P.K.Townsend - Black Holes

is RN metric in isotropic coordinates (t; ; ; ). As in Q = 0 case, there are two values for every value of r > r+, but is complex for r < r+.

r

.

This new metric covers two isometric regions (I&IV) exchanged by the geometry.

The xed points set at = M2 Q2=2 (i.e. r = r+) is a minimal

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The distance to the horizon at r = r+ along a curve of constant t; ; from r = R is

so the ER bridge separating regions I & IV becomes in nitely long in the limit as jQj ! M. In this limit, the spatial sections look like:

.

.

. 1 as jQj ! M

..

iii)M = jQj `Extreme' RN (r = M)

This is non-singular on the null hypersurface r = M.

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Proposition r = M is a degenerate (i.e. surface gravity = 0) Killing horizon of the Killing vector eld k = @=@v.

Proof From the previous calculation l = f@=@v so r = M is a Killing horizon of k, and k Dk = 0 when r+ = r = M.

Since the orbits of k on r = M are a nely parameterized they must go to in nite a ne parameter in both directions ) internal 1. This is the same internal 1 that we nd down the in nite ER bridge.

Note that k is null on r = 2M, but timelike everywhere else, so region II has disappeared and region I now leads directly to region V. The CP diagram is

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3.4.1Nature of Internal 1 in Extreme RN

The asymptotic metric as r ! 1 is Minkowski. To determine the asymptotic metric as r ! M we introduce the new coordinate by r = M(1 + ) and keep only the leading terms in , to get

of radius M

This is the Robinson-Bertotti metric. It is a kind of `Kaluza-Klein' vacuum in which two directions are compacti ed and the `e ective'

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spacetime is the two-dimensional `anti-de Sitter' (adS2) spacetime of constant negative curvature. (See Q.II.7).

3.4.2Multi Black Hole Solutions

yields the metric for N extreme black holes of masses Mi at positions xi. Note that the `points' xi are actually minimal 2-spheres. There are no-function singularities at x = xi because the lines of force continue inde - nitely into the asymptotically RB regions (`charge without charge').

Note that a static multi black hole solution is possible only when there is an exact balance between the gravitational attraction and the electrostatic repulsion. This occurs only for M = jQj.

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Chapter 4

Rotating Black Holes

4.1Uniqueness Theorems

4.1.1Spacetime Symmetries

De nition An asymptotically at spacetime is stationary if and only if there exists a Killing vector eld, k, that is timelike near 1 (where we may normalize it s.t. k2 ! 1).

i.e. outside a possible horizon, k = @=@t where t is a time coordinate. The general stationary metric in these coordinates is therefore

A stationary spacetime is static at least near 1 if it is also invariant under time-reversal. This requires g0i = 0, so the general static metric can be written as

for a static spacetime outside a possible horizon.

De nition An asymptotically at spacetime is axisymmetric if there exists a Killing vector eld m (an `axial' Killing vector eld) that is spacelike near 1 and for which all orbits are closed.

where is a coordinate identi ed modulo 2 , such that m2=r2 ! 1 as r ! 1. Thus, as for k, there is a natural choice of normalization for an axial Killing vector eld in an asymptotically at spacetime.

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Birkho 's theorem says that any spherically symmetric vacuum solution is static, which e ectively implies that it must be Schwarzschild. A generalization of this theorem to the Einstein-Maxwell system shows that the only spherically symmetric solution is RN.

But suppose we know only that the metric exterior to a star is static. Unfortunately static 6)spherical symmetry. However, if the `star' is actually a black hole we have:

Israel's theorem If (M; g) is an asymptoticallyat, static, vacuum spacetime that is non-singular on and outside an event horizon, then (M; g) is Schwarzschild.

Even more remarkable is the:

Carter-Robinson theorem If (M; g) is an asymptoticallyat stationary and axi-symmetric vacuum spacetime that is non-singular on and outside an event horizon, then (M; g) is a member of the two-parameter Kerr family (given later). The parameters are the mass M an the angular momentum J.

The assumption of axi-symmetry has since been shown to be unnecessary, i.e. for black holes, stationarity ) axisymmetry (Hawking, Wald).

Stationarity , equilibrium, so we expect the nal state of gravitational collapse to be a stationary spacetime. The uniqueness theorems say that if the collapse is to a black hole then this spacetime is uniquely determined by its mass and angular momentum (cf. state of matter in thermal equilibrium). Thus, all multipole moments of the gravitational eld are radiated away in the collapse to a black hole, except the monopole and dipole moments (which can't be radiated away because the graviton has spin 2).

These theorems can be generalized to `vacuum' Einstein-Maxwell equations. The result is that a stationary black hole spacetimes must belong to the 3-parameter Kerr-Newman family. In Boyer-Linquist coordinates the KN metric is

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where Q and P are the electric and magnetic (monopole) charges, respectively. The Maxwell 1-form of the KN solution is

Remarks

(i)When a = 0 the KN solution reduces to the RN solution.

(ii)Taking ! e ectively changes the sign of a, so we may choose a 0 without loss of generality.

(iii)The KN solution has the discrete isometry

4.2The Kerr Solution

The Kerr metric is important astrophysically since it is a good approximation to the metric of a rotating star at large distances where all multipole moments except l = 0 and l = 1 are unimportant. The only known solution of Einstein's equations for which Kerr is exact for r > R is the Kerr solution itself (for which T = 0), i.e. it has not been matched to any known non-vacuum solution that could represent the interior of a star, in contrast

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to the Schwarzschild solution which is guaranteed by Birkho 's theorem to be the exact exterior spacetime that matches on to the interior solution for any spherically symmetric star.

The Kerr metric in BL coordinates has coordinate singularities at

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In these coordinates the metric is

which shows that the spacetime is at (Minkowski) when M = 0.

The surfaces of constant t;~ r are confocal ellipsoids which degenerate at r = 0 to the disc z = 0; x2 + y2 a2.

.

= =2 corresponds to the boundary of the disc at x2 + y2 = a2 so the curvature singularity occurs on the boundary of the disc, i.e. on the `ring'

There is no reason to restrict r to be positive. The spacetime can be analytically continued through the disc to another asymptotically at region with r < 0.

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