Solution generating theorems for the TOV equation

TL;DR

This paper introduces solution generating theorems for the TOV equation, allowing the creation of new solutions by deforming existing ones based on physical observables like pressure and density profiles.

Contribution

It develops new theorems that generate solutions to the TOV equation directly from physical profiles, complementing previous geometric approaches.

Findings

New solution generating theorems for the TOV equation

Parameterization of solutions via central density and pressure shifts

Interpretation of TOV as an integrability condition

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, whereby any given solution can be "deformed" to 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. D71 (2005) 124307; gr-qc/0503007] 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 parameterized in terms of delta rho_c and delta p_c, the finite shift in the central density and central pressure. We conclude by presenting a new physical and mathematical interpretation of the TOV equation -- as an integrability condition on the density and pressure profiles.