1u 2 −u⋅v ⋅ 2 1u2 v 2 −u⋅v 1v
vu
∂ g vv =2⋅u ∂u
∂ g uu =2⋅v ∂v
∂ g vv 2⋅u2⋅v = ∂ v 1u 2v 2 2 2
2
uv
vu
2
2
∂g ∂g v⋅1−u v = =− 2 2 2 ∂u ∂u 1u v
∂g ∂g u⋅1−u v = =− 2 2 2 ∂v ∂v 1u v
∂ g vv 2⋅u⋅1v 2 =− ∂u 1u 2v 22
∂ g uu 2⋅v⋅1u 2 =− ∂v 1u 2v 2 2
u uv
=
u vu
=
v 1u 2v 2
v uv
=
v vu
=
(2.11.2)
u 1u 2v 2
∂ uuv ∂ uvu ∂ vuv ∂ v vu 2⋅u⋅v = = = =− ∂u ∂u ∂v ∂v 1u 2v 2 2 2 2 ∂ v uv ∂ v vu 1−u v = = ∂u ∂u 1u 2v 2 2
2 2 ∂ uuv ∂ uvu 1−u −v = = ∂v ∂v 1u 2v 2 2
59
(2.11.3)
2.11 Hyperbolic paraboloid
1 1u 2 u⋅v Rij =− ⋅ 2 1u 2v 2 2 u⋅v 1v
R=2⋅K =−
2 1u2 v 2 2
(2.11.4)
Far away from the signature saddle of the hyperbolic paraboloid, the curvature of the surface approaches zero: lim
u ∞ , v ∞
−
2 =0 1u 2v 22
(2.11.5)
Parametric equation in another possible coordinate system: z=
y2 x2 − 2 2 bc ac
(2.11.6)
From the arc length squared to the curvature: ds 2=
4⋅u 2 4⋅v 2 8⋅u⋅v 2 1 ⋅du 1 ⋅dv 2− 2 2 ⋅du⋅dv 4 4 ac bc a c⋅b c
2
4⋅u 4⋅u⋅v 1 − 2 2 4 ac a c⋅bc g ij = 2 4⋅u⋅v 4⋅v − 2 2 1 4 a c⋅b c bc g ij =
∣g ij∣=
2
a 4c⋅b 4c 4⋅v 2 4⋅a 2c⋅b 2c⋅u⋅v 1 ⋅ a 4c⋅b 4c 4⋅v 2 4⋅b 4c⋅u 2 4⋅a 2c⋅b2c⋅u⋅v b4c⋅a 4c 4⋅u2 ∂ g vv 8⋅v = 4 ∂v bc
∂ g uv ∂ g vu 4⋅v = =− 2 2 ∂u ∂u a c⋅bc
∂ g uv ∂ g vu 4⋅u = =− 2 2 ∂v ∂v a c⋅bc
4
4
2
2
∂ g uu 8⋅u = 4 ∂u ac
4
2
4⋅u 4⋅v 16⋅u ⋅v 1 ⋅ 4 1 − 4 4 4 ac bc a c⋅b c
4
4
4
2
uu 8⋅a ⋅b ⋅u⋅b 4⋅v ∂g =− 4 c4 c 2 c 4 2 2 ∂u ac⋅b c 4⋅v 4⋅bc⋅u
vv 8⋅a ⋅b ⋅v⋅ a 4⋅u ∂g =− 4 c4 c 2 c 4 2 2 ∂v a c⋅ bc 4⋅v 4⋅bc⋅u
32⋅a 4c⋅b 4c⋅u⋅v 2 ∂ g vv = ∂ u a 4c⋅b4c 4⋅v 2 4⋅b 4c⋅u 2 2
32⋅a 4c⋅b 4c⋅u 2⋅v ∂ g uu = ∂v a 4c⋅b4c 4⋅v 2 4⋅b 4c⋅u 22
60
2
2.11 Hyperbolic paraboloid 2
2
4
4
2
4
2
2
2
4
4
2
4
2
uv vu 4⋅a c⋅b c⋅v⋅a c⋅bc 4⋅v −4⋅b c⋅u ∂g ∂g = = 4 4 2 4 2 2 ∂u ∂u a c⋅bc 4⋅v 4⋅bc⋅u uv vu 4⋅a c⋅b c⋅u⋅b c⋅a c 4⋅u −4⋅a c⋅v ∂g ∂g = = 4 4 2 4 2 2 ∂v ∂v a c⋅b c 4⋅v 4⋅b c⋅u
uuu=
4⋅b4⋅u a 4c⋅b 4c 4⋅v 24⋅b 4c⋅u 2
v vv =
4⋅b ⋅v 4 4 2 4 2 a c⋅b c 4⋅v 4⋅bc⋅u
4
(2.11.7)
uvv=−
4⋅a 2⋅b 2⋅u a 4c⋅b4c 4⋅v 2 4⋅b 4c⋅u 2
v uu=−
4⋅a ⋅b ⋅v 4 4 2 4 2 a c⋅bc 4⋅v 4⋅b c⋅u
2
2
∂ uuu 4⋅b4c⋅a 4c⋅b 4c 4⋅v 2−4⋅b 4c⋅u 2 = ∂u a 4c⋅b 4c 4⋅v 24⋅b 4c⋅u 2 2
∂ v vv 4⋅a 4c⋅b4c⋅ ac44⋅u 2 −4⋅a 4c⋅v 2 = ∂v a 4c⋅bc44⋅v 2 4⋅b 4c⋅u 2 2
∂ v uu 32⋅a 2c⋅b 6c⋅u⋅v = 4 4 ∂u a c⋅b c 4⋅v 24⋅b4c⋅u2 2
∂ uvv 32⋅a 6c⋅b 2c⋅u⋅v = 4 4 ∂v a c⋅b c 4⋅v 2 4⋅b4c⋅u 22
∂ uvv 4⋅a 2c⋅b 2c⋅ a 4c⋅ 4⋅v 2−b 4c 4⋅b4⋅u 2 = ∂u a 4c⋅bc44⋅v 2 4⋅b 4c⋅u 2 2 v
u
4
∂ v uu 4⋅a 2c⋅b 2c⋅ ac4⋅ 4⋅v 2−b 4c −4⋅b4⋅u2 = ∂v a 4c⋅b4c 4⋅v 2 4⋅b 4c⋅u 2 2
4
∂ vv ∂ uu 32⋅a c⋅b c⋅u⋅v = =− 4 4 2 4 2 2 ∂u ∂v a c⋅b c 4⋅v 4⋅b c⋅u
4⋅a 2c⋅b2c
(2.11.8)
b4c⋅ 4⋅u 2a 4c 4⋅a 2c⋅b 2c⋅u⋅v Rij = 4 4 ⋅ a c⋅bc 4⋅v 2 4⋅bc4⋅u 22 4⋅a 2c⋅b2c⋅u⋅v a 4c⋅4⋅v 2b 4c R=2⋅K =−
8⋅a 6c⋅b 6c 4
4
2
4
(2.11.9)
2 2
a c⋅bc 4⋅v 4⋅b c⋅u
2.12 Torus It is a variable curvature surface, negative on the inner side, and positive on the outer edge. The parametric equation, where a is the main radius, b is the secondary radius: x=a c b c⋅sin⋅cos
y=a c b c⋅sin ⋅sin
61
z =bc⋅cos (2.12.1)
2.12 Torus From the arc length squared to the curvature: 2
2
2
2
2
ds =bc⋅d a c b c⋅sin ⋅d
g ij =
ij
g =
b 2c 0 0 a c b c⋅sin 2
1 b2c
0
0
1 a c bc⋅sin 2
∂ g =2⋅cos ⋅sin ⋅2⋅b2c⋅sin 2 3⋅a c⋅bc⋅sin a2c ∂
bc 2⋅cos ∂ g 1 =− ⋅ ∂ a c bc⋅sin 2⋅sin 2 sin a c bc⋅sin
(2.12.2)
a c bc⋅sin sin 2 =− ac bc⋅sin ⋅cos ⋅sin ⋅ b2c
a b ⋅sin ⋅1sin = =cos ⋅ c c a c b c⋅sin ⋅sin
∂ ac bc⋅sin sin 2 −cos 2 ⋅ a cbc⋅sin = ⋅ sin 2 −4⋅cos2 ⋅sin ∂ bc bc 2
2
−cos ⋅sin ∂ ∂ a 2c 3⋅a c⋅bc⋅sin 2⋅bc a 2c 2⋅ac⋅b c⋅sin ⋅cot 2 = = ∂ ∂ a 2c b 2c⋅sin 2 2
a c 2⋅b c⋅sin ⋅sin −2⋅bc⋅cos 2 Rij = a c b c⋅sin⋅sin 0
0 R
(2.12.3)
3⋅a c 2⋅bc⋅sin ⋅b c⋅sin 2 a 2c −2⋅b 2c⋅cos2 ⋅sin −2⋅a c⋅bc⋅cos 2 R = ⋅sin b 2c 2 2 a 2⋅bc⋅sin ⋅sin −2⋅bc⋅cos R=2⋅K = 2⋅ c a c bc⋅sin ⋅sin bc
62
(2.12.4)
2.12 Torus The surface of the torus: dA=bc⋅d ⋅ a cbc⋅sin ⋅d 2⋅
2⋅
A= ∫ b c⋅d ⋅∫ a c b c⋅sin ⋅d =4⋅a c⋅bc⋅2 0
0
63
(2.12.5)
3. Flat and general spacetime
3. Flat and general spacetime The simplest four dimensional solution of the Einstein equations is the matter free flat spacetime. In it we can neglect the gravitation of the test particles, and the geodesics they are following become everyday straight lines that can be described with simple partial derivatives. The theory of special relativity deals with this solution, described by Albert Einstein in 1905, short before the birth of general relativity. In the Kaluza model, electromagnetism also has a geometric origin, it is not part of the matter distribution, with this extension the theory describes a five dimensional spacetime. However in our experience, the gravitational interaction is more general than the electromagnetic. Thus gravitational interaction can act alone, while an electromagnetic field is not possible without gravitation, since the presence of electromagnetic energy in itself is already causing changes in the spacetime geometry. Thus we can discuss pure gravitational interaction without neglecting anything. In order to understand what is going on, it is important to keep in mind, that motion happens in the entire four dimensional spacetime. Time is a direction of motion that can be used to measure distances, just as the other three directions of space.
3.1 Proper time We use rectangular coordinates in flat space, and we write down the arc length squared in the following way: 2
2
2
2
2
ds =c ⋅dt −dx −dy −dz
2
t: time coordinate x, y, z: space coordinates c: speed of light
(3.1.1)
It is customary to use unique notation for the metric tensor:
1 0 0 0 0 = = 0 −1 0 0 0 −1 0 0 0 0 −1
(3.1.2)
It is easy to see, that this is a solution of the Einstein equations in four dimensions: 1 R − ⋅R⋅ =0 2
(3.1.3)
Two dotted lines denote the paths of light rays on the following graph, that lead away or into the event in the centre. The spacetime domains separated by the light cone have different significances. Under the lower cone sheet lay past events, that might have influenced the observer in the centre, these compose the time-like past. Events above the upper cone sheet might be influenced by the centre, therefore they compose the time-like future. Since information cannot 64
3.1 Proper time propagate faster than the speed of light, the outside domain is not influenced by this, therefore the concepts of future and past cannot be interpreted there. At the graphical representation of the coordinate system, we neglected the y and z coordinates: c∙t
x
In the spacetime using different coordinate systems the arc length squared is the same, it is an invariant quantity. Let us write it on the left side of the equation in a coordinate system, where the arc is parallel to the time coordinate, and use a generally oriented coordinate system on the right side: 2
2
2
2
2
c ⋅d =c ⋅dt −dl
(3.1.4)
Where we have grouped together the space-like coordinate differentials: 2
2
2
dl =dx dy dz
2
(3.1.5)
The space-like velocity squared: dl 2 v t = 2 dt 2
(3.1.6)
And the proper time:
d =dt⋅ 1−
v t2 dl 2 =dt⋅ 1− c 2⋅dt 2 c2
(3.1.7)
3.2 Lorentz-transformation According to the relativity principle, reference frames in state of constant, rectilinear motion with respect to one another are equivalent. We determine the transformation matrix that corresponds 65
3.2 Lorentz-transformation to the exchange between them:
t a x e 2 = i 2y m 2z 2
i
i 2∂ x ⋅x a =i a⋅x a 2x = a ∂x
b f j n
c g k o
d t h x ⋅ l y p z
(3.2.1)
Linear transformations can always be represented by matrices, and matrices always create linear transformations. The transformation matrices compose a group, because they obey the following rules: Identity element:
1 1⋅x = x
Closure:
3 2⋅ 2 1⋅x = 3 1⋅x
Associativity:
4 3⋅ 3 2⋅ 2 1 ⋅x = 4 3⋅3 2 ⋅2 1⋅x
Inverse element:
2 1⋅x=1 2⋅x
−1
(3.2.2)
We examine first the identity element, we approach with a transformation that changes the vector only slightly:
t a x = e y i z m
b f j n
c g k o
d t h⋅x l y p z
(3.2.3)
Write out the matrix operation in detail: t=a⋅tb⋅xc⋅yd⋅z
= a−1⋅tb⋅x c⋅yd⋅z
x=e⋅t f⋅xg⋅yh⋅z
=e⋅t f −1⋅xg⋅y h⋅z
y=i⋅t j⋅xk⋅y l⋅z
=i⋅t j⋅x k −1⋅y l⋅z
z =m⋅tn⋅xo⋅y p⋅z
=m⋅tn⋅x o⋅y p−1⋅z
(3.2.4)
If the small deviation approaches zero, the transformation matrix approaches the identity matrix:
0
→
a e i m
b f j n
c g k o
d 1 0 h 0 1 l 0 0 p 0 0
0 0 1 0
0 0 0 1
(3.2.5)
Since the transformation is linear, the macroscopic translations will also have the same form. The 66
3.2 Lorentz-transformation small deviation is a contravariant vector, but we going to write the macroscopic deviation as a covariant tangent vector:
u =
= ⋅u
v ⋅t c
(3.2.6)
Investigate the transformation of the unit vectors, first in the time-like direction. We drop the speed of light in the formulas, to avoid confusion with one of the matrix elements:
tt
⋅ut u t a xx ⋅u x −u x = = e i −u y yy⋅u y zz m −u z ⋅u z
b f j n
c g k o
d h⋅0 l 0 p 0
v t⋅t=a⋅ −v x⋅t=e⋅ −v y⋅t=i⋅ −v z⋅t =m⋅
(3.2.7)
From this the matrix elements: t 1v t⋅ =a
v − x ⋅ a−1=e vt
v − y⋅a−1=i vt
v − z⋅a−1=m vt
(3.2.8)
The unit vector in the x direction:
ut a x−u x = e i −u y m −u z
b f j n
c g k o
d 0 h⋅x l 0 p 0
v t⋅t=b⋅x x−v x⋅t= f⋅x −v y⋅t= j⋅x −v z⋅t=n⋅x
(3.2.9)
The matrix elements: t v t⋅ =b x
v 1− x⋅b= f vt
v − y⋅b= j vt
v − z⋅b=n vt
(3.2.10)
Spacetime is the same in every direction, therefore we expect the same form in both other spacelike directions: t v t⋅ =c y
v − x⋅c=g vt
1−
vy ⋅c=k vt
t v t⋅ =d z
v − x⋅d =h vt
v − y⋅d =l vt
v − z⋅c=o vt
(3.2.11)
v 1− z⋅d = p vt
(3.2.12)
The diagonally opposite elements look the same, and since the length of the unit vectors is the same, they coincide:
67
3.2 Lorentz-transformation t v t⋅ =b x
t v t⋅ =e
↔
v − x ⋅ a−1=e=b vt
→
e=b
v − z⋅a−1=i=c vt
v − z⋅a−1=m=d vt
(3.2.13)
Thus the matrix is symmetric:
a e i m
b f j n
c g k o
d a h = b l c p d
b f g h
c g k l
d h l p
(3.2.14)
The velocity squared in three dimensional space: ⋅v ⋅v =v 2t −v 2x −v 2y −v 2z
v 2t =v 2x v 2y v 2z=v 2
→
(3.2.15)
After this the form of the matrix elements: t 1v t⋅ =a
t v t⋅ =b x
t v t⋅ =c y
t v t⋅ =d z
v − x ⋅ a−1=b vt
v 1 x2⋅a−1= f v
v x⋅v y ⋅a−1= g v2
v x⋅v z ⋅ a−1=h v2
v − y⋅a−1=c vt
v y⋅v x ⋅a−1= g 2 v
v 2y 1 2⋅a−1=k v
v y⋅v z
v − z⋅a−1=d vt
v z⋅v x ⋅ a−1=h v2
v z⋅v y ⋅a−1=l v2
v 1 2z⋅a−1= p v (3.2.16)
2
v
2
⋅a−1=l
2
The coordinates of the centre of the standing coordinate system in the moving coordinate system:
t a −v x⋅t = b −v y⋅t c d −v z⋅t
b f g h
c g k l
d h⋅0 l 0 p 0
t =a −v x⋅a=b −v y⋅a=c −v z⋅a=d
The reverse transformation:
68
(3.2.17)
3.2 Lorentz-transformation
a −v x⋅a −v y⋅a −v z⋅a t v ⋅t −v ⋅a f g h 0 = x ⋅ x 0 v y⋅t −v y⋅a g k l 0 v z⋅t −v z⋅a h l p
(3.2.18)
Determine the „0,0” matrix element: =a⋅t⋅1−v 2 0= f −a⋅v x g⋅v y h⋅v z 0=g⋅v x k −a⋅v y l⋅v z 0=h⋅v x l⋅v y p−a ⋅v z
a=
→
1 1−v 2
(3.2.19)
The matrix of the Lorentz-transformation with SI units:
v v v − x ⋅a − y⋅a − z⋅a c c c 2 v v v x⋅v y v x⋅v z − x⋅a 1 x2⋅a−1 ⋅a−1 ⋅a−1 2 c v v v2 = vy v y⋅v x v 2y v y⋅v z − ⋅a ⋅a−1 1 2⋅a−1 ⋅a−1 2 c v v v2 vz v z⋅v x v z⋅v y v 2z − ⋅a ⋅a−1 ⋅ a−1 1 2⋅a−1 c v2 v2 v a
a=
1
1−
v2 c2
(3.2.20)
Investigate the transformation along the x axis:
c⋅2t a x 2 = b c 0 d 0
b f g h
c g k l
d c⋅t h⋅ x l 0 p 0
(3.2.21)
The transformation formulas:
2
t =a⋅t
b⋅x = c
1
1−
v ⋅t− x⋅ c v2 c2
v t− 2x⋅x c
1
x ⋅ = c v v2 1− 2 1− 2 c c
2
69
(3.2.22)
3.2 Lorentz-transformation vx ⋅ 2 x =b⋅c⋅t f ⋅x=− c
1
1−
v2 c2
⋅c⋅t 1
v 2x v
2
⋅
1
1−
v2 c2
−1 ⋅x=
x−v x⋅t
1−
v2 c2
(3.2.23)
For the reverse transformation, all we have to do is change the sign of the velocity. Substitute them into the four-distance, that is the finite variant of the arc length squared: s 2=c 2⋅t 2− x 2− y 2− z 2
2
2
2
2
v t− 2x⋅x c
c ⋅2t − 2 x =c ⋅
c 2⋅2t 2− 2 x 2=
(3.2.24) 2
v2 1− 2 c
−
x−v x⋅t 1−
v2 c2
2
2
v ⋅ c2⋅t 2 −2⋅t⋅v x⋅x 2x⋅x 2 −x 22⋅x⋅v x⋅t−v 2x⋅t 2 2 v c 1− 2 c 1
v 2x 1 2 2 2 c ⋅2t − 2 x = ⋅ c ⋅t −x ⋅ 1− 2 2 v c 1− 2 c 2
2
2
2
2
2
2
2
c ⋅2t − 2 x =c ⋅t −x
2
(3.2.25)
The coordinate transformation is therefore correct, because it preserves the invariant four-distance. By writing out the entire four-distance, we can conclude that the y and z coordinates do not transform: 2
2
2
2
2
2
2
2
2
c ⋅2t − 2 x −2 y −2 z =c ⋅t −x − y −z 2
2
2
2
y 2 z = y z
2
2
2
y= y
2
z =z
(3.2.26)
Length contraction of a moving rod; let the end points of the rod in each coordinate systems be x and y: 2
x− 2 y=
x−v x⋅t
1−
v2 2 c
−
y−v x⋅t
1−
v2 2 c
70
3.2 Lorentz-transformation 2
x− 2 y=
x− y
1−
v2 2 c
The length of the rod: l=x− y
v2 l=2l⋅ 1− 2 c
(3.2.27)
3.3 Addition of velocity and acceleration How much is the measured speed of an object moving with constant velocity from a different, moving reference frame? Let one of the coordinate system move relatively to the other in the x direction, let their relative velocity be Vx, the moving test object also moves along the x direction, and its speed in each coordinate system: v x=
dx dt
dx 2 dt
v =2
2 x
(3.3.1)
The transformation of the change in the coordinate: 2
dx =2 x B−2 x A =
x B− x A −V x⋅t B−t A dx −V x⋅dt = V2 V2 1− 2 1− 2 c c
(3.3.2)
Substitute it: dx 2dx dt dx−V x⋅dt 1 dt v −V x dt = ⋅ = ⋅ ⋅ = x ⋅ 2 2 dt 2dt dt 2 dt 2 dt V V 2dt 1− 2 1− 2 c c
v =2
2 x
(3.3.3)
The transformation of the change of time: t B −t A − 2
dt = 2t B− 2t A=
Vx ⋅ x B−x A c2 2
V 1− 2 c
dt− =
The mutual ratios of the changes of time:
71
Vx c2
⋅dx
V2 1− 2 c
(3.3.4)
3.3 Addition of velocity and acceleration
dt = dt
Vx V ⋅v ⋅dx 1− x 2 x 2 c 1 c ⋅ = 2 2 dt V V 1− 2 1− 2 c c
dt−
2
(3.3.5)
Reinsert is:
V2 v x −V x dt v −V x c2 ⋅ = x ⋅ 2v x = V x⋅v x V 2 2 dt V2 1− 2 1− 2 1− 2 c c c
1−
The transformation formula of the velocity along the x direction: v =
2 x
v x −V x V ⋅v 1− x 2 x c
(3.3.6)
We can always find a coordinate system, where the velocity of the moving object is zero, this is the co-moving coordinate system, where: V x =v x
(3.3.7)
The perpendicular velocity components are also transforming: 2
dy 2dy dt dy dt dt = ⋅ = ⋅ =v y⋅ dt dt dt dt dt 2 2 2 2 dt
v y= 2
V2 2 c v =v ⋅ 2 y y V ⋅v 1− x 2 x c 1−
(3.3.8)
dz 2dz dt dz dt dt = ⋅ = ⋅ =v z⋅ dt 2dt dt 2dt 2 dt 2dt
v =2
2 z
V2 2 c 2v z =v z⋅ V ⋅v 1− x 2 x c 1−
(3.3.9)
For the reverse transformation formulas all we have to do is reversing the sign of the relative 72
3.3 Addition of velocity and acceleration velocity between the two coordinate systems. Write down the transformation law of the change of the velocity:
2
dv x =d
V2 c2 ⋅dv x 2 V x⋅v x
1−
v x −V x = V x⋅v x 1− 2 1− 2 c c
(3.3.10)
Divide it with the change of time: 1− 2dt =
V x⋅v x c2
⋅dt
V2 1− 2 c
The transformation of the acceleration in the x direction: 2 3 2
dv x = 2dt
2 2a x =
V 1− 2 c
V ⋅v 1− x 2 x c
⋅a x
(3.3.11)
3
Let us examine the momentary state of an object with constant acceleration. In this case, its velocity in its own reference frame is zero, and the mutual velocities of the two coordinate systems coincides with the velocity of the object in the other coordinate system: a 0=
1 2 3 2 2
1−
v c
⋅a=
d ⋅ dt
v 1−
v2 2 c
(3.3.12)
Perform the integration: v=
a 0⋅t
2
(3.3.13)
a 1 0⋅t c
With further integration we get the dependence of the objects position from the coordinate-time:
2
a0 c2 x= ⋅ 1 ⋅t −1 x 0 a0 c
(3.3.14)
73
3.3 Addition of velocity and acceleration We introduce the contravariant four-velocity, the derivative of the coordinate-change according to the proper time. Calculating the time-like component: V 0=
dx 0 = d
c⋅dt
2
v dt⋅ 1− 2 c
=
c
v2 1− 2 c
(3.3.15)
Determine the space-like components of the four-velocity, from the common three-velocity: V i=
dx i dxi dt = ⋅ = d dt d
vi
1−
(3.3.16)
v2 c2
The reverse relationship: Vi v i =c⋅ 0 V
(3.3.17)
The four-velocity is a vector, therefore its square is an invariant scalar: dx dx ds 2 2 2 V = ⋅ ⋅ = 2 =c d d d
(3.3.18)
The four-acceleration is the differential of the four-velocity according to the proper time: A =
dV d
(3.3.19)
Every component of the four-acceleration of a non-moving object is zero: c 0 0 0 dV A= =d =0 0 0 0 d d
(3.3.20)
In this case the scalar product of the two vectors can be easily written down. This is however an invariant formula, therefore its true for any moving body: V ⋅A=0
(3.3.21)
3.4 Aberration of light The movement of an object in two different coordinate system can be characterized by the following velocity components:
74
3.4 Aberration of light v y =v⋅sin
v x =v⋅cos v =2 v⋅cos 2
2 x
2
v y =2v⋅sin 2
(3.4.1)
One of the coordinate systems moves in the x direction, with Vx velocity relatively to the other. The transformation laws in the x and y directions using the formulas containing angles above: v =
2 x
v x −V x v⋅cos−V x = 2v⋅cos 2 = V ⋅v V ⋅v⋅cos 1− x 2 x 1− x c c2
(3.4.2)
V2 V2 1− 2 1− 2 c c =2 v⋅sin 2 =v⋅sin ⋅ 2v y =v y⋅ V ⋅v V ⋅v⋅cos 1− x 2 x 1− x c c2
(3.4.3)
Divide them with each other and we get the transformation law of the azimuth angle of the trajectory of the moving body: cot 2 =
v⋅cos−V x
(3.4.4)
V2 v⋅sin ⋅ 1− 2 c
In the case of light, the speed v is equal to the speed of light c, this is the aberration of light: cos − cot 2 =
Vx c
(3.4.5)
V2 sin⋅ 1− 2 c
3.5 Doppler-effect Since the length and the elapsed time are also coordinate dependent quantities, therefore, while the speed of light is constant, the wavelength of light is measured to be different by different observers. Wave-fronts leaving the light source with λ wavelength distance from each other, reach the observer moving away with v velocity under the following time intervals: t=
c−v
(3.5.1)
This can be expressed also using the frequency:
75
3.5 Doppler-effect t=
1 v 1− ⋅ c
=
c
The elapsed time calculated in the proper time of the distancing observer:
v2 c2
v v c 1 = ⋅ 2t =t⋅ 1− 2 = v v c 1− 1− ⋅ c c
2
1−
1
(3.5.2)
From this the measured frequency:
v 1 c ⋅ 2 = = v 2t 1 c 1−
(3.5.3)
If the light source and the observer pass by each other, for a short period of time their distance does not change, their relative velocity along the line connecting them is zero. From the point of view of the light source, wave-fronts leaving the light source with λ wavelength distance from each other, reach the observer with the following time intervals: 1 t= = c
(3.5.4)
However the proper time of the observe differs from the light source, therefore it measures the arrival of the wave-fronts with different time intervals:
2t =t⋅ 1−
2
v = 2 c
1−
v2 2 c
(3.5.5)
The frequency Doppler shift in the perpendicular direction:
v2 = 1− ⋅ 2 2 c
(3.5.6)
3.6 Sequence of events The ordering between cause and effect can be secured only, if by observing from every possible reference frames, the moment the effect happens is later in time than the cause. We are 76
3.6 Sequence of events going to investigate, under what circumstances this condition is fulfilled. The difference between moments in time transforms between coordinate systems the following way: t B−t A− t − 2 t A=
2 B
Vx c
2
⋅ x B− x A
V2 1− 2 c
V x −x 1− 2x ⋅ B A c t B−t A =t B−t A ⋅ V2 1− 2 c
(3.6.1)
The velocity vx is the speed the information travels with, from event A (the cause) to event B (the effect): v x=
x B −x A t B −t A
(3.6.2)
Substitute it to the transformation formula: V ⋅v 1− x 2 x c 2t B − 2t A=t B−t A ⋅ 2 V 1− 2 c
(3.6.3)
It follows from our condition, that the time difference has to be positive: t −2t A0
2 B
t B −t A 0
(3.6.4)
This leads to the following inequality: 1−
V x⋅v x c2
0
2
(3.6.5)
c V x⋅v x
If vx = c, that is, the information causing the second event comes from the first event with the greatest possible velocity, the speed of light, the mutual velocity of the reference frames cannot exceed the speed of light: cV x
(3.6.6)
Returning to our original transformation formula, let us examine what effects it has, if we demand that the time differences are positive:
77
3.6 Sequence of events t B−t A− t − 2t A=
2 B
Vx c
2
⋅ x B− x A 2
V 1− 2 c
t −2t A0
2 B
t B −t A −
Vx ⋅ x B−x A 0 c2
∣V∣ 0 c
c⋅t B−t A − x B −x A0
(3.6.7)
Squaring the terms does not change the direction of the inequality: 2
2
2
(3.6.8)
c ⋅t B −t A − x B −x A 0
Thus the four-distance between the two events is always bigger than zero, its time-like.
3.7 Energy and momentum Picture an weightless floating empty box. One of the internal walls emits a photon, that is absorbed by the opposite wall later. Because of the conservation of momentum, the box slightly moves in the opposite direction and then stops, as the photon is absorbed. The photon has no rest mass, but it has momentum: p f=
E0 c
(3.7.1)
We can write the momentum of the box using its mass and velocity: p d =M⋅v
(3.7.2)
It takes Δt time for the photon to reach the other side of the box, while the box gets displaced for Δx distance, this is the velocity of the box: v=
x t
(3.7.3)
Because of the conservation of momentum, in the centre of mass system the magnitude of the momentum of the photon and the box are equal: x E0 M⋅ = t c
(3.7.4)
78
3.7 Energy and momentum If we know the l width of the box, and the speed of the photon, we can determine the time that passed while it crossed the box: t=
l c
(3.7.5)
Substitute it into the formula for the conservation of momentum: M⋅ x=
E 0⋅l
(3.7.6)
c2
In order to determine the movement of the box relative to the centre of mass, let us write the position of the centre of mass, as if a particle with mass moving inside it has caused its displacement, it will be called the effective mass of the photon: x=
M⋅x d m⋅x f M m
(3.7.7)
The position of the centre of mass is the same both at the emission and absorption of the photon: M ⋅x d m⋅x f M ⋅ x d − x m⋅l d = M m M m
(3.7.8)
If we consider the starting position of the photon (xf = 0), and the box (xd = 0) both zero, we can significantly simplify the relationship above: m⋅l d =M ⋅ x
(3.7.9)
Substitute the conservation of momentum formula: m⋅l d =
E 0⋅l d c
2
From this we can express the equivalence of the rest mass and energy: E 0=m⋅c
2
(3.7.10)
The equation of motion of an object accelerating because of a constant force: F =m⋅a=
dp dt
(3.7.11)
Let us examine the situation is a given moment, when the velocity of the moving body is zero in one of the coordinate systems, in this case its velocity looks the same as the relative velocity of the two reference frames in the other coordinate system: 79
3.7 Energy and momentum v =V
(3.7.12)
Express its acceleration:
2a x =
V2 1− 2 c
V⋅v 1− 2 c
3 2
3
⋅a x =
1 2 3 2 2
⋅a x =
1 2 3 2 2
v 1− c
v 1− c
dv d ⋅ = ⋅ dt dt
v v2 1− 2 c
(3.7.13)
Substitute it into the equation of motion: d m⋅v d F =m⋅a= ⋅ = ⋅p 2 dt dt v 1− 2 c
(3.7.14)
We can read from this the relativistic momentum: p=
m⋅v
v2 1− 2 c
(3.7.15)
Force is the negative gradient of the potential energy, we rewrite this: F =−
dE dx
F⋅v=−
dE dx dE ⋅ =− dx dt dt
(3.7.16)
Substitute the momentum, and one version of the expression of the acceleration: 2
dp d m⋅v m⋅v dv d m⋅c v⋅F=v⋅ =v⋅ ⋅ = ⋅ = ⋅ 3 2 2 2 dt dt v v 2 dt dt v 1− 2 1− 2 1− 2 c c c
(3.7.17)
From this the relativistic energy: E=
m⋅c 2
1−
(3.7.18)
v2 c2
Relationship between energy and momentum: 80
3.7 Energy and momentum
E=
p⋅c 2 v
(3.7.19)
In order to investigate the relationship between energy and momentum, we examine the contravariant velocity. Divide the change in coordinate with the proper time: dx =c⋅dt
dx =v = d
d =dt⋅ 1−
dx dy dz vx
c 1−
v2 c2
1−
vy
v2 c2
1−
vz
v2 c2
1−
v2 c2
v2 2 c
(3.7.20)
If we multiply this with the mass, we get the contravariant energy-momentum four-vector:
m⋅v =
m⋅v x
m⋅c
2
v 1− 2 c
m⋅v y
2
v 1− 2 c
m⋅v z
2
v 1− 2 c
2
v 1− 2 c
=
E c
px
py
pz
(3.7.21)
During changes between coordinate system, the four-vectors Lorentz-transform: ∂ xi a ⋅x =i a⋅x a 2x = a ∂x i
2
The equation of transformation while moving along a line:
E a c = b 2 px c 0 d 0 2
2
2
b f g h
c g k l
E =a⋅Eb⋅p x⋅c=
E d c h⋅ px l p 0 0
v ⋅E− x⋅ c v2
1
1−
c
v E p x =b⋅ f⋅p x =− x ⋅ c c
2
1
1−
v2 2 c
⋅p x⋅c=
E−v x⋅p x
1−
2 x 2
v E 1 ⋅ 1 ⋅ −1 ⋅p x = v v c v2 1− 2 1− 2 c c 1
2
81
(3.7.22)
v2 2 c v p x − 2x⋅E c
v2 1− 2 c
(3.7.23)
3.7 Energy and momentum Using the arc length squared, we can once again conclude, that the y and z components remain unchanged: s 2=
E2 − p 2x − p 2y − p 2z 2 c
E2 E2 2 − p = − p2x 2 x 2 2 c c
2
2
p y= p y
2
p z= p z
(3.7.24)
Insert the transformation of the rest mass and the momentum into each other:
2
E=
E−v x⋅p x
1−
v2 2 c
p x =2 p x⋅ 1−
2 E = E⋅ 1−
2
p x=
v p x − 2x⋅E c
1−
v2 c2
v2 vx ⋅E c2 c2
v2 −v x⋅2 p x c2
(3.7.25)
If the particle moves at the speed of light (the sign shows the direction of the movement): 2
E 0=−c⋅2 p x
(3.7.26)
Divide the arc length squared with the infinitesimal change in time: 2
2
2
2
2
ds =c ⋅dt −dx −dy −dz
2
1 /⋅ dt
ds 2 2 2 2 2 =c −v x −v y −v z dt 2
(3.7.27)
In order to keep it simple, we calculate in one dimension. First we write an obvious identity, that we reorder: 2
2
2
c −v =c −v
2
82
3.7 Energy and momentum c 2−v 2 2 =c v2 1− 2 c 2
c
2
1−
v c2
c
2
1−
v c2
m⋅c 2
2
1−
v c2
v
=
2
1−
v c2
c 2 /⋅m⋅c
2
1−
v c2
2
m⋅v
=
c 2
2
v
=
2
2
1−
v c2
⋅c 2m2⋅c 4
By substituting the energy and the momentum, we have the total energy of a moving body: E= p2⋅c 2m2⋅c 4
(3.7.28)
If its mass is zero: E 0= p⋅c m⋅c 2
2
v 1− 2 c
=
(3.7.29) m⋅v
v2 1− 2 c
⋅c
Objects with zero mass move at the speed of light in every reference frame: v=c
(3.7.30)
3.8 Relativistic rocket The rocket is a complex system that loses mass while constantly accelerating. The two main components are the payload and the fuel, that is exhausted with a constant velocity in the rocket's reference frame. The relativistic rocket equation establishes a relationship between the following quantities: M:
initial mass of the rocket 83
3.8 Relativistic rocket dm: w: U: u:
mass of the fuel that is ejected in an infinitesimally small period of time the constant exhaust velocity of the fuel relatively to the rocket velocity of the rocket in the centre of mass system velocity of the exhaust in the centre of mass system
During the movement of the rocket, the conservation of momentum is valid, the left side is the change of momentum of the rocket, the right side is of the exhausted fuel: d
M
2
1−
U c2
dm
⋅U =u⋅
1−
u2 c2
(3.8.1)
As well as the conservation of energy: d
M
2
1−
U c2
⋅c2 =−
dm
2
1−
u c2
⋅c 2
(3.8.2)
Observing from the centre of mass system, the velocity of the exhausted fuel is the sum of the exhaust velocity relative to the rocket and the rocket velocity: u=
w−U U⋅w 1− 2 c
Substitute it to the conservation of momentum formula: d
M
2
1−
U c2
⋅U =
U −w ⋅d U⋅w 1− 2 c
M
1−
U2 c2
(3.8.3)
Reorder the differential of the denominator with the square root: d
1
2
1−
U c2
U⋅dU
=
c2 −U 2⋅ 1−
U2 c2
Substitute: 2
M⋅dU U⋅dM
M⋅U ⋅dU U −w M⋅U⋅dU = ⋅ dM 2 2 2 2 U⋅w c −U c −U 1− 2 c
84
3.8 Relativistic rocket
U−
U −w M⋅U U −w ⋅dM =− M 2 ⋅ U− ⋅dU 2 U⋅w U⋅w c −U 1− 2 1− 2 c c
Simplify: dM =− M
dU
U2 w⋅ 1− 2 c
(3.8.4)
Substitute the differential of the logarithm: d log x =
log
dx x
U c c =− ⋅log 2⋅w U 1− c 1
M M start
The traditional form of the relativistic rocket equation:
U c M =M start⋅ U 1− c 1
−
c 2⋅w
(3.8.5)
The maximal exhaust velocity of the fuel: w= e⋅2−e⋅c
(3.8.6)
3.9 Faster than light particles Theoretical particles that can move faster than light are called tachyons. If we calculate their relativistic energy, we get an imaginary result: E=
m⋅c 2
1−
=−i⋅ 2
v c2
m⋅c 2 v2 −1 c2
vc
(3.9.1)
Therefore we should define their mass imaginary, thus their energy and momentum will be real numbers. From an experimental point of view, we can do this, because there are no inertial frames 85
3.9 Faster than light particles faster than the speed of light where the tachyon could be at rest, thus its mass is not measurable. m=i⋅m ' E=
m'⋅c2
v2 −1 c2
m ' =−i⋅m
p =
m'⋅v
v2 −1 2 c
(3.9.2)
From this we can see, that in the case of tachyons, and increase in energy is followed by a decrease in velocity, and vice versa. The sign of the momentum four-vector changes: E2 − p 2=−m' 2⋅c 2 2 c
(3.9.3)
The energy and momentum of an object slower than the speed of light, the energy can vary between the energy at rest and infinity, the momentum can take on any value:
0≤vc
m⋅c2 ≤E∞
0≤ p∞
(3.9.4)
The energy and momentum of an object faster than the speed of light, the energy can take on any value, however the momentum cannot decrease beyond a certain value, tachyons cannot slow down:
cv≤∞
0≤E∞
m '⋅c≤ p∞
(3.9.5)
Let us examine the velocity addition in the case of tachyons: v =
2 x
v x −V x V ⋅v 1− x 2 x c
(3.9.6)
We can always find a reference frame, where the velocity of the faster-than-light object is infinite: V x⋅v x c2 V x=
=1
c2 vx
(since v x c , it is always true that V x c
Reinserting this we can determine the limit of the velocity addition formula, where the velocity in it goes to infinity: v x −V x c2 lim = V x⋅v x V x v ∞ 1− 2 c
(3.9.7)
x
86
3.9 Faster than light particles
3.10 Circular motion and Thomas precession We determine the proper acceleration of an object in constant circular motion, this is what it measures in its proper time, and also its coordinate acceleration, that is measured by the nonmoving observer in coordinate time. Let us write down the arc length squared, metric tensor and its derivatives and the connection of the cylindrical coordinate system in flat spacetime. 2
2
2
2
2
2
ds =c ⋅dt −dr −r ⋅d −dz
1 0 0 0 0 −1 0 0 g = 2 0 0 −r 0 0 0 0 −1
2
(3.10.1)
g =
1 0 0 −1 0
0
0
0
0 0 0 0 1 − 2 0 r 0 −1
∂ g =−2⋅r ∂r
∂ g 2 = 3 ∂r r
r =−r
∂ r =−1 ∂r
∂ r ∂ r 1 = =− 2 ∂r ∂r r
r
r
=
(3.10.2)
(3.10.3)
=
1 r
(3.10.4)
(3.10.5)
Insert these values into the geodesic equation, where we partially differentiate, once according to proper time, and then according to coordinate time (we are allowed to do this, because both values increase monotonically). ∂2 x j ∂ xa ∂ xb j ⋅ ⋅ =0 ab ∂ ∂ ∂ 2
∂2 x j ∂ xa ∂ xb j ⋅ ⋅ =0 ab ∂t ∂ t ∂ t2
(3.10.6)
The equations have the same form in both cases, therefore we use the general dot notation for partial differentiation: c⋅¨t =0 r¨
r
2
2
⋅ ˙ =r¨ −r⋅ ˙ =0
2 ⋅r⋅ ¨ ˙ ˙ r˙ = ¨ ˙ ˙ ⋅ ˙ =0 r ⋅r r⋅⋅ r z¨ =0
(3.10.7)
87
3.10 Circular motion and Thomas precession The following coordinate conditions apply to objects in constant circular motion: r˙ =0
t˙ =0
z˙ =0
=const. ˙
(3.10.8)
From this the centripetal acceleration: r¨ =r⋅ ˙
2
(3.10.9)
Determine the relationship between the angular velocities according to the proper time and the coordinate time: ∂2 r ∂ =r⋅ 2 ∂ ∂
2
∂2 r ∂ =r⋅ 2 ∂t ∂t
2
(3.10.10)
Substitute the proper time into the second derivative of the radial coordinate:
2
d =dt⋅ 1−
v 2 c
2
∂ r 1 ∂ ⋅ =r⋅ 2 2 ∂ ∂t v 1− 2 c
2
(3.10.11)
Reorder it, and make it equal to the formula written with the coordinate time: 2
∂2 r ∂ v2 =r⋅ ⋅ 1− ∂ ∂t 2 c2 2
2
r⋅
∂ ∂ v2 =r⋅ ⋅ 1− 2 ∂t ∂ c
(3.10.12)
The relationship between the angular velocities using the proper time and the coordinate time: ∂ ∂ = ⋅ ∂ ∂t
1 1−
v2 c2
(3.10.13)
The difference between the two quantities shows, that a non-rotating object moving on a circular orbit, after having completed a circle it will not face in the same direction as before, this phenomenon is called the Thomas precession:
88
3.10 Circular motion and Thomas precession =
∂ ∂ − = ∂ ∂t
1
∂ −1 ⋅ ∂t v 1− 2 c 2
(3.10.14)
3.11 Gravitational redshift On the world line of an object at rest relatively to the coordinate system only the coordinate time changes. The relationship with the proper time can be determined with the four-distance: 2
2
2
c ⋅ =c ⋅g tt⋅t
2
(3.11.1)
The proper times of two different test objects: 1
= 1 g tt⋅t
2
= 2 g tt⋅t
(3.11.2)
The tt component of the metric tensor has to be positive, so that the direction of the proper time and the coordinate time coincides, and that the second assumption, the principle of the ordering between cause and effect is not violated. By substituting the coordinate time we can determine the relationship between the proper times:
1
=
g tt ⋅2 2 g tt 1
(3.11.3)
The frequency of light or any periodic phenomenon: =
1
(3.11.4)
The gravitational redshift:
1
=
g tt ⋅2 1 g tt 2
(3.11.5)
89
4. Spherically symmetric spacetime
4. Spherically symmetric spacetime The most general spherically symmetric solution of the Einstein equations was derived by Karl Schwarzschild in 1916, not long after the initial discovery of those equations. This chapter deals with the matter free version of it. We map the spacetime of the Schwarzschild solution with various coordinate systems, and investigate the trajectories of moving bodies in it. We verify our results with observations from the Solar System. Several well known phenomena get a new interpretation, once we use geometric methods to understand them, and unexpected new effects also occur. Since every phenomenon is a result of interplay between distances and angles, tempering with them has many, previously unknown impacts on the orbits of celestial bodies.
4.1 Spherically symmetric coordinate system Let us set up a spherically symmetric coordinate system in flat spacetime, that we are going to use as a basis for the subsequent general derivation. The arc length squared is created by extending the arc length squared of the sphere with radial and time coordinates: ds 2=c 2⋅dt 2−dr 2−r 2⋅ d 2sin 2 ⋅d 2
(4.1.1)
Determine the metric tensor, the connection, and their derivatives:
1 0 0 0 0 −1 0 0 g = 2 0 0 −r 0 2 0 0 0 −r ⋅sin 2
1 0 0 −1
g = 0
0
0
0
0 0 1 − 2 r 0
0 0 0 −
1 r ⋅sin2 2
(4.1.2)
∂ g =−2⋅r ∂r
∂g 2 = 3 ∂r r
∂ g =−2⋅r⋅sin 2 ∂r
∂ g 2 = 3 ∂r r ⋅sin 2
∂ g =−2⋅r 2⋅cos ⋅sin ∂
∂ g 2⋅cos = 2 ∂ r ⋅sin3
r =−r
90
r
2
=−r⋅sin
(4.1.3)
4.1 Spherically symmetric coordinate system r = r = r = r =
=
1 r
=−cos ⋅sin
(4.1.4)
=cot
r
∂ =−1 ∂r
r
∂ =−sin 2 ∂r
∂ r ∂ r ∂ r ∂ r 1 = = = =− 2 ∂r ∂r ∂r ∂r r
r
∂ =−2⋅r⋅cos ⋅sin ∂ ∂ =sin2 −cos 2 ∂
∂ ∂ = =−cot 2 −1 ∂ ∂
(4.1.5)
Every component of the curvature tensor is zero, since the spacetime is flat. We write down the geodesic equations: c⋅¨t =0 r¨ r ⋅˙ 2 r ⋅ ˙ 2= r¨ −r⋅˙ 2−r⋅sin 2 ⋅ ˙ 2=0 2 2 ¨ ˙ ˙ ˙ ⋅ ¨ 2⋅r˙⋅−cos ˙ ⋅sin ⋅ ˙ = ˙ =0 ˙⋅ r ⋅r r⋅⋅r r
2 ˙ ˙ ˙ =0 2⋅ ⋅r˙⋅2⋅cot ⋅⋅ ¨ ˙ ¨ ˙ ˙ ˙⋅2⋅ r ⋅r ⋅⋅= r
(4.1.6)
4.2 Schwarzschild coordinates We start by simplifying the Einstein equations further: 1 R − ⋅R⋅g =0 2
/⋅g
1 R− ⋅R⋅4=0 2 R=0
(4.2.1)
By reinserting this we get, that the Ricci-tensor is zero in vacuum: R =0
(4.2.2)
We assume about the shape of the resulting spacetime, that at great distances from the source of 91
4.2 Schwarzschild coordinates gravitation, it approaches the flat spacetime. Since none of the coordinates change as a function of another, there are no mixed coordinate products. Therefore we use spherically symmetric coordinates to calculate the arc length squared, and we extend it with unknown functions, that depend only on the distance from the centre, and the elapsed time: 2
2
2
2
2
2
2
2
ds = Ar , t⋅c ⋅dt −B r ,t ⋅dr −C r , t ⋅r ⋅d − Dr , t⋅r ⋅sin ⋅d
2
(4.2.3)
The position of the axis of the coordinate system can be arbitrary, by adjusting it to our liking, we have one less unknown functions: C r , t = Dr , t
(4.2.4)
Let us write an even more general form, where we allow the radial and time coordinates to mix: 2
2
2
2
2
2
2
2
2
2
ds = f ⋅c ⋅# dt 2⋅ f ⋅g⋅c⋅dt⋅dr −h ⋅dr −C⋅r ⋅d sin ⋅d
(4.2.5)
This however can be led back to the diagonal form, using substitutions where we rescale the time coordinate: A⋅c 2⋅dt 2= f ⋅c⋅# dt g⋅dr 2 2
2
2
B=g 2h2 2
2
2
2
2
ds = Ar , t⋅c ⋅dt −B r ,t ⋅dr −C r , t ⋅r ⋅d sin ⋅d
(4.2.6)
Since the radial coordinate can also be arbitrarily rescaled, arbitrary relationships can be established between the remaining functions: C r , t =1 Schwarzschild coordinates: displays distances perpendicular to the radial direction undistorted B r ,t =C r , t Isotropic coordinates: displays directions undistorted B r ,t =1 Gaussian polar coordinates: displays distances parallel to the radial direction undistorted A r ,t =1 Co-moving coordinates: coordinates of radially falling bodies are constant
(4.2.7)
The arc length squared in Schwarzschild coordinates, we determine the geometric quantities that characterize the surface, from the metric tensor to the Ricci-tensor: ds 2= Ar , t⋅c 2⋅dt 2−B r ,t ⋅dr 2−r 2⋅d 2−r 2⋅sin 2 ⋅d 2
Ar , t 0 0 0 0 −B r , t 0 0 g = 2 0 0 −r 0 2 0 0 0 −r ⋅sin2 92
(4.2.8)
4.2 Schwarzschild coordinates
1 Ar ,t 0
g =
0 −
1 B r ,t
0
0
0
0
0
0
0
0
1 r2
0
−
0
−
1 r ⋅sin2 2
(4.2.9)
We symbolize partial differentiation with respect to time with an upper point, and with respect to space with an upper apostrophe: ∂ g tt = A˙ ∂t
tt ∂g A˙ =− 2 ∂t A
∂ g rr =−B˙ ∂t
rr ∂g B˙ = 2 ∂t B
∂ g tt = A' ∂r
∂ g tt A' =− 2 ∂r A
∂ g rr =−B ' ∂r
∂ g rr B ' = 2 ∂r B
∂ g =−2⋅r ∂r
∂ g 2 = 3 ∂r r
∂ g =−2⋅r⋅sin 2 ∂r
∂ g 2 = 3 ∂r r ⋅sin 2
∂ g =−2⋅r 2⋅cos ⋅sin ∂
∂ g 2⋅cos = 2 ∂ r ⋅sin3
t tt = r
tt =
A˙ 2⋅A
t tr = t rt =
A' 2⋅B
tr = rt =
r =−
r
r
A' 2⋅A
rr =
B˙ 2⋅B
rr =
t
r
r B
r =−
93
B˙ 2⋅A B' 2⋅B
r⋅sin 2 B
(4.2.10)
4.2 Schwarzschild coordinates r = r = r = r =
=
1 r
=−cos ⋅sin
(4.2.11)
=cot
t
t
t
t
¨ A˙ 2 ∂ tt A⋅A− = 2 ∂t 2⋅A
˙ − A⋅A' ˙ ∂ tr ∂ rt ∂ tt A⋅A' = = = 2 ∂t ∂t ∂r 2⋅A
˙ B˙ ∂ t rr A⋅B− ¨ A⋅ = ∂t 2⋅A2
˙ ∂ r tt B⋅A'− A'⋅B˙ = ∂t 2⋅B2
∂ r tr ∂ rrt B⋅B− ¨ B˙ 2 = = ∂t ∂t 2⋅B2
∂ r rr ∂ rtr ∂ r rt B⋅B˙ '− B⋅B ˙ ' = = = ∂t ∂r ∂r 2⋅B 2
∂ r⋅B˙ = 2 ∂t B
r
r 2 ˙ ∂ r⋅B⋅sin = 2 ∂t B
∂ t tr ∂ t rt A⋅A' '− A' 2 = = ∂r ∂r 2⋅A2
∂ t rr A⋅B˙ ' −A '⋅B˙ = ∂r 2⋅A2
∂ r tt B⋅A' '− A'⋅B ' = ∂r 2⋅B 2
∂ r rr B⋅B' '−B ' 2 = ∂r 2⋅B 2
r
r
∂ r⋅B '−B = 2 ∂r B
∂ r⋅B '−B⋅sin 2 = 2 ∂r B
∂ r 2⋅r =− ⋅cos ⋅sin ∂ B
∂ r ∂ r ∂ r ∂ r 1 = = = =− 2 ∂r ∂r ∂r ∂r r
∂ =sin2 −cos2 ∂
∂ ∂ = =−cot 2 −1 ∂ ∂
Rt rtr =−Rt rrt = R
t
R
t
t
=−R
t
=−R
t
t
2 2 ¨ ˙ B˙ B−A' ' B˙ − A'⋅B ' A' − A⋅ − 2 2⋅A 4⋅A⋅B 4⋅A
=−
t t
Rr ttr =−Rr trt =
r⋅A' 2⋅A⋅B
=−
R
r⋅A' 2 ⋅sin 2⋅A⋅B
t r
=−R
t r
=−
r⋅B˙ 2⋅A⋅B
Rt r =−R t r =−
¨ A' ' B˙ 2− A'⋅B ' A ' 2− A⋅ ˙ B˙ B− − 2 2⋅B 4⋅A⋅B 4⋅B 94
r⋅B˙ ⋅sin2 2⋅A⋅B
(4.2.12)
4.2 Schwarzschild coordinates Rr t =−Rr t =
r⋅B˙ 2 2⋅B
Rr t =−R r t= R
tt
R
tr
=−R
t t
=−R
=R
tr
Rr r =−Rr r =
r⋅B˙ ⋅sin 2 2 2⋅B
tt
=R
=−R
rt
Rtt =
=−
A' 2⋅r⋅B
r t
=R
t r
R =−R = 1−
Rr r =−R r r =
tt
=−R
r⋅B ' 2⋅B2 r⋅B' ⋅sin 2 2 2⋅B
Rr r =−Rr r =Rr r =−Rr r=−
=−R t r =R
r t
=−R
r t
=−
B˙ 2⋅r⋅B
1 ⋅sin 2 B
R =−R = 1−
B' 2⋅r⋅B
1 B
(4.2.13)
B˙ B˙ A˙ A ' B' A' A' ' − B¨ A' ⋅ − ⋅ 4⋅B B A 4⋅B B A 2⋅B r⋅B
Rrr =− R =
B˙ B˙ A˙ A' B' A' A ' ' − B¨ B' ⋅ ⋅ − 4⋅A B A 4⋅A B A 2⋅A r⋅B
r⋅B ' r⋅A' 1 − − −1 2 2⋅B 2⋅A⋅B B
R =
r⋅B' r⋅A' 1 − − −1 ⋅sin 2 2 2⋅A⋅B B 2⋅B
Rtr =Rrt =
B˙ r⋅B
(4.2.14)
We derived from the Einstein equations, that in vacuum the Ricci-tensor is zero. The non-diagonal term shows, that the derivative of the B function with respect to time is zero: Rtr =Rrt =
B˙ =0 r⋅B
˙ B=0
(4.2.15)
Reinsert this into the tt and rr components of the Ricci-tensor. Simplify the tt component: Rtt =−
−
A' B' A' A' ' A ' ⋅ =0 4⋅B B A 2⋅B r⋅B
/⋅B
A' B ' A' A' ' A ' ⋅ =0 4 B A 2 r
(4.2.16)
95
4.2 Schwarzschild coordinates Simplify the rr component also: Rrr =
A' B ' A' A'' B' ⋅ − =0 4⋅A B A 2⋅A r⋅B
/⋅A
A' B' A' A' ' B '⋅A ⋅ − =0 4 B A 2 r⋅B
(4.2.17)
Add the two equations together: B'⋅A A' =0 r⋅B r B' A' =0 B A
(4.2.18)
Reinsert the result into the equation coming from the tt component, where now we have just the derivatives of the A function. With substitution we reduce the degree of derivatives: A' ' A' =0 2 r
f r = A'
(4.2.19)
df 2⋅ f =0 dr r
∫
df dr =−2⋅∫ f r
log f =−2⋅log r ⋅c 1=log
c1 r2
We raise to natural power both sides of the equation, and reinsert the original function: f=
c1 r
2
= A' =
dA dr
(4.2.20)
c
∫ r 12⋅dr=∫ dA Thus the A function is also time independent: −
c1 c 2= A r
(4.2.21)
Let us examine the formula, that we got when we added the two equations we made from the Riccitensor components: 96
4.2 Schwarzschild coordinates B' A' =0 B A With a little reordering we can show, that the product of the two functions is a constant: F ' = A⋅B ' A'⋅B=0
→
F =A⋅B=c3
(4.2.22)
From this the other function:
1
B=c 3⋅
c 2−
c1 r
We can reinterpret the unknown variables in a way, that the third is understood as part of the first two, in this case the two unknown functions are mutually reciprocals of each other: B=
c A=c 2− 1 r
1 c 2−
c1 r
c 3=1
(4.2.23)
The first constant is a quantity characteristic for the spherically symmetric spacetime, it is the Schwarzschild radius. In this distance from the centre there is a coordinate singularity, and its physical meaning can be found with Newtonian approximation: r g =c 1
(4.2.24)
The value for the second constant can be recovered from the condition, that the Schwarzschild solution at great distances shall approach the flat spacetime, in other words A and B in the arc length squared shall approach 1: c1 =c 2−0=1 r ∞ r
(4.2.25)
lim A=c 2 −lim r ∞
Because of the time independence of the metric, if a celestial body suffers radial changes, but does not receive or lose mass, the geometry of the surrounding spacetime will not change. For example a spherically symmetric pulsating star does not create gravitational radiation, neither a symmetric supernova explosion nor a celestial body collapsing to a black hole. Because of the slow rotation of the Sun and the minimal contribution of the planets, the spacetime of the Solar System is Schwarzschild to a great accuracy. The arc length squared and the other geometric quantities in the Schwarzschild metric:
ds 2= 1−
rg 2 2 dr 2 ⋅c ⋅dt − −r 2⋅ d 2sin 2 ⋅d 2 r r 1− g r
97
(4.2.26)
4.2 Schwarzschild coordinates
0
g =
rg r
1−
0 −
0 0
1 r 1− g r 0 0
1
0
0
0
−r 2 0 2 0 −r ⋅sin2
0
rg 1− r
g =
0
0
− 1−
0
0
0
rg r
0
0
0
0
1 r2
0
−
0
0
1 − 2 r ⋅sin2
(4.2.27)
∂ g rr ∂ g tt =− = ∂r ∂r
∂ g tt ∂ g rr r g =− = ∂r ∂ r r2
rg
∂ g =−2⋅r ∂r
∂ g 2 = 3 ∂r r
∂ g =−2⋅r⋅sin 2 ∂r
∂ g 2 = 3 ∂r r ⋅sin 2
∂ g =−2⋅r 2⋅cos ⋅sin ∂
2⋅cos ∂g = 2 3 ∂ r ⋅sin
t tr = t rt =− r rr =
r
rg 2⋅r⋅r −r g
=−r −r g
r = r = r = r =
r tt =
=
1 r
r
2
r r ⋅ 1− g r 2
(4.2.28)
r g⋅r −r g 2⋅r 3 2
=− r−r g ⋅sin
=−cos ⋅sin (4.2.29)
=cot
98
4.2 Schwarzschild coordinates t
t
r
∂ tr ∂ rt ∂ rr r ⋅ 2⋅r −r g = =− =− g 2 2 ∂r ∂r ∂r 2⋅r ⋅r −r g r
∂ r =−1 ∂r
∂ tt r ⋅2⋅r −3⋅r g =− g 4 ∂r 2⋅r
∂ r =−sin 2 ∂r
∂ r ∂ r ∂ r ∂ r 1 = = = =− 2 ∂r ∂r ∂r ∂r r ∂ r =−2⋅r −r g ⋅cos ⋅sin ∂
∂ =sin2 −cos 2 ∂
∂ ∂ = =−cot 2 −1 ∂ ∂ Rt rtr =−Rt rrt =
(4.2.30)
rg 2
r ⋅ r−r g
Rt t =−Rt t =Rr r =−Rr r =−
rg 2⋅r
Rt t =−Rt t= Rr r =−Rr r =− Rr trt =−Rr ttr =
rg ⋅sin 2 2⋅r
r g⋅r −r g r
4
Rt t =−R t t =Rt t=−Rt t =
r g⋅r −r g
Rr r =−Rr r =Rr r =−Rr r =−
2⋅r 4 rg 2
2⋅r ⋅r −r g
r R =−R = g⋅sin2 r R =−R =
rg r
(4.2.31)
This spacetime is a hypersurface of a six dimensional, flat, pseudo-euclidean space with a signature of (+ + – – – –). The parametric form:
r x 1= 1− g⋅cos c⋅t r
r x 2= 1− g⋅sin c⋅t r 99
4.2 Schwarzschild coordinates
x 3=∫
rg r ⋅ g 3 1 ⋅dr r −r g 4⋅r
x 4 =r⋅sin ⋅cos
x 5=r⋅sin ⋅sin
x 6=r⋅cos
(4.2.32)
4.3 Geodesic equations Substitute the connection coefficients of the Schwarzschild solution into the geodesic equations: t c⋅t¨ 2⋅ tr⋅c⋅t˙⋅r˙ =0
t¨
rg ⋅t˙⋅r˙ =0 r⋅r −r g r
2
2
r
(4.3.1) 2
r¨ tt⋅c ⋅˙t rr⋅r˙
r¨
r
2
⋅˙
r
2
⋅ ˙ =0
r g⋅ r−r g 2 2 rg ˙ ⋅c ⋅ t − ⋅r˙ 2−r −r g ⋅˙ 2−r −r g ⋅sin2 ⋅ ˙ 2=0 3 2⋅r⋅r −r g 2⋅r
(4.3.2)
¨ ˙ 2⋅ ˙ 2=0 r ⋅r˙ ⋅ 2 ¨ 2⋅r˙ −cos ˙ ⋅sin ⋅ ˙ =0 r
2⋅ ¨
r
⋅r˙ 2⋅ ˙
(4.3.3)
˙ =0 ⋅⋅ ˙
2 ˙ =0 ⋅r˙ 2⋅cot ⋅⋅ ¨ ˙ ˙ r
(4.3.4)
Let us investigate the third equation. If we orientate the coordinate system in such a way, that the test body is in the equatorial plane of the coordinate system, and the direction of the motion falls into this plane, then it will also stay in this plane. Substitute the longitudinal angle and the momentarily zero angular velocity along this coordinate into the third geodesic equation: =
2
˙ =0
2 2 ¨ 2⋅r˙ −cos ˙ ¨ 2⋅r˙ 0−0⋅1⋅ ⋅sin ⋅ ˙ = ˙ =0 r r
The longitudinal angular acceleration is zero, the test body stays in the equatorial plane: 100
4.3 Geodesic equations ¨ =0
(4.3.5)
Substitute into the other geodesic equations as well: t¨
rg ⋅t˙⋅r˙ =0 r⋅r −r g
(4.3.6)
r¨
r g⋅ r−r g 2 2 rg ⋅c ⋅˙t − ⋅r˙ 2−r −r g ⋅ ˙ 2=0 3 2⋅r⋅r −r 2⋅r g
(4.3.7)
¨ =0
(4.3.8)
2 ⋅r˙ =0 ¨ ˙ r
(4.3.9)
4.4 Gravitational redshift This phenomenon is one of the classical evidences for general relativity. Einstein described it already in 1907, based on the principle of equivalence, but initially he did not think it was possible to measure it experimentally. The experiment was eventually performed by R. V. Pound and G. A. Rebka, in 1959 in the laboratory of Harward University, in the United States. The measured red shift of the gamma rays emitted by radioactive iron atoms, and directed 22.5 meters upwards to the detectors confirmed Einstein's prediction within the 10% error margin. Later the error margin has been reduced to less than 1% using hydrogen masers. We substitute the Schwarzschild metric tensor components into the earlier formula:
rg 2 g tt 2r ⋅2= ⋅ 1 = rg 2 1 g tt 1− 1r
1−
(4.4.1)
If the light source is closer to the source of the gravitational field than the observer, then: 1
r ≥2r
→
1
≤2
(4.4.2)
Thus the observed frequency is higher than the emitted, the radiation in the visible spectrum is shifted towards the red, this is where the name of the phenomenon comes from. Redshift observed by an observer at a great distance, where the light source is at r distance from the centre of gravitation (like the surface of a star):
101
4.4 Gravitational redshift z=
1
r 1− g r
−1
(4.4.3)
4.5 Wormhole In order to demonstrate the shape of the spacetime, we investigate the properties of a coordinate surface. The parameters of the surface: 2
t=const.
=
dt=0
d =0
(4.5.1)
Substitute them into the Schwarzschild arc length squared:
ds 2= 1−
rg 2 2 dr 2 ⋅c ⋅0 − −r 2⋅ 0 2sin 2 ⋅d 2 r rg 2 1− r
−ds 2=dl 2 =
dr 2 r 2⋅d 2 rg 1− r
(4.5.2)
The characteristic geometric quantities on the surface:
1
r g ij = 1− g r 0 ∂ g rr =− ∂r
0
ij
g =
rg
rg r
rg 2⋅r⋅r −r g
0
(4.5.3)
1 r2
∂ g rr r g = ∂ r r2
2
∂ g =2⋅r ∂r r rr =−
rg r
0
r2
r 2⋅ 1−
1−
∂ g 2 =− 3 ∂r r
r
=− r−r g
102
(4.5.4)
r = r=
1 r
(4.5.5)
4.5 Wormhole r
r
∂ rr r g⋅2⋅r−r g = 2 ∂r 2⋅r⋅r −r g
−
Rij =
∂ =−1 ∂r
rg
0
2
2⋅r ⋅ r−r g
r − g 2⋅r
0
R=2⋅K =−
∂ r ∂ r 1 = =− 2 ∂r ∂r r
(4.5.7)
rg r
(4.5.6)
(4.5.8)
3
This coordinate surface can be embedded into the flat three dimensional space, thus its easy to visualize. We set up a spherical coordinate system, and spread the surface in it. The arc length squared: 2
2
2
2
dl =dr r ⋅d dz
2
Make it equal with the arc length squared measured on the surface: dr 2 dr r ⋅d dz = r 2⋅d 2 r 1− g r 2
2
2
dr 2dz 2=
2
2
dr r 1− g r
(4.5.9)
We do not know the relationship that describes the z coordinates of the surface, therefore we substitute it as an unknown function, and since the surface inherited circular symmetry from the Schwarzschild metric, we assume that it depends on the radius only: z = f r dz = f ' r ⋅dr dr 2 1 f ' r ⋅dr = r 1− g r 2
f ' r =
2
rg r −r g
(4.5.10)
Perform the integration. The shape of the entry of the wormhole:
103
4.5 Wormhole f r =2⋅ r g⋅r−r g c
(4.5.11)
The surface with angular and rectangular coordinates:
4.6 Newtonian approximation According to the correspondence principle, the new physical laws in limiting cases must approximate the old ones, this is no different in the case of relativity theory. In this case the previous model is the Newtonian absolute space and time, and the forces and emerging potentials acting in it. In order to establish the correspondence, we must formulate the two gravitational theories in the same language. Classical mechanics does not work with geometric methods, but with diverse terms instead, like the force, and other quantities derived from it. Using this set of tools, it is possible to describe only a limited set of gravitational phenomena, but within its limit, it provides correct results. Since this can be experimentally verified, the broadly valid geometry based theory has to approach the Newtonian model within the mentioned limits. The validity of the former is limited to those situations, where movements are much slower than the speed of light, and the proper time coincides with the coordinate time: v ≪c
d ≈dt
(4.6.1)
The Lagrange function summarizes the properties of a dynamical system. From it the equations of movement can be derived using the action principle. With the action functional in the nonrelativistic case, the following relationship is satisfied: t2
S [ x t ]=∫ L x , x˙ , t ⋅dt
(4.6.2)
t1
According to the action principle, the evolution of a mechanical system is characterized by the solution of the following functional equation:
104
4.6 Newtonian approximation S =0 x t
(4.6.3)
Let x(t) be the function describing the possible evolution of the system. In this case ε(t) is an infinitesimally small variation on it, that is zero in the starting and ending points, this is our boundary condition: t 1 = t 2=0
(4.6.4)
Use it to vary the action functional, we assume that the Lagrange function does not depend on time: t2
S =∫ L x , x˙ − ˙ L x , x˙ ⋅dt
(4.6.5)
t1
Write down the Taylor series of the Lagrange function, and we write down the variation of the action functional again, using the first order terms: ∂L ∂L L x , x˙ ˙ =L x , x˙ i⋅ i ˙i⋅ i ∂x ∂ x˙ t2
∂L ∂L S i=∫ i⋅ i ˙ i⋅ i ⋅dt ∂x ∂ x˙ t 1
(4.6.6)
Partially integrate the second term: t2
[
t2
t2
1
1
]
∂L ∂L d ∂L S i=∫ ˙ i⋅ i⋅dt= i t⋅ i −∫ i⋅ ⋅dt dt ∂ x˙ i ∂ x˙ ∂ x˙ t t t 1
Reinsert it into the variation equation: t2
t2
1
1
]
[
∂L ∂L d ∂L S i= i t ⋅ i ∫ i⋅ i −i⋅ ⋅dt i dt ∂ x˙ t t ∂x ∂ x˙
(4.6.7)
Because of the boundary condition, the first term is zero: t2
S i=∫ i⋅ t1
∂ L d ∂L − ⋅dt=0 ∂ x i dt ∂ x˙ i
According to the action principle, the variation of the action functional is zero. This is satisfied, if the expression in the parentheses is zero, that is the general equation of movement: d ∂ L ∂L − =0 dt ∂ x˙ i ∂ xi
(4.6.8)
105
4.6 Newtonian approximation In a non-relativistic conservative force field the Lagrange function is the difference of the kinetic and the potential energy: L=E k −E p
(4.6.9)
The total energy of the moving body: E=
m⋅c
2 2
1−
v 2 c
The kinetic energy is the part above the rest energy:
E k =m⋅c2⋅
1
2
1−
v c2
−1
(4.6.10)
Write down the expression in the parentheses with a binomial series: n
abn =∑ i=0
v2 1− 2 c
−
1 2
n⋅ n−1 n−2 2 n! ⋅a n −i⋅bi=a nn⋅an −1⋅b ⋅a ⋅b n−i!⋅i! 2!
1 v2 3 v4 =1 ⋅ 2 ⋅ 4 2 c 8 c
(4.6.11)
Reinsert it into the formula for kinetic energy:
1 v2 3 v4 E k =m⋅c ⋅ ⋅ 2 ⋅ 4 2 c 8 c 2
At velocities slow compared to the speed of light, the kinetic energy is approximately the following: 1 E k ≈ ⋅m⋅v 2 2
(4.6.12)
In a central force field the potential energy depends only on the mass of the test body and the distance from the centre: E p =m⋅r
(4.6.13)
In Newtonian mechanics space is flat. Write down the Lagrange function is a three dimensional spherical coordinate system. The velocity squared is calculated from the arc length squared:
106
4.6 Newtonian approximation 1 ds 2=dr 2 r 2⋅ d 2 sin2 ⋅d 2 /⋅ 2 ∂t v 2 =r˙ 2r 2⋅ ˙ 2sin2 ⋅ ˙ 2
(4.6.14)
The Lagrange function:
1 2 L=E k −E p =m⋅ ⋅ r˙ 2r 2⋅ ˙ 2sin 2 ⋅ ˙ − r 2
(4.6.15)
Substitute it into the equation of movement and divide with the mass: d ∂ L ∂L 1 − i =0 /⋅ i dt ∂ x˙ ∂ x m The first term according to the radial coordinate:
2
d ∂ 1 2 2 ˙2 1 d ∂ r˙ ⋅ r˙ r ⋅ sin2 ⋅ ˙ 2 −r = ⋅ =r¨ dt ∂ r˙ 2 2 dt ∂ r˙ The second term according to the radial coordinate:
d r 1 2 2 ˙2 2 2 2 2 −∂ ⋅ r˙ r ⋅ sin 2 ⋅ ˙ −r =−r⋅˙ −r⋅sin ⋅ ˙ ∂r 2 dr The equation of motion in the radial direction is their sum: 2 2 2 r¨ −r⋅˙ −r⋅sin ⋅ ˙
d r =0 dr
(4.6.16)
The first term of the equation of movement according to the latitude:
d ∂ 1 2 2 ˙2 1 d d 2 2 ˙ ¨ ˙ 2⋅¨ ⋅ r˙ r ⋅ sin2 ⋅ −r = ⋅ ∂ r 2⋅˙ 2 = r 2⋅r ⋅=2⋅r⋅ r˙⋅r ˙ dt ∂ ˙ 2 2 dt ∂ ˙ dt The second term according to the latitude:
1 2 2 ˙2 2 2 2 2 2 2 − ∂ ⋅ r˙ r ⋅ sin 2 ⋅ ˙ −r =− ∂ r ⋅sin ⋅ ˙ =−r ⋅cos ⋅sin ⋅ ˙ ∂ 2 ∂ The equation of motion in the latitude direction is their sum: 2 ¨ 2⋅r⋅ ˙ ⋅sin ⋅ ˙ =0 ˙ −cos r
(4.6.17)
107
4.6 Newtonian approximation The first term of the equation of motion according to the longitude:
d ∂ 1 2 2 ˙2 1 d ∂ 2 2 ⋅ r˙ r ⋅ sin 2 ⋅ r 2⋅sin 2 ⋅ ˙ −r = ⋅ ˙ = dt ∂ 2 2 dt ∂ ˙ ˙ d 2 d 2 2 d 2 r ⋅sin 2 ⋅= r ⋅sin 2 ⋅r ⋅ sin2 ⋅r ⋅sin 2 ⋅= ˙ ˙ ˙ ¨ dt dt dt 2 2 2⋅r⋅r˙⋅sin 2 ⋅r ˙ ⋅2⋅cos ⋅sin ⋅r ˙ ⋅sin 2 ⋅ ¨
The second term is zero, therefore by rewriting the first term we get the equation of motion according to the longitude: 2 ˙ =0 ⋅r˙⋅2⋅cot ⋅⋅ ¨ ˙ ˙ r
(4.6.18)
In flat space using a spherical coordinate system the radial geodesic equation: r¨
r
2
⋅˙
r
2 2 2 2 ⋅ ˙ = r¨ −r⋅˙ −r⋅sin ⋅ ˙ =0
In the presence of a central gravitational field we see a difference in the radial equation of motion: d r 2 r¨ −r⋅˙ 2−r⋅sin2 ⋅ =0 ˙ dr Only the time-like coordinate velocity is constant, therefore the new term is an exactly identifiable connection coefficient: d c⋅dt c⋅dt = r tt⋅ ⋅ =c 2⋅ r tt dr dt dt
(4.6.19)
We seek the metric that produces geodesics, like the Newtonian equations of movement. The calculation of the connection from the metric:
∂ g ∂ g ∂ g 1 = ⋅g ⋅ − 2 ∂ x ∂ x ∂ x ∂ g t ∂ g t ∂ g tt 1 r tt = ⋅g r ⋅ − 2 ∂t ∂t ∂x
(4.6.20)
Assume that the metric does not depend on time:
∂g ∂g ∂g ∂g ∂g 1 1 r tt =− ⋅g r ⋅ tt =− ⋅ g rt⋅ tt g rr⋅ tt g r ⋅ tt g r ⋅ tt 2 2 ∂t ∂r ∂ ∂ ∂x
In the spherically symmetric spacetime it also does not depend on the angular coordinates, thus the 108
4.6 Newtonian approximation equation simplifies: ∂g 1 r tt =− ⋅g rr⋅ tt 2 ∂r
(4.6.21)
∂g 1 d r 1 ⋅ =− ⋅−1⋅ tt 2 dr 2 ∂r c 2 d r ∂ g tt ⋅ = 2 dr ∂r c 2 ⋅r c 1=g tt 2 c
(4.6.22)
At great distances from the source of gravity the shape of space approaches the plain, with this we can determine the integration constant: lim r =0 r ∞
lim g tt = lim
r 0
r 0
2 ⋅ r c 1=0c 1=1 c2
(4.6.23)
The form of the gravitational potential in the Newtonian theory of gravity: r =−
⋅M r
(4.6.24)
Where γ is the gravitational constant, M is the central mass causing gravity, r is the distance of the test body from it. Substitute it into the metric tensor component, and write down the arc length squared of the spacetime, that causes the exact same geodesics, like the Newtonian equations of movement: g tt =1−
2⋅⋅M 2 r⋅c
ds 2= 1−
(4.6.25)
2⋅⋅M ⋅c 2⋅dt 2−dr 2 −r 2⋅ d 2 sin2 ⋅d 2 2 r⋅c
(4.6.26)
Compare the corresponding metric tensor components of the Schwarzschild metric and the just derived Newtonian limiting case, and with this we can determine the second integration constant of the Schwarzschild derivation, the Schwarzschild radius: g tt =1−
r 2⋅⋅M =1− g 2 r r⋅c
109
4.6 Newtonian approximation r g=
2⋅⋅M 2 c
(4.6.27)
The presently accepted value of the gravitational constant: =6.67428⋅10−11
m3 s 2⋅kg
This constant of nature is one of the hardest to measure, therefore using methods of celestial mechanics, the masses of celestial bodies can be measured only with the same error margin. Thus for precise orbit calculations the product of the two quantities is used, this is the standard gravitational parameter. With this it is possible to calculate the Schwarzschild radius of celestial bodies as well, with great accuracy. The Schwarzschild arc length squared, using pure SI units:
ds 2= 1−
2⋅⋅M dr 2 2 2 ⋅c ⋅dt − −r 2⋅ d 2 sin2 ⋅d 2 2 2⋅⋅M r⋅c 1− r⋅c 2
(4.6.28)
4.7 Circular orbit All energy present in the system causes spacetime curvature, therefore we let go a test body in it with such a small mass, that has negligible influence on events. For the sake of simplicity, it will move on a circular orbit around the gravitational centre, at a distance far greater than the Schwarzschild radius, along a force-free local line, a geodesic. In order to determine the orbit parameters, we write down the coordinate conditions first, that come from the properties of the circular orbit: t=t
∂t =const. ∂
r =const.
dr =0
2
d =0
=
=
∂ r ∂2 r = =0 ∂ ∂ 2
∂ =const. ∂
These simplify the general geodesic equations: t¨
rg ⋅t˙⋅r˙ =0 r⋅r −r g
110
(4.7.1)
4.7 Circular orbit t¨ =0
r¨
(4.7.2)
r g⋅ r−r g 2 2 rg ⋅c ⋅˙t − ⋅r˙ 2−r −r g ⋅ ˙ 2=0 3 2⋅r⋅r −r g 2⋅r
r g⋅ r−r g 2 2 ⋅c ⋅˙t − r−r g ⋅ ˙ 2=0 3 2⋅r
(4.7.3)
¨ =0
(4.7.4)
2 ⋅r˙ =0 ¨ ˙ r (4.7.5)
=0 ¨
Utilize the coordinate conditions on the arc length squared as well:
ds 2= 1−
rg 2 2 02 ⋅c ⋅dt − −r 2⋅ 02 sin2 ⋅d 2 r r 2 1− g r
ds 2= 1−
rg 2 2 2 ⋅c ⋅dt −r ⋅d 2 r
(4.7.6)
This is the point of view of the infinitely distant observer. The astronaut moving on the orbit however does not see this. He “feels” to be weightless, and by him the Minkowski arc length squared can be written down locally. We set it equal to the arc length squared seen by the distant observer:
c 2⋅d 2= 1−
rg 2 2 2 ⋅c ⋅dt −r ⋅d 2 r
rg r 2 d 2 d = 1− − 2⋅ 2 ⋅dt 2 r c dt 2
d = dt
Substitute the angular velocity along the orbit, and we get the relationship between the proper time and the coordinate time: d =
1−
rg r 2⋅ 2 − 2 ⋅dt r c
(4.7.7)
Since the ratio of the two quantities is constant, the coordinate time is also guaranteed to grow monotonically during the movement, therefore it can be used as parameter when solving the geodesic equation. The geodesic equation in the radial direction:
111
4.7 Circular orbit 2 r g⋅ r−r g 2 ∂t 2 ∂ ⋅c ⋅ − r−r g ⋅ =0 3 ∂t ∂t 2⋅r
r g⋅ r−r g 2 ⋅c − r−r g ⋅ 2=0 3 2⋅r rg 2⋅r
⋅c 2− 2=0
3
The angular frequency of a test object with negligible mass on a circular orbit:
=c⋅
rg
(4.7.8)
2⋅r 3
The orbital period calculated from it equals to the Newtonian limiting case at arbitrary orbital radii: t k=
2⋅ 2⋅r 3 ⋅ c rg
(4.7.9)
The Earth orbits on an approximately circular orbit, therefore it is a good example to demonstrate the relationship. The standard gravitational parameter of the Sun and its gravitational radius: ⋅M =1.32712440018⋅1020
3
m s2
r g=
→
2⋅⋅M 3 =2.9532500765⋅10 m 2 c
The semi-major axis of the Earth's orbit: 11
r =1.49598261⋅10 m
From these we can calculate the orbital period: 7
t k =3.15583195⋅10 s=365.258328 days
(4.7.10)
The difference between the actual and the calculated orbital period: t k2 =365.256363004 days
tk −1=5.37919714⋅10−6 t k2
(4.7.11)
The difference from the measured value is caused by ignoring that the Earth's orbit deviates from the ideal circle, the other planets also influence the Earth's movement, and the Sun's rotation also has an influence on spacetime. The ratio of the orbital periods and the radii of the circular orbits gives Kepler's third law:
112
4.7 Circular orbit 2 1 k 2 2 k
t t
=
2
r3
1
r3
(4.7.12)
Substitute the angular frequency of the orbit into the relationship of the proper time and the coordinate time:
d =
2
rg
2
r 1− g − r
d = 1−
r ⋅ c⋅
2⋅r 3
c2
⋅dt
3⋅r g ⋅dt 2⋅r
(4.7.13)
If the change of the proper time is zero, it means a light-like geodesic, thus we are speaking about light on a circular orbit:
0= 1−
3⋅r g ⋅dt 2⋅r
The radius of the orbit: r=
3⋅r g 2
(4.7.14)
Objects slower than the speed of light can orbit around the centre only at greater distances than this. The circular geodesics inside this radius are all space-like, that is shown in the fact that the number under the square root is negative, thus the proper time becomes an imaginary quantity.
4.8 Surface acceleration and hovering If an object does not move in a Schwarzschild coordinate system, for example it is at rest on the surface of a spherical planet, how much is its acceleration? We perform the calculations with respect to the coordinate time. In the case of an every-day size planet with a solid surface it is not a significant discrepancy. The coordinate conditions in this case: t=t
∂t =const. ∂
r =const.
dr =0
=const.=
2
d =0
113
4.8 Surface acceleration and hovering d =0
=const.
(4.8.1)
We are looking for the radial acceleration, we substitute into the corresponding equation of movement: r¨
r g⋅ r−r g 2 2 rg ⋅c ⋅˙t − ⋅r˙ 2−r −r g ⋅ ˙ 2=0 3 2⋅r⋅r −r 2⋅r g
Surface acceleration in the Schwarzschild solution, when the rotation of the planet is negligible: r¨ =−
r g⋅r −r g 2 2 ⋅c ⋅˙t 2⋅r 3
(4.8.2)
If we want to take the rotation of the planet into account, the coordinate conditions will expand, the observer will perform circular motion along a latitude. We continue to work with the same equation of movement (because of using the Schwarzschild metric, we neglect the effects of the rotation on the spacetime, but in the case of a small angular momentum, this is an adequate approximation): t=t
∂t =const. ∂
r =const.
dr =0
=const.
d =0
=const.
d =const. d
(4.8.3)
We start with the most general radial equation of movement: r¨
r g⋅ r−r g 2 2 rg ⋅c ⋅˙t − ⋅r˙ 2−r −r g ⋅˙ 2−r −r g ⋅sin2 ⋅ ˙ 2=0 3 2⋅r⋅r −r 2⋅r g
Surface acceleration in the Schwarzschild solution, when the planet rotates: r¨ =−
r g⋅r −r g 2 ⋅c r−r g ⋅sin2 ⋅2 3 2⋅r
(4.8.4)
The standard gravitational parameter of Earth, and the gravitational radius: ⋅M =3.986004418⋅1014
m3 s2
r g=
→
2⋅⋅M =8.870056078⋅10−3 m 2 c
The equatorial radius of Earth and the angular frequency of the rotation:
114
4.8 Surface acceleration and hovering =
6
r =6.3781366⋅10 m
Surface acceleration on the Earth's equator r¨ =−9.7982867
2⋅ 1 =7.292115⋅10−5 tk s
= 2 :
m m m 0.0339157 2 =−9.764371 2 2 s s s
(4.8.5)
The actual acceleration measured on the Earth's equator, and the difference from the calculated value: r¨2=−9.780327
m 2 s
r¨2 −1=1.634105⋅10−3 r¨
(4.8.6)
The discrepancy from the geographic value is caused by the Earth's not exactly spherical shape, this has mainly an impact on the term that takes the rotation into account. Acceleration of a hovering object with respect to distance, from the point of view of the infinitely distant observer: a=−
r g⋅ r−r g 2 ⋅c 3 2⋅r
(4.8.7)
On the graphic the distance from the gravitational centre increases from left to right, the coordinate acceleration of the hovering body is displayed on the vertical axis, the dotted line shows the position of the event horizon: a
r
Approaching the gravitational radius, the coordinate acceleration goes to zero:
115
4.8 Surface acceleration and hovering a=lim − r r g
r g⋅r −r g 2⋅r
3
⋅c 2=0
(4.8.8)
The derivative according to the radial coordinate is zero at the point of maximal acceleration: 2⋅r −3⋅r g 2 ∂a =r g⋅ ⋅c =0 ∂r 2⋅r 3 3 r = ⋅r g 2
(4.8.9)
4.9 Geodesic precession In 1916, while working on the relativistic correction of the Moon's orbit, Willem de Sitter Dutch astronomer pointed out this phenomenon. By analysing laser light reflected from prisms placed on the surface of the Moon during the Apollo program the phenomenon was confirmed to 0.7% accuracy. NASA launched Gravity Probe B in 2004 with the best mechanical gyroscopes on board ever created by mankind. The result of the experiment, that confirmed the accuracy of the theory of relativity within 1% was published in April of 2007 at the annual congress of the American Physical Society. Maybe one of the best evidences for the curvature of spacetime is the parallel displacement along a geodesic, for example a circular orbit. The direction of the vector arriving at the starting point will differ from the original. Parallel displacement along a geodesic, where v is the vector in the original point, u is already displaced: u =v − ⋅v ⋅dx
(4.9.1)
Our geodesic of choice is the circular orbit. The infinitesimal displacement vector along the orbit: dx =c⋅dt 0 0 d
(4.9.2)
The vector lays in the orbital plane, therefore its zeroth and second components remain zeroes after the displacement: u =0 u r 0 u
v =0 v r 0 v
(4.9.3)
The orbiting happens in the equatorial plane: =
2
(4.9.4)
Substitute the connection of the Schwarzschild solution into the formula of parallel displacement:
116
4.9 Geodesic precession u t=v t − t tr⋅v r⋅dt− t rt⋅v t⋅dr =v t − r
r
r
t
r
r
u =v − tt⋅v ⋅dt− rr⋅v ⋅dr−
r
r
rg ⋅v r⋅dt 2⋅r⋅r −r g
⋅v ⋅d −
u =v − r ⋅v ⋅dr − r⋅v ⋅d −
r
(4.9.5) r
⋅v ⋅d =v r −r g ⋅v ⋅d
(4.9.6) (4.9.7)
⋅v ⋅d =0
1 u =v − r ⋅v ⋅dr − r⋅v r⋅d − ⋅v ⋅d − ⋅v ⋅d =v − ⋅v r⋅d r
(4.9.8)
Rearrange the second and fourth displacement equations, substitute the difference between the original and the displaced vector, as well as the angular velocity: r
dv r −r g ⋅v ⋅d =0 v¨ rr −r g ⋅v˙ ⋅=0
1 /⋅ 2 dt
←
v −u =dv
←
d = dt
(4.9.9)
1 r 1 dv − ⋅v ⋅d =0 /⋅ r dt 1 r v˙ = ⋅v ⋅ r
(4.9.10)
Express the change of the displaced vector, and substitute it into the radial displacement equation: v¨ r
r −r g 2 r ⋅ ⋅v =0 r
v¨ r=−
(4.9.11)
r −r g 2 r ⋅ ⋅v r
This is the differential equation of the harmonic oscillator, that looks like this in the general case: d2 sin⋅t =−2⋅sin ⋅t 2 dt
r
v =sin ⋅t We can see its angular frequency: =
r −r g ⋅ r
(4.9.12)
The geodesic precession is the difference between this and the angular frequency:
117
4.9 Geodesic precession
=−= 1−
r −r g ⋅ r
(4.9.13)
In the case of weak gravitational field this effect is small, but accumulates over several revolutions. The orbital period and frequency of Gravity Probe B, that was in Earth orbit for 50 weeks between 2004 and 2005 at an altitude of 642 km: t k =5850 s=1 h 37 min 30 s
r =7013000 m =
2⋅ 1 =1.074⋅10−3 tk s
The standard gravitational parameter of Earth, and the gravitational radius: ⋅M =3.986004418⋅1014
3
m 2 s
r g=
→
2⋅⋅M −3 =8.870056078⋅10 m 2 c
From these the angular velocity of the geodesic precession: =6.792⋅10−13
1 s
(4.9.14)
Rotation in a year: −5
(4.9.15)
=⋅t year =2.143⋅10 rad =4.421 ' '
The de Sitter effect is the precession of the lunar orbit in the gravitational field of the Sun. The orbital frequency of the Earth from the orbital period: t k =365.256363004 days
=
→
2⋅ −7 1 =1.99098659277⋅10 tk s
We substitute the gravitational radius of the Sun (from the standard gravitational parameter) and the radius of the Earth orbit into our derived equation: ⋅M =1.32712440018⋅1020
m3 s2
r g=
→
2⋅⋅M =2.9532500765⋅103 m 2 c
The semi-major axis of the Earth orbit: 11
r =1.49598261⋅10 m The angular velocity of the lunar orbit's precession: =1.96522383⋅10−15
1 s
(4.9.16)
118
4.9 Geodesic precession Rotation in a year: −8
(4.9.17)
=⋅t year =6.20188278⋅10 rad =0.0127923015' '
4.10 Stability of circular orbits In the spacetime of the Schwarzschild solution, circular orbits are obviously geodesics, but this is just a theoretical situation, the trajectories of real objects always deviate from this, if only a little bit. The question is, will the spacetime geometry generated by the central gravitating celestial body correct their movement if they stray from the ideal path? Will they move on stable orbits, or will the geometry increase the perturbation and make them leave the system forever, or turn in the wrong direction and increase the mass of the central celestial object? Therefore we want to find out, what orbital radius belongs to what angular frequency, and in the vicinity of the orbit, in what direction will the geometric potential herd the orbiting bodies. Calculating the elapsed proper time on time-like geodesics in a single plane, around the gravitational centre (we use the metric functions as a shorthand): =
2
d =0
1 c 2⋅d 2=A⋅c 2⋅dt 2−B⋅dr 2−r 2⋅d 2 /⋅ 2 c ⋅d 2
(4.10.1)
dt 2 B dr 2 r 2 d 2 1= A⋅ 2 − 2⋅ 2 − 2⋅ 2 d c d c d We determine the components of the covariant tangent vector from the arc length squared, in the time-like and horizontal direction they are the following: 2
2
ds t = A⋅c ⋅dt
2
dt u t= A⋅c 2⋅ d
2
2
ds =r ⋅d
u =r 2⋅
2
d 2 =r ⋅ d
1 /⋅ d (4.10.2)
These quantities are constants of movement, because the metric tensor does not depend on the coordinates they are directed to (see chapter 1). Substitute the relationship between the tangent vectors and the two metric functions, and we determine the dependence of the radial velocity from the selected velocity vectors depending on the distance: B=
1 A
119
4.10 Stability of circular orbits 2
2
2 u u 1 dr 1= t 2 − ⋅ − 22 2 2 A⋅c A⋅c d r ⋅c
/⋅A⋅c 2
u 2 2 dr 2 2 =u − A⋅ c t 2 2 d r
(4.10.3)
Substitute the function from the Schwarzschild solution:
r g u2 2 dr 2 2 =u t − 1− ⋅ 2 c 2 r d r
(4.10.4)
The second term on the right is the geometric potential:
U eff = 1−
2
r g u 2 ⋅ 2 c r r
(4.10.5)
We are discussing the behaviour of this function. Where the derivative according to the radial coordinate is zero, the geometric potential is horizontal, this means a potential orbit around the gravitational centre:
2
2
dU eff r g u 2 r 2⋅u = 2⋅ 2 c − 1− g ⋅ 3 =0 dr r r r r r g⋅c 2⋅r 2−2⋅u 2⋅r3⋅u2⋅r g =0
(4.10.6)
Solve the quadratic equation, the canonical form and the quadratic formula: a⋅x 2 b⋅xc=0 −b± b2−4⋅a⋅c x 1,2= 2⋅a Substitute into the quadratic formula: r 1,2=
2⋅u2 ± 2⋅u2 2−4⋅r g⋅c 2⋅3⋅u 2⋅r g 2⋅r g⋅c 2
The geometric potential is horizontal at the following distances, the radii of possible orbits at a given horizontal velocity: r 1,2=
u 2±u⋅ u 2−3⋅r 2g⋅c 2
(4.10.7)
r g⋅c 2
120
4.10 Stability of circular orbits The piece of the geometric potential that is interesting to us, on a logarithmic diagram, with the maxima and minima noted: log(Ueff) r2
r1 log(r) We get circular orbits only if the quantity under the square root, the discriminant is not negative. In the other case, there can be no circular orbit, the test object falls on a spiral path, or leaves forever in the opposite direction. If the discriminant is zero: u 2=3⋅r 2g⋅c 2
(4.10.8)
Reinsert into the quadratic formula: r 1,2=
3⋅r 2g⋅c 2± 3⋅r 2g⋅c 2⋅ 0 r g⋅c 2
r 1=r 2=3⋅r g
(4.10.9)
In the general case, r1 is always greater and r2 is always less than this limit separating the two kinds of circular orbits. Since the geometric radius of the Sun vastly exceeds this limiting case, only r1 orbits occur in the Solar System. By substituting these two values into the second derivative of the geometric potential, it turns out that the r1 orbits are stable, r2 orbits are unstable geodesics, as we can see it on the graph.
4.11 Perihelion precession After Urbain le Verrier – by examining the orbit of Uranus – discovered Neptune on paper, he did similar calculations in 1859 regarding the movement of Mercury. After he determined the contributions by every other planet, a discrepancy remained, greater than the error margin between the measured and calculated values of the perihelion precession. Contemporary explanations failed, until Einstein in 1915 was able to explain this anomaly easily using general relativity, this problem became one of the classical tests of his theory. At this time, none of the exact solutions of his 121
4.11 Perihelion precession equation were known (except for the flat spacetime), therefore Einstein used a different version than the one presented here. We examine the movement of a body on an orbit that is slightly different from a circle. It turned out when we investigated the stability of the orbit, that the orbiting distance oscillates around a medium value with a definite period: Tr, and because of the revolution the angle of the position around the centre changes also of course, with the following period: Tφ. The shape of the resulting orbit approximates a rotating ellipse in the simplest case, where a angular turn with respect to the coordinate time is:
=et⋅T r −T = et⋅
2⋅ 2⋅ − =2⋅⋅ 1− et e
(4.11.1)
Arbitrary orbits around the Sun are characterized by the following relationship, we met previously: dr 2 2 =u t −U eff 2 d
(4.11.2)
If we approximate the geometric potential with its second derivative around the equilibrium point, then we can ultimately replace it with the differential equation of the harmonic oscillator, where we can identify the angular frequency of the periodic movement: 1 d2 U eff r ≈ ⋅ 2⋅U r ⋅ r−r 2 2 dr dr 2 1 d2 2 =u t − ⋅ 2⋅U r ⋅ r−r 2 2 2 dr d
(4.11.3)
We identify the angular frequency of the angular turn, this time with respect to the proper time: 2e =
U ' ' eff 2
(4.11.4)
Calculate the second derivative of the geometric potential: d 2 U eff dr
2
u2
r g 2⋅u2 r g 2⋅u2 r g 6⋅u 2 =− 3 ⋅ 2 c − 2⋅ 3 − 2⋅ 3 1− ⋅ 4 r r r r r r r r 2⋅r g
2
d 2 U eff 2⋅r 6 12⋅r =− 3 g⋅c2 4 − 5 g ⋅u 2 2 dr r r r u =r 2⋅
(4.11.5)
d 2 ≈r ⋅ d
Determine the angular frequency of the rotation of the ellipse, we recognize in the first term the orbital frequency according to the coordinate time of the orbiting body, and we substitute the horizontal velocity in a form that is valid for circular movement: 122
4.11 Perihelion precession
r 3 6⋅r 2e =− g3⋅c 2 4 − 5 g ⋅u 2 r r r
2e =−2⋅ 2 3−
6⋅r g 6⋅r ⋅ 2= 1− g ⋅ 2 r r
(4.11.6)
The relationship between the angular frequencies according to the coordinate time and the proper time can be derived from the relationship between the coordinate time and the proper time:
d = 1− d = d
e=
3⋅r g ⋅dt 2⋅r
d 3⋅r 1− g⋅dt 2⋅r et
→
3⋅r g 1− 2⋅r
2
et= 1−
3⋅r g 2 ⋅e 2⋅r
(4.11.7)
We discuss distant orbits, where the ratio of the Schwarzschild radius and the distance is very small, therefore we can allow ourselves a small inaccuracy, that keeps us within the error margin, but we can arrange to a more comfortable form the relationship we are looking for:
2e = 1−
6⋅r g 3⋅r g ⋅ 2≈ 1− ⋅ 2 r 2⋅r
et= 1−
3⋅r g ⋅ 2⋅r
(4.11.8)
By reinserting we get the perihelion precession of orbits:
=2⋅⋅ 1−
et r =3⋅⋅ g r
(4.11.9)
The explanation for the perihelion precession of Mercury is a famous confirmation of general relativity. The standard gravitational parameter of the Sun, and the gravitational radius: ⋅M =1.32712440018⋅1020
3
m s2
r g=
→
2⋅⋅M =2.9532500765⋅103 m 2 c
From the semi-major axis and the orbital period of Mercury the angular turn in a single revolution and in a century: 123
4.11 Perihelion precession 6
10
t k =7.60053024⋅10 s
r =5.79091⋅10 m
=4.80645⋅10 7=0.0991402 ' ' −4
(4.11.10)
century =1.99565⋅10 =41.1633 ' '
The total measured perihelion precession of Mercury in a century is 5599,7”, where 5028,83” is a coordinate effect due to the precession of the equinoxes, 530” is caused by the gravitational tug of the other planets, and 0,0254” is also caused by the oblateness of the Sun. The difference is: measured =40.8446 ' ' Based on this the difference is most likely caused by the curvature of spacetime.
4.12 Bending of light Based on the Newtonian particle model of light, Johann Georg von Soldner suggested already in 1801, that light rays are deflected if influenced by gravitation, and simply considering the light particles as bodies moving on orbits, he determined their deflection near the Sun. His result was half of the actual value. Einstein used relativity theory, and after an unsuccessful attempt, he correctly predicted the angle of light deflection, as it was confirmed by the British expedition led by Arthur Eddington in 1919. They travelled to Brazil and Equatorial Guinea, and determined the coordinates of stars with known positions near the dark disk of the eclipsed Sun. Later during the 1960s, radio astronomical measurements confirmed the calculations with a few times of 0.01% error margin. The method presented here differs from the traditional approach, we essentially search the shape of a geodesic from four dimensional spacetime in a subspace. In our case, in the three dimensional speacetime determined by the coordinate condition = , the path of the light rays 2 are also geodesics, as we have already seen when we wrote down the general geodesics. This is the reason for the success of the following procedure. We examine the paths of light rays for a general case in the gravitational field. Except in the case of the photon sphere, these will not be closed curves, they will either avoid the celestial body on an arched trajectory, or cross the event horizon while falling. Since the metric is spherically symmetric, we are not losing anything if we restrict ourselves to a coordinate surface. The Schwarzschild arc length squared:
ds 2= 1−
rg 2 2 dr 2 ⋅c ⋅dt − −r 2⋅ d 2sin 2 ⋅d 2 r r 1− g r
(4.12.1)
The arc length squared is zero along light-like geodesics. The coordinate conditions in the equatorial plane of the coordinate system:
124
4.12 Bending of light ds 2=0
=
2
d =0
(4.12.2)
Substitute into the arc length squared. By rearranging it, the radial coordinate and the original vertical angular coordinate describes a two dimensional surface, where the coordinate time measures the distance:
0= 1−
c 2⋅dt 2 =
rg 2 2 dr 2 ⋅c ⋅dt − −r 2⋅d 2 r r 1− g r 2
dr 2 r 1− g r
2
r 2⋅d 2 r r3 = ⋅dr 2 ⋅d 2 r r−r g r −r g 1− g r
(4.12.3)
This surface is a projection of the original spacetime, that however preserved the mutual dependence of the coordinates. If we consider the previous relationship an arc length squared, we can calculate the usual geometric quantities from the metric tensor to the connection:
g ij =
r r−r g
2
0 r3 r −r g
0
g ij =
∂ g rr 2⋅r r = ⋅ 1− 2 ∂ r r −r g r −r g
2 ∂ g r r = ⋅ 3− ∂r r −r g r−r g
r rr =−
rg r⋅ r−r g
r = r=
r−r g r
2
0 r −r g
0
r3
(4.12.4)
r−r g ∂ g rr 2⋅r −r g = ⋅ 1− 2 ∂r r r
3⋅r −r g ∂g 1 = 3⋅ 1− ∂r r r
r =−
2⋅r−3⋅r g 2⋅r⋅ r−r g
(4.12.5)
2⋅r −3⋅r g 2 (4.12.6)
The pictures of the light rays on this projection are geodesics, that can be parametrized by the distance that is valid on the surface: (1)
∂2 r ∂ r ∂r ∂ ∂ r rr⋅ ⋅ r ⋅ ⋅ =0 2 ∂ t ∂t ∂t ∂ t ∂t
125
4.12 Bending of light 2 2 rg 2⋅r −3⋅r g ∂ ∂ r ∂r = ⋅ ⋅ 2 r⋅r −r g ∂ t 2 ∂t ∂t
(2)
2
(4.12.7)
∂2 ∂ r ∂ 2⋅r ⋅ ⋅ =0 2 ∂t ∂t ∂t 2⋅r −3⋅r g ∂ r ∂ ∂2 =−2⋅ ⋅ ⋅ 2⋅r⋅r −r g ∂t ∂t ∂ t2
(4.12.8)
Determine the coordinate changes, or in other words, the velocities: v =
∂ ∂t
∂v 2⋅r −3⋅r g ∂r =− ⋅ ⋅v ∂t r⋅ r−r g ∂t 2⋅r−3⋅r g 1 ⋅dv =− ⋅dr v r⋅r −r g log v =log r −r g −3⋅log r C r −r v =C⋅ 3 g r
(4.12.9)
The integration constant can be determined, if we rearrange the arc length squared of the surface, and determine the angular velocity at an extremal case: 2
r r3 1 c ⋅dt = ⋅dr 2 ⋅d 2 /⋅ 2 r −r g r−r g dt 2
2
2
r dr 2 r 3 d 2 c= ⋅ 2 ⋅ r −r g dt r−r g dt 2 2
(4.12.10)
The change of the radial coordinate is zero at the closest proximity of orbits that avoid the celestial body on an arched trajectory: dr 20 =0 dt 2
r −r d =v =c⋅ 0 3 g dt r0
(4.12.11)
Substitute it into the angular velocity measured in this extremal position, and determine the 126
4.12 Bending of light integration constant:
r −r r −r v =C⋅ 0 3 g =c⋅ 0 3 g r0 r0
r 30 C=c⋅ r 0−r g
(4.12.12)
The angular velocity:
r 30 r−r g d =v =c⋅ ⋅ dt r 0−r g r 3
(4.12.13)
The radial velocity can also be determined, if we substitute the above formula into the arc length squared: 2
2
3
2
r dr r d c= ⋅ 2 ⋅ 2 r −r g dt r−r g dt 2
dr r −r g r 3 d 2 = ⋅ c 2− ⋅ dt r r −r g dt 2
r−r g r 30 r−r g dr =v r=c⋅ ⋅ 1− ⋅ dt r r 0−r g r 3
(4.12.14)
The ratio of the two velocities determines the change of the angular coordinate with respect to the distance. By integrating the relationship we can determine the total angular turn performed by the light ray in the proximity of the celestial body, between the closest approach and infinity:
r 30 v d 1 r 0−r g = = 2⋅ vr dr r r 30 r−r g 1− ⋅ r 0−r g r 3
r 30 ∞ r 0−r g 1 =∫ 2⋅ ⋅dr r 30 r −r g r r 1− ⋅ r 0−r g r 3 0
(4.12.15)
We get a formula that is easier to handle, if we introduce a new parameter, that changes between zero and one:
127
4.12 Bending of light =
r0 r 1
=∫ 0
1 ⋅ 1−2
1 r 1−3 1− g⋅ r 0 1−2
⋅d
(4.12.16)
This integral cannot be written in a closed form, but the integrand can be broken up to a sum of terms, that can be individually integrated:
1
2
3
3 3 3 1 1 r g 1− 3 r g 1− 5 r g 1− =∫ ⋅ 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅d 2 2 r 0 1− 2 8 r 0 1−2 16 r 0 1−2 0 1−
(4.12.17)
The first term characterizes the light ray that moves in flat spacetime: 1
1=∫ 0
1 ⋅d =arcsin 1−arcsin 0= 2 2 1−
(4.12.18)
The deviation from this is called the bending of light in the presence of gravitation. This phenomenon could be observed during solar eclipses for the first time, when the coordinates of stars with known positions was determined near the dark disk of the Sun. These measurements have an error margin, that is greater than that of the second term in the formula: 1
2=∫ 0
1 1 r g 1−3 ⋅ ⋅ ⋅ ⋅d 1−2 2 r 0 1− 2
∣
0 r 1 r 1− 2= ⋅ g⋅ − − 1−⋅1 = g 2 r0 1 r0 1
(4.12.19)
Since this angle is valid only from the perihelion to infinity, the total turn is two times this value: r =2⋅ g r0
(4.12.20)
By evaluating the other terms, we get a more precise relationship for light bending in a gravitational field, verified by radio astronomical measurements:
2
3
r r 15 15 61 r g =2⋅ g ⋅−1 ⋅ g − ⋅ ⋅ r0 16 r0 16 12 r 0
(4.12.21)
We calculate the deflection of the light rays that graze the surface of the Sun. Since the error margin of the observation does not exceed that of the first term, we consider only this. The standard gravitational parameter of the Sun, and the gravitational radius: 128
4.12 Bending of light
⋅M =1.32712440018⋅1020
m3 2 s
r g=
→
2⋅⋅M =2.9532500765⋅103 m 2 c
The radius of the Sun: 8
r 0=6.955⋅10 m
The deflection of light rays near the Sun: r =2⋅ g =8.492⋅10−6 =1.752' ' r0
(4.12.22)
Light rays coming from the same direction are focused by the Sun into an opposite area, thus it behaves like a gravitational lens:
The paths of the parallel light rays approaching the Sun (that is represented with a vertical line) hold together on the other side, however unlike in the case of optical lenses, they slightly diverge, not all of them are focused into the same point. Despite this, the pictures of objects on the other side are enlarged and brightened. This phenomenon can be utilized in practice by astronomers. The graph is strongly distorted by the way, the parallel Sun-grazing light rays incoming from infinity meet again at a great distance from the Sun, this is called the gravitational focal distance of the Sun: f =r 0⋅cot =8.19⋅10 13 m
(4.12.23)
4.13 Tides Back in the year of 1616, Galilei considered it a superstition, that according to Johannes Kepler, the tides on Earth are caused by the Moon's gravitational pull, however history verified the latter scientist. Already in the Newtonian theory of gravitation, the Moon and the Sun are responsible for the appearance of the tidal bulges, according to Kepler's suspicion. The theory of 129
4.13 Tides relativity can describe this phenomenon to an even greater accuracy, that is the deviation of geodesics: ∂2 x ∂ x ∂x R ⋅dx ⋅ ⋅ =0 ∂ ∂ ∂ 2
(4.13.1)
Apply this in the Schwarzschild coordinate system. The question is, what kind of tides are caused by the central mass in the surrounding extended objects that orbit it? At a given point on the circular orbit, where according to our choice: x =0 x r
∂ x dt = c⋅ ∂ d
x 0 0 0
d dt = c⋅ dt d
0 0
(4.13.2)
Substitute into the general formula, and use the coordinate time as the parameter. Under these conditions only four components of the curvature tensor play a role: (1)
∂2 r ∂t ∂ t ∂ ∂ Rr ttr⋅dr⋅c⋅ ⋅c⋅ Rr r⋅dr⋅ ⋅ =0 2 ∂t ∂ t ∂t ∂ t ∂t 2 r ∂ r r g⋅r −r g 2 − ⋅c ⋅dr g ⋅sin2 ⋅dr⋅ 2=0 2 4 2⋅r ∂t r
(4.13.3)
We are in the equatorial plane, and we substitute the angular frequency also: =
2
=c⋅
rg 2⋅r
3
r r ∂2 r r g⋅r −r g 2 − ⋅c ⋅dr g ⋅dr⋅c 2⋅ g 3 =0 2 4 2⋅r ∂t r 2⋅r The coordinate acceleration along the orbital radius: ∂ 2 r r g⋅ 4⋅r −5⋅r g 2 = ⋅c ⋅dr 2 4 ∂t 4⋅r (2)
(4.13.4)
∂2 ∂t ∂t ∂ ∂ Rtt ⋅d ⋅c⋅ ⋅c⋅ R ⋅d ⋅ ⋅ =0 2 ∂t ∂t ∂t ∂ t ∂t r ∂2 r g⋅r −r g 2 − ⋅c ⋅d − g⋅sin2 ⋅d ⋅2=0 2 4 r ∂t 2⋅r
We perform the same substitutions:
130
(4.13.5)
4.13 Tides 2 r r ∂ r g⋅r −r g 2 − ⋅c ⋅d − g⋅d ⋅c 2⋅ g 3 =0 2 4 r ∂t 2⋅r 2⋅r
The coordinate acceleration perpendicular to the orbital radius: 2 ∂ rg 2 = ⋅c ⋅d ∂t 2 2⋅r 3
(4.13.6)
The two acceleration components above distort a spherical planet – like Earth – as seen on the graph on the left side, and it shows in what direction the points of the surface are accelerated. At distances with the same magnitude like the Schwarzschild radius, the normally weaker component becomes significant, the shape of the ellipse changes. The elongation of the falling body is called spaghettification. At the right side the graph displays the acceleration vectors distorting a sphere with a diameter of 5 meters, positioned 5000 meters away from a solar mass black hole, where time-like circular orbits are still possible:
When will the tides become destructive to the celestial body? Pieces start to detach from the surface, when the tidal acceleration exceeds the surface acceleration. When doing the comparison we must keep in mind, that the two formulas do not apply to the same spacetime curvature; the tidal acceleration is calculated in the spacetime of the central star, the surface acceleration is calculated in the spacetime of the orbiting planet. The approximation method has two limitations: first, the Einstein equation is non-linear, therefore gravitational acceleration of the two bodies is added with a certain error. Second, we did not take into account, that the celestial bodies under investigation have distorted each other, neither their shape, nor their gravitational field is spherically symmetric. Since the tidal acceleration on the surface grows linearly with the size of the object (if the mass does not change), this phenomenon creates an upper limit to the sizes of the celestial objects orbiting gravitational sources: 2
r g⋅ 4⋅r−5⋅r g 2 r ⋅ r − r ∂2 r ∂ b r ⋅c ⋅ r = = 2 =− b g b 3b g ⋅c2 b 4 2 4⋅r ∂t ∂t 2⋅b r
131
4.13 Tides r g⋅ 4⋅r−5⋅r g 2⋅r
4
=− b
r g⋅ br −br g b
r
4
r g⋅ 4⋅r−5⋅r g 4 ⋅br −br g⋅b r b r 2g =0 4 2⋅r
(4.13.7)
The solution of this equation of the fourth degree gives the upper limit for the size of the celestial body, that not yet gets peeled by the tides. We rewrite it first: 2 r br g ⋅ r =0 br − Dr b Dr 4
b g
Dr =
r g⋅4⋅r−5⋅r g 2⋅r
4
(4.13.8)
The first step towards the solution of the equation of the fourth degree is to write down the resulting equation of the third degree: x 4 b⋅x3 c⋅x 2d⋅xe=0 y 3c⋅y 2b⋅d −4⋅e ⋅y4⋅c⋅e−d 2 −b2⋅e⋅d =0
(4.13.9)
Substituting: y 3−4⋅e⋅y−d 3=0 3
4⋅br 2g r y− ⋅y b g =0 Dr Dr 3
(4.13.10)
We have to solve this equation in the second step. The special form of the equation of the third degree, and the solution formula (during substitution, let us be careful about the signatures): y 3− p⋅y−q=0
3 q q2 p3 3 q q2 p3 y 1,2,3= − − − 2 4 27 2 4 27
(4.13.11)
Finally, we substitute the result into the following expressions, and get the solutions of the equation of the fourth degree:
b2 −c y 4
R=0
→
R=
D=
2
3⋅b 2 −2⋅c±2⋅ y −4⋅e 4
132
4.13 Tides R≠0
→
D=
3⋅b2 b⋅c−8⋅d −b3 −R2 −2⋅c± 4 4⋅R
b R D x 1,2 ,3,4 =− ± ± 4 2 2
(4.13.12)
Substitute from the Earth's standard gravitational parameter the gravitational radius: m3 ⋅M =3.986004418⋅10 2 s 14
r g=
→
2⋅⋅M =8.870056078⋅10−3 m 2 c
The semi-major axis of the Moon's orbit: 8
r =3.84399⋅10 m
From the Moon's standard gravitational parameter its gravitational radius: r =1.091020268509284⋅10−4 m
b g
Among the solutions of the equation of the fourth degree, the first result can be physical, this is the greatest possible size the Moon can have. With a greater radius than this, the Earth attracts the rocks on the surface more than the Moon itself: 7
−4
r =7.04273⋅10 m
b 2
r =1.09248⋅10 m
r =−3.52137⋅107 m−i⋅6.09918⋅107 m
b 4
b 1
b 3
r =−3.52137⋅10 7 mi⋅6.09918⋅10 7 m (4.13.13)
The distance within which a celestial body will disintegrate due to the tides caused by the central celestial body is called the Roche radius, and it is the solution of the following equation of the fourth order: r g⋅ 4⋅r−5⋅r g 2⋅r
4
=− b
r g⋅ br −br g b
r4
r ⋅ b r −b r g 4 5⋅r 2g ⋅r −2⋅r ⋅r =0 g 4 2 br
b g
(4.13.13)
Rewrite and write down the resulting equation of the third degree: 2
r 4−
2⋅r g 5⋅r g ⋅r =0 Dr 2⋅Dr
Dr = b
r g⋅ br −br g b
r4
(4.13.14)
3
10⋅r 2g 2⋅r g y− ⋅y =0 Dr Dr 3
(4.13.15) 133
4.13 Tides Just like in the previous case, we write down the solution formulas, substitute the variable, among them the radius of the Moon: 7
1.73814⋅10 m
Among the solutions of the equation of the fourth degree, the first result again can be physical, this is the smallest possible distance the Moon can approach the Earth. With a smaller distance than this, the Earth attracts the rocks on the surface more than the Moon itself: 7
r 2=0.0110876 m
r 1=9.48694⋅10 m 7
7
7
r 3=−4.74347⋅10 m−i⋅8.21593⋅10 m
7
r 4=−4.74347⋅10 mi⋅8.21593⋅10 m (4.13.16)
4.14 Falling orbit A test body moves along a geodesic also when it falls into the black hole directly, this is characterized by the following coordinate conditions: t=t
=t
r =r
r =r t
=const.=
2
=const.
d =0 d =0
(4.14.1)
The equations of movement of the trajectory: t c⋅t¨ 2⋅ tr⋅c⋅t˙⋅r˙ =0
t¨
rg ⋅t˙⋅r˙ =0 r⋅r −r g
(4.14.2)
r¨ r tt⋅c 2⋅˙t 2 r rr⋅r˙ 2 r ⋅˙ 2 r ⋅ ˙ 2=0 r¨
r g⋅ r−r g 2 2 rg ⋅c ⋅˙t − ⋅r˙ 2=0 3 2⋅r⋅r −r 2⋅r g
(4.14.3)
2 ¨ ˙ 2⋅ ˙ =0 r ⋅r˙ ⋅
¨ =0
(4.14.4) 134
4.14 Falling orbit
2⋅ ¨
r
⋅r˙ 2⋅ ˙
˙ =0 ⋅⋅ ˙
(4.14.5)
=0 ¨ Substitute the coordinate conditions into the arc length squared:
ds 2= 1−
2 rg 2 2 dr ⋅c ⋅dt − −r 2⋅ d 2sin 2 ⋅d 2 r r 1− g r
c 2⋅d 2= 1−
rg 2 2 dr 2 ⋅c ⋅dt − r r 1− g r
The relationship between the proper time and the coordinate time is velocity dependent:
r d = 1− g − r
v 2r
⋅dt
vr=
rg c ⋅ 1− r 2
dr dt
(4.14.6)
We make the arc length squared along a time-like infalling geodesic equal to the arc length squared of the co-moving coordinate system, then divide with the change in proper time, and write down the equation with the tangent vectors: A⋅c 2⋅ t
u=
dt 2 dr 2 c 2⋅d 2 2 −B⋅ = =c d 2 d 2 d 2
dt d
r
u=
dr d
A⋅c 2⋅u t 2−B⋅ ur 2=c 2
(4.14.7)
We have derived in the mathematical introduction, that if the partial derivative of the metric tensor along a coordinate is zero, then the corresponding covariant tangent vector is a constant of movement: ∂ g =0 ∂t
→
∂u t =0 ∂t
(4.14.8)
We calculate the time-like covariant tangent vector from the contravariant one with index lowering: u t= g t ⋅u= g tt⋅u t =A⋅u t
(4.14.9)
Rearrange the arc length squared and express the square of the time-like covariant tangent vector: 135
4.14 Falling orbit 2
2
2
2
2
2
t 2
2
r 2
c ⋅u t =A ⋅c ⋅u = A⋅ c B⋅u 2
r 2
because:
c ⋅u t =A⋅c u
B=
1 A
(4.14.10)
At the beginning of the fall, the radial velocity is zero: 2
2
c ⋅u t =A r 0 ⋅c
2
(4.14.11)
We make the two results equal, and express the radial velocity. We pick the negative root, because the numeric value of the radial coordinate has to decrease, we are looking for the infalling solution. r0 is the radial coordinate of the starting point: r
u =−c⋅ Ar 0− Ar
r r −r dr =−c⋅ g⋅ 0 d r0 r
r 1 r0 ∫ d =− c ⋅ r ⋅∫ r r−r' ' ⋅dr ' g r 0 0
0
The time dependence of the fall:
r 1 r 2⋅r = ⋅ 0⋅ r⋅ r 0−r 0⋅ arcsin 1− c rg 2 2 r0
(4.14.12)
The proper time passes from the left to the right on the graph, the vertical axis is the radius. The horizontal dotted line is the event horizon. The trajectory of the infalling body apparently crosses it as if it were not there: r
τ 136
4.14 Falling orbit The test body falling into the black hole reaches the centre in a finite proper time:
r r max = ⋅ 0⋅ 0 2 rg c
(4.14.13)
In order to form an idea of the magnitudes, let us replace the Sun with a black hole of the same mass, and we jump into the depth from a distance that corresponds to the surface of the Sun. The radius of the Sun: 8
r 0=6.955⋅10 m
The time that passes until the impact, from the point of view of the falling astronauts: max =29 min 28.5 s
(4.14.14)
Calculate the movement with respect to the coordinate time: dr dr d dr 1 = ⋅ = ⋅ dt d dt d u t
(4.14.15)
The contravariant time-oriented tangent vector changes during the movement, its covariant counterpart however does not, therefore we substitute the latter: dr dr A = ⋅ dt d ut
u t=
ut A
(4.14.16)
Substitute the time-oriented covariant tangent vector: dr dr A = ⋅ dt d Ar 0
u t= Ar 0
(4.14.17)
The complete expression cannot be integrated in a closed form:
∫
r 1 r ⋅r r r dt= ⋅ 0 g⋅ 0⋅ ⋅ ⋅dr c r0 r g r −r g r 0−r r 1 r ⋅r r' r' t= ⋅ 0 g⋅ ⋅ ⋅dr ' c r g r r ' −r g r 0−r '
(4.14.18)
0
This integral asymptotically approaches the event horizon, but it reaches it after an infinite time. The situation inside the event horizon is similar, if we track backwards the geodesics with respect to the coordinate time, we experience the same when approaching the Schwarzschild radius. The distant observer cheering the afore mentioned brave astronauts will see, that his peers get slower as they approach the black hole, and never reach it, or rather they do but in an infinite time. 137
4.14 Falling orbit The coordinate time passes from the left to the right on the graph, the vertical axis is the radius. The horizontal dotted line is the event horizon. The trajectory of the infalling body breaks on this graph: r
t The function describing the movement of the body asymptotically approaches the event horizon from both sides, but reaches it only at infinity.
4.15 Isotropic coordinates The spherically symmetric spacetime can be mapped with other kind of coordinate systems as well. We look at the general form of the arc length squared again, and modify the arbitrary functions: 2
2
2
2
2
2
2
2
ds = A I r ⋅c ⋅dt −B I r ⋅dr I r I⋅d sin ⋅d
(4.15.1)
Compare it with the arc length squared in the Schwarzschild coordinate system: 2
2
2
2
2
2
2
2
ds = AS r ⋅c ⋅dt −B S r ⋅dr S −r S⋅d sin ⋅d
(4.15.2)
The arc length squared is an invariant quantity, therefore the two are equal. This is also true, if we just measure it along one of the coordinates, that has always the same direction in each cases. Therefore the following equations can be written down with the arc length squares along the timelike, radial, and horizontal coordinates: A I =AS 2
2
B I⋅dr I =B S⋅dr S
B I⋅r 2I =r 2S
(4.15.3)
138
4.15 Isotropic coordinates Divide the two equations with each other, and square root: dr I dr = B S⋅ S rI rS dr I = rI
1
r r S⋅ 1− g rS
⋅dr S
Integrate and resolve the logarithm: log r I =log
1−
rg 1 rS
1−
rg −1 rS
r I =C⋅
1−
r I =C⋅ r S −
rg 1 −log rS
1−
rg −1 C rS
rg r S⋅r S −r g 2
(4.15.4)
The unknown multiplier can be determined from a geometric condition: infinitely distant from the gravitational centre, or by turning off the gravity completely, the two coordinate systems should coincide, in this case the extent of the event horizon is zero:
0 r I =C⋅ r S − r S⋅r S −0 2 C=
1 2
(4.15.5)
Express the radial coordinate of the Schwarzschild coordinate system, and determine with it the first unknown function of the isotropic coordinate system's arc length squared:
r r S =r I⋅ 1 g 4⋅r I
2
r A I =AS =1− g =1− rS
(4.15.6)
rg
r r I⋅ 1 g 4⋅r I
4⋅r I −r g = 2 4⋅r I r g
139
2
(4.15.7)
4.15 Isotropic coordinates The other unknown function:
BI =
r 2S r
2 I
=
r r I⋅ 1 g 4⋅r I r
2
2 I
4⋅r I r g = 4⋅r I
4
(4.15.8)
The gravitational radius appearing in the equations continues to be measured in the Schwarzschild coordinate system, here it is a constant independently from the coordinate system, a quantity that characterizes the mass. The arc length squared in the isotropic coordinate system: 2
4
4⋅r −r g 4⋅rr g ds = ⋅c 2⋅dt 2− ⋅ dr 2r 2⋅ d 2sin 2 ⋅d 2 4⋅r r g 4⋅r 2
(4.15.9)
The geometric quantities from the metric tensor to the curvature tensor:
g ij =
ij
g =
4⋅r−r g 4⋅rr g
2
0
4⋅r r g − 4⋅r
0 0
4⋅rr g 4⋅r−r g 0
0
0
0
4
4
4⋅r r g − ⋅r 2 4⋅r
0
0
0
4⋅rr g − ⋅r 2⋅sin 2 4⋅r
0
0
0
0
0
0
0
0
4
2
4⋅r − 4⋅r r g
0 0
4
0
4
4⋅r 1 − ⋅ 2 4⋅r r g r
0
0
4⋅r 1 − ⋅ 2 4⋅r r g r ⋅sin2
0
∂ g tt 16⋅r g⋅4⋅r −rg = ∂r 4⋅rr g 3
∂ g rg rg =− 2⋅r⋅ 1 −r g ⋅ 1 ∂r 4⋅r 4⋅r
4
∂ g rr r g⋅4⋅r r g 3 = ∂r 64⋅r 5 3
140
(4.15.10)
4.15 Isotropic coordinates
3
∂ g r r =− 2⋅r⋅ g 1 −r g ⋅ g 1 ⋅sin 2 ∂r 4⋅r 4⋅r ∂ g 4⋅r r g 4 =− ⋅cos ⋅sin ∂ 128⋅r 2 3
tt 16⋅r g⋅4⋅r r g ∂g =− ∂r 4⋅r −r g 3
rr 1024⋅r ⋅r g ∂g =− 5 ∂r 4⋅r r g
512⋅r⋅ 4⋅r −r g ∂g = 5 ∂r 4⋅r r g
512⋅r⋅4⋅r −r g ∂g = 5 2 ∂r 4⋅rr g ⋅sin
∂ g 512⋅r 2⋅cos = ∂ 4⋅rr g 4⋅sin 3 t tr = t rt =
(4.15.11)
8⋅r g 2
2
16⋅r −r g 4
2048⋅r ⋅r g⋅ 4⋅r−r g tt = 7 4⋅rr g r
r =−
r⋅ 4⋅r−r g 4⋅r r g
=
2⋅r g r⋅ 4⋅rr g
r =−
r = r = r = r =
r rr=−
4⋅r−r g r⋅4⋅r r g
r⋅4⋅r −r g ⋅sin2 4⋅rr g
=−cos ⋅sin
(4.15.12)
=cot
∂ t tr ∂ t rt 256⋅r⋅r g = =− ∂r ∂r 16⋅r 2 −r 2g 2 ∂ r tt 8192⋅r 3⋅r g⋅8⋅r 2 −8⋅r⋅r g r 2g = ∂r 4⋅r r g 8 r
2
∂ r rr 2⋅r g⋅8⋅rr g = 2 ∂r r ⋅4⋅r r g 2
2
r
∂ 16 r 8⋅r⋅r g −r g =− 2 ∂r 4⋅rr g
2
2
∂ 16 r 8⋅r⋅r g −r g =− ⋅sin2 2 ∂r 4⋅rr g
2
2
∂ r ∂ r ∂ r ∂ r 16⋅r −8⋅r⋅r g−r g = = = =− 2 2 ∂r ∂r ∂r ∂r r ⋅ 4⋅r r g
141
4.15 Isotropic coordinates r
∂ 2⋅r⋅ 4⋅r−r g =− ⋅cos ⋅sin ∂ 4⋅r r g ∂ ∂ 1 = =− 2 ∂ ∂ sin
∂ =−cos 2⋅ ∂ Rt rtr =−Rt rrt =
16⋅r g 2
r⋅ 4⋅rr g
Rt t =−Rt t =Rr r =−Rr r =−
8⋅r⋅r g 2
4⋅r r g
Rt t =−Rt t= Rr r =−Rr r =−
Rr ttr =−Rr trt =
R
tt
=−R
tt
(4.15.13)
8⋅r⋅r g 4⋅rr g 2
⋅sin2
4096⋅r 3⋅r g⋅4⋅r −r g 2 4⋅rr g 8
=R
tt
=−R
tt
2048⋅r 3⋅r g⋅4⋅r −r g 2 =− 4⋅r r g 8
Rrr =−Rr r =Rrr =−Rr r =
R =−R =
16⋅r⋅r g
8⋅r g r⋅4⋅r r g 2
⋅sin2
R =−R =−
2
4⋅rr g
16⋅r⋅r g 4⋅r r g
2
(4.15.14)
Substitute the connection coefficients of the isotropic coordinate system into the geodesic equations: t c⋅t¨ 2⋅ tr⋅c⋅t˙⋅r˙ =0
t¨
16⋅r g 16⋅r 2−r 2g
⋅t˙⋅r˙ =0
(4.15.15)
r¨ r tt⋅c 2⋅˙t 2 r rr⋅r˙ 2 r ⋅˙ 2 r ⋅ ˙ 2=0 4
2048⋅r ⋅r g⋅4⋅r −r g 2 2 2⋅r g r⋅ 4⋅r−r g 2 r⋅4⋅r−r g 2 r¨ ⋅c ⋅˙t − ⋅r˙ 2− ⋅˙ − ⋅sin2 ⋅ ˙ =0 7 r⋅4⋅r r g 4⋅r r g 4⋅r r g 4⋅r r g (4.15.16) 2 ¨ ˙ 2⋅ ˙ =0 r ⋅r˙ ⋅
142
4.15 Isotropic coordinates
¨
8⋅r −r g ˙ ⋅r˙ −cos ⋅sin⋅ ˙ 2=0 r⋅4⋅r r g
2⋅ ¨
¨
r
⋅r˙ 2⋅ ˙
(4.15.17)
˙ =0 ⋅⋅ ˙
8⋅r −r g ˙ =0 ⋅r˙ 2⋅cot ˙ ⋅⋅ ˙ r⋅4⋅r r g
(4.15.18)
Gravitational redshift in isotropic coordinates:
1
=
g 00 ⋅2 = 1 g 00 2
4⋅2 r −r g 4⋅2 r r g 4⋅1r −r g 4⋅1r r g
2
⋅ =
2 2
4⋅2r −r g ⋅ 4⋅1r −r g 2
(4.15.19)
If the light source is closer to the source of the gravitational field than the observer, then the mutual ratios of the radii and the frequencies are the same as in the Schwarzschild case: 1
r ≥2r
→
1
≤2
(4.15.20)
To get the orbital frequency of the test body on a circular orbit, insert the exchange formula between the coordinate systems into the result from the Schwarzschild coordinates:
I =c⋅
rg 2048⋅r 3I =c⋅ 2⋅r 3S r 5g
(4.15.21)
4.16 Gaussian polar coordinates We try out further unknown functions in the general formula for the arc length squared: 2
2
2
2
2
2
2
2
ds = AG r ⋅c ⋅dt −dr G −C G r ⋅r G⋅d sin ⋅d
(4.16.1)
Compare it with the arc length squared from the Schwarzschild coordinate system: 2
2
2
2
2
2
2
2
ds = AS r ⋅c ⋅dt −B S r ⋅dr S −r S⋅d sin ⋅d
Arc length squared along the time-like, radial, and horizontal coordinates: AG = AS
143
(4.16.2)
4.16 Gaussian polar coordinates 2
2
dr G= BS⋅dr S 2
2
(4.16.3)
C G⋅r G =r S The second and third equations are related, we perform the integration: r G=∫ B S⋅dr S = r G=∫
1
1−
rg rS
rS CG
⋅dr S =
rS CG
Exchange between the radial coordinates of the Gaussian polar and the Schwarzschild coordinate systems:
r r G= g⋅ log 2 r g =0
r G=
1−
1−
1−
rg 1 −log rS
rg r −1 r S⋅ 1− g K rS rS
r G=0r S⋅ 1−
→
rg ⋅ log 2
rg 1 −log rS
1−
0 K rS
K =0
→
rg r −1 r S⋅ 1− g rS rS
(4.16.4)
Compare the Gaussian polar coordinates with the isotropic coordinates as well: 2
2
2
2
2
2
2
2
ds = AG r ⋅c ⋅dt −dr G −C G r ⋅r G⋅d sin ⋅d
ds 2= A I r ⋅c 2⋅dt 2−B I r ⋅dr 2I r 2I⋅d 2 sin2 ⋅d 2
(4.16.6)
Arc length squared along the time-like, radial, and horizontal coordinates: AG = AI 2
2
dr G= B I⋅dr I 2
2
(4.16.7)
C G⋅r G =B I⋅r I
Integrate the second relationship: 2
2
dr G= B I⋅dr I
144
4.16 Gaussian polar coordinates dr G= B I⋅dr I =
r G=∫
r G=
4
4⋅r I r g ⋅dr I 4⋅r I
2
4⋅r I r g ⋅dr I 4⋅r I
rg r ⋅ log r I − g r I C 2 8⋅r I
r g =0
r G=0r I C
→
C=0
→
Exchange between the radial coordinates of the Gaussian polar and the isotropic coordinate systems:
r r r G= g⋅ log r I − g r I 2 8⋅r I
(4.16.8)
We get transcendent equations in both cases that lack an analytic solution, therefore we are satisfied with the relationships we have found.
4.17 Rotating Schwarzschild coordinates It is useful to discuss several problems in rotating coordinate systems. We transform from the usual Schwarzschild coordinate system the following way: (4.17.1)
⋅t This is how the arch length squared changes:
2
r⋅ ds = Ar − ⋅sin2 ⋅c 2⋅dt 2−2⋅⋅r 2⋅sin 2 ⋅dt⋅d c 2
2
2
2
2
(4.17.2)
2
−B r ⋅dr −r ⋅ d sin ⋅d The domain of validity extends until it preserves the signature of the original metric. For example the sign of one of the metric tensor component changes, when its value is zero: 2
A r −
r⋅ ⋅sin 2 =0 c
2 rg r⋅ 1− − ⋅sin2 =0 r c
145
4.17 Rotating Schwarzschild coordinates
r 3−
r g⋅c 2 c2 ⋅r− =0 2⋅sin 2 2⋅sin2
(4.17.3)
The special form of the equation of the third degree and the solution formula: x 3− p⋅x−q=0
3 q q2 p3 3 q q2 p3 x 1,2,3= − − − 2 4 27 2 4 27
(4.17.4)
Since there is a negative number under the square root, the arithmetic rules of the complex numbers apply when using the solution formula. The equation of the third degree always has a real number solution. We set up a rotating coordinate system in the Solar System, with the same angular frequency like the orbit of the Earth: t k2 =365.256363004 days
=
→
2⋅ 1 =1.99098659277⋅10−7 tk s
The standard gravitational parameter of the Sun and the gravitational radius: ⋅M =1.32712440018⋅1020
m3 s2
r g=
→
2⋅⋅M =2.9532500765⋅103 m 2 c
The result: 15
(4.17.5)
r =1.5057509⋅10 m
This is a circle with a radius of one lightyear. At greater distances than this the constant coordinate points of the coordinate system move with a speed greater than the speed of light, therefore they are not suitable to describe the time-like paths of moving bodies. The condition for the validity of the solution is, that the value under the square root is negative. This stops to be the case, when the angular frequency becomes so big, that the mentioned expression become zero or positive: q 2 p3 ≥0 4 27
r g⋅c2
2
3
c2 − 2 − ⋅sin 2 2⋅sin2 ≥0 4 27
r 2g c2 − ≥0 4 27⋅2⋅sin 2
146
4.17 Rotating Schwarzschild coordinates ≥
2⋅c 27⋅sin ⋅r g
(4.17.6)
The twice covariant metric tensor of the rotating Schwarzschild coordinate system:
g =
2 rg r⋅ 1− − ⋅sin 2 r c
0
0
−
0 2 −⋅r ⋅sin 2
1 r 1− g r 0 0
0
−⋅r 2⋅sin 2
0
0
−r 0
2
0 −r ⋅sin 2 2
(4.17.7)
The metric tensor has non-zero non-diagonal components. We write down a partial matrix using the rows and columns where these components appear:
2 rg r⋅ 2 2 2 1− − ⋅sin −⋅r ⋅sin g ij = r c 2 −⋅r ⋅sin2 −r 2⋅sin2
The determinant of the partial matrix: g =g tt⋅g −g t ⋅g t
2 rg r⋅ 2 2 2 2 4 4 g =− 1− − ⋅sin ⋅r ⋅sin − ⋅r ⋅sin r c
After this the components of the twice contravariant metric tensor: g tt =
g = g
g t =
1
2
1−
g t g =g t = t =− g g
g =
g tt = g
rg r⋅ − ⋅sin 2 −2⋅r 2⋅sin 2 r c
2
1−
rg r⋅ − ⋅sin 2 −2⋅r 2⋅sin2 r c
2 rg r⋅ 1− − ⋅sin 2 r c
2 rg r⋅ − 1− − ⋅sin 2 ⋅r 2⋅sin2 − 2⋅r 4⋅sin4 r c
147
(4.17.8)
4.17 Rotating Schwarzschild coordinates
rr
g =
rg 1 =− 1− g rr r
g =
1 1 =− 2 g r
(4.17.9)
The derivatives of the metric tensor and the connection: ∂ g tt 2⋅ 2⋅r⋅sin 2 r g =− − 2 2 ∂r c r
∂ g tt 2⋅2⋅r 2⋅cos ⋅sin =− ∂ c2
∂ gt ∂ g t = =−2⋅⋅r⋅sin 2 ∂r ∂r
∂ g t ∂ g t = =−2⋅⋅r 2⋅cos ⋅sin ∂ ∂
∂ g rr rg = ∂ r r −r g 2
∂ g =−2⋅r ∂r
∂ g =−2⋅r⋅sin 2 ∂r
∂ g =−2⋅r 2⋅cos ⋅sin ∂
t tr = t rt =
2⋅ c 2−1⋅ 2⋅r 3⋅sin 2 c 2⋅r g 2⋅c 2−1⋅ 2⋅r 4⋅sin2 2⋅c 2⋅r⋅ r−r g
t t = t t = 2
c 2−1⋅2⋅r 3⋅cos ⋅sin c 2−1⋅ 2⋅r 3⋅sin 2 c 2⋅ r−r g 2
3
2
c ⋅r g −2⋅ ⋅r ⋅sin ⋅r −r g tt = 2 3 2⋅c ⋅r r
r rr =−
(4.17.10)
rg 2⋅r⋅r −r g
r
=−r −r g
r
r
t
t
=
r
2
t
=−⋅r −r g ⋅sin
2
=− r−r g ⋅sin
2
tt =− 2 ⋅cos ⋅sin c r = r = r = r =
1 r
t
=−⋅cos ⋅sin
=−cos ⋅sin
2
c ⋅⋅2⋅r−3⋅r g = = 2 2 4 2 2 2⋅c −1⋅ ⋅r ⋅sin 2⋅c ⋅r⋅ r−r g tr
=
rt
148
4.17 Rotating Schwarzschild coordinates 2
t
=
t
=
c ⋅⋅r −r g = 2 ⋅cot 2 3 2 2 c −1⋅ ⋅r ⋅sin c ⋅ r−r g
(4.17.11)
=cot
4.18 Kruskal-Szekeres coordinates When we investigated the infalling path, it turned out that from the test body's point of view, the event horizon poses no obstacle. Our goal is to map the spacetime of the Schwarzschild black hole with such a coordinate system, that can be interpreted on the event horizon, and certainly covers the entire spacetime. For this we determine the trajectories of the infalling and outward heading light rays. We multiply the general spherically symmetric arc length squared on the falling light-like geodesics with a monotonic changing parameter: 1 A r ⋅c 2⋅dt 2− Br ⋅dr 2=0 /⋅ 2 d dt 2 dr 2 A r ⋅c 2⋅ 2 −B r ⋅ 2 =0 d d
(4.18.1)
Substitute the time-oriented tangent vector, and that the second function is the reciprocal of the first: dt u t= A⋅c 2⋅ d
B=
1 A
u2t dr 2 − =0 /⋅A A⋅c 2 A⋅d 2 ut dr =± c d
(4.18.2)
The left side of the equation is constant, therefore the change of the velocity relates linearly to the monotonic changing parameter, therefore it is also a monotonic changing parameter. Thus we can use the radius as a parameter as well: d ⋅±
ut =dr c
d =±dr
ut dr =± =±1 c d
(4.18.3)
Substitute into the tangent vector and integrate:
149
4.18 Kruskal-Szekeres coordinates ut dt =A⋅c⋅ =±1 c dr c⋅dt 1 =± =± dr A
c⋅t=±∫
1 r 1− g r
1 r 1− g r ⋅dr
Because of the logarithm we have to differentiate between two cases, outside and inside the gravitational radius. The positive and negative sign makes a distinction between the outward and inward going light rays: c⋅t=±r ±r g⋅log
r −1 C rg
r r g
c⋅t=±r ±r g⋅log r g −r K
r r g
(4.18.4)
Write down the two light paths separately outside the gravitational radius, and choose dimensionless integration constants: c⋅t=rr g⋅log
r −1 r g⋅u rg
c⋅t=−r −r g⋅log
(4.18.5)
r −1 r g⋅v rg
(4.18.6)
The u and v parameters, together with the angular coordinates describing the spherical coordinate surface, are suitable to represent the Schwarzschild solution, and eliminate the coordinate singularity. These are the Eddington-Finkelstein coordinates u v :
(4.18.7)
(4.18.8)
1 r u= ⋅ c⋅t−r−r g⋅log −1 rg rg 1 r v = ⋅ c⋅tr r g⋅log −1 rg rg
These coordinates are not mutually affine parameters, this also means for example, that if we move along the values of u from -∞ and +∞, we would not be able to explore the entire extent of the v geodesic:
150
4.18 Kruskal-Szekeres coordinates
u=v−2⋅
r r log −1 rg rg
(4.18.9)
(4.18.10)
v =−u2⋅
r r log −1 rg rg
Therefore these geodesics are not complete, they leave the u and v coordinate plane. The Eddington-Finkelstein coordinates cover the same manifold as the Schwarzschild coordinates. However this is apparently not the entire spacetime, since we have found geodesics that leave the area we have mapped so far. In order to obtain the entire map, we need coordinates describing lightlike geodesics, that are mutually affine parameters. Perform the following modification: u 2
v 2
r
r r U =e =e ⋅ −1 ⋅e rg −
−
v 2
u 2
r
r r V =e =e ⋅ −1 ⋅e rg
g
(4.18.11)
g
By substitution we obtain the dependence from the Schwarzschild coordinates, these are the Kruskal-Szekeres coordinates U V : r−c⋅t
r 2⋅r U= −1⋅e rg r V= −1⋅e rg
(4.18.12)
g
r c⋅t 2⋅r g
(4.18.13)
Since they are monotonic increasing functions of the radial coordinate, that turned out to be an affine parameter earlier, these coordinates are mutually affine parameters from the point of view of the light-like geodesics they represent. Combinations of the two coordinates: U⋅V =
r
r r −1 ⋅e rg
g
c⋅t
V r =e U
(4.18.14)
g
We express from the product the Schwarzschild radial coordinate. Rearrange the exponential equation to a general form: −
e
r rg
=
1 r 1 ⋅ − U⋅V r g U⋅V
(4.18.15)
The general form and the solution formula, where W(x) is the Lambert function:
p a⋅xb=c⋅x d
W − x=−
151
a⋅log p b − ⋅p c a⋅log p
a⋅d c
−
d c
4.18 Kruskal-Szekeres coordinates Substitute into the solution formula:
r U⋅V =W 1 rg e
r =r g⋅W
U⋅V r g e
(4.18.16)
If r = 0, then U⋅V =−1, here is the true singularity of the surface, that is independent of the choice of coordinate system. The U⋅V −1 however is not only satisfied when both of them are sufficiently large positive numbers, but also when they are both negative numbers of the same magnitude. We have discovered a whole new portion of the spacetime of the Schwarzschild singularity, that was hidden from us because of the unfortunate choice of coordinate system: V
U
The hyperbolic surfaces connect points with the same distance from the centre, the linear surfaces connect points with the same time coordinate. The coordinate axes themselves satisfy these conditions, they connect points with -∞ and Schwarzschild radius distances, and with +∞ and -∞ time coordinates. According to this, the black hole has two “entrances”, in the lower left and the upper right quarters, separated from them by event horizons. The interior domains are represented in the upper left and lower right quarters. Negative r coordinates also can be defined (let us not forget that the meaning of r is not distance but coordinate), and its hyperboles fill the interior domain. The coordinates that spanned the wormhole surface are also present here, the time coordinate is defined both in the first and third coordinate quarters, and the roles of the angular coordinates did not change. The curvature of the surface, thus its shape is a coordinate independent quantity, now we can depict the whole geometric shape. The parameters of the coordinate surface: 2
t=const.
=
dt=0
d =0
(4.18.17)
152
4.18 Kruskal-Szekeres coordinates The full wormhole surface in polar and rectangular coordinates:
The real significance of the wormhole is even more apparent on these graphs, that is seemingly connecting two asymptotically flat universes. However every world line that starts on one side and continues on the other, always has a space-like section at the neck of the funnel, therefore this wormhole is not traversable for massive objects slower than the speed of light. We determine the arc length squared in the U⋅V −1 domain. Since U and V are light-like geodesics, therefore gUU = 0 and gVV = 0. Their product is however not zero, therefore we can expect that gUV would not be either: ds 2=2⋅g UV⋅dU⋅dV −r 2⋅d 2sin 2 ⋅d 2
(4.18.18)
The unknown component of the metric tensor can be calculated with the help of the Schwarzschild metric tensor, we apply the transformation formula: g UV =
∂U ∂ V ∂ U ∂V ⋅ ⋅S g tt ⋅ ⋅g ∂t ∂t ∂ r ∂ r S rr
(4.18.19)
Substitute:
∂ g UV = ∂t ∂ − ∂r
r −1⋅e rg
r −1⋅e rg
r −c⋅t 2⋅r g
r −c⋅t 2⋅r g
∂ ⋅ ∂t
∂ ⋅ ∂r
r −1⋅e rg
r −1⋅e rg
r c⋅t 2⋅r g
rc⋅t 2⋅r g
153
⋅ 1−
rg r
1 ⋅ r 1− g r
4.18 Kruskal-Szekeres coordinates
r c⋅ −1⋅e rg g UV =− 2⋅r g −
r⋅e
r c⋅ −1⋅e rg ⋅ 2⋅r g
r−c⋅t 2⋅rg
rc⋅t 2⋅rg
⋅ 1−
rg r
r c⋅t 2⋅r g
r⋅e 1 ⋅ ⋅ rg r r 2⋅r 2g⋅ −1 2⋅r 2g⋅ −1 1− r rg rg
2
g UV =−
r −c⋅t 2⋅r g
3 g
2 g
2
3
2
2
2
2 g
c ⋅r g⋅ r −4⋅r⋅r 6⋅r ⋅r g −4⋅r r ⋅c ⋅r −r ⋅e 4⋅r 4g⋅r −r g 2
r rg
2⋅r 3g − rr g UV =− ⋅e r
(4.18.20)
g
The arc length squared of the Kruskal-Szekeres coordinate system: 3
r
4⋅r g − r ds =− ⋅e ⋅dU⋅dV −r 2⋅d 2sin 2 ⋅d 2 r 2
(4.18.21)
g
Substitute the transformation formula between the Schwarzschild and Kruskal-Szekeres coordinates into the arc length squared, that can be used already to calculate the usual geometric quantities: 2
U⋅V −W −1 4⋅r 2g U⋅V e ds =− ⋅e ⋅dU⋅dV − r g⋅W r g ⋅d 2 sin2 ⋅d 2 e U⋅V W 1 e (4.18.23)
2
g UV =g VU =−
U⋅V −W −1 2⋅r 2g e ⋅e U⋅V W 1 e
W
U⋅V g =− r g⋅W r g e
g UV =g VU =−
g =−
2
U⋅V 1 U⋅V W 1 e e ⋅e 2⋅r 2g
1
g =−
2
U⋅V g =− r g⋅W r g ⋅sin 2 e
2
U⋅V r g⋅W r g e 1
2
U⋅V r g⋅W r g ⋅sin 2 e (4.18.24)
154
4.18 Kruskal-Szekeres coordinates
∂ g UV ∂ g VU = = ∂U ∂U
2⋅r 2g⋅V⋅ W
e 2⋅ W
U⋅V 1 e
U⋅V 2 e
U⋅V ⋅W 1 e
∂ g 2⋅r 2g⋅V =− U ⋅V ∂U W 2 e e
∂ g UV ∂ g VU = = ∂V ∂V
2
∂ g 2⋅r 2g⋅V =− ⋅sin 2 U⋅V ∂U W 2 e e
2⋅r 2g⋅U⋅ W
e 2⋅ W
U⋅V 1 e
U⋅V 2 e
U⋅V ⋅W 1 e
∂ g 2⋅r 2g⋅U =− U ⋅V ∂V W 2 e e
2
∂ g 2⋅r 2g⋅U =− ⋅sin 2 U⋅V ∂V W 2 e e 2
∂ g U⋅V =− r g⋅W r g ⋅cos ⋅sin ∂ e
e
W
W
V
UUU =−
⋅
U⋅V 1 e
U⋅V 2 e
2
U⋅V W 1 e
V U⋅V V =− ⋅ W 1 2 e
V U⋅V V =− ⋅ W 1 ⋅sin 2 2 e
U U⋅V U =− ⋅ W 1 2 e
U U⋅V U =− ⋅ W 1 ⋅sin 2 2 e
U
VVV =− e
W
(4.18.25)
W
⋅
U⋅V 1 e
U⋅V 2 e
U⋅V W 1 e
2
V
U = U = U = U = e
U⋅V W 1 e
2
U⋅V ⋅W 1 e
155
4.18 Kruskal-Szekeres coordinates U
V = V = V = V =
U⋅V W 1 e
e
U⋅V ⋅W 1 e
=−cos ⋅sin
2
= =cot
(4.18.26)
4.19 Kruskal-Szekeres spacetime We can introduce coordinates, where one of them is time-like and the other three are spacelike, similar to the Schwarzschild coordinates, however they cover the entire spacetime of the singularity. This is the Kruskal-Szekeres spacetime c⋅T R . The transformation formulas: R=r g⋅V U
c⋅T =r g⋅V −U U=
1 ⋅ R−c⋅T 2⋅r g
V=
1 ⋅ Rc⋅T 2⋅r g
(4.19.1)
Some useful combinations of the coordinates: R2−c 2⋅T 2=4⋅r g⋅U⋅V =4⋅r g⋅r −r g ⋅e
U R−c⋅T = V Rc⋅T
r rg
(4.19.2)
We express the Schwarzschild radial coordinate from the first. Bring the exponential equation to the general form: −
e
r rg
2
2
4⋅r g
4⋅r r = 2 2 2⋅ − 2 2g 2 R −c ⋅T r g R −c ⋅T
(4.19.3)
The general form and the solution formula, where W(x) is the Lambert function:
a⋅log p b − W − ⋅p c x=− a⋅log p
p a⋅xb=c⋅x d
a⋅d c
−
d c
Substitute it into the solution formula:
r R 2−c 2⋅T 2 =W 1 2 rg 4⋅e⋅r g
r =r g⋅W
156
R2−c 2⋅T 2 r g 2 4⋅e⋅r g
(4.19.4)
4.19 Kruskal-Szekeres spacetime It is the same graph, but according to the coordinates of the Kruskal-Szekeres spacetime: R
T
The arc length square of the Kruskal-Szekeres spacetime: r g − rr 2 ds = ⋅e ⋅c ⋅dT 2−dR 2−r 2⋅d 2sin 2 ⋅d 2 r 2
(4.19.5)
g
Substitute the transformation formula between the Schwarzschild and Kruskal-Szekeres spacetime into the arc length squared: −W
2
2
2
R −c ⋅T −1 4⋅e⋅r 2g
2
R 2−c 2⋅T 2 ds = ⋅c ⋅dT −dR − r ⋅W r g ⋅ d 2sin 2 ⋅d 2 g 2 2 2 2 4⋅e⋅r g R −c ⋅T W 1 2 4⋅e⋅r g (4.19.6) 2
e
2
2
2
−W
e
g TT =−g RR= W
2
2
2
R −c ⋅T 4⋅e⋅r 2g 2
2
R −c ⋅T 4⋅e⋅r 2g
−1
2
g TT =−g RR=
1
R 2−c 2⋅T 2 g =− r g⋅W r g 4⋅e⋅r 2g
W
R2−c 2⋅T 2 1 2 4⋅e⋅r g −W
e
2
g =−
2
R2−c2⋅T 2 g =− r g⋅W r g ⋅sin 2 2 4⋅e⋅r g
2
2
R −c ⋅T −1 2 4⋅e⋅r g
1
g =−
2
2
R2−c 2⋅T 2 r g⋅W r g 4⋅e⋅r 2g 1
2
R 2−c 2⋅T 2 r g⋅W r g ⋅sin2 2 4⋅e⋅r g (4.19.7)
157
5. Spacetime of the rotating black hole
5. Spacetime of the rotating black hole Celestial bodies influence spacetime not only with their mass, but also with their rotation. The structure of the surrounding spacetime around a rotating black hole and a rotating massive body does not coincide like in the spherically symmetric case. However those simpler phenomena that appear in the spacetime of a black hole also appear around a rotating body, although their effect depends slightly differently on the mass and rotation of the gravitational source. We are going to encounter new phenomena, that are alien to the Newtonian theory of gravitation. The spacetime marks an axis in space, that breaks the symmetry between the two orbiting directions around the axis, and it also influences rotations pointing in the direction of the axis.
5.1 Axially symmetric spacetime The spacetime around uniformly rotating bodies is called stationary. The metric does not change, therefore it does not depend on time, and when rotating it returns into itself, therefore neither on the longitudinal angular coordinates. The arc length squared of the axially symmetric metric in the general case, where the unknown functions depend only on x and y, that are arbitrary coordinates t x y : 2
2⋅
2
2
2⋅
2
2⋅
2
2⋅
ds =e ⋅c ⋅dt −e ⋅d −⋅c⋅dt −e ⋅dx −e ⋅dy
2
(5.1.1)
We multiplied the coordinates with four unknown functions and used the angular frequency. The metric tensor:
2⋅
2
e − ⋅e 0 g = 0 ⋅e 2⋅
2⋅
2⋅
0 0 ⋅e 2⋅ −e 0 0 2⋅ 0 −e 0 0 0 −e 2⋅
(5.1.2)
The metric tensor has non-zero non-diagonal components. We write down a partial matrix using the rows and columns where these components appear: g ij =
e 2⋅− 2⋅e 2⋅ ⋅e 2⋅ ⋅e 2⋅ −e 2⋅
(5.1.3)
The determinant of the partial matrix: g =g 00⋅g − g t ⋅g t=e 2⋅− 2⋅e 2⋅ ⋅−e 2⋅ −⋅e 2⋅ ⋅⋅e 2⋅ =−e 2⋅− Invert the partial matrix and extend with it the twice contravariant metric tensor:
158
(5.1.4)
5.1 Axially symmetric spacetime
e−2⋅ 0 0 ⋅e−2⋅ 0 −e−2⋅ 0 0 g = −2⋅ 0 0 −e 0 −2⋅ −2⋅ ⋅e 0 0 −e 2⋅e−2⋅
(5.1.5)
Keep the angular frequency on a constant value for now, in this case for example we restrict ourselves to a single direction of revolution on an equatorial circular orbit. After we calculated the geometric quantities, we obtain the components of the simplified Ricci tensor: P=
∂ ∂ ∂ ∂ − − − ∂y ∂y ∂ y ∂ y
Rtt =2⋅e 2⋅ −⋅
∂ ∂ ∂ ∂ − − − ∂x ∂ x ∂ x ∂ x
∂ ∂2 ∂ ∂2 ⋅P− 2 2⋅e 2⋅−⋅ ⋅Q− 2 ∂y ∂x ∂y ∂x
2 ∂2 ∂ ∂ 2⋅− ∂ − ⋅Q e ⋅ − ⋅P 2 2 ∂x ∂ x ∂y ∂y
e 2⋅−⋅
Rt =R t =⋅e 2⋅ −⋅
R xx=e
Q=
∂ ∂2 ∂ ∂2 ⋅P− 2 ⋅e 2⋅−⋅ ⋅Q− 2 ∂y ∂x ∂y ∂x
2
2
∂ ∂2 ∂ ∂ ∂2 ∂ ∂2 ∂ ∂2 ∂ ⋅ ⋅P− 2 −Q ⋅ − 2 − − − − 2− 2 ∂y ∂x ∂x ∂x ∂x ∂x ∂x ∂y ∂x ∂x
2⋅ −
R yy =e
R xy=R yx=
2
∂ ∂ ∂2 ∂ ⋅ Q ⋅ − − ∂ x ∂ x ∂ x2 ∂ x
2⋅−
2
2
2
∂ ∂ ∂2 ∂ ∂2 ∂ ∂2 ∂ −P ⋅ − 2 − − 2− − 2− ∂y ∂y ∂y ∂y ∂y ∂y ∂y ∂y
2
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂2 ∂ ∂ ∂2 ⋅ ⋅ − ⋅ − − ⋅ − ∂ y ∂ y ∂x ∂ x ∂ x ∂ y ∂ x ∂ y ∂ x⋅∂ y ∂ x ∂ y ∂ x⋅∂ y
R =e 2⋅ −⋅
2
2
∂ ∂ ∂ ∂ ⋅P− 2 e 2⋅−⋅ ⋅Q− ∂y ∂x ∂y ∂x2
(5.1.6)
We set the general Ricci tensor and the Einstein tensor to zero, and we obtain the system of equations for the rotationally symmetric stationary vacuum solutions: e Rtt = 0:
2
2
∂ ∂ ∂ ∂ ∂ ∂ ⋅ ⋅ − e−2⋅⋅ ⋅ − = 2 ∂x ∂x ∂x ∂ y2 ∂ y ∂ y
−2⋅
2
1 2⋅− −2⋅ ∂ ∂ ⋅e ⋅e ⋅ e −2⋅⋅ 2 ∂x ∂y
159
2
5.1 Axially symmetric spacetime
e−2⋅⋅ Rφφ = 0:
2 ∂2 ∂ ∂ ∂ ∂ −2⋅ ∂ ⋅ − e ⋅ ⋅ − = 2 2 ∂x ∂x ∂y ∂y ∂x ∂y
2
1 ∂ ∂ − ⋅e 2⋅−⋅ e −2⋅⋅ e−2⋅⋅ 2 ∂x ∂y
2
Rtφ = 0:
∂ e 3⋅−−⋅∂ ∂ e3⋅−−⋅∂ =0 ∂x ∂x ∂y ∂y
Rxy = 0:
∂2 − ∂ ⋅∂ − ∂ ⋅∂ ∂⋅∂ ∂ ⋅∂ = ∂ x⋅∂ y ∂x ∂y ∂y ∂x ∂x ∂y ∂x ∂ y 1 2⋅− ∂ ∂ ⋅e ⋅ ⋅ 2 ∂x ∂y e
2 ∂ ⋅ ∂ 2 ∂ ⋅ ∂ − ∂y ∂y ∂y ∂y ∂ ∂ ∂ ∂ e−2⋅⋅ ⋅ ⋅ = ∂ x ∂x ∂x ∂ x
−2⋅
Gxx = 0:
2
1 2⋅− −2⋅ ∂ ∂ ⋅e ⋅e ⋅ −e −2⋅⋅ 4 ∂x ∂y
2
2 ∂ e−2⋅⋅ ∂ 2 ∂ ⋅ ∂ − ∂ x ∂ x ∂x ∂x ∂ ∂ ∂ ∂ e−2⋅⋅ ⋅ ⋅ = ∂ y ∂y ∂y ∂y
Gyy = 0:
2
2
(5.1.7)
2
1 2⋅ − −2⋅ ∂ ∂ ⋅e ⋅e ⋅ −e−2⋅⋅ 4 ∂x ∂y
2
Introduce a new notation, that can be used to rewrite the equations into a symmetric form, and we rewrite the equations originating from the tt and φφ components of the Ricci tensor: =
2
∂ e −⋅∂ ∂ e −⋅∂ = 1⋅e 3⋅−⋅ e−⋅ ∂ e −⋅ ∂ ∂x ∂x ∂y ∂y 2 ∂x ∂y
2
2
∂ e −⋅∂ ∂ e −⋅∂ =− 1⋅e 3⋅−⋅ e−⋅ ∂ e −⋅ ∂ ∂x ∂x ∂y ∂y 2 ∂x ∂y
2
(5.1.8) The sums and differences of these and the Einstein tensor components:
160
5.1 Axially symmetric spacetime
∂ e ∂ e Rtt R =G xxG yy= ∂ e −⋅ ∂ e−⋅ =0 ∂x ∂x ∂y ∂y
2
Rtt − R =−e
3⋅−
⋅e
∂ ∂ ⋅ e−⋅ ∂x ∂y
2
−
2
G x −G yy =−e 2⋅−⋅ e−⋅
∂ ∂ −e−⋅ ∂x ∂y
2
(5.1.9)
We can make a coordinate condition because of the gauge freedom: e 2⋅−= x , y
(5.1.10)
Rewrite the arc length squared by substituting new functions: =e 2⋅ −
=e −
=
1 e 2 ds =e ⋅ ⋅c ⋅dt − d −⋅c⋅dt − ⋅ dx 2⋅dy 2 2
2
2
(5.1.11)
Insert them into the equations originating from the Ricci tensor:
2 2 2 2 ∂ e 3⋅−−⋅∂ − ∂ e 3⋅−−⋅∂ − =0 ∂x ∂x ∂y ∂y
(5.1.12)
Thus ω and 2− 2 satisfy the same equation. With this new solutions can be created from the axially-symmetric stationary solutions. For example the conjugate metric – that will become important later for the derivation of the Kerr solution – with the following transformation: t i⋅
−i⋅t
The arc length squared changes with the substitution: 1 ⋅c 2⋅dt 2− ⋅d −⋅c⋅dt 2
1 2 2 2⋅ 2− 2 ⋅c ⋅dt ⋅c⋅dt⋅d − ⋅d 2 (5.1.13)
→
That can also be interpreted as the result of the following transformation: 2⋅dt 2− 1 ⋅d −⋅c⋅dt ⋅c 2 =
− 2 2
=
− 2 2
With the choice of a proper gauge, we can rewrite the arc length squared: 161
(5.1.14)
5.1 Axially symmetric spacetime =
=1
1 ds 2=e ⋅ ⋅c 2⋅dt 2 − ⋅ d −⋅c⋅dt2 −e 2⋅⋅ dx 2dy 2
(5.1.15)
∂2 e =0 ∂ x⋅∂ y
(5.1.16)
Because of this:
We perform a coordinate transformation, where we use the exponential expression as a coordinate: e =
x , y , z
∂ ∂ z = ∂x ∂ y
∂ ∂z =− ∂y ∂x
(5.1.17)
Substitute it into the arc length squared, where now the unknown functions depend on ρ and z, this is the Papapetrou metric t z :
1 ds 2=⋅ ⋅c 2⋅dt 2− ⋅ d −⋅c⋅dt 2 −e 2⋅⋅ d 2dz 2
(5.1.18)
5.2 Ernst equation It is possible to make the metric more clear without loosing generality, and the equations can be reduced to standard form. We assume that there exist a light-like surface in the metric, this is the first crucial distinction between the flat spacetime and the rotating black hole. We introduce spherical polar coordinates t r . The equation of the event horizon: N x , y=N r , =0 The condition for being light-like is that the four-distance is zero on it: ∂N ∂N g ⋅ ⋅ =0 x x
(5.2.1)
In our choice of metric: 2
e 2⋅−⋅
2
∂N ∂N =0 ∂r ∂
(5.2.2)
A choice of gauge, and then using it for the equation of the surface: e 2⋅−= r =0
(5.2.3) 162
5.2 Ernst equation On the surface, we can assume the general form of our exponential expression to be the following: e = ⋅ f r , = ⋅ f =0
(5.2.4)
Insert all this into the sum of the tt and φφ components of the Ricci tensor:
2
1 2⋅− −2⋅ ∂ ∂ ⋅e ⋅e ⋅ e −2⋅⋅ 2 ∂x ∂y ∂ ∂r
2
=0
∂ 1 ∂2 f ⋅ 2 =0 ⋅ ∂r f ∂
(5.2.5)
The solutions of the equation:
∂2 =2 ∂ r2
Solution for Δ:
=r −r g⋅r a
f =sin
2
2
(5.2.6)
Where a and rg are constants of integration, and our choice of symbols is not accidental of course. The following expressions are transformation invariant, here p and q are real constants, we will use these relationships later:
2 2r g = ⋅ p
2
rg −a 2 2
r r − 2a = g −a 2 2
2 g
2
rg q −a 2 2a = ⋅ p 2 2
2
(5.2.7)
p q =1
Return to the Papapetrou metric, write down the solutions of the metric functions, and from them the transformations between the coordinates: e
−
=e = ⋅sin
=
z = r−
rg ⋅cos 2
(5.2.8)
Introduce new coordinates, substitute them into the components of the Ricci tensor, and their combinations t r : 2
=cos
2
=1− =sin
2
∂− ∂− ∂ ∂ 2⋅− Rtt − R = ∂ ⋅ ∂ ⋅ =−e ⋅ ⋅ ⋅ ∂r ∂r ∂ ∂ ∂r ∂
163
2
5.2 Ernst equation
∂ ∂ Rt =R t = ∂ ⋅e 2⋅−⋅ ∂ ⋅e 2⋅ −⋅ =0 ∂r ∂r ∂ ∂
(5.2.9)
The same equations with the substitution of =e − : 2
∂ ∂ Rtt − R = ∂ ⋅ ∂ ⋅ = ∂r ∂r ∂ ∂
2
⋅
∂ ∂ ⋅ ∂r ∂ 2
∂ ∂ Rt =R t = ∂ ⋅ ∂ ⋅ =0 2 ∂r ∂r ∂ 2 ∂
(5.2.10)
Rearrange the equations:
2
∂ ∂ ∂ ∂ ⋅ ∂ ⋅ ∂ ⋅ =⋅ ∂r ∂r ∂ ∂ ∂r ∂r
2
2
⋅
∂ ∂ ∂ ∂
2
∂ ∂ ∂ ∂ ∂ ∂ ⋅ ∂ ⋅ ∂ ⋅ =2⋅ ⋅ ⋅ ⋅ ⋅ ∂r ∂r ∂ ∂ ∂r ∂r ∂ ∂
(5.2.11)
If we perform these substitutions, we obtain two symmetric equations:
X =
Y =−
1 ∂X ∂X ⋅ X Y ⋅ ∂ ⋅ ∂ ⋅ 2 ∂r ∂r ∂ ∂
2
∂X =⋅ ∂r
1 ∂Y ∂Y ∂Y ⋅ X Y ⋅ ∂ ⋅ ∂ ⋅ =⋅ 2 ∂r ∂r ∂ ∂ ∂r
∂X ⋅ ∂
2
⋅
∂Y ∂
2
2
(5.2.12)
The following equations will help to determine μ and η at the (5.3.13) relationship: rg ∂ 2 ∂ 2 ∂ X ∂Y ∂ X ∂ Y R xy=R yx =− ⋅ ⋅ = ⋅ ⋅ ⋅ 2 ∂r ∂ X Y ∂r ∂ ∂ ∂ r r−
r g ∂ ∂ ⋅ 2⋅⋅ = 2 ∂r ∂ r r− g 2 4 ∂ X ∂Y ∂ X ∂Y 2 ⋅ ⋅ ⋅ −⋅ ⋅ −3⋅ ∂r ∂r ∂ ∂ X Y 2 G xx −G yy =2⋅ r−
(5.2.13)
We introduce new coordinates again, and use the transformation rules we have introduced earlier 164
5.2 Ernst equation t : r− =
rg 2
2
=
2
rg −a 2 2
rg −a 2 ⋅2 −1 2
(5.2.14)
Write down all four previous equations with them:
1 ∂X 2 2 ∂ X ⋅ X Y ⋅ ∂ −1⋅ ∂ 1− ⋅ 2 ∂ ∂ ∂ ∂
∂X = −1⋅ ∂ 2
2
2
2
∂X 1− ⋅ ∂ 2
2
1 ∂Y ∂Y ∂Y ∂Y ⋅ X Y ⋅ ∂ 2−1⋅ ∂ 1− 2 ⋅ =2−1⋅ 1− 2 ⋅ 2 ∂ ∂ ∂ ∂ ∂ ∂
(5.2.15) −
∂ ∂ 2 ∂ X ∂Y ∂ X ∂Y ⋅ 2 ⋅ = ⋅ ⋅ ⋅ 2 2 ∂ ∂ 1− −1 X Y ∂ ∂ ∂ ∂
∂ ∂ 2⋅⋅ 2⋅⋅ = ∂ ∂ 4 ∂ X ∂Y ∂ X ∂Y 3 1 ⋅ 2−1⋅ ⋅ −1− 2 ⋅ ⋅ − 2 2 ∂r ∂r ∂ ∂ X Y −1 1− 2
(5.2.16)
The following transformations are also solutions of the equations describing X and Y, where c is an arbitrary constant, this will also become useful later: 2
X=
X 1c⋅X
2
Y=
Y 1−c⋅Y
(5.2.17)
We express new functions from the old ones and substitute them into the symmetric equations: X= =
1F 1−F
1− F⋅G 1− F ⋅1−G
Y=
1G 1−G
=
F−G 1−F ⋅1−G
∂F ∂F 1−F⋅G⋅ ∂ 2−1⋅ ∂ 1− 2 ⋅ ∂ ∂ ∂ ∂ ∂G 2 2 ∂G 1−F⋅G⋅ ∂ −1⋅ ∂ 1− ⋅ ∂ ∂ ∂ ∂
(5.2.18)
2
2
∂F ∂F =−2⋅G⋅ −1⋅ 1− 2 ⋅ ∂ ∂
165
2
2
=−2⋅F⋅ −1⋅
2
2
∂G ∂G 2 1− ⋅ ∂ ∂ (5.2.19)
5.2 Ernst equation The solutions of the equations, where p and q are real constants: p 2−q2=1
G=− p⋅q⋅
F =− p⋅−q⋅
(5.2.20)
The angular frequency can be derived from a coordinate potential in the following way:
∂ ∂ Rt =R t = ∂ ⋅ ∂ ⋅ =0 2 ∂r ∂r ∂ 2 ∂ ∂ ∂ = ⋅ ∂ 2 ∂
∂ ∂ = ⋅ ∂ 2 ∂ r
(5.2.21)
The potential is determined by the following equation:
2 2 ∂ ⋅∂ ∂ ⋅∂ =0 ∂ ∂ ∂ ∂
(5.2.22)
The other equation can also be written down with a potential:
2
∂log ∂ log 2 ∂ 2 ∂ Rtt − R = ∂ ⋅ ∂ ⋅ = ⋅ ⋅ ∂ ∂ ∂ ∂ ∂ ∂
2
(5.2.23)
We introduce yet another potential, with it we can write down equations with the same form like (5.2.11), here Ψ corresponds to χ, Φ to ω, and κ to r: =
⋅
∂ ∂ ⋅ ∂ ⋅ ∂ ⋅ ∂ ∂ ∂ ∂
2
=⋅
∂ ∂ ∂ ∂
2
2
⋅
∂ ∂ ∂ ∂ ∂ ∂ ⋅ ∂ ⋅ ∂ ⋅ =2⋅ ⋅ ⋅ ⋅ ⋅ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂
2
(5.2.24)
If we consider the potentials to be the components of a single complex quantity, we can write down an equation like (5.2.15): Z =i⋅
∂Z ∂Z ℜZ ⋅ ∂ ⋅ ∂ ⋅ ∂ ∂ ∂ ∂
2
∂Z ∂Z =⋅ ⋅ ∂ ∂
2
(5.2.25)
We can write down a transformation relationship with the same form like previous one here too:
166
5.2 Ernst equation 2
Z=
Z 1i⋅c⋅Z
(5.2.26)
Since the equation has the same form, we can use a function substitution of the same form, thus we obtain the Ernst equation: Z =−
1E 1−E
∂E 1−E⋅E ∗⋅ ∂ ⋅ ∂ ∂
∂E ∂ ⋅ ∂ ∂
2
=−2⋅E∗⋅ ⋅
∂E ∂E ⋅ ∂ ∂
2
(5.2.27)
Conjugate potentials, using the conjugate metric functions: =
− 2
=
2
− 2 2
2 2 ⋅ =e ⋅ − =e 2⋅− 2⋅e 2⋅ =
∂ 2 ∂ ∂ = 2⋅ = ⋅ ∂ ∂ ∂
2 ∂ ∂ ∂ =− 2⋅ =− ⋅ ∂ ∂ ∂
1 E Z =i⋅ =− 1− E
(5.2.28)
Write down the conjugate Ernst equation, and notice two more relationships between the conjugate potentials:
E ∗⋅ ∂ ⋅∂ E 1− E⋅ ∂ ∂
∂ E ∂ ⋅ ∂ ∂
E ∗ 1− E⋅ =ℜ Z =− 2 ∣1− E∣
2
∂ E ∂ E =−2⋅E∗⋅ ⋅ ⋅ ∂ ∂
E ∗ E− =ℑ Z =−i⋅ 2 ∣1− E∣
2
(5.2.29)
5.3 The derivation of the Kerr solution If we investigate every possible symmetry in the axially-symmetric vacuum spacetime, we will arrive at a single analytic expression about the metric, the first time found by Roy Kerr in 1963. The conjugate Ernst equation has a similar form to (5.2.19), therefore the following match is possible, and we directly obtain the solution: F = E
G= E ∗ 167
5.3 The derivation of the Kerr solution E=− p⋅−i⋅q⋅
p 2q2=1
(5.3.1)
Express the complex potential: 1− p⋅−i⋅q⋅ Z =i⋅ =− 1 p⋅i⋅q⋅ p 2⋅ 2−1−q 2⋅1− 2 = p⋅12q2⋅ 2
=
2⋅q⋅ p⋅12q2⋅ 2
(5.3.2)
Return to the r coordinate, substitute p and q, and also introduce ρ t r :
2 p= ⋅ rg
2
rg −a 2 2
q=
2⋅a rg
2 =r 2a 2⋅ 2=r 2a 2⋅cos 2
(5.3.3)
Write down the two potentials with them: 2
−a2 ⋅ =
=
a⋅r g⋅ 2
(5.3.4)
Determining unknown conjugate metric functions from the potentials: 2 2 2 ∂ 2⋅a⋅r⋅r g⋅ ∂ = −a ⋅ ⋅∂ =− = ⋅ 4 4 ∂ ∂r ∂ ⋅
a⋅r g 2 2 2 2 ∂ ∂ −a 2⋅2 ∂ = 4 ⋅r −a ⋅ =− ⋅ =− ⋅ 4 ∂ ∂r ∂r ⋅ a⋅r g⋅ r 2−a 2⋅ 2 ⋅ ∂ =− 2 2 ∂r −a ⋅
2⋅a⋅r⋅r g⋅⋅ ∂ =− ∂ −a2⋅2
(5.3.5)
Conjugate angular frequency, and from it the angular frequency: =
a⋅r⋅r g⋅ = 2− 2 −a 2⋅
(5.3.6) 2
−a ⋅ 2⋅ =e ⋅2 −2 =e 2⋅− 2⋅e 2⋅ = 2 =
a⋅r⋅r g⋅ 2 a⋅r⋅r g⋅ −2⋅ ⋅ −2 = ⋅e 2 −a ⋅ 2 168
(5.3.7)
5.3 The derivation of the Kerr solution Combine the two upper equations and substitute the product of the two deltas: e 2⋅ =⋅ −a 2⋅ 2⋅ 2⋅ 2 4⋅ 4 2 2 2 ⋅e =e − ⋅e = 4⋅⋅ −a ⋅r ⋅r g⋅ 2
(5.3.8)
Write down a few algebraic identities, and introduce yet another metric function: r 2a 2∓a⋅ ⋅⋅ ±a⋅ =2⋅ ±a⋅r⋅r g⋅ 2⋅−a 2⋅=4⋅−a 2⋅r 2⋅r 2g⋅ 2 = r 2a 22−a 2⋅⋅
(5.3.9)
Using this the metric functions: e 2⋅ =
⋅ 2 2
=
e 2⋅=e 2⋅ −e 2⋅ =
2⋅ 2
a⋅r⋅r g 2
=e −=
2 ⋅ 2
(5.3.10)
Using the identity we can express X and Y, and their derivatives: X ==
a⋅ r a 2 a⋅ ⋅⋅
X =−=
−a⋅ r a 2 −a⋅ ⋅⋅
2
∂ X ∂Y = = ∂r ∂r
(5.3.11)
2
2⋅ r −
rg −2⋅r⋅ a⋅ ⋅ 2 2
r 2a 2a⋅ ⋅ ⋅ ⋅
∂ X ∂Y ⋅ ⋅r 2 a 2 a 2⋅2⋅a⋅ ⋅ = = 2 ∂ ∂ r 2a 2a⋅ ⋅ ⋅ 3
(5.3.12)
Substitute the results above into (5.2.13): rg r ∂ 2 ∂ − ⋅ ⋅ = 2 ⋅ r − g ⋅ 22⋅a 2⋅−2⋅r⋅ ∂r ∂ 2 ⋅⋅ r−
169
5.3 The derivation of the Kerr solution 2
r r− g r ∂ 2 ∂ r− g ⋅ ⋅ =2− 2 ∂r ∂
(5.3.13)
r⋅r g 2
The solution of the system of equation: e =
2
(5.3.14)
In our choice of notation the metric functions are the followings: e
e 2⋅=
−
=
2
e 2⋅ =2
(5.3.15)
We have expressed every metric function, therefore the only thing left to do is to substitute them into the original arc length squared: 2
a⋅r⋅r g 2 2 2 2 2 ds = ⋅ 2⋅c 2⋅dt 2 − 2 ⋅ d − ⋅c⋅dt ⋅sin − ⋅dr − ⋅d 2 2 2
2
2
Where:
2
2 2
2
2
2 2
2
2
2
(5.3.16)
2
= r a −a ⋅⋅=r a −a ⋅r −r g⋅r a ⋅sin
Write down the spacetime of the black hole in rotational ellipsoid coordinates: x= r 2a 2⋅sin ⋅cos y= r 2a 2⋅sin ⋅sin z =r⋅cos
(5.3.17)
The arc length squared of the Kerr solution in Boyer-Lindquist coordinates:
ds 2= 1−
r⋅r g
2⋅r⋅r g⋅a 2 2 2 2 ⋅sin ⋅d ⋅c⋅dt− ⋅dr − ⋅d 2 2 2 r⋅r ⋅a − r 2a2 g2 ⋅sin 2 ⋅sin 2 ⋅d 2 ⋅c2⋅dt 2
Where:
2
=r −r g⋅r a
2
(5.3.18)
2 =r 2a 2⋅cos 2
If a approaches zero, we get back the spherically symmetric Schwarzschild solution, with this we managed to identify this quantity, the geometric angular momentum:
170
5.3 The derivation of the Kerr solution
ds 2= 1−
a=0 →
rg 2 2 dr 2 ⋅c ⋅dt − −r 2⋅ d 2sin 2 ⋅d 2 r r 1− g r
Schwarzschild radius:
r g=
Kerr angular momentum:
a=
2⋅⋅M c2
J 2⋅⋅J = 3 m⋅c c ⋅r g
(5.3.19)
If the mass of the black hole approaches zero, we get back the rotational ellipsoid coordinate system in flat spacetime: 2 ds 2=c 2⋅dt 2− 2 2⋅dr 2−2⋅d 2−r 2 a 2 ⋅sin 2 ⋅d 2 r a
(5.3.18)
The geometric quantities from the metric tensor to the connection:
g =
1−
r⋅r g 2 0
0 a⋅r⋅r g⋅sin 2
2
0
a⋅r⋅r g⋅sin 2 2
0
2 0 2 0 −
0
−
0
0
(5.3.19)
0 r⋅r ⋅a − r 2a 2 g2 ⋅sin 2 ⋅sin2
The metric tensor has non-zero non-diagonal components. We write down a partial matrix using the rows and columns where these components appear:
r⋅r g a⋅r⋅r g⋅sin2 2 2 g ij = a⋅r⋅r g⋅sin 2 r⋅r ⋅a − r 2a2 g2 ⋅sin 2 ⋅sin 2 2 1−
(5.3.20)
The determinant of the partial matrix: a⋅r⋅r g⋅2a−1⋅r⋅r g ⋅sin 2 r 2a 2⋅ 2⋅ 2−r 2⋅r g 2 g =g tt⋅g −g t ⋅g t =− ⋅sin 4 (5.3.21) Invert the partial matrix and extend with it the twice contravariant metric tensor:
171
5.3 The derivation of the Kerr solution
g =
r 2a 22−a 2⋅⋅sin 2 ⋅2 0
0
0
a⋅r⋅r g ⋅2
2
0
0
−
0
0
a⋅r⋅r g ⋅2
1 − 2
0
0
−a 2⋅sin2 − ⋅ 2
0
∂ g tt r ⋅2⋅r 2−2 =− g 2 ∂r
∂ g t ∂ g t a⋅r g⋅2⋅r 2−2 2 = =− ⋅sin 2 ∂r ∂r
∂ g rr 2⋅r−r g ⋅2−2⋅⋅r = ∂r 2
∂ g =−2⋅r ∂r
(5.3.22)
∂ g a⋅r g⋅2⋅r 2− 2⋅sin 2 −2⋅r 2⋅4 = ⋅sin2 2 ∂r ∂ g tt 2⋅a 2⋅r⋅r g =− ⋅cos ⋅sin 4 ∂ ∂ g t ∂ g t 2⋅a⋅r⋅r g⋅a 2⋅sin 2 2 = = ⋅cos ⋅sin ∂ ∂ 4 ∂ g rr 2⋅a 2 = ⋅cos ⋅sin ∂
∂ g =2⋅a 2⋅cos ⋅sin ∂
∂ g 2⋅a⋅r⋅r g⋅a 2⋅sin 2 −2⋅2 ⋅sin 2 −2⋅a 2r 2 ⋅4 = ⋅cos ⋅sin ∂ 4
(5.3.23)
r 2a 2 2−a 2⋅r⋅r g ⋅sin 2 tr = rt =r g⋅2⋅r − ⋅ 2⋅⋅6 t
t
2
2
a 2⋅ r⋅r g ⋅sin r⋅r g⋅2−r 2a 2 2 t = t=a ⋅r⋅r g⋅ ⋅cos ⋅sin 6 ⋅ t
t
2
2⋅r 2− 2⋅r 2a2 2−a⋅r⋅r g a⋅⋅sin2 2⋅r 2⋅4 2 r = r=−a⋅r g⋅ ⋅sin 6 2⋅⋅ t
t
t = t =−a⋅r⋅r g⋅cos ⋅sin ⋅ a 3⋅ r⋅r g a⋅⋅sin 4 a⋅2⋅r⋅r g a⋅⋅ 2−a⋅r 2 a 2 2⋅sin2 2⋅r 2a 2 ⋅ 2−r 2−a 2 6
⋅
172
5.3 The derivation of the Kerr solution
r tt = r rr =
⋅r g⋅2⋅r 2−2 2⋅6 r 2⋅r −r g − 2 2⋅
r =−
tt =− rr =
r t = r t =
a2 r r = r r = r =− 2⋅cos ⋅sin a⋅r g⋅2⋅r 2−2 ⋅sin 2 −2⋅r⋅4 2 =⋅ ⋅sin 6 2⋅
r⋅ 2
a 2⋅r⋅r g 6
a⋅⋅r g⋅2−2⋅r 2 ⋅sin 2 6 2⋅
r
⋅cos ⋅sin
a2 ⋅cos ⋅sin 2 ⋅
t = t = r = r =
a⋅r⋅r g⋅a 2⋅sin 2 2 6
⋅cos ⋅sin
r 2
−a 2⋅sin 2 ⋅sin 2 r⋅r g = =a⋅r g⋅2⋅r − ⋅ 6 2⋅⋅ tr
rt
t = t=a⋅r⋅r g⋅
r
=
r
=
2
2
a 2⋅a 2⋅sin 2 2−⋅sin 2 −r⋅r g −⋅2 6
⋅
⋅cos ⋅sin
a⋅r g⋅ a2⋅sin 2 −⋅sin 2 −a⋅r⋅r g ⋅2⋅r −2 2⋅r⋅4⋅−a 2⋅sin 2 2 ⋅sin 6 2⋅⋅
cos ⋅sin 2 ⋅ a ⋅r⋅r g⋅−a 3⋅sin6 a⋅−2⋅2 ⋅sin 4 r⋅r g⋅ 2 6 ⋅ 4 2 2 2 3 2 4 2 2 a⋅−a⋅ ⋅ r a r⋅r g⋅2⋅ ⋅a ⋅r⋅r g ⋅sin ⋅⋅r a
= =
(5.3.24)
The partial derivatives of the connection get very complicated, just like the other quantities that follow from them, therefore we do not write down all of them. The geodesics of the Kerr solution: t t t t ˙ ˙ ˙ c⋅t¨ 2⋅ tr⋅c⋅˙t⋅r˙ t ⋅c⋅t˙⋅ ˙ ˙ r ⋅r⋅ ⋅⋅=0
r ˙ r¨ r tt⋅c 2⋅˙t 2 r rr⋅r˙ 2 r ⋅˙ 2 r ⋅ ˙ 22⋅ rt ⋅c⋅˙t⋅ ˙ ˙⋅=0 r ⋅r 2 2 2 ¨ ˙ ˙ ˙⋅=0 tt⋅c ⋅t˙ rr⋅r˙ 2⋅ t ⋅c⋅˙t⋅ r ⋅r t t t t ˙ ˙ ˙ c⋅t¨ 2⋅ tr⋅c⋅˙t⋅r˙ t ⋅c⋅t˙⋅ ˙ ˙ r ⋅r⋅ ⋅⋅=0
173
(5.3.25)
5.4 Coordinate singularities
5.4 Coordinate singularities The singularities in the Kerr metric are much more diverse than in the spherically symmetric case. We examine the tt component of the metric tensor: g tt =1−
r⋅r g
(5.4.1)
2
It becomes meaningless in the following case: r⋅r g = 2=r 2a 2⋅cos2 2
2
2
r −r⋅r g a ⋅cos =0
The solution of the quadratic equation gives the place of the infinite redshift: r 1,2=r g
r ±
2 g
−4⋅a 2⋅cos 2 2
(5.4.2)
The rr component of the metric tensor: 2
g rr =−
(5.4.3)
2
2
=r −r g⋅r a =0
The solutions give the places of the event horizons, where the metric changes signature: r 1,2=r g
r ±
2 g
−4⋅a 2 2
(5.4.4)
For real results, the discriminant of the quadratic formula has to be greater than zero, that creates a condition for the angular momentum: 2
2
r g −4⋅a ≥0 r g ≥2⋅∣a∣
⋅M 2 ≥J c
(5.4.5)
For the sake of example, we examine these surfaces on the longitudinal section of a black hole with an extreme angular momentum – mass ratio:
174
5.4 Coordinate singularities
a 84 = r g 41
Infinite redshift occurs on the black surfaces with variable shapes. The grey, spherically symmetric surfaces are the exterior and interior event horizons. The most outward, grey, unfinished spherical surface would be the Schwarzschild radius, if the black hole would not rotate. The domain between the external redshift limit and the exterior event horizon is the ergosphere.
5.5 Redshift This time we substitute the components of the Kerr metric tensor into the previous equation:
r⋅r g r⋅r 1− 2 22 g 2 2 2 g tt 2 2r a ⋅cos 2 ⋅2= ⋅2 = ⋅2 1 = 1 g tt 1r⋅r g 1 r⋅r g 1− 2 1− 2 2 2 1 1r a ⋅cos 1
1− 2
(5.5.1)
If the light source is closer to the source of the gravitational field than the observer, then: 1
r ≥2r
→
1
≤2
(5.5.2)
There is a discrepancy also, when the source and the observer have the same distance from the gravitating centre, but they are on different latitudes: 1
≤2
→
1
≤2
2
(5.5.3)
5.6 Frame dragging NASA launched in 1976 and in 1992 a LAGEOS (Laser Geodynamics Satellite) each, that are passive metal spheres with diameters of 60 cm and masses of 411 kg, therefore the upper 175
5.6 Frame dragging atmosphere of the Earth has practically no effect on them. By examining the laser light reflected from mirrors on their surface, it is possible to measure their highly regular orbits with great accuracy. The are used to accurately determine the shape of the Earth and the velocities of the tectonic plates, while relativistic effects cumulate in their orbital parameters. By analysing observations lasting for decades, it was possible to show the effects of spacetime dragged by the rotating Earth, with 20% accuracy. The existence of non-diagonal tensor components has interesting consequences on the relationship between the contravariant and the covariant velocities: t
t
tt
t
v =g ⋅v =g ⋅c⋅v t g ⋅v
(5.6.1)
v =g ⋅v =g t⋅v t g ⋅v
(5.6.2)
For example in the second case, if the horizontal momentum of the test body is zero, it can still have non-zero velocity and vice versa, and in the first case, it can have momentum without rest energy. Furthermore a test body at infinite distance with zero orbital velocity falling radially into the gravitational field of the rotating black hole is forced to orbit, it obtains the following angular frequency: t
g ⋅v g ⋅v v f = t = tt t t v g ⋅v t g ⋅v
v =0
(5.6.3)
Angular frequency caused by the frame dragging of the rotating black hole: f=
c⋅a⋅r⋅r g t = 2 22 2 g 2 tt g r a −a ⋅⋅sin
(5.6.4)
We approach the Earth's spacetime with the Kerr metric, our instruments are not yet precise enough to distinguish between the effects of a rotating body and a rotating black hole. The angular momentum is a product of the angular frequency and the moment of inertia:
J =⋅
(5.6.5)
We approximate the Earth with a rigid rotating sphere: 2 2 = ⋅M ⋅R 5
→
2 ⋅R2 a= ⋅ 5 c
(5.6.6)
From the Earth's radius and rotational angular frequency, the geometric angular momentum: 6
R=6.371⋅10 m
t f =86164.1 s
→
=7.292115⋅10−5
a=3.949 m
1 s (5.6.7)
Earth's standard gravitational parameter and the gravitational radius: 176
5.6 Frame dragging
⋅M =3.986004418⋅1014
m3 2 s
r g=
→
2⋅⋅M =8.870056078⋅10−3 m 2 c
We assume for simplicity, that the LAGEOS-1 satellite is on an equatorial orbit. The semi-major axis and orbital period of the orbit, and the frame dragging: t k =3.758 h=13528.8 s
r =1.227⋅10 7 m f =5.685⋅10−15
1 s
(5.6.8)
The accumulated displacement along the orbit in a year: year =0.037' '
→
s=2.201 m
(5.6.9)
We assume the same about the LAGEOS-2 satellite as well. The semi-major axis and orbital period of the orbit, and the frame dragging: t k =223 min=13380 s
r =1.2163⋅107 m −15
f =5.836⋅10
1 s
(5.6.10)
The accumulated displacement along the orbit in a year: year =0.038' '
→
s=2.24 m
(5.6.9)
5.7 Equatorial circular orbit The coordinate conditions coincide with the spherically symmetric case: t=t
∂t =const. ∂
r =const.
dr =0
2
d =0
=
=
∂ r ∂2 r = =0 ∂ ∂ 2
∂ =const. ∂
Because of the coordinate conditions, the geodesics simplify: 177
(5.7.1)
5.7 Equatorial circular orbit r
2
2
tt⋅c ⋅t˙
r
2 r ⋅˙ 2⋅ t ⋅c⋅t˙⋅=0 ˙
tt⋅c 2⋅t˙ 22⋅ t ⋅c⋅t˙⋅=0 ˙
(5.7.2)
The arc length squared also simplifies:
ds 2= 1−
r g 2 2 2⋅r g⋅a r ⋅a ⋅c ⋅dt ⋅d ⋅c⋅dt− r 2a 2 g ⋅d 2 r r r
(5.7.3)
This of course is equal to the arc length measured in the coordinate system of the moving observer: =
d dt
c 2⋅d 2= 1−
r g 2 2⋅r g⋅a r ⋅a ⋅c ⋅c⋅− r 2a 2 g ⋅ 2 ⋅dt 2 r r r
The relationship between the proper time and the coordinate time:
d =dt⋅ 1−
rg 2⋅r g⋅a r ⋅a 1 ⋅− 2⋅ r 2a 2 g ⋅ 2 r c⋅r r c
(5.7.4)
This equation is satisfied by two different angular momenta, therefore it is valid on two different circular orbits. Since the ratio of the two quantities is constant, the coordinate time also can be used as a parameter for the geodesic equations. In this case the tangent vectors can be identified, the equation can be solved, and the possible angular frequencies can be determined: c⋅t˙ =c
r
= ˙ 2
r
r
2
(5.7.5)
⋅ 2⋅ t ⋅c⋅ tt⋅c =0
Apply the quadratic formula: −b± b2−4⋅a⋅c x 1,2= 2⋅a 1,2 =
−2⋅ r t ⋅c± 2⋅ rt ⋅c2−4⋅ r⋅ r tt⋅c2 2⋅ r
− rt ± r t 2− r ⋅ rtt 1,2 =c⋅ r
(5.7.6)
178
5.7 Equatorial circular orbit We obtain two possible orbital frequencies, each corresponding to an orbital direction The affected connection coefficients are simplified by the coordinate conditions: r tt = r tt =
⋅r g⋅2⋅r 2−2 2⋅6 ⋅r g
(5.7.7)
2⋅r 4
a⋅⋅r g⋅2−2⋅r 2 2 t = t = ⋅sin 6 2⋅ r
r
r t = r t =−
a⋅⋅r g 2⋅r
(5.7.8)
4
a⋅r g⋅2⋅r 2−2 ⋅sin 2 −2⋅r⋅4 2 =⋅ ⋅sin 6 2⋅ r
a⋅r g −2⋅r 3 =⋅ 2⋅r 4 r
(5.7.9)
Substituting: a⋅⋅r g 1,2 =c⋅
2⋅r 4
±
2
a⋅r g −2⋅r 3 ⋅r g − −⋅ ⋅ 4 2⋅r 4 2⋅r 4 2⋅r 3 a⋅r −2⋅r ⋅ g 4 2⋅r a⋅⋅r g
Two orbital frequencies are possible on equatorial circular orbits, depending on the orbital direction: a⋅r ± a⋅r g 2 − a⋅r g −2⋅r 3 ⋅r g 1,2 =c⋅ g 3 a⋅r g −2⋅r
(5.7.10)
5.8 Kerr-Schild metrics We write down their general form. The Kerr spacetime also belongs to this group, actually Roy Kerr was also looking for the solution in this form, ηηκ is the metric tensor of flat spacetime here, and lη is a light-like vector: g = l ⋅l
g = −l ⋅l
179
5.8 Kerr-Schild metrics ⋅l =l
l ⋅l =0
(5.8.1)
Since the Schwarzschild spacetime is a special case of the Kerr spacetime, it is also of this form. Reorder the Kerr arc length squared: ds 2=
sin 2 2 2 ⋅dt−a⋅sin ⋅d − ⋅r 2a 2 ⋅d −a⋅dt 2 − 2⋅dr 2−2⋅d 2 2 2
(5.8.2)
Introduce new coordinates and substitute them: du=dt −
r 2a 2 ⋅dr
a d =d − ⋅dr
(5.8.3)
ds 2= 2 sin 2 2 ⋅ du−a⋅sin ⋅d − ⋅r 2 a 2 ⋅d −a⋅du22⋅du−a⋅sin2 ⋅d ⋅dr −2⋅d 2 2 2 (5.8.4) Rearrange metric tensor and arc length squared:
g =
1−
r⋅r g
2
1 0
1
0
0 0
0 −2
a⋅r⋅r g ⋅sin 2 −a⋅sin 2 2
0
a⋅r⋅r g 2
⋅sin 2
−a⋅sin2 0 2 − 2 ⋅sin 2
ds 2=dudr 2 −dr 2−2⋅d 2−r 2a 2 ⋅sin 2 ⋅d 2−2⋅a⋅sin 2 ⋅d ⋅dr −r⋅r g 2 2 ⋅du−a⋅sin ⋅d 2
(5.8.5)
Write down the light-like vector and substitute it: l =0 1 0 0
l =1 0 0 −a⋅sin 2
ds 2=dudr 2−dr 2− 2⋅d 2− r 2a 2 ⋅sin 2 ⋅d 2−2⋅a⋅sin 2 ⋅d ⋅dr −r⋅r g ⋅l ⋅l ⋅dx⋅dx 2 Substitute rectangular coordinates into it: t=ur
x=r⋅cos a⋅sin⋅sin
y=r⋅sin −a⋅cos ⋅sin
z =r⋅cos 180
(5.8.6)
(5.8.7)
5.8 Kerr-Schild metrics x 2 y 2= r 2a 2⋅sin 2
(5.8.8)
The original metric written down by Kerr: ds 2 =c 2⋅dt 2−dx 2−dy 2−dz 2 2
2
ds =ds −
r 3⋅r g r 4a 2⋅z
⋅ c⋅dt − 2
r⋅ x⋅dx y⋅dya⋅ x⋅dx− y⋅dy z⋅dz − r r 2a 2
2
(5.8.9)
The equation expressing the radial coordinate: r 4− x 2 y 2 z 2−a 2⋅r 2−a2⋅z 2=0
(5.8.10)
The circular singularity is the true singularity of the Kerr spacetime: 2
2
2
x y z =a
2
z =0
(5.8.11)
The new coordinates can be introduced with another sign as well: du=dt
r 2a 2 ⋅dr
a d =d ⋅dr
(5.8.12)
5.9 Tomimatsu-Sato spacetimes They are the models of the axially symmetric spacetimes of rotating bodies, however they do not cover every possible solution. Write down the complex Ernst potential in the following form: =
(5.9.1)
Where α and β are polynomials in the x and y coordinates, and posses the following properties: (a)
always real:
∂ ∂ ⋅ −⋅ ∂x ∂x
always imaginary:
∂ ∂ ⋅ −⋅ ∂y ∂y
(5.9.2)
(b)
the coefficients of the even powers of y are real, of the odd powers are imaginary.
(c)
the degree of the α polynomial is δ2, of the β is δ2 – 1, where δ is a natural number called the deformation parameter.
(d)
α and β are δ-degree polynomials in the p and q real parameters, where: 181
5.9 Tomimatsu-Sato spacetimes p 2q2=1 (e)
(5.9.3)
in the q = 1 case, let ξ be the potential of the static Zipoy-Voorhees spacetimes. Therefore in the case of q t1
(8.5.2)
It propagates along one of the coordinates, therefore the coordinate conditions: =const.
(8.5.3)
=const.
Write down the arc length squared along the light-like geodesic: c 2⋅dt 2 −a 2 t ⋅d 2=0
(8.5.4)
The elapsed time: dt=±
a t⋅d c
dt d =± a t c t0
0
∫ adtt =± 1c⋅∫ d =± c1 t 1
(8.5.5)
1
The next light maximum: departs:
t 1 t 1
arrives:
t 0 t 0 t 0 t 0
∫ t 1 t 1 t0 t 0
∫ t1 t 1 t1
∫ t 1 t 1
(8.5.6) t0
dt dt = 1 =∫ a t c t a t 1
t1
t0
dt dt dt = ∫ ∫ a t t t a t t a t 1
dt a t
1
t 0 t 0
∫ t0
1
t 0t 0
∫ t0
t0
dt dt =∫ a t t a t 1
dt =0 a t
(8.5.7)
Since the δt change in time is small, therefore during this time the time dependent a(t) function does not change significantly: −
t1 t0 =0 a t 1 a t 0
0 1 t 0 a t 1 = = = 1 0 t 1 a t 0
(8.5.8)
247
8.5 Cosmological redshift The z parameter characterizes the ratios of the distances of distancing objects from us, without knowing their absolute distance: z=
0 a t 0 −1= −1 1 a t 1
(8.5.9)
We observe this value to be positive, and we interpret this as the expansion of the Universe.
8.6 Hubble law Distance along one of the coordinates: l t=a t⋅
(8.6.1)
Two celestial bodies with constant coordinates: l t 1 l t = a t 1 at
l l t= ⋅a t a
(8.6.2)
We examine with derivatives according to time, how fast the distance changes because of the expansion: a l˙ = ˙ ⋅l=H⋅l a
a H = ˙ ⋅c a
(8.6.3)
Where H is the current value of the Hubble constant: H 0=73,8±2,4
km 1 1 ⋅ =2.39⋅10−18 s MPc s
(8.6.4)
The Hubble time is approximately in the same magnitude with the age of the Universe: tH =
1 =4.18⋅1017 s=1.32⋅1010 year H0
(8.6.5)
It is possible to determine by approximation from the cosmic redshift and the Hubble constant, how long time it took, until the light has reached us. Series expansion around the state of the observer: ∞
1 1 dn 1 n =∑ ⋅ n ⋅− t a t 0 n! dt a t
1 1 a 1 2⋅a 2 a 1 a 6⋅a⋅a 6⋅a 3 ≈ ˙ 2⋅ t ⋅ ˙3 − ¨2 ⋅ t 2− ⋅ − 2 ˙ 3 ¨ − ˙4 ⋅ t 3 a t a a 2 a 6 a a a a
248
8.6 Hubble law
2
3
a a a a a a⋅a a ≈1 ˙ ⋅ t ˙ 2 − ¨ ⋅ t 2 ˙ 3 − ˙ 2¨ ⋅ t 3 a t a 6⋅a a 2⋅a a a
(8.6.6)
Substitute the H Hubble constant and the Q deceleration parameter, and write down with them the Taylor series of the cosmological redshift: H a˙ = c a
a Q= ¨ a
a H2 Q H3 H a 2 3 z= −1≈H⋅ t 2 − ⋅ t − 3 − ⋅Q ⋅ t a t 2 c 6⋅a c c
(8.6.7)
The distance of the light source in the case of a small z: l≈
H ⋅c z
(8.6.8)
8.7 Flat geometry According to our observations, the space in the Universe is not curved on the large scale. We determine the critical density, that characterizes this universe-model. To write down the flat arc length squared, we start with Minkowskian coordinates and extend them with the time dependent scale factor: ds 2=c 2⋅dt 2−a 2 t ⋅ dx 2dy 2dz 2
(8.7.1)
The geometric quantities from the metric tensor to the Ricci scalar:
1
1 0 0 0 2 0 −a 0 0 g = 0 0 −a 2 0 0 0 0 −a 2
0 1 0 − 2 a
g =
xx
0
0
0
0
yy
0
0
0
0
1 a2
0
−
0
−
(8.7.2)
1 a2
zz
∂ g xx ∂ g yy ∂ g zz = = =−2⋅a⋅a˙ ∂t ∂t ∂t
∂g ∂g ∂g 2⋅a = = = 3˙ ∂t ∂t ∂t a
(8.7.3)
t xx= t yy =t zz =a⋅a˙
a xtx = xxt = yty = y yt = ztz = z zt = ˙ a
(8.7.4)
249
8.7 Flat geometry t
t
t
∂ xx ∂ yy ∂ zz 2 = = =a⋅a ¨ a˙ ∂t ∂t ∂t x tx x xt yty yyt z tz z zt a¨ a˙ 2 = = = = = = − ∂t ∂t ∂t ∂t ∂t ∂t a a 2
(8.7.5)
Rt xtx =−Rt xxt =Rt yty =−Rt yyt =Rt ztz =−R t zzt =a⋅a¨ a x x y y z z R ttx=−R txt= R tty=−R tyt=R ttz =−R tzt = ¨ a R x yxy =−R x yyx =R xzxz =−R xzzx =R y xyx=−R yxxy =R y zyz=−R yzzy =R z xzx=−R z xxz =R z yzy =−R z yyz =a˙ 2 (8.7.6)
R =
−
3⋅a¨ a 0 0 0
0
0
0
2 a⋅a 0 0 ¨ a˙ 2 0 a⋅a 0 ¨ a˙ 2 0 0 a⋅a ¨ a˙
a⋅aa R=−6⋅ ¨ 2 ˙ a
(8.7.7)
2
(8.7.8)
The normalized form of the energy-momentum tensor:
⋅c 2
T =
0
0 p − 2 a
0
0
0
0
0
0
0
0
p a2
0
−
0
−
(8.7.9)
p a2
Substitute them into the Einstein equations: 1 8⋅⋅ R − ⋅R⋅g =− 4 ⋅T 2 c
(8.7.10)
The time-like component: 1 8⋅⋅ Rtt − ⋅R⋅g tt =− 4 ⋅T tt 2 c
(8.7.11)
250
8.7 Flat geometry −
2
3⋅a¨ 1 a⋅a a 8⋅⋅ − ⋅ −6⋅ ¨ 2 ˙ ⋅1=− 4 ⋅⋅c 2 a 2 a c 2
a 8⋅⋅ 3⋅ ˙ 2 = 2 ⋅ a c The first Friedmann equation gives the density of the Universe: 2
3⋅H = 8⋅⋅
(8.7.12)
Substitute and determine the numerical value: =1.02⋅10−26
kg 3 m
(8.7.13)
This means an average 6.11 hydrogen atoms in every cubic meters.
8.8 General Friedmann equations We express the three possible cases with a single equation:
{
0 k 0 =0
k⋅ x 21 x 22x 23 x 24 =a 2
(8.8.1)
Where we differentiate with k between the solutions that are either flat, or have positive or negative curvature. The spatial arc length squared: dl 2=dx 21dx12dx 21k⋅dx 24
(8.8.2)
Switch to the usual coordinate system and write down the arc length squared: d =
d 1−k⋅ 2
d 2 2 2 2 2 ds =c ⋅dt −a t⋅ ⋅ d sin ⋅d 2 1−k⋅ 2
2
2
2
The geometric quantities from the metric tensor to the Ricci scalar:
251
(8.8.3)
8.8 General Friedmann equations
0 0 0 2 a t 0 − 0 0 2 g = 1−k⋅ 2 2 0 0 −a t ⋅ 0 2 2 2 0 0 0 −a t ⋅ ⋅sin
g =
1
1
0 1−k⋅ 2 0 − 2 a t 0
0
0
0
−
0
0
0
0
1 a t⋅ 2
0
2
0
−
1 a t ⋅ 2⋅sin 2 2
(8.8.4)
∂ g 2⋅a⋅a˙ =− ∂t 1−k⋅ 2
∂ g 2⋅1−k⋅ 2⋅a˙ =− ∂t a3
∂ g =−2⋅ 2⋅a⋅a˙ ∂t
∂ g 2⋅a˙ = 2 3 ∂t ⋅a
∂ g 2 =−2⋅ 2⋅a⋅a⋅sin ˙ ∂t
∂ g 2⋅a = 2 3 ˙ ∂t ⋅a ⋅sin
∂ g 2⋅k⋅⋅a 2 =− ∂ 1−k⋅ 22
∂ g 2⋅k⋅ = 2 ∂ a
∂ g =−2⋅⋅a 2 ∂
∂g 2 = 3 2 ∂ ⋅a
∂ g =−2⋅⋅a 2⋅sin 2 ∂
∂ g 2 = 3 2 ∂ ⋅a ⋅sin 2
∂ g =−2⋅ 2⋅a 2⋅cos ⋅sin ∂
2⋅cos ∂ g = 2 2 ∂ ⋅a ⋅sin3
t
t
=
a⋅a˙ 1−k⋅ 2
=
t
=
t
=
t
=
t
t
2
=
= ⋅a⋅a˙
t
=
a˙ a
252
t
(8.8.5) 2
2
= ⋅a⋅a⋅sin ˙
8.8 General Friedmann equations =
k⋅ 2 1−k⋅
=
=
=−⋅1−k⋅ 2
=
=
=−⋅1−k⋅ 2 ⋅sin 2
1
=−cos ⋅sin
= =cot t
(8.8.6) ∂ t 2 = 2⋅ a⋅a ¨ a˙ ∂t
2 a⋅a a = ¨ ˙2 ∂t 1−k⋅
∂ t 2 2 = 2⋅a⋅a ¨ a˙ ⋅sin ∂t
∂ t ∂ t ∂ t ∂ t ∂ t ∂ t a⋅a− a˙ 2 ¨ = = = = = = ∂t ∂t ∂t ∂t ∂t ∂t a2 ∂ t 2⋅k⋅⋅a⋅a˙ = ∂ 1−k⋅ 22
t
t
∂ =2⋅⋅a⋅a˙ ∂
∂ k⋅1k⋅ 2 = 2 2 ∂ 1−k⋅
∂ 2 =2⋅⋅a⋅a⋅sin ˙ ∂
∂ =−1−3⋅k⋅ 2 ∂ ∂ ∂ ∂ ∂ 1 = = = =− 2 ∂ ∂ ∂ ∂
∂ =−1−3⋅k⋅ 2 ⋅sin 2 ∂ t
∂ =2⋅ 2⋅a⋅a⋅cos ⋅sin ˙ ∂
∂ =−2⋅⋅1−k⋅ 2 ⋅cos ⋅sin ∂
∂ =sin2 −cos2 ∂
∂ ∂ 1 = =− 2 ∂ ∂ sin
Rt t =−Rt t = R
t
R
t
t
=−R
t
=−R
t
a⋅a¨ 2 1−k⋅ 2
t
= ⋅a⋅a¨
t
2
t
2
= ⋅a⋅a⋅sin ¨
a Rtt =−Rt t= Rtt =−R t t =Rtt =−Rt t = ¨ a R
=−R
=R
=−R
2
2
= ⋅ a˙ k
R =−R =R =−R = 2⋅ a˙ 2k ⋅sin 2
253
(8.8.7)
8.8 General Friedmann equations R
=−R
R =
−
=R
3⋅a¨ a
=−R
2
a k = ˙ 2 1−k⋅
0
(8.8.8)
0
0
a⋅a2⋅ a˙ 2k ¨ 0 0 1−k⋅ 2 0 2⋅a⋅a2⋅ a˙ 2k 0 ¨ 2 0 0 ⋅a⋅a2⋅ a˙ 2k ⋅sin 2 ¨
0 0 0
(8.8.9) R=−
6 2 ⋅ a⋅a ¨ a˙ k 2 a
(8.8.10)
The normalized form of the energy-momentum tensor:
T =
⋅c
2
0 0 0 a t − ⋅p 0 0 1−k⋅ 2 0 −a 2 t ⋅ 2⋅p 0 2 2 0 0 −a t⋅ ⋅sin 2 ⋅p 2
0 0 0
(8.8.11)
Substitute them into the Einstein equation: 1 8⋅⋅ R − ⋅R⋅g −⋅g =− 4 ⋅T 2 c
(8.8.12)
The time-like component: 1 8⋅⋅ Rtt − ⋅R⋅g tt −⋅g tt =− 4 ⋅T tt 2 c −
(8.8.13)
3⋅a¨ 1 6 8⋅⋅ 2 2 − ⋅ − 2⋅a⋅a ¨ a˙ k ⋅1−⋅1=− 4 ⋅⋅c a 2 a c
a˙ 2⋅c 2 k⋅c 2 ⋅c 2 8⋅⋅ 2 − =− ⋅ 2 3 3 a a The first general Friedmann equation: H 2
2
2
k⋅c ⋅c 8⋅⋅ − =− ⋅ 2 3 3 a
(8.8.14)
254
8.8 General Friedmann equations The spatial components: 1 8⋅⋅ Rii − ⋅R⋅g ii −⋅g ii =− 4 ⋅T ii 2 c
(8.8.15)
1 8⋅⋅ R − ⋅R⋅g −⋅g =− 4 ⋅T 2 c
a⋅a2⋅ a˙ 2k 1 ¨ 6 a2 a2 8⋅⋅ a2 2 − ⋅ − ⋅a⋅ a a k ⋅ − −⋅ − =− ⋅ − ⋅p ¨ ˙ 2 1−k⋅ 2 a2 1−k⋅ 2 1−k⋅ 2 c4 1−k⋅ 2
a⋅c 2 a 2⋅c 2 k⋅c 2 8⋅⋅ −2⋅ ¨ − ˙ 2 − 2 ⋅c 2= 2 ⋅p a a a c We get the second general Friedmann equation in the two other cases as well: 2
2
a⋅c k⋅c 8⋅⋅ −2⋅ ¨ −H 2− 2 ⋅c 2= ⋅p a a c2
(8.8.16)
8.9 World models Write down the relationship between the density and the scale factor in two theoretical scenarios, one of them corresponds to the matter dominated universe, and in the second case the energy is present mostly in the form of radiation. We introduce dimensionless quantities: 3
p=0
4
p=
m t⋅a t =const.
s t⋅a t=const.
K m=
⋅c2 3
K s=
8⋅⋅ ⋅m⋅a 3=const. 2 3⋅c
8⋅⋅ ⋅ s⋅a 4 =const. (8.9.1) 2 3⋅c
Write down the Friedmann equations of movement with them: a˙ 2−
K m K s ⋅a 2 − 2− =a˙ 2V a =−k a 3 a
(8.9.2)
In it the Friedmann potential: V a=−
K m K s ⋅a 2 − 2− a 3 a
(8.9.3)
The time dependence of the scale factor while neglecting the cosmological constant: 255
8.9 World models
a˙ 2≈
Km 3 a t ∝ t 2 a
a˙ 2≈
Km 3 a t ≈ t 2 a
(8.9.4)
In a static Universe the scale factor does not change, its derivatives according to time are zeroes. In the present state the energy of radiative origin can be neglected, therefore the cosmological constant necessary for a static situation can be determined from the equation of movement: 2
−K m k⋅a − =−1 a 3 k =
3 3⋅K m − 3 a2 a
(8.9.5)
Friedmannian world models: k=–1
k=0
k=1
0
closed, periodic
closed, periodic
closed, periodic
=0
open, expanding
open, asymptotic
closed, periodic
0 k
open, expanding
open, expanding
periodic / open
= k
open, expanding
open, expanding
open, static, unstable
k
open, expanding
open, expanding
open, expanding
Introduce the following dimensionless variables: x =
a t a0
= H 0⋅t
x 0=0
(8.9.6)
Characterizing the present state: a˙ 2−
K m ⋅a 2 − =−k a 3
x˙ 2−
m −⋅x˙ 2=k a
2
/⋅
c H 20⋅a 20 (8.9.7)
Where dimensionless constants characterize the state of the Universe: m=
8⋅⋅ ⋅ m= m 2 k 3⋅H 0
=
⋅c 2 3⋅H 20
256
k =−
k⋅c2 H 20⋅a 20
8.9 World models m k =1
(8.9.8)
Their values according to our present knowledge: m=0.273
k ≈−0.023
=0.727
(8.9.9)
Distance of a point from the centre of the coordinate system:
D=∫ g ⋅d =R t⋅
(8.9.10)
0
Distance of the particle horizon: t0
t0
D0 =R0⋅∫ d =R0⋅∫ 0
0
c⋅dt Rt
(8.9.11)
Cosmic escape velocity: v k=
d dR R⋅= ⋅= H 0⋅R0⋅ dt dt
lim v k =∞
(8.9.12)
∞
Calculating the dimensionless time: x
= H 0⋅t=∫ 0
dx
(8.9.13)
m ⋅x 2 K x
The age of the Universe: x=1
t 0= ∫ 0
x=1
dx 1 = ⋅∫ x˙ H 0 0
dx
m ⋅x 2 K x
=
0.9897 =4.14⋅10 17 s=1.31⋅1010 years H0 (8.9.14)
The radius of the observable part of the Universe: t0
x=1
c⋅dt c D0 =a 0⋅∫ = ⋅∫ a t H 0 0 0 26
dx c = ⋅3.433 4 2 m⋅x⋅x K⋅x H 0 10
D0 =4.475⋅10 m=4.73⋅10 lightyears
The distancing velocity of the border of the visible region: 257
(8.9.15)
8.9 World models t0
x=1
c⋅dt v D0 =c⋅a˙ 0⋅∫ =H 0⋅D0 =c⋅∫ a t 0 0
dx m =3.433⋅[c ] 4 2 s m⋅x⋅x K⋅x
(8.9.16)
This value exceeds the speed of light significantly. However this has no consequences from the point of view of the bodies moving in the spacetime.
258
Summary
Summary We have peaked into the world of deterministic physics. It was apparently possible to geometrize this part of science, however we always have to keep in mind that like every model, this also has limits. Some of them follow from our conditions earlier, when we determined what phenomena interest us, and what do not; others occurred on the way, and it happens that we cannot satisfy certain expectations; and in the worst case our conclusions can be rebutted by experimental data. First and foremost in the beginning of the 20th century an old philosophical debate was concluded: the world is essentially indeterministic. It does not mean that its is unpredictable, it means only that we cannot predict events with arbitrary accuracy. The problem is not that we do not have enough information about the states, like many have thought initially. They assumed hidden variables, that we cannot measure, but they unambiguously determine the flow of events. It turned out that such variables do not exist, nature has been determined to be probabilistic. These phenomena under a certain size limit and above a certain energy density make the results described in this book useless. Several philosophical expectations are not satisfied by the results in the book. One of the most famous of them is the vaguely defined Mach principle, that would mean that the Einstein equations cannot have a solution in empty space. Further complications arise because particles with spin cannot be properly discussed within the framework of general relativity, it has to be expanded with torsion besides curvature to geometrize the concepts of force and torque. In the resulting model additional effects manifest that were not yet confirmed experimentally. A more serious problem than these is the appearance of naked singularities, that have to be dealt with because of the insufficient definition of the appearing complex quantities in the model. The validity of the model is questioned time to time, alternative theories predict different outcomes for various phenomena. We can however point out two things: within the current boundaries of measurement, considering the shortcomings of our astronomical knowledge (and if we stay within the postulated limits of validity), there is no result that would contradict the general theory of relativity. On the other hand, the competing models that predict with high accuracy in some problematic phenomena (like the problem of dark matter), gravely err in completely everyday situations. The Kaluza theory is a generalization of general relativity as much as general relativity is an extension of special relativity. It addresses several gaps in Einstein's original theory, it finally gets rid of the idea of force, thus erases such shortcomings like the possibility of a force that could accelerate objects to become faster than the speed of light, and several solutions of the Einstein equations, that are although completely valid, are also completely unphysical. Their combined model can handle the most complete deterministic limiting case with solely mathematical tools.
259
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Bibliography
Index 1616...............................................................129 1801...............................................................124 1859...............................................................121 1907...............................................................101 1915...............................................................121 1916..........................................................90, 116 1918...............................................................229 1919.......................................................124, 229 1925...............................................................224 1927...............................................................224 1928...............................................................226 1959...............................................................101 1963...............................................................167 1976...............................................................175 1992...............................................................175 1993...............................................................229 1998...............................................................243 2004........................................................116, 118 2005................................................................118 2007............................................................3, 116 20th century...................................................259 accelerated expansion....................................243 action functional.................................104p., 212 action principle.....................................39, 104p. adiabatic index............................................196p. affine parameters.........................................150p. age of the Universe................................248, 257 Albert Einstein.....................................9, 64, 229 algebraic identities.........................................169 algebraic minor................................................40 American Physical Society............................116 amplitude...............................................217, 239 angular frequency..112pp., 117, 119, 122p., 130, 146, 158p., 166, 168, 176, 178 angular turn.........................................122p., 127 antique world...................................................12 antisymmetric..................17, 19, 24, 26, 33p., 36 Apollo program..............................................116 Arthur Eddington...........................................124 astronomical unit..............................................11 average mean path.........................................201 averaging..................................................17, 243 Bianchi identity..........................................36, 40 binary system.................................................229 binomial series...............................................106
black hole97, 131, 134, 137, 149, 152, 158, 162, 170p., 174pp., 185, 196 boundaries.............................................39p., 259 boundary condition................................105, 196 Boyer-Lindquist coordinates..........................170 Brazil..............................................................124 central force field...........................................106 centre of mass.........................................78p., 84 centripetal acceleration....................................88 charge......................................9, 212p., 223, 228 charged particle......................212pp., 223p., 228 choice of gauge..............................................162 circular orbit.........88, 110, 112p., 116, 119, 121, 130p., 143, 159, 177pp., 241 co-moving coordinate system. .72, 135, 202, 207 Co-moving coordinates....................................92 collapsing dust cloud......................................211 collapsing star................................................211 commutator.........................................32pp., 216 complex Ernst potential.................................181 complex function...........................................224 complex potential...........................................168 conjugate Ernst equation................................167 conjugate metric..................................161, 167p. conjugate potential.........................................167 conservation of energy.............................84, 224 conservation of momentum....................78p., 84 conservative force field..................................106 conservative force law...................................212 constant circular motion................................87p. constant of nature...........................................110 constants of integration..................................163 constants of movement..................................119 constellation Aquila.......................................229 continuity condition.......................................226 continuous fluid model..................................188 contract.......................................39, 70, 187, 216 contravariant four-velocity...............................74 contravariant vector. .14, 16, 22, 28, 30, 32p., 67 coordinate acceleration.........87, 115, 130p., 206 coordinate condition..........88, 110p., 113p., 124, 134p., 161, 177, 179, 201p., 247 coordinate conditions........88, 110p., 113p., 124, 134p., 177, 179, 201p., 247 coordinate effect.....................................124, 229
263
Index coordinate plane.............................................151 coordinate potential........................................166 coordinate singularity..............................97, 150 coordinate surface. 54, 102p., 124, 150, 152, 185 coordinate surfaces........................................186 coordinate time 87pp., 104, 111, 113, 122p., 125, 130, 135, 137p., 178, 187, 202, 204p. correspondence principle...............................104 Cosmic escape velocity..................................257 cosmological constant.........................243, 255p. cosmological principle...................................243 covariant vector.....13pp., 22, 25pp., 30, 32, 34p. critical density................................................249 cross polarized...............................................238 current density.....................................213p., 216 cylinder condition.....................................219pp. deceleration parameter...................................249 density...185, 188, 192, 195pp., 206, 213p., 216, 243, 249, 251, 255, 259 determinant....10, 38, 40, 43, 147, 158, 171, 183 deterministic physics......................................259 diagonal...17, 26, 67, 92, 95, 147, 158, 171, 176, 192 differential geometry........................................12 dimensionless constant..................................256 dipole approximation.....................................237 discriminant...........................................121, 174 distancing velocity.........................................257 divergence theorem..................................40, 186 Earth. .4, 9, 12, 112, 114p., 118, 129, 131, 133p., 146, 176, 195 Earth's orbit....................................................112 Eddington-Finkelstein coordinates.............150p. eigenvalue equation..................................225pp. Einstein...4, 9, 40, 64, 90p., 95, 101, 121p., 124, 131, 159p., 185pp., 198, 209p., 214, 218, 229, 232, 234, 236, 238, 243, 250, 254, 259 Einstein equation...40, 64, 90p., 95, 131, 185pp., 193, 209p., 214, 218, 232, 234, 236, 238, 243, 250, 254, 259 electric field strength.....................................214 electricity.......................................................218 electromagnetic energy-momentum tensor.215p. electromagnetic field..........64, 188, 212pp., 218, 223p., 228 electromagnetic force..................................214p. electromagnetic four-force.............................213 electromagnetic four-potential.......................218
electromagnetic interaction............212, 214, 223 electromagnetic tensor...................213p., 219pp. electromagnetic wave...............................215pp. electron...................................................224, 226 electron spin...................................................224 elliptic orbit....................................................241 energy current.............................................238p. energy density................................185, 192, 259 energy operator..............................................225 energy-momentum four-vector........................81 energy-momentum tensor.....185pp., 192, 206p., 214pp., 236, 238, 250, 254 energy-momentum vector..............................185 entropy...............................................................9 equation of motion.............79p., 107p., 213, 223 equation of movement...105, 107, 114, 186, 214, 223, 256 equation of the fourth degree....................132pp. equation of the straight line.............................17 equation of the third degree................132p., 146 equations of movement......104, 108p., 134, 201, 255 Equatorial Guinea..........................................124 equatorial plane......................100, 116, 124, 130 ergosphere......................................................175 Erwin Schrödinger.........................................224 event horizon....115, 124, 136pp., 149, 152, 162, 174p., 184 expansion of the Universe..............243, 246, 248 falling astronauts............................................137 field equation.................................................186 fine structure..................................................224 finite volume...............................................245p. first quantization............................................224 five dimensional geodesic equation...............222 five dimensional metric tensor.......................218 five dimensional spacetime..............................64 focus...............................................................129 four-acceleration..............................................74 four-distance................................70, 78, 89, 162 four-potential..........................................218, 228 four-velocity..................74, 188p., 192, 206, 213 free index...............................................217, 220 free indices.................................................10, 23 free movement...............................................212 free particle.................................................224p. frequency......75p., 89, 101, 112pp., 117pp., 122, 130, 143, 146, 158p., 166, 168, 176
264
Index Friedmann equation............................251, 254p. Friedmann potential.......................................255 Friedmannian world models..........................256 functional equation........................................104 G. A. Rebka....................................................101 Galileo Galilei....................................................9 gauge freedom................................................161 gauge invariance....................................214, 216 Gaussian polar coordinates...................92, 143p. geodeses...........................................................12 geodesic equation....18, 30p., 87, 91, 100p., 108, 110p., 142, 178, 185, 201, 205, 214, 218, 222 geometric angular momentum...............170, 176 geometric condition.......................................139 geometric potential....................................119pp. geometric radius.....................................121, 197 Gravitation. . .9, 11, 64, 89, 92, 101, 104, 108pp., 112, 114p., 118pp., 123p., 128p., 131, 133, 139p., 143, 146, 150, 158, 175p., 185pp., 189, 192, 195, 197, 205p., 212, 214, 218, 223, 229, 231, 235pp., 240, 243 gravitational constant............................11, 109p. gravitational focal distance............................129 gravitational lens............................................129 gravitational potential....................109, 186, 192 gravitational radius.....112, 114p., 118, 123, 128, 133, 140, 146, 150, 176, 195, 197, 205p. gravitational wave detector............................229 Gravity Probe B......................................116, 118 harmonic condition........................................234 harmonic oscillator.................................117, 122 Harward University........................................101 hidden variables.............................................259 homogeneity...................................................243 homogeneous...................47, 188, 195, 198, 243 homogeneous fluid.........................................195 Hubble constant..........................................248p. Hubble time....................................................248 hydrogen maser..............................................101 hydrostatic equilibrium...............................194p. hypersurface..........................................99, 236p. Ibn al-Haytham..................................................9 identity matrix..................................................66 indeterministic...............................................259 index lowering.......................................135, 203 index notation......................................4, 10, 189 inertial force...................................................212 Infinite redshift...........................................174p.
infinite volume............................................245p. infinitely distant observer...............111, 115, 205 infinitesimal quantities..................................22p. infinitesimally small quantities........................21 integration constant.....................109, 126p., 150 internal friction..............................................188 invariant parameter..........................................41 invariant quantity.........................17, 30, 65, 138 invariant scalar.....................................15, 39, 74 isotropic. . .47, 92, 138pp., 142pp., 188, 194, 243 isotropy..........................................................243 iteration............................................................41 James Clerk Maxwell.........................................9 Johann Georg von Soldner.............................124 Johannes Kepler.............................................129 Joseph Hooton Taylor Jr.................................229 Julian year........................................................11 Karl Schwarzschild..........................................90 Kepler's third law...........................................112 kinetic energy.........................................106, 239 Kronecker delta...............................10, 14pp., 21 Kronecker-delta....................................24, 27, 29 LAGEOS................................................175, 177 Lagrange function.....................................104pp. Lambert function..............................10, 151, 156 laser.....................................................116, 175p. Legendre polynomials....................................182 light cone..........................................................64 light-like geodesic. .113, 124, 149, 151, 153, 247 light-like surface............................................162 lightcone.........................................................236 lightyear...........................................11, 146, 243 limiting case of zero mass..............................226 linear transformation........................................66 linearised conservation law............................237 luminosity...................................................239p. lunar orbit.......................................................118 Mach principle...............................................259 magnetic induction.........................................214 magnetism................................................64, 218 mass distribution............................186, 195, 212 mathematical language....................................12 maximal exhaust velocity................................85 Maxwell equations.................214, 216, 218, 226 Maxwell stress tensor.....................................215 mechanical gyroscopes..................................116 Mercury...............................................121, 123p. meter..................................9, 101, 131, 189, 251
265
Index metric function......119, 163, 167pp., 183, 193p., 197p. metric functions.....119, 163, 167pp., 183, 193p. momentum density.........................................185 momentum operator.......................................225 monochromatic planar wave..........................235 monotonic changing parameter......................149 Moon...........................................116, 129, 133p. Moon's orbit...........................................116, 133 multiple indices................................................15 NASA.....................................................116, 175 Natural constants..............................................10 negative gradient..............................................80 Neptune..........................................................121 neutrino..........................................................228 neutron star....................................................196 Newtonian approximation................97, 104, 187 Newtonian limiting case........................109, 112 Nobel Prize in Physics...................................229 numerical methods...........................................41 observable part of the Universe.....................257 ocean liner..........................................................9 orbital period..........112, 118, 123, 177, 229, 242 orbital plane...................................................116 orbiting distance.............................................122 ordering between cause and effect.........9, 76, 89 Oscar Klein....................................218, 221, 224 oscillation.......................................................231 Papapetrou metric.......................................162p. parametric equation...43, 47, 49, 51p., 56, 58pp., 211 parsec...............................................................11 partial matrix..................................147, 158, 171 particle horizon..............................................257 Paul Adrien Maurice Dirac............................226 Pauli matrices.................................................227 perfect fluid.........................188p., 192, 196, 206 perihelion....................................121, 123p., 128 Permute..........................................24, 26, 29, 35 phase equation.....................................216p., 226 photon...........................................78p., 124, 225 planet with a solid surface..............................113 plus polarized.................................................238 polar coordinates.46p., 49p., 92, 143p., 162, 244 polarisation unit tensor...................................238 polarisation unit vector..................................238 potential energy................................80, 106, 192 Poynting vector..............................................215
present state....................................................256 pressure..........185p., 188, 194pp., 206, 209, 211 principle of equivalence.............................9, 101 principle of least action..................................213 principle of relativity.........................................9 product of arbitrary tensors..............................15 products of infinitesimal quantities..................23 proper time......64p., 74, 76, 81, 87pp., 104, 111, 113, 119, 122p., 135pp., 178, 187, 202, 205 PSR B1913+16..............................................229 quadratic equation..................................120, 174 quadratic formula........................120p., 174, 178 quadrupole formula................................236, 239 Quadrupole moment.........................182, 238pp. quadrupole moment tensor.............................238 quantum mechanical operator........................224 quark..............................................................226 R. V. Pound....................................................101 radial acceleration..........................................114 radial change............................................97, 212 radial changes..................................................97 radiated performance into a unit solid angle..239 radio astronomical measurements..........124, 128 raise the index.....................................217, 220p. rank of a tensor................................................15 real force........................................................212 rearranged Einstein equation.........................187 reciprocal of the differential.............................14 reduced mass..................................................241 reference frame. 65, 71, 73, 76p., 79, 83, 86, 212 relativistic energy.......................................80, 85 relativistic momentum.....................................80 relativity principle............................................65 rest mass..........................................78p., 82, 187 resulting equation of the third degree.........132p. Robertson-Walker metric............................245p. Roche radius..................................................133 rocket equation...........................................83, 85 rotating ellipse................................................122 rotating rod.....................................................240 rotation..........26, 29, 97, 112, 114p., 118p., 122, 158p., 170p., 176, 182, 189 Roy Kerr................................................167, 179 Rudolf Clausius..................................................9 Russel Alan Hulse..........................................229 scalar density....................................................38 scalar field.........................................12p., 25, 28 scale factor..........................................249, 255p.
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Index Schur theorem................................................243 Schwarzschild radius.. .97, 109p., 123, 131, 137, 152, 171, 175, 197, 240 semi-major axis......112, 118, 123, 133, 177, 241 senses.................................................................9 separation constant......................................210p. Series expansion............................................248 SI......................................................4, 9, 69, 110 singularity....97, 150, 152, 156, 174, 181p., 184, 198, 259 solar eclipse............................................128, 229 Solar System..............................90, 97, 121, 146 solution formula..........132, 134, 146, 151p., 156 space-like velocity squared..............................65 spaghettification.............................................131 spatial component..........................185, 189, 255 special relativity...............................64, 224, 259 speed of sound............................................196p. spring.............................................................239 standard gravitational parameter...110, 112, 114, 118, 123, 128, 133, 146, 176 static Universe................................................256 stress.......................................................186, 215 summation indices.....................................10, 23 Sun.......9, 97, 112, 118, 121pp., 128p., 137, 146 supernova explosion........................................97 surface acceleration...........................113pp., 131 tachyon..........................................................85p. tangent vector.....17, 30pp., 37, 41, 67, 119, 135, 137, 149, 178, 202pp. Taylor series.........................23, 25, 36, 105, 249 test body.....100, 106, 109p., 134, 137, 143, 149, 176, 214 test particles.....................................................64 Theodor Kaluza..........................................9, 218 three-vector of the plane wave....................238p. three-velocity...................................................74 tidal acceleration............................................131 tidal bulges.....................................................129
time independence...........................................97 time-like component................74, 189, 250, 254 time-like direction............................................67 time-like future................................................64 time-like geodesic..........................................119 time-like past....................................................64 time-like path.................................................146 torsion.....................................19, 33p., 220, 259 total derivative...........................................13, 21 trajectories of real objects..............................119 transcendent equations...................................145 transformation matrix............13pp., 27, 38, 65p. transformation rule............15, 19, 25pp., 38, 164 transversal term..............................................231 transversal wave.....................................229, 235 two-body problem..........................................240 unit vector................................................67, 238 Uranus............................................................121 Urbain le Verrier............................................121 variation of the connection..............................20 vector potential..............................212, 214, 219 vibrating rod...................................................240 viscosity.........................................................188 vomit comet.......................................................9 Walter Gordon................................................224 wave equation................................................216 wave function...........................216p., 225, 227p. wave number vector............................216p., 236 weak interaction.............................................226 white dwarf....................................................196 Willem de Sitter.............................................116 wormhole.........................................102p., 152p. Zipoy-Voorhees spacetimes...........................182 ..39, 41, 102, 110pp., 114, 118, 123, 127p., 130, 133, 146, 149pp., 176, 178, 181, 241, 246, 248p. γ-matrices.......................................................227 π-meson..........................................................224
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This series presents modern physics through examples and derivations. The organization of topics does not follow the traditional historical approach, it was determined by practical considerations instead. We build up the mathematical framework of the models first, and then completely derive the most important consequences and compare them to recent experimental results. The volumes can be used in the specialized fields as reference materials, and are suitable for self-study in each topic. They may also serve the needs of university lectures as well. The first volume deals with the general theory of relativity. This description has only historical relevance by now, since during the last century, much experimental evidence was found for the consequences of Einstein's theory. It is a classical field, where centuries-old scientific and philosophical ideas got mathematical formulation, and the experimental confirmation. It is important to point out, that the traditional mechanical worldview, that is often considered to be easier to grasp, is in fact an incomplete intellectual achievement. The basic assumptions of general relativity draw from everyday experiences, and the recognition of the curved nature of spacetime follows naturally. It is comparable to the understanding of Earth's spherical shape, and if we are familiar with the mathematical foundations, it does not even require too much imagination. The Reader is assumed to have some basic knowledge in higher mathematics and among the more traditional subjects in physics, but there is only as much mathematical depth in this book, as absolutely necessary. We use the traditional symbols of differential calculus and index notation. The SI system of measurement is used in all physical derivations.
