General Relativity: An Introduction for Physicists · Problem 12.6

Further spherically symmetric geometries — Hobson, Efstathiou & Lasenby

Physics · uniform-density-star-pressure-profile

Problem

For a spherical star of uniform density

\rho

and central pressure

p_0

, verify that the Oppenheimer–Volkoff equation requires

p(r)

to satisfy

\frac{\rho c^2 + 3p(r)}{\rho c^2 + p(r)} = \frac{\rho c^2 + 3p_0}{\rho c^2 + p_0}\left(1 - \frac{8\pi G}{3c^2}\rho r^2\right)^{1/2},

and hence show that

p(r) = \rho c^2 \frac{(1 - 2\mu r^2/R^3)^{1/2} - (1 - 2\mu/R)^{1/2}}{3(1 - 2\mu/R)^{1/2} - (1 - 2\mu r^2/R^3)^{1/2}},

where

R

is the coordinate radius of the star.