Past exam of the general relativity and cosmology course of the University of Oxford / 2017 / 2

Set $m=GM/c^2$ and $f(r)=1-2m/r$. A photon follows a null geodesic, with an affine parameter $\lambda$ satisfyingThe Christoffel symbols follow from the supplied Schwarzschild spacetime metric tensor. An efficient equivalent procedure is to use the geodesic Lagrangianwhere dots denote $d/d\lambda$. Its Euler-Lagrange equations give the affinely parametrized geodesic equations. The condition $\mathcal{L}=0$ is imposed after variation; setting the Lagrangian identically to zero before varying would discard the equations.

By spherical symmetry the ray's angular momentum fixes an orbital plane through the center. Rotate the angular coordinates so that plane is $\theta =\pi /2$. The $\theta$ Euler-Lagrange equation isso $\theta =\pi /2$, $\dot{\theta }=0$ is preserved once imposed initially. A radial ray has no unique orbital plane, but any plane through that radial line can be chosen.

The coordinates $t$ and $\varphi$ are cyclic. Thus their canonical momenta obeyThe two conserved quantities can be chosen asThese are the geodesic conserved quantities from Killing vectors ${\partial }_t$ and ${\partial }_{\varphi }$. With the chosen parameter normalization the conserved energy is $E=c^2\epsilon$ and axial angular momentum is $\ell$; multiplying the affine tangent by a constant multiplies both quantities by that constant.

The additional invariant is the squared norm of the affine tangent. Metric compatibility and the geodesic equation make it constant, and a photon has the null conditionOn the equatorial plane this givesand the conserved quantities of part (a) yieldFor a nonradial ray $\ell \ne 0$, put $u=1/r$. Sincethe orbit equation becomesHere $b=|\ell |/(c\epsilon )$ is the positive impact parameter for a future-directed ray arriving from infinity. A strictly radial ray has $\ell =0$ and must be described by the radial equation rather than by $\varphi$ as parameter.

The observer is static at $r_0$, with four-velocity $U=f_0^{-1/2}{\partial }_t$, where $f_0=1-2m/r_0$. Angles are measured in the observer's rest frame, using the positive spatial metric tensoron the equatorial plane. One cannot directly normalize a photon tangent with the full Lorentzian metric to define an angle, because that tangent is null.

At the observer the spatial direction of an outward-moving grazing ray is proportional to $w=A{\partial }_r+{\partial }_{\varphi }$, with $A=(dr/d\varphi {)}_{r_0}$. Its angle $\psi =\alpha /2$ to the outward radial direction equals the corresponding sky angle to the inward center of the image. Taking the spatial inner product with $e_r={\partial }_r$,ConsequentlyThe limiting value $A=0$ is understood by continuity.

For a grazing ray the areal radius has a turning point at the stellar surface $R$, so $\dot{r}=0$ there. The radial equation givesSince $R=4m>3m$, the surface lies outside the photon sphere. The function $f/r^2$ decreases for $r>3m$, so this tangential ray can escape to every $r_0>R$ and correctly defines the image's rim. For the specified surface,Also $dr/d\varphi =-r^{2}du/d\varphi$, henceIts sign depends on the orientation of the ray and of $\varphi$, and does not affect the angular diameter. Substitution givesThis apparent angular radius of a spherical star in Schwarzschild spacetime has useful limiting checks. As $r_0\downarrow R$, the star occupies a hemisphere, $\alpha \to \pi$. Far away, $\alpha \simeq 2b/r_0=8\sqrt{2}m/r_0$: Schwarzschild light deflection increases the apparent radius by a factor $\sqrt{2}$ relative to the flat-space value $R/r_0$.