CHARGED AND ROTATING BLACK HOLES IN GENERAL RELATIVITY PHY-501A END SEMESTER PROJECT REPORT Sudarshana Laha 151169
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Supervisor:Dr. Gautam Sengupta Dept. of Physics
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Indian Institute of Technology, Kanpur November 5, 2016 Abstract One profound prediction of Einstein’s General Theory of Relativity is that a sufficiently compact mass can deform spacetime to form a black hole.The black hole solutions are obtained from solving the Einstein’s Field Equations, set of nonlinear PDEs by approximation method,which specify how the geometry of space and time is influenced by whatever matter and radiation are present.In this report,we have discussed about such black hole solutions(mostly the charged and rotating case) and their properties.
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A Brief Discussion of the General Theory of Relativity
Introduction GTR is basically based on The Principle of Equivalence which Einstein proposed in 1907.The principle of equivalence of Inertia and Gravitation tells us how an arbitrary physical system responds to external gravitational field.The principle of Equivalence is divided into two sub principles namely Strong Equivalence Principle and Weak Equivalence Principle. Weak principle is generally the equality of Inertial and Gravitational mass. The statement goes like :the local effects of motion in a curved space (gravitation) are indistinguishable from those of an accelerated observer in flat space, without exception. On the other hand,the strong equivalence principle suggests the laws of gravitation are independent of velocity and location. In particular,The outcome of any local experiment (gravitational or not) in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime. In other words,at any point in an arbitrary gravitational field it is possible to choose a locally inertial frame such that within a sufficiently small region of the point in question,the laws of nature take the same form as in
General theory of Relativity is the geometric theory of gravitation published by Albert Einstein in 1915 which generalizes special relativity and Newton’s law of universal gravitation, providing a unified description of gravity as a geometric property of space and time. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.A few months after Einstein developed his theory, Karl Schwarzschild found a solution to the Einstein field equations, which describes the gravitational field of a point mass and a spherical mass.In the same year, the first steps towards generalizing Schwarzschild’s solution to electrically charged objects were taken, which eventually resulted in the Reissner Nordstrom solution, now associated with electrically charged black holes.In 1963, Roy Kerr found the exact solution for a rotating black hole. Two years later, Ezra Newman found the axisymmetric solution for a black hole that is both rotating and electrically charged. 1
an unaccelerated cartesian coordinate system in the is given by: absence of gravitation. ds2 = −(1 −
2.1
Concept of a Geodesic
In Euclidean Geometry,the shortest distance between two points is a unique straight line but in Riemannian Geometry the shortest distance between two points is not unique and is known as a geodesic.Geodesics are not parallel as they intersect at the poles.In curved spacetime the equation of geodesic is given by dxν dxλ d2 xµ + Γµνλ =0 2 d τ dτ dτ with τ as parameter.
2.2
3.1
(1)
• Singularities and Event Horizon: The two radial points where the metric diverges are r=0 and r=2GM.The former is a true singularity as it holds in all coordinate systems but the latter is a coordinate singularity and can be removed by choice of suitable coordinates.The best check for a true singularity is divergence of at least one scalar(invariant under coordinate transformation) obtained from contraction of the curvature tensor.r = 0 is a spacelike surface.
The Einstein field equations are a set of 10 equations that describes the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy.Published in 1915, it is a tensorial non-linear equation, equating the curvature tensor to the stress-energy tensor.In the (- + + +)convention it is given as, 1 8πG Rik − Rgik = 4 Tik + Λgik (2) 2 c where Λ is the cosmological constant,Rik is the Ricci tensor,R = Rik Rik is the Ricci scalar and Tik is the energy-momentum tensor. i Rklm is the curvature tensor of spacetime and contains second derivatives of the metric tensor gij . δΓijl δΓijk i m i − + Γm jk Γlm − Γjl Γkm δxl δxk
• rS =2GM is the null surface,known as the event horizon of the Schwarzschild black hole.The properties of the spacetime differ significantly for r < rS from r > rS ,where rS is known as the Schwarzschild Radius.Any gravitating body whose radius is less than or equal to rS is called a black hole.For an asymptotic obsrver all events outside rS freeze at the event horizon,which represents the point past which light can no longer escape the gravitational field.
(3)
where Γijk is the Affine connection given by, Γijk =
1 im δgmj δgmk δgjk g [ k + − m] 2 δx δxj δx
(4)
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Properties of the metric:
• Static,Stationary and Spherically Symmetric: The metric is independent of time and thus stationary and also invariant under time reversal, hence static.Since the metric is only a function of radial coordinate r(g = g(r)), it is invariant under rotation and spherically symmetric.But as r → ∞,the metric becomes asymptotically flat and defined by the Minkowski metric.
Einstein field equations
i Rjkl =
2GM 2 2GM −1 2 )dt + (1 − ) dr + r2 dω 2 r r (5)
Schwarzschild Solution
Reissner-Nordstorm Black Hole:
This is a charged black hole,solution of coupled Maxwell-Einstein equation.In this case too the cosmological constant, Λ = 0 .However,unlike the Schwarzschild case, Tij 6= 0 .In reality,the black holes with charge get neutralized fast due to interaction with matter and hence not physically realisable.They have two event horizons.
The Schwarzschild solution is the most general,spherically symmetric vaccum solution to Einstein equations, describing the gravitational field outside a spherical mass on the assumption that the electric charge ,angular momentum of the mass and universal cosmological constant are all zero.The metric 2
4.1
Properties of the metric:
• Description: The metric is spherically symmetric,static and stationary.In this case,charge is also present in the black hole along with mass and energy.The R-N metric is given by, ds2 = −∆dt2 + ∆−1 dr2 + r2 dω 2
(6)
where ∆=1−
G(Q2 + P 2 ) 2GM + r r2
(7)
,M is the mass of the black hole,Q is the electric charge and P is the magnetic charge(the black hole is considered to posses magnetic monopole). • Singularity: Computing the scalar Rijkl Rijkl ,we find that the metric has a true singularity at r=0 due to the divergence of the above scalar. Figure 1: Fig:∆(r)-r plot : Zeroes indicate the location of an event horizon (Sean Carroll,pg-256)
• Event Horizon: If we set grr = 0 ,then we get, 1−
G(Q2 + P 2 ) 2GM + =0 r r2
(8)
4.3
Second Case: G2 M 2 > G(Q2 + P 2 )
or r± = GM ±
p
G2 M 2 − G(Q2 + P 2 )
• Singularity and Event Horizon:This black hole has two event horizons at p r± = GM ± G2 M 2 − G(Q2 + P 2 ) (10)
(9)
Thus we can have three solutions for G2 M 2 − G(Q2 + P 2 ) > 0, = 0, < 0.
4.2
2
2
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Outside r+ and inside r− ,the metric components are positive. These two singularities are coordinate singularities and the true singularity occurs at r=0. The outer event horizon is like the event horizon for Schwarzschild black hole.In the region r > r+ and also r < r− ,the spatial coordinate is spacelike.So,the r=0 is a timelike line.And for r− < r < r+ , the spatial coordinate is timelike and the time coordinate is spacelike.So an ingoing particle is bound to move in the direction of decreasing r once it crosses r+ .But it may not reach r=0. After crossing r− ,the particle has a choice to continue towards decreasing r or reverse its path and move towards increasing r. On crossing r+ ,it is like emerging from a white hole.On going back to r+ ,its like going back to the black hole but this time a different one from the previous.
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First Case: G M < G(Q + P )
• Nature of Singularity:The singularity here is naked singularity.As,the metric is always regular at r=0,the tangent to the hypersurface is timelike.So,the singularity at r=0 is a timelike line.The coordinate t and coordinate r are always timelike and spacelike respectively.So there is no apparent event horizon.
• Asymptotic Flat Nature: At r → ∞, the metric becomes Minkowski metric. • Drawback: The concept of naked singularity is not universally accepted ,as this solution demands that the electromagnetic contribution is greater than the total energy of the hole which is unnatural.
• Kruskal Coordinates:The coordinates (t, r) are singular at r+ and therefore we require 3
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Kruskal coordinates to extend beyond this surface.However,these coordinaates are horizon specific and we need multiple coordinate patches to describe the R-N manifold and these patches are joined by tessellation.So we have a specific set,say (U+ , V+ ) for the region r+ < r < ∞.But this set is singular at r− and moreover can’t probe the region r− < r < r+ for which we require the (t, r) coordinates which render the r coordinate timelike as mentioned before.Then we need to define (U− , V− ) to describe region r < r− where the r coordinate regains its spacelike property,and the constant r surfaces are timelike surfaces,thus making r = 0 surface timelike and hence avoidable.
Axially Symmetric Solution of Field Equation: Kerr Black Hole:
The black holes which posses angular momentum are called rotating black holes.They rotate about one of its axis of symmetry.The Kerr metric is given by, 2GM r 2 2GM arsin2 θ )dt − (dtdφ + dφdt) ρ2 ρ2 ρ2 sin2 θ + dr2 + ρ2 dθ2 + 2 [(r2 + a2 )2 − a2 ∆sin2 θ]dφ2 ∆ (11)
ds2 = −(1 −
where ∆(r) = r2 − 2GM r + a2 ,ρ2 (r, θ) = r2 + J a2 cos2 θ,a = M is the angular momentum of the black hole per unit mass.M and a parametrize the possible solutions.
5.1
Properties of the metric
• Stationary,Axially Symmetric and Rotating:The metric is stationary but not static.This metric has axial symmetry along the rotation axis and it is also neutral.The same case with presence of charge is the Kerr-Newmann Solution.
Figure 2: Fig:Kruskal patches for Reissner Nordstorm spacetime(E.Poisson,pg-137)
• Boyer-Lindquist coordinates:The metric can also be expressed in terms of the above coordinates (t, r, θ, φ) given by,
4.4
Third Case: G2 M 2 = G(Q2 + P 2 )
r2 + a2 cos2 θ 2 dr r2 + a2 (12) +(r2 + a2 cos2 θ)dθ2 + (r2 + a2 )sin2 θdφ2 ds2 = −dt2 +
This solution is called extreme Reissner Nordstorm Solution.But this solution is not stable,as maintaining equilibrium of matter with the charges is not easy and adding a small amount of matter will destroy the symmetry.
If a → 0 the metric approaches Schwarzschild metric and as M → 0 it becomes the flat metric in ellipsoidal coordinate.The transformation relation between Cartesian and Ellipsoidal coordinate in terms of spatial coordinates is given by,
• Singularity and Event Horizon:There are two singularities.r = 0 is a timelike line and r = GM is a null surface.Any hypersurface at r < GM will be timelike,so the singularity r = 0 is timelike.The r = GM is the event horizon of this metric where the metric components vanish. The region on both sides of r = GM is spacelike.And r = 0 being a timelike line we can avoid hitting the singularity and continue to move to the future to extra copies of the asymptotically flat region,but the singularity is always to the left.
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x = (r2 + a2 ) 2 sinθcosφ 1
y = (r2 + a2 ) 2 sinθsinφ
(13)
z = rcosθ • Singularity:Calculating the curvature scalar we can see the metric has a true singularity at ρ2 = 0. ρ2 = r2 + a2 cos2 θ = 0 (14) So,when r = 0 and θ = lipsoidal coordinate 4
π 2
then ρ = 0.In the el-
x2 + y2 = (r2 + a2 )sin2 θ z = rcosθ So r = 0 is a disk in the equatorial plane z = 0 and x2 + y 2 6 a2 and θ = π2 corresponds to the points on the edge of the disk.So the Kerr solution ring singularity at r = 0 and θ = π2 .An observer can avoid the singularity if he passes along θ 6= π2 and can cross the r = 0 non equatorial plane and can reach another asymptotically flat spacetime that is not like ours. • Event Horizon:The event horizon is at that is ∆ = r2 − 2GM r + a2 p r = GM ± G2 M 2 − a2
∆ ρ2
angular momentum when it comes close to a rotating black hole.Angular velocity of the inertial frame increases as its distance from the black hole decreases.The observer goes in the direction of the rotation of the hole.When close enough angtφ gular velocity ω = dφ dt = − gφφ . At large distance ω≈
2J r3
and at infinity this effect disappears.
• Static limit surface and the Ergosphere: For static observer force must be applied to hold them in its place.So,the motion is not geodesic.Static observer cannot remain at every point inthe Kerr black hole.There is a limit called the stationary limit surface rsl inside which no static observer can exist even after applying infinite force.Stationary limit surface is governed by the condition
= 0,
(15)
There are three cases: 1. GM < a 2. GM = a 3. GM > a The last case is similar to the R-N black hole case.There is a naked singularity in this case.The second case also has similarity with the R-N case but the solution is unstable.
gtt = r2 − 2GM r + a2 cos2 θ = 0 (16) √ Thus, rsl = GM + M 2 − a2 cos2 θ From the expression of outer event horizon r+ and stationary limit surface rsl ,we can see that the event horizon and stationary limit surface coincide only at θ = π2 .There exists a region between them called ergosphere. Interesting things appear in the ergosphere.However we can exit the ergosphere moving out.But there is an upper limit of the angular velocity of the black hole as a 6 GM or J 6 GM 2 .The a = GM corresponds to extremal solution.
Figure 3: Fig:Concept of ring singularity of Kerr metric in ellipsoidal coordinates.The points on the boundary of the 2-d disk with coordinates r = 0, θ = π2 are the region of singularity.(Sean Carroll,pg-262)
5.2
Figure 4: Fig: Different regions of Kerr black hole (Sean Carroll,pg-264)
Third Case: GM > a
The location of 2 event horizons,i.e, two null surfaces corresponding to grr = 0 are at r± = GM ± √ G2 M 2 − a2 .
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Kerr-Newman Black Hole:
• Dragging of Inertial Frames:An inertial ob- The black hole is defined by a metric which is the soserver with zero angular momentum acquires an lution of the coupled Einstein-Maxwell equations in 5
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GR that describes the spacetime geometry in the region surrounding a charged,rotating mass.The KerrNewman metric is given by,
• Important aspects of General theory of Relativity have been studied.
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ds2 = −(
∆ dr + dθ2 )ρ2 + (cdt − asin2 θdφ)2 2 ∆ ρ (17) 2 sin θ 2 2 2 −((r + a )dφ − acdt) ρ2
• Einstein Field Equations have been summarized. • Solutions to the field equations have been studied with a detailed study of charged and rotating black hole.
where the coordinates (r, θ, φ) are Boyer-Lindquist coordinates and the length scales introduced are, a = MJ c , ρ2 = r2 + a2 cos2 θ, 2 ∆ = r2 − rs r + a2 + rQ rQ is the length scale corresponding to the electric charge Q of the mass and defined as, Q2 G 2 rQ = 4π 2 0c
6.1
Conclusion
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Acknowledgement
I acknowledge the discussion sessions with my supervisor,Dr.Gautam Sengupta who suggested me the topic of the project in the first place.I am very grateful for the motivation I received from my supervisor to work out the problems that I encountered while doing the project.
Properties of the metric
• Special cases and generalizations:The KerrNewman metric is the generalization of other exact solutions in GR.It is the Kerr metric with Q = 0,the RN metric with angular momentum,J = 0 and Schwarzschild metric with mass M ,charge Q and rotation parameter a all zero.
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References 1 https://en.wikipedia.org/wiki/General relativity
• Other aspects: Stationary, axisymmetric, asymptotically flat solution of Einstein’s equations in the presence of an electromagnetic field in four dimensions. It is sometimes referred to as an ”electrovacuum” solution of Einstein’s equations. Any KerrNewman source has its rotation axis aligned with its magnetic axis.Thus, a KerrNewman source is different from commonly observed astronomical bodies, for which there is a substantial angle between the rotation axis and the magnetic moment. Like the Kerr metric for an uncharged rotating mass, the KerrNewman interior solution exists mathematically but is probably not representative of the actual metric of a physically realistic rotating black hole due to stability issues. Although it represents a generalization of the Kerr metric, it is not considered as very important for astrophysical purposes since one does not expect that realistic black holes have an important electric charge.
2 Sean Caroll, Spacetime and Geometry:An Introduction to General Relativity 3 S.Weinberg,Gravitation and Cosmology 4 E.Poisson,An advanced course in general relativity 5 https://en.wikipedia.org/wiki/KerrNewmann metric
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