Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 311 / 1

a

An isolated collapse emits transient matter and gravitational waves. The exterior is expected to settle to an asymptotically flat stationary spacetime. Because the star is uncharged, the black-hole uniqueness theorem identifies the regular final state with a Kerr black hole. Its higher multipole moments are fixed by its conserved charges: the black-hole no-hair theorem leaves only the total mass

M

J

. Thus the late-time spacetime is characterized by $M$ and $J$ .

Factor

f (r) = (1 - \frac{r _{+}^{2}}{r ^{2}}) (1 - \frac{r _{-}^{2}}{r ^{2}}), r_{\pm}^{2} = M \pm M^{2} - Q^{2} .

Introduce

v = t + r_{*}

with

d r_{*} / d r = f^{- 1}

. Since

d t = d v - d r / f

d s^{2} = - f (r) d v^{2} + 2 d v d r + r^{2} d Ω_{3}^{2}, d Ω_{3}^{2} = d θ^{2} + sin^{2} θ d ϕ^{2} + cos^{2} θ d ψ^{2} .

(v, r)

- 1

, so the metric and its inverse are nonsingular at

r = r_{+}

. For

M \geq Q

r_{+}

is real and the metric extends analytically through it. The original potential becomes

A = - Q d v / r^{2} + Q d r / (r^{2} f)

; the radial term is locally exact, and a gauge transformation gives the horizon-regular potential

A = - Q d v / r^{2}

k = \partial_{t} = \partial_{v}

, with

k^{2} = - f

. The normal to

r = constant

has squared norm

g^{rr} = f

, so

r = r_{+}

k

is both tangent and normal. It is therefore a Killing horizon.

For a static metric of this form, the surface gravity is

κ = f^{'} (r_{+}) /2

. Since

r_{+}^{2} r_{-}^{2} = Q^{2}

and

r_{+}^{2} + r_{-}^{2} = 2 M

κ = \frac{r _{+}^{2} - r _{-}^{2}}{r _{+}^{3}} = \frac{2 M ^{2} - Q ^{2}}{r _{+}^{3}} .

M = Q

, the horizons coincide and $κ = 0$ .

The zeros of

f

If $M > Q$ , there are two simple horizons $r_{+} > r_{-} > 0$ . The maximally extended Penrose diagram is the repeating subextremal charged-black-hole diagram: $r = r_{+}$ is an event horizon, $r = r_{-}$ is a Cauchy horizon, and $r = 0$ is a timelike singularity.
If $M = Q$ , the roots coincide. The extremal diagram has one degenerate horizon, no bifurcation surface, and an infinite throat between the exterior and interior.
If $M < Q$ , $f$ has no real zero. The timelike $r = 0$ singularity is visible from infinity, so there is a naked singularity and no black-hole region.