Past exam of the mathematical physics course of the University of Oxford / 2007 / B

Use metric signature $(+---)$ and the curvature sign fixed by the stated Newtonian limit. In flat spacetime the source-free Maxwell equations are ${\partial }_a{F}^{ab}=0$ and ${\partial }_{[a}{F}_{bc]}=0$. The stress-energy tensor in mixed form is $T^a{}_b=F^{ac}{F}_{cb}+{\delta }_b^a{F}_{cd}{F}^{cd}/4$. Its divergence isContracting the cyclic Maxwell equation with $F^{ac}$ gives $F^{ac}{\partial }_a{F}_{cb}=-{\partial }_b(F_{cd}{F}^{cd})/4$. The first term also vanishes, proving $\boxed{{\partial }_a{T}^{ab}=0}$. The component $b=0$ expresses local electromagnetic energy conservation, and the spatial components express momentum conservation. Integrating converts them into the usual energy and momentum flux balances.

In curved spacetime the tensor equations areThey agree with the flat equations in coordinates of a local inertial frame at a point, where the connection vanishes, and covariance then extends them to arbitrary coordinates. Antisymmetry makes the homogeneous equation equivalent to $dF=0$. Metric compatibility permits the same contraction proof, now giving $\boxed{{\nabla }_a{T}^{ab}=0}$. In general,which need not vanish. Covariant local balance remains valid; ordinary coordinate components do not provide a universal conserved global energy-momentum vector in a curved geometry. Additional spacetime symmetries can furnish conserved currents. A locally inertial frame removes the connection at a point, not the curvature or the global obstruction.

The alternating array with fixed component ${ϵ}_{0123}=1$ is appropriate in oriented inertial special-relativistic coordinates, whose proper Lorentz transformations have determinant one. It is not itself a covariant tensor under general coordinate changes. For an oriented Lorentzian metric, replace it by the metric metric volume tensor ${\epsilon }_{abcd}=\sqrt{-\det g}[abcd]$. The genuine dual isThis is the Hodge star; the determinant factor, orientation and raising conventions are essential. Even in flat spacetime arbitrary curvilinear coordinates require the metric volume tensor, so the distinction is about coordinate covariance rather than curvature alone.

The isolated equation for $R_{00}$ singles out a coordinate component and cannot be a generally covariant law. Replacing it by a proportionality $R_{ab}\propto T_{ab}$ still fails: the contracted Bianchi identity gives ${\nabla }^a{R}_{ab}={\nabla }_{b}R/2$, whereas stress-energy conservation requires ${\nabla }^a{T}_{ab}=0$. Such a law would force $R$ constant and hence impose an unphysical restriction on a varying relativistic dust density. Missing units or a coupling constant in the proposed proportionality would not repair this structural problem.

The Einstein tensor $G_{ab}=R_{ab}-R{g}_{ab}/2$ is divergence-free. Thereforeis consistent with covariant stress-energy conservation in the source's curvature convention. Its trace gives $R=8\pi GT$ and hence $R_{ab}=-8\pi G(T_{ab}-T{g}_{ab}/2)$. For slowly moving relativistic dust with $V^a{V}_a=1$, $T_{00}\simeq \rho$, $T\simeq \rho$, $g_{00}\simeq 1$, so $R_{00}\simeq -4\pi G\rho$, exactly matching $-\Delta \varphi$ and Poisson's equation.

Finally, electromagnetic stress-energy is trace-free: $T^a{}_a=F^{ac}{F}_{ca}+F_{cd}{F}^{cd}=0$ by antisymmetry. The traced field equation then gives $\boxed{R=0}$ for a purely electromagnetic source. This does not say $R_{ab}=0$ or that spacetime is flat; its trace-free Ricci curvature can still be nonzero.

Under a smooth coordinate change $x\mapsto x^{\prime }$, a contravariant vector obeysThe scalar-contraction hypothesis gives $U_a^{\prime }(\partial x^{\prime a}/\partial x^b)T^b=U_b{T}^b$ for every choice of $T^b$. Equality of all coefficients and multiplication by the inverse Jacobian matrix give $\boxed{U_a^{\prime }=(\partial x^b/\partial x^{\prime a})U_b}$, the covector law.

Let $V^a=d{x}^a/ds$ be a timelike geodesic tangent. Metric compatibility and the geodesic equation giveA Killing vector therefore gives a conserved contraction. Conversely, if that contraction is conserved along every timelike geodesic, then at any point its symmetric tensor $S_{ab}={\nabla }_{(a}{T}_{b)}$ satisfies $S_{ab}{V}^a{V}^b=0$ for every unit timelike initial tangent. Homogeneity extends this to the entire timelike cone, an open set of tangent vectors. A quadratic polynomial vanishing on an open set has every coefficient zero, so $\boxed{{\nabla }_{(a}{T}_{b)}=0}$. Considering a single selected geodesic would not suffice. This is the Converse geodesic characterization of a Killing vector.

For the given exterior transformation, differentiating at fixed $r$ givesThe transformation also gives $UV=(2m-r)e^{r/(2m)}$. This determines $r$ smoothly for $r>0$ throughout $UV<2m$. The flow of the displayed vector field rescales $U\mapsto e^{-\lambda /(4m)}U$, $V\mapsto e^{\lambda /(4m)}V$. It leaves $UV$, $r$, $dUdV$ and the angular metric unchanged. Thus it preserves the full metric everywhere in that region and extends as a global Killing field, not merely by a formal exterior coordinate substitution.

Its squared norm isIt is timelike for $UV<0$ ($r>2m$), spacelike for $0<UV<2m$ ($0<r<2m$), and a nonzero null vector on either horizon $UV=0$ away from $U=V=0$. At $U=V=0$ it is the zero vector, on the whole bifurcation surface of angular points. Zero must be distinguished from a nonzero null vector. This is the Schwarzschild time-translation Killing field in Kruskal coordinates.

The coordinate derivatives of the smooth two-parameter map commute. With a torsion-free Levi-Civita connection,because ${\Gamma }^a{}_{bc}$ is symmetric in its lower indices. Each member of the family is a geodesic, so $DV/Ds=0$. ConsequentlyThe commutator of covariant derivatives is the action of the Riemann curvature tensor. With the source's index and sign convention this givesThis is geodesic deviation: free fall relative acceleration measures curvature, even where an individual observer can eliminate the connection at one point.

For Earth's exterior Newtonian field, $\varphi =-GM/r$ and acceleration is $a_i=-{\partial }_i\varphi$. Nearby freely falling particles have relative acceleration $\delta a_i=-{\varphi }_{,ij}{Y}^j$. At the observation point align $z$ radially upward and $x,y$ horizontally. Since $GM/R_E^2=g$,The weak-field geodesic-deviation equation therefore identifiesIn these units time is measured in seconds as in the stated Newtonian comparison; if the zeroth coordinate is $ct$ instead, the curvature components acquire the corresponding $c^{-2}$ factor. A small freely falling cloud is squeezed in the two horizontal directions and stretched radially. These are tidal effects, not the uniform acceleration removed by a freely falling frame.

Their characteristic inverse-square time is $g/R_E$, so $\boxed{t_{\mathrm{tide}}=\sqrt{R_E/g}\simeq 8.1\times 10^2\mathrm{s}\simeq 13.4\min }$. One concrete thought experiment is an ideal satellite moving in a circular orbit just above a spherical Earth, ignoring atmosphere and terrain. Its orbital angular frequency is $\sqrt{GM/R_E^3}=\sqrt{g/R_E}$ and its period is $2\pi t_{\mathrm{tide}}\simeq 84$ minutes. Equivalently, initially nearby free fall trajectories develop order-one tidal changes on this characteristic scale. This is the surface tidal time scale of a spherical gravitating body.

Write $f(r)=1-2m/r$ and restore $c$ for the numerical estimate. A stationary clock at radius $r$ has $d\tau =\sqrt{f(r)}dt$. Let radii $r_1$ and $r_2=r_1-h$ denote the upper and lower computers. The metric is stationary and the same two fixed spatial points exchange both signals, so each signal has the same coordinate propagation time. Therefore the interval between reception events equals the interval between emission events in coordinate time: the common delay cancels.

For upper proper running time ${\tau }_1$, the lower computer has run forIn the weak gravitational field near Earth's surface, $m=GM/c^2$ andThe lower clock runs more slowly, so its remaining work isThus about 660 micro-operations remain; the proper-time shortfall is only about $6.6\times 10^{-10}$ seconds.

Successful comparison tests the gravitational dependence of stationary proper-clock rates, specifically the radial variation of $g_{tt}$ and its weak-field gravitational redshift prediction, assuming the computers execute equally per unit proper time. It also checks that this clock effect is independent of the chosen physical clock mechanism. It does not separately verify the Schwarzschild radial metric coefficient: although light propagation uses the null metric, the identical start and finish propagation delays cancel from this particular observable. This is Gravitational redshift measured by a computation counter.

Put $f=1-2m/r$, use proper time $s$, and denote its derivatives by dots. The timelike geodesic equations, away from the Schwarzschild coordinate singularity, arewith normalization $f{\dot{t}}^2-f^{-1}{\dot{r}}^2-r^2{\dot{\theta }}^2-r^2{\sin }^2\theta {\dot{\varphi }}^2=1$. Spherical symmetry permits an equatorial plane, $\theta =\pi /2$, and the cyclic coordinates give $\boxed{E=f\dot{t},J=r^2\dot{\varphi }}$. Normalization then gives ${\dot{r}}^2=E^2-f(1+J^2/r^2)$.

For $J\ne 0$ and $u=m/r$, $du/d\varphi =-m\dot{r}/J$. ThusAt $E=1$, $J=4m$, the right side factors as $2u(u-1/4{)}^2$. Set $q=2\sqrt{u}$, so $u=q^2/4$ and $(q^{\prime }{)}^2=(q^2-1{)}^2/8$. Integrating the regions $0<q<1$ and $q>1$ gives, up to angular shifts and time reversal,The constant $u=1/4$, or $r=4m$, is the corresponding circular timelike orbit. The hyperbolic-tangent branch lies at $r>4m$ and approaches that orbit from outside as $|\varphi |\to \infty$; the hyperbolic-cotangent branch lies at $0<r<4m$ and approaches it from inside. Near the limit $u-1/4$ decays exponentially as $e^{-|\varphi |/\sqrt{2}}$. Since $\dot{\varphi }$ tends to $J/(4m{)}^2=1/(4m)$, the limiting approach also takes infinite proper time. Neither nonconstant branch reaches and joins the circular orbit at a finite angular parameter.

Differentiating the orbit equation gives $u^{\prime \prime }=m^2/J^2-u+3u^2$. A perturbation about $u=1/4$ obeys $\delta u^{\prime \prime }=\delta u/2$, showing that the circular orbit is unstable. The two explicit curves are its separatrix branches, not ordinary small stable oscillations. For $E=1$ the energy is marginally bound, so the outer critical branch is the limiting orbit between scattering and Schwarzschild marginally bound capture from infinity. Time-reversed inner and outer branches describe the corresponding approach or departure from that unstable orbit. These are the Explicit separatrices of the marginally bound Schwarzschild orbit.