On Schwarzschild’s Interior Solution and Perfect Fluid Star Model

Abstract

We solve the boundary value problem for Einstein’s gravitational field equations in the presence of matter in the form of an incompressible perfect fluid of density ρ and pressure field p(r) located in a ball r≤r0. We find a 1-parameter family of time-independent and radially symmetric solutions ga,ρa,pa:−2m<a<a1 satisfying the boundary conditions g=gS and p=0 on r=r0, where gS is the exterior Schwarzschild solution (solving the gravitational field equations for a point mass M concentrated at r=0) and containing (for a=0) the interior Schwarzschild solution, i.e., the classical perfect fluid star model. We show that Schwarzschild’s requirement r0>9κM/(4c2) identifies the “physical” (i.e., such that pa(r)≥0 and pa(r) is bounded in 0≤r≤r0) solutions {pa:a∈U0} for some neighbourhood U0⊂(−2m,+∞) of a=0. For every star model {ga:a0<a<a1}, we compute the volume V(a) of the region r≤r0 in terms of abelian integrals of the first, second, and third kind in Legendre form.