P.K.Townsend - Black Holes

Chapter 6

Black Hole Mechanics

6.1Geodesic Congruences

De nition A congruence is a family of curves such that precisely one curve of the family passes through each point. It is a geodesic congruence if the curves are geodesics.

The equations of a geodesic congruence may be written as x = x (y ; ) where the parameters y ; = 0; 1; 2 label the geodesic and is an a ne parameter on the geodesic, i.e.

is a ne, t2 1 for timelike geodesics (while t2 0 for null geodesics). The vectors

may be considered as a basis of `displacement' vectors across the congruence:

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Note that t and commute (since we could choose coordinates x s.t. t = @=@ and = @=@y ), so

measures the failure of the displacement vectors to be paralelly-transported along the geodesics, i.e. it measures geodesic deviation.

A geodesic nearby some ducial geodesic may now be speci ed by a displacement vector , but this speci cation is not unique because 0 = +at (a = constant) is a displacement vector to the same geodesic.

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.

.

For timelike geodesics we can remove this ambiguity by requiring to be orthogonal to t, i.e.

Strictly, speaking we can only make such a choice at a given value of , by choosing the origin of across the congruence. However

since t2 1 for timelike congruences, so if t is chosen to vanish at one value of it will do so for all .

For null congruences the condition t = 0 is not su cient to eliminate

when t2 = 0, which means that 0 t = 0 whenever t = 0. The problem is that the 3-dim space of vectors orthogonal to t now includes t itself, so the displacement vectors orthogonal to t specify only a two-parameter family of geodesics. Displacement vectors to the other null geodesics in the congruence have a component in the direction of a vector n that is not

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orthogonal to t. The choice of n is otherwise arbitrary (it is analogous to the choice of gauge in electrodynamics), but it is convenient to choose it such that

e.g. if t is tangent to an outgoing radial null geodesic, then n is tangent to an ingoing one.

r

Consistency of the choice of n requires that n2 and n t be independent of, which is satis ed if

i.e. we choose n to be parallely-transported along the geodesics.

Having made a choice of the vector n, we may now uniquely specify a twoparameter subset of geodesics of a null geodesic congruence by displacement vectors orthogonal to t by requiring them to also satisfy

The vectors now span a two-dimensional subspace, T?, of the tangent space, that is orthogonal to both t and n, i.e. P = , where

projects onto T?.

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so t is normal to a family of hypersurfaces by Frobenius' theorem. (In this case we can take t = l).

Conversely, if t is normal to a family of null hypersurfaces, then Frobenius' theorem implies t[ D t ] = 0. Then, reversing the previous steps we

If !^ = 0 we have a family of null hypersurfaces. The family is parameterized by the displacement along n

6.1.1Expansion and Shear

Two linearly independent vectors (1) and (2) orthogonal to n and t determine an area element of T?. The shear ^ determines the change of shape of this area element as increases. The magnitude of the area element de ned by (1) and (2) is

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i.e. measures the rate of increase of the magnitude of the area element. If> 0 neighboring geodesics are diverging, if < 0 they are converging.

Raychaudhuri's equation for null geodesic congruences

This is Raychaudhuri's equation for null geodesic congruences.

Some consequences of Raychaudhuri's equation for null hypersurfaces

Proposition The expansion of the null geodesic generator of a null hypersurface, N, obeys the di erential inequality

provided the spacetime metric solves Einstein's equations G = 8 GT and T satis es the weak energy condition.

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Proof ^2 0 because the metric in the orthogonal subspace T? (to l and n) is positive de nite. !^2 0 also, but this comes in with wrong sign, however !^ = 0 for a hypersurface. Thus Raychaudhuri's equation implies

Corollary If = 0 < 0 at some point p on a null generator of a null hypersurface, then ! 1 along within an a ne length 2= j 0j.

Proof Let be the a ne parameter, with = 0 at p. Now

If 0 < 0 the right-hand-side ! 1 when = 2= j 0j, so ! 1 within that a ne length.

Interpretation When < 0 neighboring geodesics are converging. The attractive nature of gravitation (weak energy condition) then implies that they must continue to converge to a focus or a caustic.

Proof Let be the Killing vector s.t. = fl (l Dl = 0) on N for some non-zero function f. Then

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^

Proof Using d =d = 0 and B = 0 in Raychaudhuri's equation.

6.2The Laws of Black Hole Mechanics

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:

6.2.1Zeroth law

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

i.e. J = ( T ) @ is tangent to H+ . It follows that J can be expanded on a basis of tangent vectors to H+

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

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

) is constant on H+.

6.2.2Smarr'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.

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

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

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

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