Blackholes and Gravitational Collapse

Birla Institute of Technology and Science Goa Campus Dept.

of Physics

Blackholes and Gravitational Collapse

Author: K Pratyush Kumar

Dr.

Supervisor: Kinjal Banerjee

Sept 2018 - Nov 2018

Acknowledgements I am profoundly grateful to Prof. Kinjal Banerjee for his expert guidance and continuous encouragement throughout to see that this project is true to its target since its commencement to its completion. I would like to express my deepest appreciation towards Mr. Padamata for his help in finding appropriate resources.

Surendra

At last I must express my sincere heartfelt gratitude to my friends, family members and teachers who helped me directly or indirectly through out the project.

- Pratyush

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Abstract The project involved a review of Schwarzschild Blackholes, The Chandrasekhar Limit and Pressure Free Gravitational Collapse.

1 1.1

Introduction Motivation

Singularities are an essential features of General Relativity as proved by Penrose and Hawking and many interesting phenomena can be observed around singularities. One of the widely studied phenomena are blackholes, the aim of the project was to review features of Schwarzchild Blackholes, conditions for formation of Blackholes (Chandrasekhar Limit) and process of gravitational collapse from a star to Blackhole.

1.2

Schwarzschild Metric

Karl Schwarzschild proposed a solution of Einstein Field Equations in 1916. The solution described an asymptotically flat space-time around a spherically symmetric, non rotating and chargeless but static massive body.[2] 2GM −1 2 2GM  2  dt + 1 − dr + r2 dΩ ds = − 1 − r r 2



Since the body is static, there exists a time symmetry. Outside the body the Rµν = 0. To derive this metric we start from a metric of a very general form with the following features : spherical symmetry

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and time symmetry. ds2 = −e−2α(t) dt2 + e2β(r) dr2 + e2γ(r) r2 dΩ r¯ = eγ(r) r  dγ  d¯ r = eγ 1 + r dr  dγ −2 2 ds2 = e=2α dt2 + e2β e−2r 1 + r d¯ r + r2 Ω . dr   dγ −2 → e2β r¯ → r e2β e−2r 1 + r dr ds2 = e−2α(r) dt2 + e2β(r) dr2 + r2 dΩ2

Since outsde the star Rµν is zero. Making each component of the tensor zero yeilds us the following : Rtt = Rrr = 0 ⇒ exp 2(β − α)Rtt + Rrr = 0 ⇒ α = −β Rθθ = 0 ⇒ exp 2α = 1 − Rs /r Rs is called the Schwarzschild radius. Metric equation becomes :  Rs  2  Rs −1 2 dt + 1 − dr + r2 dΩ ds2 = − 1 − r r In the weak field limit, the tt component of metric around a point mass gives R = 2GM . This parameter M can interpreted as the Newtonian mass of the star. The mass here is what can be measured by observing orbits of bodies around the star. 1.2.1

Singularities

We notice that the tt component of the metric blows up at r=0 and rr component blows up at Rs . However this doesn’t necessarily mean that the metric or the space-time has blown up there, they could be coordinate singularities. Curvature is a reliable measure of whether sapce-time has been blown up , so if the Ricci scalar is finite it means that there is no singularity there. The Ricci scalar 2 2 is 48Gr6M . r = 0 is a singularity but Rs is not. Nonetheless Rs 3

has very interesting features associated with it. A singularity is the situation when the observables derived from the equations of the theory at hand take arbitrarily large values and hence can not be measured. The singularity at the center of the star is one such singularity, they also have the characteristics that the geodesics through it are incomplete i.e. they have a terminate at the point. One of the contemporary objectives of physics is to remove singularities from General Relativity because it breaks down at singularities and a better description of nature is required for such cases.

1.3

Geodesics of Schwarzschild Metric

The Python package ’gravipy’ can been used to obtain Christoffel connections, Riemann and Ricci tensors.[1] The geodesics of the Schwarzschild Metric are : 2GM dr dt d2 t + =0 2 dλ r(r − 2GM ) dλ dλ  dr 2 h dθ 2  dφ 2 i d2 r GM  dt 2 GM 2 + =0 − −(r−2GM ) +sin (θ) dλ2 r2 dλ r(r − 2GM ) dλ dλ dλ  dφ 2 d2 θ 2 dθ dr + − sin θ cos θ =0 dλ2 r dλ dλ dλ dθ dφ d2 φ 2 dφ dr + + 2 cot(θ) =0 dλ2 r dλ dλ dλ dλ There could be 4 Killing Vectors; one for time symmetry and 3 for spherical symmetry.

We know that Kµ T µ = const.

This implies Kµ

c. Since there exists a time symmetry, it implies K µ = (∂t )µ = (1, 0, 0, 0).   ν Kµ = gµν K = 2GM/r − 1, 0, 0, 0

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dxµ = dλ

Schwarzschild Blackholes

The calculations done above were valid for r > 2GM which is an apparent singularity. Now, let us consider radial null curves i.e. 4

θ = c1 , φ = c2 and ds2 = 0.c1 and c2 are constants.   2GM −1 2 dt 2GM −1 2GM  2  dt + 1 − dr ⇒ =± 1− ds2 = 0 = − 1 − r r dr r r → 2GM,

dt → ±∞ dr

dt → ±1/∞ dr However this is a result of inappropriate coordinates. coordinate transformation : r → 0,

We make a

dt 1 =± ⇒ t = ±(r + a ln(r − 2GM )) + C dr 1 − 2GM/r t = ±r∗ + C r∗ = r + 2GM ln(r/2GM − 1) At r = 2GM , r∗ goes to −∞. That is the coordinate singularity has been pushed to infinity. This new coordinate is called the tortoise coordinate. By differentiating the above equation, we get dr∗ = dr

 r/2GM  r/2GM − 1

(1)

and the metric can be written in terms of the tortoise coordinate as :  2GM  ∗ 2 2GM  2  2 dt + 1 − dr + r2 dΩ ds = − 1 − (2) r r We observe that the metric is non singular at r = 2GM . Since, dt we were looking at radial null geodesics = ±1. Integrate this dr∗ and we get t ± r = C, C is a constant. Now, we define the two quantities t ± r as our new coordinates. v = t + r∗ u = t − r∗ dv = dt + dr∗

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And the metric (1) can be written as :  2GM  2 ds = 1 − (−dt2 + dr∗ 2 ) + r2 dΩ r dt2 = (dv − dr∗ )2 = dr∗ 2 + dv 2 − 2dr∗ dv  2GM  2 ds2 = − 1 − dv + (drdv + dvdr) + r2 dΩ r These are the Eddington-Finkelstein coordinates. Even in these coordinates r = 2GM is not a singularity however r= 0 is. For null radial curves  2GM  = drdv + dvdr dv 2 1 − r  dv 2 dv α=2 dr dr α = 1 − 2GM/r dv Using the above equation we have two solutions for . We can construct dr space-time diagrams from them to find their physical meaning.  dv  dv α −2 =0 dr dr dv =0 dr or dv 2 = dr 1 − 2GM/r

Since we started of with a radial null curve, our two solutions correspond to an infalling and outgoing path. The outgoing path is the second solution  > 0 for r > 2GM dv  = ∞ (3) for r = 2GM dr   < 0 for r < 2GM dv The first solution = 0 always holds while the second one changes dr with radial coordinate, these are the equations describing the slopes 6

of the light cone. Outside r = 2GM Going inside as well as going outside is possible. At r = 2GM only going inside is possible dr = 0. Inside r = 2GM since the other slope is infinity or dv the angle of the light cone becomes even smaller and the only possible direction of motion is towards r = 0.

captionTilting of light cone

2.1

A Note on Event Horizon

R = 2GM is called the event horizon. Black Holes contrary to the popular perception do not suck everything in, neither the statement that their escape velocity is c entirely accurate. Outside the event horizon the gravitational pull (in Newtonian terms) is just like a star or any other massive body. However once the event horizon has been crossed, it is impossible to get out on timelike or null like paths. For an ordinary star, planet or massive body an object can escape to infinity even without attaining escape velocity i.e by a consistent acceleration however inside the event horizon even 7

dv = 0. This is what makes escaping this is not possible since dr from a blackhole different from escaping from other bodies like stars or planets.

2.2

Maximally Extended Schwarzschild Solution

In the Eddington-Finkelstein coordinates we saw that only future directed paths are allowed inside R = 2GM . Now, we take a look at the other coordinate u = t − r∗ .  2GM  (−dt2 + dr∗ 2 ) ds2 = 1 − r u = t − r∗ du = dt − dr∗ dt2 = du2 + dr∗2 + dudr∗ + dr∗ du  2GM  2 (−du2 − dudr∗ dr∗ du) ds = 1 − r Using eqn(1.4.1)  2GM  2 2 du − (dudr + drdu) ds = − 1 − r dv du Previously we solved for here we solve for for by a similar dr dr approach and get two solutions. du =0 dr du −2 = dr 1 − 2GM/r Here, the light cones have the  < 0 du  = −∞ dr   >0

opposite behaviour. for r > 2GM for r = 2GM for r < 2GM

(4)

Outside the Event Horizon there is no restriction, however to cross the event horizon we need to go along the past directed curves or the past directed light cone allows crossing the horizon. 8

Figure 1:

Tilting lightcones

We have extended the spacetime in two directions t → ∞ and t → −∞. We now look at space like curves to explore other regions

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of the spacetime.

Redefining coordinates as :

v 4GM u 0 u = − exp 4GM In terms of r and t : r r v0 = − 1 exp (r + t)/4GM 2GM r r 0 u =− − 1 exp (r − t)/4GM 2GM v0 dv 0 = dv 4GM u0 du0 = du 4GM We had the metric in terms of u and v, substitute the above equations to get 1 2GM  ds2 = − 1 − (dvdu + dudv) + r2 dΩ ⇒ 2 r 1 2GM  16G2 M 2 ds2 = 1− (dv 0 du0 + du0 dv 0 ) 2 r  u0 v 0 r − 1 exp (r/2GM ) ⇒ u0 v 0 = − 2GM G3 M 3 ds2 = −16 (du0 dv 0 + dv 0 du0 ) exp (−r/2GM ) + r2 dΩ r v 0 = exp

The coordinate singularity has been totally removed at 2GM and we have the extended Schwarzschild solution as stated in previous section. 2.2.1

Kruskal-Szekeres Coordinates

Since, our common coordinate system is one timelike and three spacelike coordinates, we will make a further transformation to the above coordinates

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even though the coordinate singularity has been entirely removed. 1 (5) T ≡ (v 0 + u0 ) 2 1 R ≡ (v 0 − u0 ) (6) 2 r  r t T = − 1 er/4GM sinh (7) 2GM 4GM r  r t R= − 1 er/4GM cosh (8) 2GM 4GM T + R = exp (v/4GM ) = v 0 (9) T − R = exp (−u/4GM ) = u0 (10) Let us see the event horizon in these new coordinates r → 2GM ⇒ r∗ → −∞ ⇒ v → −∞ ⇒ v 0 → 0 Similarly,u0 → 0 ⇒ T = ±R T 2 − R2 = 0 For a constant radius surface we get  t + r∗   −t + r∗  2 2 0 0 T − R = v u = exp exp = exp (r∗ /2GM ) = const. 4GM 4GM T t = tanh R 4M T = ±1 For t → ±∞ ⇒ R Since the only singularity present now is r = 0, T and R should take all values except when they correspond to r = 0. Let R vary from (−∞, ∞). From eqn.(9) and (10) we get T 2 − R2 . T 2 − R2 = (1 − r/2GM ) exp(r/2GM )

(11)

This function is always less than or equal to unity and at r = 0 the function becomes unity therefore (T, R) can take the following values : −∞ ≤ R ≤ ∞ T 2 − R2 < 1 11

(12) (13)

If we construct a diagram of (T, R) coordinates, we will have certain allowed regions due to eqn.(12) and (13) and points in this region will correspond to a given sphere or surface of constant r due to eqn.(11).

Figure 2:

Kruskal Diagram

The above image is caled the Kruskal Diagram with angular coordinates not taken into account. The Hyperbolae are curves of constant r and straight lines of constant t. It is divided into 4 regions. Region 1 is for R > 2GM , region 2 can be explored by following a future directed path from 1 to 2. Region 3 can be accessed only by past directed curves and region 4 is accesible only by spacelike curves which is a physical impossibility. Region 2 represents the blackholes, once entered one will evenetually hit r = 0. While 12

Figure 3:

Kruskal regions

region 3 is its opposite, things can only escape from 3 to 1, but it can not be entered from region 1. This is called a Whitehole.

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The Chandrasekhar Limit

S. Chandrasekhar derived an expression in 1930 for the maximum mass of a star that could become a white dwarf by balancing electron degeneracy pressure with gravitational attraction. Since electrons are Fermions, they follow Fermi-Dirac distribution and therefore their degeneracy is limited. So if a sufficient number of electrons are forced into a small volume of space we will experience a pressure because we are essentially forcing all electrons to a single state. This pressure is called LEctron-Degenracy pressure. Once a collapsing star overcomes this, the next stage could be a blackhole or Neutron star depending on the Neutron degeneracy pressure. [3]

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< Nr > = Probability of a state being occupied. gr 1 < Nr > = f (Er ) = gr 1 + exp Ekr −µ T

(14) (15)

b

The energy is given by special relativity : p Er = p2 c2 + m2 c4

The number density of states :

dΠ d3 xd3 p

Number of states in a cell is density times volume : h3 . g

Notice that eqn.

(16)

f (p) =

dΠ d3 xd3 p

×

(16) is constraint relating energy and momentum.

Density in coordinate space is : Z Z g × f (p) 3 dΠ 3 d p= dp n= 3 3 d xd p h3 Energy density in coordinate space is : Z Z Z dΠ 3 gf (p) 3 E × P (E)d p = E d p = E 3 d3 p 3 h .3xd p Since, pressure is associated with momentum flux Z Z 1 1 dΠ 3 1 pv 3 3 d p = pvgf (p)/h3 d3 p P = n < vp >= 3 3 d xd p 3 We assume that at temperature zero we have ideal Fermions and g = 2, the equations can be integrated with the upper limit Fermi momentum given by Ef2 = p2f c2 + m2 c4 . g ne = 3 h

Z 0

pf

g d p= 3 h 3

Z 0

pf

4πgp3f 8πp3f p dp(4π) = = 3h3 3h3 2

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Since a star is not entirely composed of electrons, thye can be considered as embedded in Baryonic matter. We assume that each Baryon is associated with an average number of electrons. Baryon rest mass : mB . Baryon volume : VB Mean number of electrons per Baryon : Ye . Electron number density : ne . mB VB VB = Ye /ne ne m B Rest mass density of Baryon. ρ0 = Ye 8πp3f 8πmB p3f Ye ρ 0 ne = = ⇒ ρ0 = 3h3 mB 3h3 Ye Pressure : Z Z pf Z pf 2 2 2 3 3 pvd p = 3 4πp3 vd3 p P = 3 pvf (p)d p = 3 3h 3h 0 3h 0 Integral limits from Inverse Transform Sampling | pf = f −1 (f (p)) ρ0 =

The integral will have different values for non relativistic and ultra relativistic case. 1. 8π p = me v ⇒ P = p5f 3 15h me 2. 2πc v ∼ c ⇒ P = 3 p4f 3h 8πmB p3f 3h3 Ye ⇒ P = Kργ0 5/3 h2 Ye  3 2/3 1 5 Non relativistic K = γ = 5/3 8π 5me 3 mB 4/3 4 hYe  3 1/3 c Relativistic K = 4/3 γ= 8π 4 3 mB We had :ρ0 =

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(17) (18) (19) (20)

Now, we take a look at the different forces in action inside a star, the gravitational pull and force due to degeneracy pressure. Gm(r) 4πr2 ρ0 dr 2 r dFP = −4πr2 dP dFg =

(21) (22)

dP In equilbrium they will be equal = −m(r)Gρ0 /r2 (23) dr Z r dm ρ0 4πr2 dr ⇒ = 4πr2 ρ0 (24) m(r) = dr 0  dm d  dP 2 =− r /(ρ0 G) = 4πr2 ρ0 (25) dr dr dr Since we assumed a polytropic equation of state P = Kργ0 (26) dP rho0 = Kγργ−1 (27) 0 dr dr 1 d 2 dρ0  Using this in (25) we get 2 r Kγργ−1 = −4πGρ0 0 r dr dr (28) We have an equation in density and radius, (29) so we choose appropriate boundary conditions (30) ρ0 = ρc f or r = 0 and ρ0 = 0 at r = R. (31) We apply the following transformations in sequence : (32) γ − 1 = 1/n ρ0 = ρc θn q 1−n a = (n + 1)Kρc n /(4πG) r = aξ d 2θ ⇒ ξ = −θn dξ dξ

(33) (34) (35) (36) (37)

This is called Lame Emden Equation of index n. For our values of n it can only be solved numerically. The roots of θ as a function of ξ are given below. We are looking at the function’s roots because there ρ0 will be zero i.e surface of the star. 16

γ = 5/3 γ = 4/3

n = 3/2 n=3

ξ1 = 3.65375 ξ1 = 6.89685

ξ12 |θ0 (ξ1 )| = 2.71406 ξ12 |θ0 (ξ1 )| = 2.01824

To find the mass of the star, Z ξ1 Z R 2 a3 ξ 2 θn dξ 4πr ρ0 dr = 4πρc M= 0 0  dθ i dθ i  = −4πρc a3 ξ 2 − ξ2 = −4πρc a3 ξ12 θ0 (ξ1 ) dξ ξ1 dξ 0 = 4πρc a3 ξ12 |θ0 (ξ1 )| It can be seen from numerical solutions that the derivative is negative, hence its modulus is used in the last step. Now for the Non relativistic Case : γ = 5/3 n = 3/2 r r 1−3/2 (n + 1)K ρc 3/2 a= 4πG −1/6 = ω 1 ρc Since R = aξ ⇒ R = ω2 ρ−1/6 c r 1 √ = R−3 M ∼ a3 ρc ∼ ρc ∼ R6 As we keep adding mass to the star its radius will decrease. This means that the space available for electrons to move about is decreasing. From uncertainity principle it follows that ∆x ∼

1 ∆p

The uncertainity in momentum will increase, and this will cause more and more electrons to achieve near light speed. So as the star becomes

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more massive we need to switch to the Relativistic case. γ = 4/3 n = 3 r (n + 1)K √ −2/3 a= a r 4πG K −1/3 ρ ⇒ a= πG c − 13

a ∼ ρc−1/3 ⇒ R ∼ ρc −1

M ∼ a3 ρc ∼ (ρc 3 )3 ρc = const. As electrons approach light speed M tends to become constant w.r.t R. M is not very sensitive to changes in R or R is highly sensitive to changes in M . It implies that by adding more and more mass a critical mass is reached where R shrinks to zero. This is critical mass is the Chandrasekhar limit. We had an expression for M from the Lame-Emden equation for Relativistic case. Mch = 4πρc a3 ξ12 |θ0 (ξ1 )| ⇒  K  23 Mch = 4π2.01824 πG If the Baryonic matter is Helium then Ye = 5

h2 Ye3  3 2/3 1 K = 5/3 8π 5me mB mB = 1.67 × 10−24 c = 3 × 1010 G = 6.67 × 10−8 h = 6.63 × 10−27 M◦ = 1.989 × 1033 Mch = 1.46 M◦

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1 2

4

Spherically Symmetric Gravitational Collapse

Take G = 1 and c = 1. The Schwarzschild Metric reads  2M −1 2 2M  2  dt + 1 − ds2 = − 1 − dr + r2 dΩ r r The metric is valid for outside the surface of the star/body in vacuum. The metric that describes the inside of the star will be different, however at the surface of the star both the metrics should be same if we assume continuity. Take r = R(t) h 2M −1 dR
2 i 2M  2  dt − 1 − + R(t)2 dΩ (38) ds2 = − 1 − r r dt On the surface there is zero pressure and spherical symmetry, so a given point on the surface will follow radial timelike curve i.e the angular component can be neglected. dΩ2 = 0

ds2 = −dτ 2 h 2M  2  2M −1 dR
2 i ds2 = −dτ 2 = − 1 − dt − 1 − r r dt h 2M  2  2M −1 dR
2 i dt 2 dt − 1 − 1= 1− r r dt dτ µ If K is a Killing Vector ∂ K µ T µ = c1 ⇒ K µ = = ∂t ∂t  2M  Kµ = gµ0 K 0 ⇒ K0 = − 1 − R  2M  dt =− 1− R dτ From eqn.(41) h i 1 = (1 − 2M/R) − (1 − 2M/R)−1 R˙ 2 2 (1 − 2M/R)−2 1 2M 2 R˙ 2 = 2 1 −  R  2M  R˙ 2 = − 1 + 2 R 2 ˙ R = 0 ⇒ R = 2M andR = 2M/(1 − 2 )

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(39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51)

These are the points at which the Collapse stops or starts. So, it starts at the second root and ends at the first root. A plot of R˙ 2 shows that the second root is the maximum allowed radius and it goes to R = 2M at a point of zero slope, i.e R goes to 2M as t = ∞. The above analysis is from the point of view of an observer at ∞. If we switch to a time coordinate that is local to the surface of the star we see a different scenario. dτ d 1 2M  d d = = 1− dt dt dτ  R dτ  dR 2 h 1  2M  dR i2 1− = dt  R dτ Use eqn.49   dR 2  2M = − 1 + 2 = 0 dτ R 2M Rmax ≡ 1 − 2  dR 2 R  max = (1 − 2 ) −1 dτ R The plot of the last equation shows that the rate of change of radius is zero only at Rmax . R = 2M is not a root of the equation, i.e. it reaches R = 2M in a finite time. [4]

4.1

A Note on Naked Singularities and Cosmic Censorship

English physicist and mathematician Roger Penrose had conjectured that a singularity must necessarily be cloaked by an event horizon, therefore any information about the singularity can never be obtained. However there exist some models of gravitational collapse which lead to naked singularities. These models have not been verified yet, but also the Cosmic Censorship conjecture is yet to be proven.

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References [1] T. Birkandan et al. “Symbolic and Numerical Analysis in General Relativity with Open Source Computer Algebra Systems”. In: ArXiv e-prints (Mar. 2017). arXiv: 1703.09738 [gr-qc]. [2] Sean Carroll. “Spacetime and geometry: An introduction to general relativity”. In: (2004). [3] Dave Gentile. Chandrasekhar Limit Derivation. http : / / www . davegentile.com/thesis/white_dwarfs.html. Accessed: 2018-11-10. [4] P. K. Townsend. “Black Holes”. In: ArXiv General Relativity and Quantum Cosmology e-prints (July 1997). eprint: gr-qc/ 9707012.

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