Solution generating theorems for the Tolman-Oppenheimer-Volkov equation

Abstract

The Tolman-Oppenheimer-Volkov (TOV) equation constrains the internal structure of general relativistic static perfect fluid spheres. We develop several ``solution generating'' theorems for the TOV equation, whereby any given solution can be deformed into a new solution. Because the theorems we develop work directly in terms of the physical observables---pressure profile and density profile---it is relatively easy to check the density and pressure profiles for physical reasonableness. This work complements our previous article [Phys. Rev. D 71, 124037 (2005)] wherein a similar algorithmic analysis of the general relativistic static perfect fluid sphere was presented in terms of the spacetime geometry---in the present analysis the pressure and density are primary and the spacetime geometry is secondary. In particular, our deformed solutions to the TOV equation are conveniently parametrized in terms of $\ensuremath{\delta}{\ensuremath{\rho}}_{c}$ and $\ensuremath{\delta}{p}_{c}$, the finite shift in the central density and central pressure. We conclude by presenting a new physical and mathematical interpretation for the TOV equation---as an integrability condition on the density and pressure profiles.