What is the Metric of the Gravitational Field of the Sun?

Question

What metric determines the "geometry" of the gravitational field generated by the mass of the sun? Is there a general metric that incorporates arbitrary mass and devolves into the Schwarzchild metric for black holes that conforms with the Einstein Field Equations?

Answer 1

The spacetime around the Sun is very well approximated by the Schwarzschild metric, which is appropriate outside any constant, spinless, spherically symmetric mass distribution.

The Sun is almost perfectly spherical - the polar and equatorial diameters differ by only about 1 part in $10^5$. It also spins slow enough that one can usually ignore the spin for all but the most precise of calculations.

If one wishes to incorporate spin, then there are approximations of increasing precision. The two that I am reasonably familiar with are the Lense-Thirring metric, which is exact for a spherical body with constant density, and reduces to the Schwarzschild metric when the angular momentum is small. The next level of approximation would be the Kerr metric. This introduces the dimensionless spin parameter $a = Jc/(GM^2)$ (in SI units), where $a=0$ would correspond to the Schwarzschild metric. However, the Kerr metric is only an exact solution for a black hole with spin.

For an arbitrary mass distribution, with spin and some asphericity, there are a host of ever more complex metrics, often written as perturbations to the Kerr metric, which feature the multipole moments of the mass distribution (e.g. Frutos-Alfaro et al. 2015). A good example is the Hartle-Thorne metric.

Answer 2

The Schwarzschild-Droste metric (widely called simply the Schwarzschild metric) describes spacetime outside any static spherically symmetric body. Therefore, in the approximation that the Sun is spherically symmetric and the gravitation of other bodies can be neglected, this is the metric of spacetime around the Sun.

To go beyond that approximation you would have to bring in more details of the shape of the Sun. A multipole expansion would be suitable. You could then use the Newtonian theory to give an accurate estimate of the gravitational potential $\Phi$, and then use the metric $$ ds^2 = -(1 + 2 \Phi/c^2) c^2 dt^2 + (1 - 2 \Phi/c^2) (dx^2 + dy^2 + dz^2). $$ This is not exact but it is sufficiently accurate for most calculations.

This neglects the effect of the motions of the Sun (rotation and vibration) and of further 'post-Newtonian' effects, but these are very very small.