Is an exact solution to Einstein's Field Equations known for the interior of a sphere of uniform density (to approximate a star or planet, for example?)
Asked
Active
Viewed 1,789 times
1 Answers
17
Yes, the exact solution is known. The general spherically symmetric metric is $$g=-B(r)\mathrm{d}t^2+A(r)\mathrm{d}r^2+r^2\mathrm{d}\Omega^2.$$
The solution for $A(r)$ is $$A(r)=\left[1-\frac{2G\mathcal{M}(r)}{r}\right]^{-1},\quad\mathcal{M}(r)=\int^r \rho \,\mathrm{d}V=\int_0^r 4\pi r'^2\rho(r')\,\mathrm{d}r.$$ The solution for $B(r)$ is $$B(r)=\exp\left\{-\int_r^\infty \frac{2G}{r'^2}[\mathcal{M}(r')+4\pi r'^3 P(r')]A(r')\,\mathrm{d}r'\right\}.$$ In these equations $\rho$ is the density of the star and $P$ its pressure.
The derivation can be found in e.g.
S. Weinberg, Gravitation and Cosmology (1973), Sect. 11.1
R.M. Wald, General Relativity (1984), Sect. 6.2
N. Straumann, General Relativity (2013), Sect. 7.4
Ryan Unger
- 8,813
-
So taking $ r >> 2GM (r) $ you get the Schwarzschild solution – ibnAbu Jun 17 '19 at 09:01
-
Weinberg p. 302 has $2G/r^2$ not $2G/r'^2$. Who is right? OK, my copy is from 1972, so maybe it has a typo? – Harald Jun 20 '19 at 18:43