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

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.