Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 72 (General Relativity)

Use sequential semicolon indices: $X^a{}_{;kl}={\nabla }_l({\nabla }_k{X}^a)$, where the second covariant derivative also differentiates the lower index of ${\nabla }_k{X}^a$. For a torsion-free connection, direct expansion givesSubtract the expression with $k,l$ exchanged. Commuting partial derivatives cancel the first term; the first-derivative terms cancel in pairs; symmetry of the lower Christoffel symbol indices cancels both terms containing ${\Gamma }^p{}_{kl}$. What remains isThis is the Ricci identity with the curvature convention printed on the PDF's first page. In terms of derivative operators it is $[{\nabla }_l,{\nabla }_k]X^a=R^a{}_{bkl}{X}^b$; writing the operator order explicitly avoids reversing the sign.

The lower-index rule follows by applying the commutator to the scalar contraction $Y_a{X}^a$. Its second derivatives commute, so the product rule implies $Y_{a;kl}-Y_{a;lk}=-R^p{}_{akl}{Y}_p$. The derivative commutator is a derivation on tensor products, since the cross terms cancel. It therefore acts once on every index, with a positive sign on each upper index and a negative sign on each lower index. For an arbitrary tensor of valence $(r,s)$,Local sums of decomposable tensors establish the formula for every tensor field, rather than only for a single tensor product.

The first derivative of a scalar is a covector, soCommuting partial derivatives and the torsion-free connection give ${\phi }_{;lm}={\phi }_{;ml}$, the symmetric Hessian of a scalar field. Its next covariant derivative isIn particular, tensor index symmetrization over $l,m$ adds nothing: ${\phi }_{;(lm)n}={\phi }_{;lmn}$. This does not make the third index interchangeable with the first two. Applying the lower-index Ricci identity to the covector ${\phi }_{;l}$ yields the requested tensor index antisymmetrization:Parentheses and brackets include the usual averaging factors. To make the remaining symmetry explicit, let $S_{lmn}={\phi }_{;(lmn)}$. Since the first two derivative indices are already symmetric,The two commutator identities give ${\phi }_{;lnm}={\phi }_{;lmn}+R^p{}_{lmn}{\phi }_{;p}$ and ${\phi }_{;mnl}={\phi }_{;lmn}+R^p{}_{mln}{\phi }_{;p}$. Thus the third covariant derivatives of a scalar can also be writtenThe right-hand side displays exactly the obstruction to total symmetry. At a point where the scalar gradient vanishes, or where these curvature contractions vanish, the third derivative is totally symmetric.

For the final contraction use $R_{ab}=R^c{}_{acb}$. The arbitrary-valence formula givesThe Ricci tensor of the Levi-Civita connection is symmetric, whereas the last factor is antisymmetric. Their contraction is zero, provingNo symmetry of $T^{ik}$ was assumed. This double divergence of a contravariant tensor identity uses the metric-compatible connection of general relativity: zero torsion alone for an arbitrary affine connection does not guarantee a symmetric Ricci tensor and would not suffice for this final step.

The Pound-Rebka experiment measures a small gravitational redshift by comparing nuclear emission and resonant absorption at different heights in Earth's field. The $14.4\mathrm{keV}$ transition of iron-57 supplies the gamma-ray frequency standard. The Mössbauer effect permits recoil-free emission and absorption by nuclei bound in solids, giving a sufficiently narrow resonance. A source and absorber separated vertically by about $22.6\mathrm{m}$ have a predicted fractional frequency difference of order $2.5\times 10^{-15}$. A calibrated Doppler effect from slow source motion measures the shift through its effect on resonant absorption. Interchanging the upper and lower arrangements reverses the gravitational contribution and helps separate it from intrinsic source-absorber offsets; temperature-dependent frequency shifts must also be controlled. The observed change agrees with the weak-field prediction.

The prediction follows either from an accelerating laboratory and the equivalence principle, or directly from a static metric. With $x^0=ct$, write $g_{00}=N^2\simeq 1+2\Phi /c^2$. The time-translation Killing vector field gives a conserved photon energy $E_{\xi }=p_a{\xi }^a$, while a static observer has four-velocity $u^a={\xi }^a/N$ and measures $h\nu =E_{\xi }/N$. ThusFor an upward ray over height $H$, $\Delta \nu /\nu \simeq -gH/c^2$; downward propagation has the opposite sign. The equivalent compensating Doppler speed has magnitude $gH/c$, about $7.4\times 10^{-7}\mathrm{m}{\mathrm{s}}^{-1}$ for this baseline. Identical clocks at rest have $d\tau =Ndt$, so a clock at higher potential runs faster relative to the common static coordinate time. This does not mean that the nuclear transition changes its locally measured frequency with height: the comparison involves transporting a photon between different local proper-time standards.

The weak equivalence principle asserts universal free fall for test bodies. The Einstein equivalence principle adds local Lorentz invariance and local position invariance for nongravitational experiments in a freely falling laboratory. The redshift comparison tests the clock aspect of local position invariance and supports a metric description of gravity. The strong equivalence principle extends these requirements to self-gravitating bodies and local gravitational experiments, including the contribution of gravitational binding energy. Their outcomes should be independent of the external location and motion of a freely falling laboratory when tidal effects are negligible. A nuclear redshift experiment is therefore not a test of the whole strong equivalence principle, and its success does not uniquely establish the gravitational field equations.

The geometrical realization uses the Levi-Civita connection of a spacetime metric. At any event, normal coordinates make $g_{ab}={\eta }_{ab}$ and the Christoffel symbols vanish. Freely falling test particles follow geodesics; local nongravitational equations take their special-relativistic form at that event. The Riemann curvature tensor need not vanish, so finite laboratories can measure tidal geodesic deviation. The strong principle concerns removing a uniform external gravitational field locally, not removing curvature throughout an extended region.

For a perfect fluid, local rest-frame isotropy and the absence of heat flux or viscosity give $T^{\hat{a}\hat{b}}=\mathrm{diag}(\rho ,p,p,p)$, where $\rho$ is rest-frame energy density. With the PDF's signature $(+---)$, $u^a{u}_a=1$ and $h^{ab}=g^{ab}-u^a{u}^b$ projects spatially. Its covariant form isThe pressure sign follows from the negative spatial metric; it gives positive diagonal stresses in the rest frame. Local stress-energy conservation, ${\nabla }_a{T}^{ab}=0$, produces the two projectionsThe first is the energy equation, including pressure work; the second is the relativistic Euler equation. Its rest-frame spatial pressure force is $-\nabla p$.

Further assumptions are needed to derive the Einstein field equations. Take a local generally covariant metric equation, linear in curvature and of second differential order, and demand compatibility with stress-energy conservation. The available rank-two tensors are $R_{ab}$, $R{g}_{ab}$ and a constant multiple of $g_{ab}$. Defining $R_{ab}=R^c{}_{acb}$, the contracted Bianchi identity gives ${\nabla }^a{R}_{ab}=\frac{1}{2}{\nabla }_{b}R$. Hence the combination $R_{ab}+\alpha R{g}_{ab}$ is identically divergence-free only when $\alpha =-\frac{1}{2}$. This determines the Einstein tensor $G_{ab}=R_{ab}-R{g}_{ab}/2$; a cosmological constant term remains allowed.

The coupling sign must be matched to the actual curvature convention. Here $R^a{}_{bcd}$ represents $[{\nabla }_d,{\nabla }_c]$, and a static weak field has $R_{00}\simeq -{\nabla }^2\Phi /c^2$. With negligible pressure and cosmological term, a trial equation $G_{ab}=C{T}_{ab}$ implies $R_{00}\simeq C{\rho }_{\mathrm{mass}}{c}^2/2$. Comparing with the Poisson equation ${\nabla }^2\Phi =4\pi G{\rho }_{\mathrm{mass}}$ fixes $C=-8\pi G/c^4$. Thus the Einstein equation in the sequential-derivative curvature convention isEquivalently, $R_{ab}=\Lambda g_{ab}-(8\pi G/c^4)(T_{ab}-T{g}_{ab}/2)$, with $T=\rho -3p$. Pressure therefore gravitates as well as supplying a material force. The cosmological constant is not fixed by the local Newtonian matching.

Setting $c=1$ for the action normalization, the same metric equation follows from the Einstein-Hilbert action $S_g=(2\kappa {)}^{-1}\int \sqrt{-g}(R-2\Lambda )d^{4}x$, where $\kappa =8\pi G$, together with a matter action obeying $\delta S_m=\frac{1}{2}\int \sqrt{-g}{T}_{ab}\delta g^{ab}{d}^{4}x$. After the metric boundary term is removed or supplied separately, the bulk variation is proportional to $G_{ab}+\Lambda g_{ab}+\kappa T_{ab}$. This also makes the sign convention explicit. General covariance, the field equation and the contracted Bianchi identity are mutually consistent with the fluid conservation laws. The experiment motivates the local metric description; the curvature and conservation assumptions, plus the Newtonian limit, supply the additional route to the field equation.

Let $Q=g^{ab}{\chi }_{,a}{\chi }_{,b}$ and $\mathcal{L}=p(ϵ)+n(Q-{ϵ}^2)/(2ϵ)$. Vary $n,ϵ,\chi$ independently, taking compactly supported variations. The multiplier variation is $\partial \mathcal{L}/\partial n=(Q-{ϵ}^2)/(2ϵ)=0$. With $u_a={\chi }_{,a}/ϵ$, it gives the unit four-velocity condition $u^a{u}_a=1$. The enthalpy variation isUsing $Q={ϵ}^2$ yields $p^{\prime }(ϵ)=n$. Finally,The integration by parts boundary term vanishes. Thus the three matter equations areThe last is particle number conservation in a relativistic fluid. Off shell $n,ϵ$ are independent variational fields; on shell the thermodynamic relation ties them to the specified function $p(ϵ)$.

For the Hilbert stress-energy tensor, keep these matter fields fixed while varying the inverse metric. The volume variation is $\delta \sqrt{-g}=-\frac{1}{2}\sqrt{-g}{g}_{ab}\delta g^{ab}$, and $\delta Q={\chi }_{,a}{\chi }_{,b}\delta g^{ab}$. ThereforeOn shell, $\mathcal{L}=p$ and $nϵ=\rho +p$. With signature $(+---)$,For variation of the covariant metric the same definition reads ${\delta }_{g}S=-\frac{1}{2}\int \sqrt{-g}{T}^{ab}\delta g_{ab}{d}^{4}x$. In a local rest frame the tensor has entries $\mathrm{diag}(\rho ,p,p,p)$, which checks both the energy density and pressure sign.

Define a derivative along the flow by $\dot{F}=u^a{\nabla }_{a}F$, and set $\theta ={\nabla }_a{u}^a$, $a^b=u^a{\nabla }_a{u}^b$. The number equation immediately gives $\dot{n}+n\theta =0$. The specific enthalpy per particle relation and $dp=ndϵ$ giveConsequently the energy equation follows from the number equation: $\dot{\rho }=ϵ\dot{n}=-(\rho +p)\theta$.

The velocity equation follows directly from the potential, rather than requiring an independent assumption about the fluid force. Since $ϵu_a={\nabla }_a\chi$ and the connection has zero torsion, the canonical vorticity of a relativistic fluid vanishes:Contract with $u^a$ and use $u^a{u}_a=1$ and $u^a{\nabla }_b{u}_a=0$. This gives $ϵa_b={\nabla }_bϵ-u_b\dot{ϵ}$. Multiplication by $n$ and use of ${\nabla }_{b}p=n{\nabla }_bϵ$ produces the relativistic Euler equation. The evolution system is thereforeIt is equivalent to stress-energy conservation together with number conservation, subject to the potential-flow restriction. The action describes an isentropic barotropic sector of a perfect fluid in general relativity; it does not provide general vortical or entropy-carrying fluid motions. The constraint on canonical vorticity is stronger than the Euler equation alone.

For the weak static metric, use $c=1$. To the retained order, $\sqrt{-g}=1+\varphi$, $u^0=1-\varphi +O({\varphi }^2,|\mathbf{v}{|}^2)$ and $u^i=v^i$. The correction to $u^0$ is required by normalization, even though its zeroth-order value is one. The relevant Christoffel symbols arewith no spatial connection at this order. The cancellation $\sqrt{-g}{u}^0=1+O({\varphi }^2,|\mathbf{v}{|}^2)$ makes the leading number equation particularly simple:The corresponding leading energy transport equation is ${\partial }_t\rho +\mathbf{v}\cdot \nabla \rho +(\rho +p)\nabla \cdot \mathbf{v}=0$. The spatial Euler equation contains the gravitational acceleration $-\nabla \varphi$ and pressure force $-\nabla p/(\rho +p)$. In a strict first-order expansion, convective velocity acceleration and products of the small quantities are discarded.

To exhibit a fully linear system, perturb a homogeneous rest state: $n=n_0+\delta n$, $\rho ={\rho }_0+\delta \rho$, $p=p_0+\delta p$, and $ϵ={ϵ}_0+\delta ϵ$, with the perturbations, $\mathbf{v}$ and $\varphi$ all first order. Let $w_0={\rho }_0+p_0=n_0{ϵ}_0$. The weak-field slow-motion equations for a relativistic fluid areThe equation of state gives $\delta \rho ={ϵ}_0\delta n$ and $\delta p=n_0\delta ϵ$. Where $p^{\prime \prime }({ϵ}_0)\ne 0$, it also gives the barotropic relativistic sound speedWithout this invertibility, the primitive conservation equations remain valid and this sound-speed parametrization must not be used.

The potential furnishes a further check on the pressure and gravity signs. Write $\chi ={ϵ}_{0}t+\delta \chi$. Its time and spatial components implyTheir compatibility reproduces the linear Euler equation. Pressure gradients oppose gravitational acceleration, and hydrostatic balance is $\nabla \delta p=-w_0\nabla \varphi$. Combining the energy and Euler equations gives the forced acoustic equationFor positive $c_s^2$, disturbances propagate as sound waves; the prescribed potential supplies a static forcing. A nonrelativistic equation of state gives $p\ll \rho \simeq mn$, $ϵ\simeq m$, and reduces the inertia $\rho +p$ to mass density, recovering the ordinary linearized continuity and Euler equations. Slow velocity alone does not justify dropping $p$ from the inertia of a relativistically hot fluid. The given metric supplies an external gravitational potential; the matter action by itself supplies no Poisson equation for it.

A horizon separates regions according to a specified causal, dynamical or geometrical criterion. The criterion must be stated: an event horizon, an apparent horizon, a Killing horizon and a Cauchy horizon answer different questions. Some horizons describe a black hole, others an observer's acceleration or the universe's expansion. A vanishing coordinate metric component alone does not establish any of these definitions.

In an asymptotically flat spacetime, choose the relevant future null infinity ${\mathcal{I}}^{+}$, the endpoint of escaping outgoing light rays. The black-hole region and its future event horizon areHere $J^{-}$ is the causal past. Events in $\mathcal{B}$ cannot send a causal signal to the chosen infinity. This is a global definition: identifying the horizon requires the full future development, so a local measurement need not reveal whether an event lies on it. Smooth portions of an event horizon are null hypersurfaces generated by null geodesics. Generators may join at past caustics or crease sets, so the horizon need not be smooth everywhere. A past event horizon is defined by the time-reversed escape criterion; in the maximally extended Schwarzschild solution it bounds a white-hole region.

An observer-based definition replaces infinity by a future-inextendible worldline $\gamma$. Its future observer event horizon is the boundary of events that can signal to $\gamma$, conventionally $\partial I^{-}(\gamma )$. A past observer horizon similarly bounds events reachable from its past worldline. Such horizons depend on the observer and need not imply a black hole.

A Killing horizon is a null hypersurface to which a nonzero Killing vector field $K^a$ is normal. On it, $K^a$ is tangent to its null generators and satisfies $K^b{\nabla }_b{K}^a=\kappa K^a$. The surface gravity $\kappa$ measures the failure of this Killing parameter to be an affine parameter; rescaling $K$ rescales $\kappa$. A degenerate Killing horizon has $\kappa =0$. Stationary black-hole horizons often are Killing horizons, whereas a general evolving event horizon need not have any Killing symmetry. The condition $K^a{K}_a=0$ alone is insufficient: the null vector must also be normal to a null hypersurface.

The Schwarzschild spacetime illustrates several of these notions. In units $G=c=1$, let $f=1-2M/r$. Its metric is $d{s}^2=fd{t}^2-f^{-1}d{r}^2-r^{2}d{\Omega }^2$, and $r=2M$ is the future event horizon of the black-hole region in the appropriate extension. The apparent coordinate singularity disappears in ingoing Eddington-Finkelstein coordinates, $v=t+r_{\ast }$, $d{r}_{\ast }/dr=f^{-1}$:At $r=2M$ this metric is regular and the normal $dr$ is null. Future outgoing radial rays satisfy $dr/dv=f/2$; inside the horizon their areal radius decreases even on the outgoing family. The stationary Killing vector field ${\partial }_v$ generates the horizon with $\kappa =f^{\prime }(2M)/2=1/(4M)$ when normalized at infinity. The Kretschmann scalar is $48M^2/r^6$, finite at the horizon and divergent at $r=0$. Thus the event horizon is not the curvature singularity. The photon sphere at $r=3M$, which supports circular null orbits, is also not the horizon.

Quasi-local horizon notions use null expansions. For a closed spacelike two-surface, choose future outgoing and ingoing null normals $\ell ,n$, with $\ell \cdot n=1$ in signature $(+---)$. Each expansion is the logarithmic rate of its positive area element along the corresponding null normal. A future trapped surface has ${\theta }_{\ell }<0$ and ${\theta }_n<0$: both future light fronts initially shrink. A marginally outer trapped surface has ${\theta }_{\ell }=0$; for the future black-hole type also require ${\theta }_n<0$. On a chosen spacelike slice, the outer boundary of the future trapped region is an apparent horizon, ordinarily an outermost marginally outer trapped surface when regularity permits this characterization. It depends on the slice, and a slicing need not display an apparent horizon even when the global spacetime contains an event horizon.

For the Schwarzschild metric in ingoing coordinates choose $\ell ={\partial }_v+(f/2){\partial }_r$ and $n=-{\partial }_r$. The spherical null expansion in ingoing coordinates givesThe round spheres inside $2M$ are future trapped, while $r=2M$ is marginal. On standard horizon-penetrating slices its sections give the apparent horizon, coinciding with the stationary event and Killing horizon. This coincidence in a symmetric stationary example does not identify the notions in a general spacetime.

A trapping horizon is a hypersurface foliated by marginal surfaces. Its future outer type obeys ${\theta }_{\ell }=0$, ${\theta }_n<0$ and ${\mathcal{L}}_n{\theta }_{\ell }<0$. Unlike an event horizon, this can be diagnosed from local expansion data and their variation. The null energy condition explains its causal character. At a marginal surface, the Null Raychaudhuri equation, with the present curvature sign and vanishing null twist, givesin units $c=1$. A possible nonaffinity term vanishes here because it is proportional to the zero expansion. Write a horizon evolution vector as $V=\ell -Cn$. Preserving ${\theta }_{\ell }=0$ gives $C=({\mathcal{L}}_{\ell }{\theta }_{\ell })/({\mathcal{L}}_n{\theta }_{\ell })\ge 0$. Hence $V^2=-2C\le 0$: the future outer horizon is spacelike or null. Its area expansion is ${\theta }_V=-C{\theta }_n\ge 0$. These conclusions use the outer condition and the energy condition, not merely marginality.

For the global event horizon, the future-complete horizon focusing proof of the area theorem uses the same Null Raychaudhuri equation. Under the null energy condition, a negative generator expansion forces a future caustic in finite affine parameter. Future completeness and the usual regularity or predictability assumptions exclude such a caustic on the future event-horizon boundary. Its generator expansion is therefore nonnegative and its classical area cannot decrease. These assumptions matter: this is not an unconditional result for every null surface or every matter model.

Rotating and charged solutions provide further distinctions. A nonextremal Kerr black hole has $r_{\pm }=M\pm \sqrt{M^2-a^2}$, with $a=J/M$. Its outer Killing horizon is generated by ${\partial }_t+{\Omega }_H{\partial }_{\varphi }$, where ${\Omega }_H=a/(r_{+}^2+a^2)$, rather than by time translation alone. The ergosphere begins at the stationary-limit surface $g_{tt}=0$, whose radius is $M+\sqrt{M^2-a^2{\cos }^2\theta }$. Away from the axis this lies outside the event horizon: it forbids static observers but does not forbid escape to infinity. In Reissner-Nordstrom spacetime, the nonextremal radii are $r_{\pm }=M\pm \sqrt{M^2-Q^2}$. The outer surface is an event horizon, whereas the inner surface is a Cauchy horizon in the exact analytic extension.

A Cauchy horizon concerns predictability from initial data. The future domain of dependence $D^{+}(\Sigma )$ of a partial Cauchy surface consists of events for which every past-inextendible causal curve intersects $\Sigma$. Its future Cauchy horizon isBeyond it, causal influences can arrive without meeting the specified initial surface; data on that surface no longer determine the solution there. This is distinct from the inability to send signals to infinity. The inner horizon of the exact charged black hole supplies an example, and the definition necessarily specifies which initial surface is being used.

Flat spacetime itself has acceleration horizons. In the right Rindler wedge, set $t=\xi \sinh (a\tau )$ and $x=\xi \cosh (a\tau )$, with $\xi >0$. The Minkowski spacetime metric becomes $d{s}^2=a^2{\xi }^{2}d{\tau }^2-d{\xi }^2-d{y}^2-d{z}^2$. The null boundaries $t=x$ and $t=-x$ are Rindler horizons of the boost Killing flow. A uniformly accelerated observer cannot receive signals from one side of its future horizon, even though the curvature vanishes and inertial observers encounter no such boundary. These are observer and Killing horizons, not a black-hole event horizon defined using the full Minkowski null infinity.

In a spatially flat Friedmann-Lemaître-Robertson-Walker metric with noncompact spatial sections, radial light rays have $d\chi /dt=\pm 1/a(t)$. The particle horizon measures how far signals could have travelled since the initial cosmic time; the cosmological event horizon measures how far signals emitted now could ever travel toward a comoving observer. Their comoving and proper radii areThe respective horizon is present only if its integral is finite. The event-horizon integral uses the actual future endpoint, often $t_f=\infty$, so it is global and observer-dependent. For $a(t)=t^q$ on $0<t<\infty$, a finite particle horizon exists for $0<q<1$, while a finite event horizon exists for $q>1$. Neither integral is finite for $q=1$. In the expanding flat patch of de Sitter spacetime, $a=e^{Ht}$ and the proper event-horizon radius is $H^{-1}$. The static form $d{s}^2=(1-H^2{r}^2)d{t}^2-(1-H^2{r}^2{)}^{-1}d{r}^2-r^{2}d{\Omega }^2$ identifies it as a cosmological Killing horizon; its surface-gravity magnitude is $H$ with the central static time normalization.

Finally, the Hubble radius must not automatically be called an event horizon. For a flat expanding cosmology, the two future radial null expansions of a sphere of proper radius $R$ are ${\theta }_{\pm }=2(H\pm 1/R)$. At $R=H^{-1}$ the ingoing expansion vanishes, while outside it both expansions are positive: this is the cosmological marginal, past-trapped type rather than a future black-hole trapped surface. A dust universe with $a\propto t^{2/3}$ has such a Hubble sphere but no future event horizon. Global causal boundaries, slice-dependent trapped-surface boundaries, symmetry horizons and initial-data boundaries must therefore be distinguished, even when several coincide in a special example.