Past exam of the mathematical physics course of the University of Oxford / 2007 / B / 8

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.