 Solution

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.