Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 58 (Black Holes)

Use metric signature $(-,+,+,+)$ and a future-oriented worldline. Variation of the worldline einbein gives the mass-shell condition; variation of momentum givesEliminating $p$ first leaves $L={\dot{x}}^2/(2e)-e{m}^2/2$. Its $e$ equation is ${\dot{x}}^2=-e^2{m}^2$. Choose the positive lapse branch $e=\sqrt{-{\dot{x}}^2}/m$; substituting gives $L=-m\sqrt{-{\dot{x}}^2}$. Thus $\boxed{S=-m\int \sqrt{-g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }}=-m\int d\tau }$, the proper time action. The lapse branch fixes the sign convention. Eliminating auxiliary variables on their algebraic equations preserves the worldline equations, which are timelike geodesics up to parametrization.

Locally write the null hypersurface as $u=0$, $du\ne 0$, with normal ${\ell }_{\mu }=f{\partial }_{\mu }u$, $f\ne 0$. A vector $V$ is tangent exactly when $V(u)=0$. Nullness gives $\ell (u)={\ell }^2/f=0$ on the hypersurface. Hence $\boxed{\ell \in T\mathcal{N}}$: its normal direction lies inside its tangent space.

Put $n=du$ and $q=n_{\mu }{n}^{\mu }$. Torsion freedom gives $n^{\mu }{\nabla }_{\mu }{n}_{\nu }=\frac{1}{2}{\partial }_{\nu }q$. Since $q$ vanishes on $\mathcal{N}$, all tangential derivatives vanish there and $dq=Cdu$ there for a scalar $C$. ConsequentlyThis is the unparametrized null geodesic equation. The integral curves are therefore generators of $\mathcal{N}$. An affine rescaling of a null normal removes the proportionality coefficient locally: if ${\nabla }_{\ell }\ell =\kappa \ell$, set $K=h\ell$ with $\ell (\log h)=-\kappa$.

The future domain of dependence $D^{+}(\Sigma )$ consists of points through which every past-inextendible causal curve meets the partial Cauchy hypersurface $\Sigma$. Data on $\Sigma$ determine evolution there for suitable hyperbolic equations. Its future boundary is the Cauchy horizon $H^{+}(\Sigma )$; one precise definition is $\overline{D^{+}(\Sigma )}\setminus I^{-}(D^{+}(\Sigma ))$.

For a nonextremal Reissner-Nordstrom spacetime, $0<|Q|<M$, the metric function is $F=1-2M/r+Q^2/r^2$, with $r_{\pm }=M\pm \sqrt{M^2-Q^2}$. The outer event horizon lies at $r_{+}$. The future trapped region leads toward the inner null surface $r_{-}$, a future Cauchy horizon for bridge data on the illustrated $\Sigma$. Beyond it some past causal curves arrive from other analytic blocks or timelike singularities without meeting $\Sigma$. The CP diagram repeats these analytic blocks; $r=0$ is a timelike curvature singularity.

Near the inner horizon, arbitrarily late exterior radiation is compressed into finite infaller proper time. Its frequency is exponentially blueshifted, with scale $e^{|{\kappa }_{-}|v}$, so even a decaying flux can have diverging measured stress. Generic ingoing and outgoing perturbations produce mass inflation, growing internal mass and curvature. The smooth RN Cauchy horizon is expected to be unstable to back-reaction. This is a generic instability argument, not a universal theorem from every specially tuned test trajectory; the approximation of a harmless test particle becomes inconsistent.

A null geodesic congruence is a smooth family of null geodesics filling a region without intersections there. For affine tangent $k$, the null expansion is the trace of the optical tensor: $\theta =d(\log A)/d\lambda$ for an infinitesimal pencil's area. Positive expansion means spreading; negative expansion means focusing.

For hypersurface-orthogonal generators in four dimensions, the Null Raychaudhuri equation readsThe null twist vanishes. The null energy condition implies $R_{ab}{k}^a{k}^b\ge 0$ through the Einstein field equations. An initial ${\theta }_0<0$ would force a focal point within affine distance at most $2/|{\theta }_0|$. A future-complete generator cannot develop such a point while remaining on an achronal event horizon. Thus $\boxed{\theta \ge 0}$ under the energy and global predictability/future-completeness hypotheses of Hawking's area theorem. Quantum energy-condition violations or failure of those global assumptions remove the conclusion.

The event horizon is the global boundary $\partial J^{-}({\mathcal{I}}^{+})$ and depends on the whole future. An apparent horizon on a chosen spacelike slice is the outermost marginally outer trapped surface, ordinarily ${\theta }_{\mathrm{out}}=0$, ${\theta }_{\mathrm{in}}<0$. It is quasi-local and slice-dependent; its tube need not be null.

For a shell arriving at $v=0$, the final horizon is $r=2M$. Before arrival, outgoing flat-space rays obey $dr/dv=1/2$. Tracing this horizon backward gives $r=2M+v/2$ until its beginning at $r=0$, $v=-4M$. No apparent horizon exists in that earlier flat region, explicitly illustrating the event horizon's dependence on future collapse.

The qualification concerns gravitational energy. Local matter energy exists: an observer $u$ measures $T_{\mu \nu }{u}^{\mu }{u}^{\nu }$. What is absent is a generally covariant local gravitational stress-energy density with the required universal conservation properties. The equivalence principle allows the connection to vanish at a point in freely falling coordinates; standard gravitational energy expressions depend on coordinates. Curvature does remain, but supplies no unique gravitational energy tensor of the required kind. A nonstationary geometry also lacks a preferred time-translation Killing vector field. For a trial timelike $t$, stress-energy conservation gives ${\nabla }_{\mu }(T^{\mu \nu }{t}_{\nu })=T^{\mu \nu }{\nabla }_{(\mu }{t}_{\nu )}$, not generally zero.

An internal charge instead has a current ${\nabla }_{\mu }{j}^{\mu }=0$ independently of spacetime time translations. Its hypersurface flux is conserved with suitable boundary conditions. For electric charge, the Gauss law expresses it as a surface flux. Observer-dependent charge density does not prevent a well-defined total charge.

For an asymptotically flat spacetime, take asymptotically Cartesian slice coordinates with ${\gamma }_{ij}={\delta }_{ij}+h_{ij}$ and appropriate falloff. In $c=1$ units, the ADM energy isIt is an asymptotic gravitational energy. The dominant energy condition requires $-T^{\mu }{}_{\nu }{u}^{\nu }$ to be future-directed nonspacelike or zero for every future timelike $u$, in particular $T_{\mu \nu }{u}^{\mu }{u}^{\nu }\ge 0$. Together with the constraint equations, completeness and appropriate asymptotic/boundary hypotheses, it yields $E_{\mathrm{ADM}}\ge |{\mathbf{P}}_{\mathrm{ADM}}|\ge 0$. The energy condition alone is not the complete positive-energy theorem.

For a Killing vector field $\xi$, set $K^{\mu \nu }={\nabla }^{\mu }{\xi }^{\nu }=-K^{\nu \mu }$. Choose surface orientation conventions so that the Komar charge, up to overall normalization, isThis is the Komar current from trace-reversed stress-energy construction. The equality is the spacetime Stokes theorem. With $[{\nabla }_{\mu },{\nabla }_{\nu }]V^{\rho }=R^{\rho }{}_{\sigma \mu \nu }{V}^{\sigma }$, the Killing equation and ${\nabla }_{\nu }{\xi }^{\nu }=0$ give $J_{\xi }^{\mu }=R^{\mu }{}_{\nu }{\xi }^{\nu }$. In four dimensions, at zero cosmological constant, the Einstein field equations give the currentReversing orientation changes the common overall sign, not conservation. The contracted Bianchi identity givesAn isometry preserves scalar curvature, and the second term contracts symmetric and antisymmetric tensors. Equivalently stress-energy conservation, the Killing equation, and ${\mathcal{L}}_{\xi }T=0$ cancel the matter expression; its trace derivative needs this last observation. Hence $\boxed{{\nabla }_{\mu }{J}_{\xi }^{\mu }=0}$. Nonzero cosmological constant adds $\Lambda {\xi }^{\mu }$ to the geometric current, also divergence-free.

A Killing horizon is a null hypersurface with a Killing vector field $\xi$ as its normal and generator. Write $\xi =d/d\alpha$ along its orbits. These are null geodesics, but their Killing parameter need not be affine:The coefficient is the surface gravity for the chosen normalization; nondegeneracy means $\kappa \ne 0$. If $\lambda$ is affine and $q=d\lambda /d\alpha$, then $\xi =qd/d\lambda$ and $d(\log q)/d\alpha =\kappa$. For constant $\kappa$, $\lambda ={\lambda }_0+C{e}^{\kappa \alpha }$.

The Killing equation yields ${\partial }_{\mu }({\xi }^2)=2{\xi }^{\nu }{\nabla }_{\mu }{\xi }_{\nu }=-2{\xi }^{\nu }{\nabla }_{\nu }{\xi }_{\mu }$. On the horizon, $\boxed{{\partial }_{\mu }{\xi }^2=-2\kappa {\xi }_{\mu }}$. Multiplying $\xi$ by a positive constant multiplies $\kappa$ by that constant; physical temperature therefore needs a stated time normalization.

A static observer has four-velocity $u=F^{-1/2}{\partial }_t$. Its only nonzero acceleration component is $a^r={\Gamma }^r{}_{tt}/F=F^{\prime }/2=-r/R^2$. Using $g_{rr}=F^{-1}$, the magnitude isIt vanishes only at $r=0$, where $F=1$ and $d\tau =\sqrt{F}dt=dt$. Thus $t$ is proper time for the central inertial static observer. Other static observers need inward acceleration to resist de Sitter separation and have a redshifted proper time.

For the static patch of de Sitter spacetime, $r_{\ast }=\int dr/F=(R/2)\log |(R+r)/(R-r)|$. With $v=t+r_{\ast }$ or $u=t-r_{\ast }$, respectively,Both radial blocks have determinant $-1$ and smooth coefficients at $r=R$. The normal has norm $g^{rr}=F$, so that surface is null. There $k={\partial }_v$ has covector $dr$, or $k={\partial }_u$ has covector $-dr$, making it normal as well as tangent: it is a Killing horizon.

For the signed surface gravity of a de Sitter horizon, since ${\partial }_r{k}^2=-F^{\prime }(R)=2/R$, part a gives $\kappa =-1/R$ in the ingoing chart on the past branch, and $\kappa =+1/R$ in the outgoing chart on the future branch, with $k$ future-directed in the static patch. The physical magnitude is $\boxed{|\kappa |=1/R}$, normalized by central proper time. A negative signed $F^{\prime }(R)/2$ must not become a negative temperature.

In $c=k_B=\hslash =1$ units, the Hawking temperature is $T_H=|\kappa |/(2\pi )$ for the stated time normalization. The inertial central observer sees the de Sitter horizon temperature $T_0=1/(2\pi R)$. The Tolman temperature law states $T\sqrt{-k^2}=\text{constant}$ in static thermal equilibrium. HenceWith part b, $T^2=(a^2+R^{-2})/(4{\pi }^2)$ and $T/(a/(2\pi ))=R/r\to 1$ at the horizon. This tends to the Unruh effect temperature for the increasingly accelerated observer.

Let $s$ be proper distance inward from the horizon. Then $R-r\simeq s^2/(2R)$, $F\simeq s^2/R^2$, givingThe radial factor has Rindler coordinates: $T_{\mathrm{M}}=s\sinh (t/R)$ and $X_{\mathrm{M}}=s\cosh (t/R)$ make it Minkowskian. Fixed $s$ has acceleration $1/s$ and temperature $1/(2\pi s)$, agreeing with the leading behavior. At the center $a=0$ but the finite de Sitter temperature remains; $T=a/(2\pi )$ is only the near-horizon limit here.

With $r=R\sin \chi$, the static metric isThe radial coefficient is regular at $\chi =\pi /2$, but the time coefficient vanishes and these coordinates cease to be a chart. Part c supplies smooth horizon-crossing coordinates; the local Rindler coordinates also prove the degeneracy is a coordinate one. Extending $\chi$ while keeping the same static $t$ alone is not a regular coordinate extension.

The complete analytic extension is the hyperboloid $-Y_0^2+{\sum }_{i=1}^4{Y}_i^2=R^2$. Static coordinates are $Y_0=\sqrt{R^2-r^2}\sinh (t/R)$, $Y_4=\sqrt{R^2-r^2}\cosh (t/R)$, and $(Y_1,Y_2,Y_3)=r\mathbf{n}$. Global coordinates giveWith $\tan \eta =\sinh (T/R)$ this isThe Penrose diagram is a rectangle, with spacelike past/future infinity, regular pole lines $\psi =0,\pi$, and radial null rays at $45$ degrees. Observer horizons divide static patches from inaccessible regions; they are not curvature boundaries.

Axisymmetry preserves the horizon, so the axial Killing vector field $m$ is tangent to it, so its null normal $\xi =k+\Omega m$ obeys $\xi \cdot m=0$ and ${\xi }^2=0$. Therefore $k\cdot m=-\Omega m^2$ andThus $\boxed{k^2={\Omega }^2{m}^2}$ there. A generator has $dt/d\alpha =1$, $d\phi /d\alpha =\Omega$, making $\Omega =d\phi /dt$ its angular velocity relative to the nonrotating stationary frame at infinity. For Kerr it is the Kerr horizon angular velocity $a/(r_{+}^2+a^2)$; the argument itself is not Kerr-specific.

For nonzero $a$, $\Sigma =0$ requires $r=0$, $\theta =\pi /2$. Oblate Cartesian coordinates satisfy $x^2+y^2=(r^2+a^2){\sin }^2\theta$, $z=r\cos \theta$, so this is the Kerr ring singularity of radius $|a|$. Curvature components and generic invariants diverge there. The whole $r=0$ surface is not singular: away from the ring it is a disk through which the extension continues to a sheet with $r<0$. This oblate radial parameter is not a nonnegative Euclidean distance.

In $G=c=1$ units, $\boxed{J=Ma}$. Horizon candidates solve $\Delta =0$, giving $r_{\pm }=M\pm \sqrt{M^2-a^2}$. For $J^2>M^4$, equivalently $|a|>M$, there are no real roots. The over-rotating Kerr solution has no event horizon hiding its naked ring singularity. The term black hole in this regime names the Kerr family, not a hidden-singularity black hole.

The periodic axial orbits have norm $m^2=g_{\phi \phi }$. On the equator,Thus a neighborhood on the negative sheet has $m^2<0$. A fixed-$(t,r,\theta )$ curve running once around periodic $\phi$ is a closed timelike curve: its timelike tangent returns to the same event.

No horizon in the over-rotating solution prevents passage from positive-$r$ infinity through the nonsingular disk to this region. Timelike periodic axial orbits provide the time-machine mechanism. This property of the analytic solution does not establish that such a naked geometry can be manufactured stably from generic regular collapse, as envisaged by the weak cosmic censorship conjecture.

An ergoregion of a stationary spacetime is where the stationary Killing vector field, normalized as time translation at infinity, is spacelike; a fixed-coordinate stationary observer cannot remain timelike there. For the Kerr metric, $k^2=-(1-2Mr/\Sigma )$, so the outer stationary-limit surface isFor a subextremal rotating hole $r_{\mathrm{E}}\ge r_{+}=M+\sqrt{M^2-a^2}$, with equality only at the poles. At the equator $r_{\mathrm{E}}=2M>r_{+}$. The ergosphere touches the horizon on the rotation axis and is outside it elsewhere. Its boundary is not the horizon, whose generator is $k+{\Omega }_{H}m$.

The drawing uses oblate Cartesian coordinates to display the intersection and is not an isometric embedding of the spatial geometry.

Choose spin orientation so that $0<a<M$; the printed $M/a$ threshold assumes nonzero positive $a$. Put $H=2Mr/(r^2+a^2)$ and $F=1-H$. The axial metric has components $(-F,H,1+H)$ and determinant $-1$. Conserved energy and timelike normalization giveEliminating $\dot{\tilde{t}}$ gives the Kerr axial timelike effective potentialIt extends through the zeros of $F$ in the regular original coordinates. Since $V_{\mathrm{eff}}(0)=1$, reaching zero requires ${\epsilon }^2\ge 1$. At equality the infalling orbit reaches zero with zero radial velocity and turns back, because the nearby negative sheet is forbidden. Along the axis zero is regular, unlike the off-axis ring.

For $r<0$, $V_{\mathrm{eff}}^{\prime }=2M(r^2-a^2)/(r^2+a^2{)}^2$ and the global maximum is $V_{\mathrm{eff}}(-a)=1+M/a$. If ${\epsilon }^2>1+M/a$, an inward particle has no negative-sheet turning point. It passes through zero and continues toward $\boxed{r\to -\infty }$, another asymptotically flat end, with $\dot{r}\to -\sqrt{{\epsilon }^2-1}$ and infinite proper time to infinity. It first crosses the outer and inner horizons. Perturbative Cauchy-horizon instability can invalidate this ideal Kerr axial analytic extension physically.

This is the two-dimensional axis restriction. The dashed $r=0$ curve is regular; the four-dimensional ring lies off-axis and is not an axial singular boundary.

The laws of black-hole mechanics establish the classical analogy with thermodynamics. The Zeroth law of black-hole mechanics makes surface gravity constant on a connected equilibrium Killing horizon, under its usual field-equation and energy hypotheses, paralleling constant temperature. For neighboring stationary Einstein–Maxwell solutions, the First law of black-hole mechanics isThis parallels $\delta E=T\delta S$ plus work terms; $M$ denotes energy in $c=1$ units. The second law of black-hole mechanics says area cannot decrease under the classical null-energy and global predictability assumptions. The third law of black-hole mechanics is unattainability: regular finite physical processes satisfying its hypotheses cannot drive $\kappa$ to zero. It parallels unattainability of absolute zero, not a universal assertion of zero extremal entropy.

Classically a hole absorbs without emitting, so the analogy alone does not identify a measured temperature. Quantum Hawking radiation supplies $T_H=\hslash \kappa /(2\pi )$, with $c=k_B=1$ and future-horizon normalization at infinity. Comparing the area terms in the first laws gives the Bekenstein-Hawking entropyup to a conventionally fixed additive constant. In ordinary units it is $k_B{c}^{3}A/(4G\hslash )$. Entropy scales with area, not interior volume.

For the massless scalar in quantum field theory in curved spacetime, global hyperbolicity ensures well-posed Cauchy evolution. The conserved Klein-Gordon inner product isIts current has zero divergence by the wave equation, making it slice-independent with appropriate boundary behavior. Positive frequency relative to the asymptotic past and future Minkowski times defines complete bases $u_j,v_i$, with $(u_i,u_j)={\delta }_{ij}$, $(u_i^{\ast },u_j^{\ast })=-{\delta }_{ij}$ and vanishing mixed products. Expand the field asContinuous labels replace sums by integrals; wave packets avoid artificial normalization infinities. Particle creation by a nonstationary spacetime occurs when evolution mixes frequency signs through a Bogoliubov transformation:The inner product gives $\alpha {\alpha }^{†}-\beta {\beta }^{†}=I$ and $\alpha {\beta }^{\mathsf{T}}=\beta {\alpha }^{\mathsf{T}}$. For the initial vacuum annihilated by all $a_j^{\mathrm{in}}$, $\boxed{⟨N_i^{\mathrm{out}}{⟩}_{\mathrm{in}}=\sum _j|{\beta }_{ij}{|}^2}$. Thus nonzero negative-frequency mixing makes the in-vacuum non-vacuum for final observers. The vacuum state in a stationary spacetime depends on its positive-frequency splitting; this is not inconsistent with deterministic mode evolution. A unitary implementation on one fixed infinite-mode Fock space further needs the Hilbert–Schmidt condition on $\beta$.

For collapse forming a Schwarzschild black hole, a late outgoing ray traced back to past null infinity obeys the Hawking exponential ray mapPulling $e^{-i\omega u}$ back gives $(U_H-U{)}^{i\omega /\kappa }$ for $U<U_H$. Its past Fourier transform has both frequency signs. For $y=\omega /\kappa$ and damping $ϵ>0$, the Gamma function evaluates the Fourier integrals asAs $ϵ\downarrow 0$, the two denominator arguments approach $\mp \pi /2$, so $|I_{+}|/|I_{-}|=e^{-\pi y}$. Thus the thermal ratio of Hawking Bogoliubov coefficients is $|\beta {|}^2/|\alpha {|}^2=e^{-2\pi \omega /\kappa }$. Together with the bosonic normalization difference, this yields $1/(e^{2\pi \omega /\kappa }-1)$. Potential scattering adds a greybody factor:The collapse state has outgoing flux at future infinity without an incoming thermal bath and is regular for infall. It is not the eternal-hole equilibrium state. Horizon-entering modes complete the future basis; tracing over their correlated partners gives the approximately thermal exterior state. The collapsing geometry is not literally Minkowskian everywhere in the far future: asymptotically flat null-infinity modes are the relevant application of the earlier in/out construction.

Emission reduces the isolated hole's mass. Schwarzschild temperature is proportional to $M^{-1}$, giving the negative heat capacity of a Schwarzschild black hole: losing energy makes it hotter. Dimensional estimates give luminosity proportional to $M^{-2}$ and evaporation time proportional to $M^3$, with species and greybody factors determining coefficients. Area may shrink because quantum stress violates the energy assumptions of Hawking's area theorem. The generalized second law instead concerns $S_{\mathrm{gen}}=A/(4G\hslash )+S_{\mathrm{outside}}$, incorporating radiation entropy and preventing ordinary thermodynamic violations by disposal of entropy into a hole.

Pair correlations also distinguish thermal reduced states from the complete pure state. The black hole information paradox asks whether complete evaporation preserves quantum information: an exactly thermal final exterior with no remaining partners would appear inconsistent with unitary pure-state evolution. The semiclassical calculation controls late-time flux, not the Planck-scale endpoint, and does not itself settle this question. Quantum emission gives the temperature–entropy identification physical meaning; generalized entropy replaces the classical area alone when radiation back-reacts.