Hartle–Thorne metric

The metric was found by James Hartle and Kip Thorne in the 1960s to study the spacetime outside neutron stars, white dwarfs and supermassive stars. It can be shown that it is an approximation to the Kerr metric (which describes a rotating black hole) when the quadrupole moment is set as ${\displaystyle q=-a^{2}a{M}^3}$, which is the correct value for a black hole but not, in general, for other astrophysical objects.

Metric

Up to second order in the angular momentum ${\displaystyle J}$, mass ${\displaystyle M}$ and quadrupole moment ${\displaystyle q}$, the metric in spherical coordinates is given by

${\displaystyle \begin{array}{cc}{g}_{tt}& =-\left(1-\frac{2M}{r}+\frac{2q}{r^3}{P}_2+\frac{2Mq}{r^4}{P}_2+\frac{2q^2}{r^6}{P}_2^2-\frac{2}{3}\frac{J^2}{r^4}(2P_2+1)\right),\\ g_{t\varphi }& =-\frac{2J}{r}{\sin }^2\theta ,\\ g_{rr}& =1+\frac{2M}{r}+\frac{4M^2}{r^2}-\frac{2q{P}_2}{r^3}-\frac{10Mq{P}_2}{r^4}+\frac{1}{12}\frac{q^2\left(8P_2^2-16P_2+77\right)}{r^6}+\frac{2J^2(8P_2-1)}{r^4},\\ g_{\theta \theta }& =r^2\left(1-\frac{2q{P}_2}{r^3}-\frac{5Mq{P}_2}{r^4}+\frac{1}{36}\frac{q^2\left(44P_2^2+8P_2-43\right)}{r^6}+\frac{J^2{P}_2}{r^4}\right),\\ g_{\varphi \varphi }& =r^2{\sin }^2\theta \left(1-\frac{2q{P}_2}{r^3}-\frac{5Mq{P}_2}{r^4}+\frac{1}{36}\frac{q^2\left(44P_2^2+8P_2-43\right)}{r^6}+\frac{J^2{P}_2}{r^4}\right),\end{array}}$

where ${\displaystyle P_2=\frac{3{\cos }^2\theta -1}{2}.}$