Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 37D (General Relativity)

WriteFor a static observer, $d\tau =\sqrt{f(r)}dt$. Conservation of the photon's Killing energy therefore gives the gravitational redshift between static Schwarzschild observersFor the stated distant observer, $r\gg r_{\mathrm{st}}$ and $f(r)\simeq 1$, soAs $r_{\mathrm{st}}\to 2M^{+}$, the observed wavelength tends to infinity: light emitted from a static source arbitrarily near the Schwarzschild event horizon is infinitely redshifted at infinity.

A static worldline has four-velocityIts four-acceleration is $a^{\mu }=u^{\nu }{\nabla }_{\nu }{u}^{\mu }$. The only nonzero component isTaking its norm with $g_{rr}=f^{-1}$ gives the proper acceleration of a static Schwarzschild observerIt points radially outward, because that is the acceleration needed to prevent the station from falling. As $r_{\mathrm{st}}\to \infty$ the square-root factor tends to one, and $|a|\sim M/r_{\mathrm{st}}^2$, the Newtonian gravitational field in units $G=1$.

The station's proper rest mass isA static local energy $d{E}_{\mathrm{loc}}$ at radius $r_{\mathrm{st}}$ contributes the redshifted energyto the mass measured at infinity. To first order in $M_{\mathrm{st}}/M$, the exterior mass parameter is thereforeTerms quadratic in $\rho$ describe the shell's self-gravity and are neglected.

The exterior of the shell is vacuum and spherically symmetric. The Birkhoff theorem therefore makes it a Schwarzschild spacetime with mass $M_{\mathrm{out}}$:This is the redshifted gravitational mass of a static thin shell.