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6.2 The Laws of Black Hole Mechanics

Back

Previously we showed that _2 is constant on a bifurcate Killing horizon. The

proof fails if we have only part of a Killing horizon, without the bifurcation

2-sphere, as happens in gravitational collapse. In this case we need the:

6.2.1 Zeroth law

If T__ obeys the dominant energy condition then the surface gravity _ is constant on the

future event horizon.

Proof Let _ be the Killing vector normal to H+ (here we use the theorem

that H+ is a Killing horizon). Then since R__ ____ = 0 and _2 = 0 on H+,

Einstein's equations imply

0 = 􀀀 T______jH+ _ J___jH+ (6.55)

i.e. J = (􀀀T_

___ ) @_ is tangent to H+. It follows that J can be expanded

on a basis of tangent vectors to H+

J = a_ + b1_(1) + b2_(2) on H+ (6.56)

But since _ _ _(i) = 0 this is spacelike or null (when b1 = b2 = 0), whereas it

must be timelike or null by the dominant energy condition. Thus, dominant

energy ) J / _ and hence that

0 = _[_J_]

H+ = 􀀀 _[_T _

_] __

___

H+

(6.57)

= _[_R _

_] __

___

H+

(by Einstein's eq.) (6.58)

= _[_@_]_

H+ (by result of Question IV.3) (6.59)

(6.60)

109

) @__ / __ ) t _ @_ = 0 for any tangent vector t to H+

) _ is constant on H+.

6.2.2 Smarr's Formula

Let _ be a spacelike hypersurface in a stationary exterior black hole spacetime

with an inner boundary, H, on the future event horizon and another

boundary at i0.

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

. ...... ...

...... ....

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

. ...... ...

..... .....

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

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

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

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

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

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

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

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

..........

........

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

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

_ i0

H+

The surface H is a 2-sphere that can be considered as the `boundary' of the

black hole.

Applying Gauss' law to the Komar integral for J we have

J =

8_G

dS_D_D_m_ +

16_G

dS__D_m_ (6.61)

8_G

dS_R_

_m_ + JH by Killing vector Lemma (6.62)

where JH is the integral over H. Using Einstein's equation,

J =

dS_

_m_m_ 􀀀

Tm_

+ JH (6.63)

In the absence of matter other than an electromagnetic _eld, we have T__ =

T__ (F), the stress tensor of the electromagnetic _eld. Since g__T__(F) =

T(F) = 0 we have

J =

dS_T_

_ (F)m_ + JH (6.64)

for an isolated black hole (i.e. T__ = T__(F)).

110

Now apply Gauss' law to the Komar integral for the total energy (=

mass).

M = 􀀀

4_G

dS_R_

_k_ 􀀀

8_G

dS__D_k_ (insert _ = k + Hm)

dS_ (􀀀2T_

_k_ + Tk_) 􀀀

8_G

dS__ (D___ 􀀀 HD_m_()6.65)

(6.66)

since H is constant on H. For T__ = T__ (F) (T(F) = 0)we have

M = 􀀀2

dS_T_

_ (F)k_ + 2HJH 􀀀

8_G

dS__D___ (6.67)

for an isolated black hole. Using (6.64) we have

M = 􀀀2

dS_T_

_ (F)__ + 2HJ 􀀀

8_G

dS__D___ (6.68)

For simplicity, we now suppose that T__(F) = 0, i.e. the black hole has zero

charge (see Questions III.7&8 for general case). Then

M = 2HJ 􀀀

8_G

dS__D___ (6.69)

Lemma

dS__ = (__n_ 􀀀 __n_)dA on H (6.70)

where n is s.t. n _ _ = 􀀀1.

Proof n and _ are normals to H, so we have to check coe_cients. In

coordinates such that

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

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􀀀 􀀀

􀀀􀀀

􀀀 􀀀

􀀀􀀀 􀀀

􀀀 􀀀

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􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 �� 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀

􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀 􀀀􀀀

􀀀􀀀

􀀀􀀀􀀀

􀀀􀀀

􀀀

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

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

n _

111

__ =

(1; 1; 0; 0) (6.71)

n_ =

(1;􀀀1; 0; 0) (6.72)

we should have jdS01j = dA. We do if dS__ is as given. [There is still a sign

ambiguity. Fix by requiring sensible results].

Thus

􀀀

8_G

dS__D___ = 􀀀

4_G

dA (_ _ D_)_

| {z }

___

n_ (6.73)

= 􀀀

4_G

dA_|{_zn}

􀀀1

(_ is constant by 0th(6l.a7w4)

4_G

A (6.75)

where A is the \area of the horizon" (i.e. H).

Hence

M =

+ 2HJ (6.76)

This is Smarr's formula for the mass of a Kerr black hole. [Exercise: Check,

using previous results for _, H, and A]. In the Q 6= 0 case, this formula

generalizes to

M =

+ 2HJ + _HQ (6.77)

where _H is the co-rotating electric potential on the horizon (see Question

III.6&7).

6.2.3 First Law

If a stationary black hole of mass M, charge Q and angular momentum J, with future

event horizon of surface gravity _, electric surface potential _H and angular velocity H, is

perturbed such that it settles down to another black hole with mass M+_M charge Q+_Q

and angular momentum J + _J, then

dM =

dA + HdJ + _HdQ (6.78)

1) De_nition of _H and proof for Q 6= 0 in Q. III.6&7.

112

2) This statement of the _rst law uses the fact that the event horizon of

a stationary black hole must be a Killing horizon.

`Proof' for Q = 0 (Gibbons) Uniqueness theorems imply that

M = M(A; J) (6.79)

But A and J both have dimensions of M2 (G = c = 1) so the function

M(A; J) must be homogeneous of degree 1=2. By Euler's theorem for homogeneous

functions

+ J

M (6.80)

A + HJ by Smarr's formula (6.81)

Therefore

@A 􀀀

+ J

@J 􀀀 H

= 0 (6.82)

But A and J are free parameters so

;

= H (6.83)

6.2.4 The Second Law (Hawking's Area Theorem)

If T__ satis_es the weak energy condition, and assuming that the cosmic censorship hypothesis

is true then the area of the future event horizon of an asymptotically at spacetime is

a non-decreasing function of time.

Technically the cosmic censorship assumption is that the spacetime is

`strongly asymptotically predictable' which requires the existence of a globally

hyperbolic submanifold of spacetime containing both the exterior spacetime

and the horizon. A theorem of Geroch states that in this case there

exists a family of Cauchy hypersurfaces _(_) such that _(_0) _ D+ (_(_))

if _0 > _.

113

_(_0)

_(_)

We can choose _ to be the a_ne parameter on a null geodesic generator of

H+. The \area of the horizon" A(_) is the area of the intersection of _(_)

with H+. The second law states that A(_0) _ A(_) if _0 > _.

Idea of proof To show that A(_) cannot decrease with increasing _ it is

su_cient to show that each area element, a, of H has this property. Recalling

that

= _a (6.84)

we see that the second law holds if _ _ 0 everywhere on H+. To see that

this is true, recall that if _ < 0 the geodesics must converge to a focus or

caustic, i.e. nearby geodesics to a given one passing through a point p must

intersect at _nite a_ne distance along it. The _rst point q for which this

happens is called the point conjugate to p on .

..........

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

..........

....

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

. . . . . . . . . . . . . . . . . . . . conjugate

point

Points on beyond q are no longer null separated. They are timelike separated

from p. An example illustrating this is light rays in a at 2-dim

cylindrical spacetime.

114

..........

......... .

..........

.... ......

........ ..

...... ....

... .......

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

.. ..... ...

. ..... ....

..... .....

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

. ...... ...

..... .....

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

..........

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

..........

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

..........

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

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

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

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

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

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

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

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

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

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

p space

point on beyond q

can be reached by timelike curve

from p

conjugate point to p on

(on far side of cylinder)

The existence of a conjugate point to the future of a null geodesic generator

in H+ would mean that this generator of H+ has a _nite endpoint, in contradiction

to Penrose's theorem, so the hypothetical conjugate point cannot

exist. Thus it must be that _ _ 0 everywhere on H+ and hence the second

law.

_ = 0 only for stationary spacetimes.

115

Example Formation of black hole from pressure-free spherically-symmetric

gravitational collapse. Illustrate by a Finkelstein diagram

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

past endpoint of

all generators of H+

H+ H+

_(_0)

_(_1)

_(_2 _ _1)

Star

A(_0) = 0

A(_1) 6= 0

A(_2) _ 16_M2

A = 0 on _(_0). A 6= 0 on _(_1) and it has increased to its _nal value of

A = 16_M2 for a stationary Schwarzschild black hole on _(_2).

116

Consequences of 2nd Law

(1) Limits to e_ciency of mass/energy conversion in black hole collisions.

Consider Finkelstein diagram of two coalescing black holes.

gravitational

radiation

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

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

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

M1 M2

H+ H+

new generators enter

the horizon H+

at this caustic

Then energy radiated is M1 +M2 􀀀 M3, so the e_ciency, _, of mass

to energy conversion is

_ =

M1 +M2 􀀀M3

M1 +M2

= 1 􀀀

M1 +M2

(6.85)

Assuming that the two black holes are initially approximately stationary,

so A1 = 16_M2

1 and A2 = 16_M2

2 the area theorem says that

A3 _ 16_

􀀀

1 +M2

(6.86)

But 16_M2

3 _ A3 (with equality at late times), so

M3 _

1 +M2

2 (6.87)

Thus

_ _ 1 􀀀

1 +M2

M1 +M2 _ 1 􀀀

(6.88)

The radiated energy could be used to do work, so the area theorem

limits the useful energy that can be extracted from black holes in the

same way that the 2nd law of thermodynamics limits the e_ciency of

heat engines.

117

(2) Black holes cannot bifurcate. Consider M3 ! M1 +M2 (with M1 > 0

and M2 > 0). The area theorem now says that

M3 _

1 +M2

2 _ M1 +M2 (6.89)

but energy conservation requires M3 _ M1+M2 (with M3 􀀀M1 􀀀M2

being radiated away). We have a contradiction so the process cannot

occur.

118

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6.2 The Laws of Black Hole Mechanics

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