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Московский государственный университет им. М.В. Ломоносова

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Физика

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P.K.Townsend - Black Holes

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Causal structure

Because we now have only axial symmetry we really need a 3-dim spacetime diagram to encode the causal structure, but the = 0; =2 submanifolds are totally-geodesic, i.e. a geodesic that is initially tangent to the submanifold remains tangent to it, so we can draw 2-dim CP diagrams for them.

= =2

			i+
			..
			....
			....
			....
			....
			....
	...	..	.
...	...		... ...
...			.
..
..			....
..			....
naked
singularity			...
singularity			i
at r = 0			i

(boundary of disk)

= 0

	.i+
	.
	.
	.
	.
	.
	.
	.
	.
	.
	.					=			+
	.								+
	.
	.
	.
	.
	.
	.
	.
	.
	.
	.
	.
	.
	.
	.
r < 0	.			r > 0 .
r < 0	.			r > 0 .
	.
	.									.
	.									.
	.									.
	.									.
	.								.
	.							.
	.							.
	.						.
	.						.
	.					.
	.					.	=
	. ...	..
	.	..	...
	. .		...	....
	.			....	.
	.				.	...
	.					...
	.						.	..
	.							..	..
	.								..
	.								.
	.									...
	.									...
	.									.
	.									..
	i									..
	i									r.

= 0

For = =2 each point in the diagram represents a circle (0 < 2 ). Each ingoing radial geodesic hits the ring singularity at r = 0, which is clearly naked. For = 0 we are considering only geodesics on the axis of symmetry. Ingoing radial null geodesics pass through the disc at r = 0 into the other region with r < 0. We can summarize both diagrams by the single one.

.............

.....

....

...

.. ..

......

...

....

.....=...

...

....

.. .

r < 0

r > 0

. i0

.. ..

.. .

......

....

.....

ring-singularity

at r = 0

The spacetime is unphysical for another reason. Consider the norm of the Killing vector eld m = @=@ :

					Ma2		r		!
m2	= g = a2 sin2	1 +	r2	+		2 sin4				(4.21)
			a2		r	1 +	a2	cos2
							2

Let r=a = (small) and consider = =2 + . Then

m2 =	a2 +	Ma	+ O( ); for 1	(4.22)


<	0 for su ciently small negative

So m becomes timelike near the ring-singularity on the r < 0 branch. But the orbits of m are closed, so the spacetime admits closed timelike curves (CTCs). This constitute a global violation of causality.

Moreover because of the absence of a horizon these CTCs may be deformed to pass through any point of the spacetime (Carter). They also miss the singularity by a distance M, for M a, and M can be arbitrarily large. Since the ring singularity would be naked for M2 < a2, then even if the white hole region is replaced by a collapsing star, we can invoke cosmic censorship to rule out M2 < a2.

(ii)M2 > a2. We still have a ring-singularity but now the metric (in BL coordinates) is singular at r = r+ and r = r . These are coordinate

singularities. To see this we de ne new coordinates v and by

r2 a2

dt +

(4.23)

d +

(4.24)

This yields the Kerr solution in Kerr coordinates (v; r; ; ) which are

analogous to ingoing EF for Schwarzschild:

ds2

a sin

dv2 + 2dv dr

2a sin2 r + a

dv d

2a sin2 d dr +

r2 + a2 2

a2 sin2 i

sin2 d 2 + d 2

(4.25)

This metric is non singular when = 0, i.e. when r = r+ or r = r .

Proposition

The hypersurfaces r = r are Killing horizons of the Killing vector elds

= k +

(4.26)

+ a2

with surface gravities

r r

(4.27)

r2 + a2

Proof Let N be the hypersurfaces r = r . The normals are

f grjN @ ;

for some non-zero functions f

(4.28)

r2 + a2

(Exercise)(4.29

+ a2 cos2

+ a2

}

First

/ gvv + r2 + a2 gv + (r2 + a2)2 g =0 = 0

(4.30)

cauchy horizon at r = r

so N are null hypersurfaces. Since jN / l , they are Killing horizons of . It remains to compute D . This gives the result for(Exercise).

This result can be used to nd KS type coordinates that cover 4 regions around a BK axis of each Killing horizon, and the = 0 and = =2 CP diagram of the maximal analytic extension of M2 > a2 Kerr can be found. Note that the diagram can be extended in nitely in both time directions.

																						. ...
																					. .
	CTCs .....							.....						..							. .. .																	..	..
								.....			..							.		. .																..		..
											.	.						.		.																..
												.	..						.																.
													..	.					.															...
r =	1													.	... .. ..																			..
r =		.... .				........									... .. ..															.. .			. ..
						........				.. .							..... .													.. .				.
										.. .								.											..					.				...... .....
										.. .								.										..						.				...... .....							.....
										...								.	.					. .		.								.....											.....
										.								.	.				.	. .						..		....																	.
								.										.				.	.					.	..	..		.
										r < 0 .												.						.														. ..
										r < 0 .												. .. .....												.								. ..
ring																	.... .				. .. .													.							. .. .
ring															... .						. .													. . .
singularity .... .... ......																		. . .																. . .. .
																		. .																. .																.
r = 0																	. .. .					II								+.. . i+																			...	.
r = 0									....			...... .										II								+.. . i+																			...
								.	....								.										H..				. .																	.	..
							.....	.									.														. ..												+				...	.
					..		.....						.	.																		.....					...						+			..	...
				.	..								.					IV			r < r+.																...		.					.		..
		...	.	.														IV							.		.							I					.		..... ...			.
		...								.	.													. .										I								=
		....								.											. .. .... .						r > r+															. .. ... i0
internal 1																				. .. .. .																		.. .. .
																		. .		.	III										H .. .. ..											=
																		. .			III										..		.. .
														.			. .												..		..			.
													.	.				.									..		..					.
											.		.					.							..		..							.
										.	.							.						..	..									.
								.		.								.				. . ..		..										.								. .. i0
								.										.			... .	. . ..												.								. .. i0
																		.			... .													.						..		.. .
																		.			..													.						..
																		.		. . . .														.		.. ..						=
																		. .																.
																		...																. .
																		...															. .
																		..											. ..				.			....
													..			.....													. ..							....				.
													..												. ..				.											.
																									. ..

. ..

future event horizon at r = r+

4.2.1Angular Velocity of the Horizon

The event horizon is a Killing horizon of

= k + H m

(4.31)

where
H =	a	=	J			(4.32)

	r+2 + a2		2M hM2 + p		i
				M4 J2
In coordinates for which k = @=@t and m = @=@ we have that
@ ( H t) = 0						(4.33)

i.e. = H t+constant, on orbits of , whereas is constant on orbits of k. Note that k is unique. Consider

(k + m)2 = gtt + 2 gt + 2g

(4.34)

As long as gt is nite and g r2 as r ! 1, we have (k + m)22r2 > 0 (if 6= 0) as r ! 1. So there can be only one Killing vector

k that is timelike at 1 and normalized s.t. k2 ! 1 as r ! 1.

Thus particles on orbits of rotate with angular velocity H relative to static particles, those on orbits of k, and hence relative to a stationary frame at 1. Since the null geodesic generators of the horizon follow orbits of the black hole is rotating with angular velocity H .

Lemma k = 0 on a Killing horizon, N, of .

Proof
			jN	mjN	N
kjN = 2	N	H		mjN			(4.35)
=	H		m	(since 2 = 0 on		)	(4.36)

Now, N is a xed point set of m, since m is Killing (Choose coordinates s.t. m = @=@ . The metric is independent, so the position of the horizon is independent of ). So m must be tangent to N or l m = 0 where l is normal to N. But / l on N, so mjN = 0. Hence result.

Consistency checks (See Question III.3)

2 = 0 implies that

k2 + 2 H m k m2 H = 0; on N

But k = 0 implies that

k2 + H m k = 0; on N

Consistency requires

D (k m)2

k2m2 N = 0

For Kerr, D = sin

= 0 on N .

Also

H =

gtt

in BL coordinates

a sin

a sin2

+ a2

r+2 + a2

(4.37)

(4.38)

(4.39)

(4.40)

(4.41)

(4.42)

(iii)M2 = a2 Extreme Kerr

In this case we have a degenerate ( = 0) Killing horizon at r = M of the Killing vector eld

= k + H m; H =	a	(4.43)
	2M

The CP diagram is

																																	.
																															. ..... .
																														.. .			.	.. .
																										. ..							.			.			..
																										. ..							.						..
																								.. ..		.							.									.. .
																								.. ..									.									.. .
																				. ..																								.	..
																				. ..													.												..
																		.	.. .														.													..		.
																	.. .																.															.. .
													. ..																				.																.. ..
											...		.																				.																							...
											...																																													...
												.																					.																							. .
														..																			.																						.		..
														..																			.																			. .					..		.
																	..																.																		.								.
																	..																																. .										. .
																		..															.															. ..												. ..
																																															.
																			..														.													. .																..	.
																						..																							.																		.
																						..											.												. .																			. .
																								..									.											.																					.	.
																								..									.										.	.																						.
																								.																			.																							.
																									...																	.																										..
																									...								.								...																											..		.
																											..												.	.																														.
																											..						.						.																																.	.
																														..			.					.																																		.
																														..	. .					. .																																					. .
r =			1..																												. .				..																																							. .
r =																																. . ...																																												. .
																																	.																																												.
																															..		.....	.																																										..
					.																										..			.																																								..		.
					..																									..	.		.	..		.																																						..
						.																								..						.																																					.	.
							.																				..		.							.																																					.
							..																				..						.						.																															.		..
								.	...																	.													.																															.
									...				....							.				.	...								.									.																									..		..
													....							.				.									.									.	.																								..
																	.. .... .....																.										.																							..
																						.																						.																					.
																						.											.												.																			.	.
																					..												.												.																			.
																			...																										.																		.
																			...														.													.																. .
																		..																													.	.													.
																		..																														.												..	.
																		.															.															..												..
					.......					.. ..............													......... .										.		r > 0														...										.. .																....		..
				.	.......							.	...										......... .										.		r > 0																	..				.. .																			....
		......		.								.																		..																										. .			i+													.....
	.	......									... .																					.. .																					.		.....		.		i+												.	.....
.	.												.		.																		.																				.				.		.										..		.
.															.																		.																		..								.									.	..
																.					r < 0																													.									.								.	.
ring																	..																																.											..					...		.
																	..	.															.																.											..					...
																		.															.															.. ..... .... ...........
singularity																		. .. .															.		r < M... .																										. .. .									+
																				.													.												..																			.						+
at r = 0																						.. .. ..				..							.									.. ... .					+																		.. ..=..
																										..		.					.						. .						H		+																							..		.
																												.					.					..																																		.
																												. .. .					.	.....																																							. .. .
																															. . . . .																																													. .
														.....				...... .... ..... ...... .																																						I																					.
												..		.....																		. ... ..																								I																			.	...
												..																			..		.		.																																								.
											.	.																		.			.		.																																							.
									.		.																			.						.	.																																				.	.
									.																				..				.				.																																				.
								.	.																			.					.					.																																		.
							..	.																				.											.	.																														.		.
						.	..																			..							.							.																														.
internal						.																			..								.								.																									...			..
internal					.																			.. .									.									. ..																								...
																		. ...			... .												.											..						r > M													..		...				=
																		. ...															.												.. ..															..	... .								=
																	.	..															.															.												..
																	.																																.										.	.
															.	..																	.																		..								.
															.																		.																		..			.			.	..
												.	..																				.																					.			.
												.																					.																						.. ..
											..																																													.
											. ..																						.																				. ..
															..																		.																			.
															..																		.																	..
																	.																																.
																	.																															..	.
																	. .																.															..
																		. ..															.													.. .
																				.. .													.												... .
																						. .. .											.									. ...
																									.	.. .. .							.			.. ...					..
																										.. .. .							.			.. ...
																														.	.		.		..
																															.			.
																															. . . . .
																																	. .

future event horizon

(and future Cauchy horizon) at r = M

So there can be only one Killing vector k for which k k ! 1 as r ! 1.

N.B. If you change the sign of r in the Kerr metric this e ectively changes the sign of M.

4.3The Ergosphere

Although k is timelike at 1 it need not be timelike everywhere outside the horizon. For Kerr,

a2 sin2

= 1

k2 = gtt =

(4.44)

r2 + a2 cos2

so k is timelike provided that

r2 + a2 cos2 2Mr > 0

(4.45)

For M2 a2 this implies that

+ M

a2 cos2

r > M pp

(4.46)

(or r < M M2 a2 cos2 , but this is not physically relevant).

The boundary of this region, i.e. the hypersurface p

r = M + M2 a2 cos2

(4.47)

is the ergosphere. The ergosphere intersects the event horizon at = 0; , but it lies outside the horizon for other values of . Thus, k can become spacelike in a region outside the event horizon. This is called the ergoregion.

ergosphere .

event . horizon

.... ... .....

... ...

........

.....

... ...

.... ........ .......

........

...

.. . ..

...

. ..

...

= 0

.	.. ..	.			.
.	.	.		.	.																							.
	..			.	...																							.	r = r+
	. ...				...	.	. .																						r = r+
			...		..	.			..					..				..
			.		..				..				.	..				..			.
					. . .						..		.				.		..		.
					..						..				..				..			.
					..			...				...			..		...				...			.
					..					..					..			..					..			.
									.	..								.. ..					.					.
								......................																					.	r = M +	p		2	a	2	cos	2
								. . .												. . .										r = M +		M		a		cos
							..... ....................
									..		..		..		..				.. ..						..		.
					...				..				..		.							..	...		..
					...					..						.. ...							...
					..		.... ....										...			....
					..		.. ..										.
					. . .													.. .											. ergoregion
		..	..... .............																							.			. ergoregion
	...	..			..			. .
	...		...		..
	. .

= =2

4.4The Penrose Process

Suppose that a particle approaches a Kerr black hole along a geodesic. If p is its 4-momentum we can identify the constant of the motion

E = p k

(4.48)

as its energy (since E = p0 at 1). Now suppose that the particle decays into two others, one of which falls into the hole while the other escapes to

						..	..........horizon
						.
						.
E			.			.....
	.
	.
	..... .				BH .
	.
		..	.			. 1
						. 1
						.
					.	E1		= p1	k
			.	..
			.	.
				.
						. ..
						...
								.	E2 = p2 k
								2

By conservation of energy

E2 = E E1

(4.49)

Normally E1 > 0 so E2 < E, but in this case

E1 = p1 k

(4.50)

which is not necessarily positive in the ergoregion since k may be spacelike there. Thus, if the decay takes place in the ergoregion we may have E2 > E, so energy has been extracted from the black hole.

4.4.1Limits to Energy Extraction

For particles passing through the horizon at r = r+ we have

p 0

(4.51)

Since is future-directed null on the horizon and p is future-directed timelike or null. Since = k + H m,

E H L 0

(4.52)

where L = p m is the component of the particle's angular momentum in the direction de ned by m (only this component is a constant of the motion). Thus

L	E	(4.53)
	H

If E is negative, as it is for particle 1 in the Penrose process then L is also negative, so the hole's angular momentum is reduced. We end up with a hole of mass M + M and angular momentum J + J where M = E andJ = L so


				2M M2 + p
J	M	=		2M M2 + p		M4 J2	M	(4.54)
J		=					M	(4.54)
	H				J
from formula for H . This is equivalent to (Exercise)
M2 + p					0
M2 + p			M4 J2		0			(4.55)

(This quantity must increase in the Penrose process).

Lemma A = 8 hM2 + pM4 J2i is the `area of the event horizon', of a

Kerr black hole (i.e. area of intersection of H+ with partial Cauchy surface, e.g. area of v = constant, r = r+ in Kerr coordinates (See Question III.5).

Corollary Energy extraction by Penrose process is limited by the requirement that A 0. This is a special case of the second law of black hole mechanics.

4.4.2Super-radiance

The Penrose process has a close analogue in the scattering of radiation by a Kerr black hole. For simplicity, consider a massless scalar eld . Its stress tensor is

g (@ )2

T = @ @

(4.56)

Since D T = 0 we have

D (T k ) = T D k = 0

(4.57)

so we can consider

= T

= @

k @ +

(@ )

(4.58)

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