4.1 Uniqueness Theorems

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4.1.1 Spacetime 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

ds2 = g00(~x)dt2 + 2g0i(~x)dt dxi + gij(~x)dxi dxj (4.1)

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

ds2 = g00(~x)dt2 + gij(~x)dxi dxj (4.2)

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.

We can choose coordinates such that

m =

@

@_

(4.3)

where _ is a coordinate identi_ed modulo 2_, such thatm2=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 asymptotically-at, 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 asymptotically-at 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

ds2 = 􀀀

􀀀

_ 􀀀 a2 sin2 _

_

_

dt2 􀀀 2a sin2 _

􀀀

r2 + a2 􀀀 _

_

_

dt d_

+

􀀀

r2 + a2_2

􀀀 _a2 sin2 _

_

!

sin2 _d_2 +

_

_

dr2 + _d_2

(4.4)

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where

_ = r2 + a2 cos2 _

_ = r2 􀀀 2Mr + a2 + e2 (4.5)

The three parameters are M, a, and e. It can be shown that

a =

J

M

(4.6)

where J is the total angular momentum, while

e =

p

Q2 + P2 (4.7)

where Q and P are the electric and magnetic (monopole) charges, respectively.

The Maxwell 1-form of the KN solution is

A =

Qr

􀀀

dt 􀀀 a sin2 _d_

_

􀀀 P cos _

_

adt 􀀀

􀀀

r2 + a2_

d_

_

_

(4.8)

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

t ! 􀀀t; _ ! 􀀀_ (4.9)

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