Hartle-Thorne metric

The Hartle-Thorne metric is a spacetime metric in General Relativity that describes the exterior of a slowly and rigidly rotating, stationary and axially symmetric body. It is an approximate solution of the vacuum Einstein equations.
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 formula_1, 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 formula_2, mass formula_3 and quadrupole moment formula_4, the
metric in spherical coordinates is given by
formula_5
where
formula_6