Newtonian view of general relativistic stars

A M Oliveira,I G Salako,J C Fabris,H E S Velten

doi:10.1140/epjc/s10052-014-3170-2

Abstract

Although general relativistic cosmological solutions, even in the presence of pressure, can be mimicked by using neo-Newtonian hydrodynamics, it is not clear whether there exists the same Newtonian correspondence for spherical static configurations. General relativity solutions for stars are known as the Tolman-Oppenheimer-Volkoff (TOV) equations. On the other hand, the Newtonian description does not take into account the total pressure effects and therefore can not be used in strong field regimes. We discuss how to incorporate pressure in the stellar equilibrium equations within the neo-Newtonian framework. We compare the Newtonian, neo-Newtonian and the full relativistic theory by solving the equilibrium equations for both three approaches and calculating the mass-radius diagrams for some simple neutron stars equation of state.

Highlights

General relativity is the usual theory for dealing with gravitational phenomena [1]
It is a remarkable fact that the general relativity predictions for expanding backgrounds can be mimicked by simple Newtonian models
The Newtonian cosmology fails in describing epochs of the universe where the pressure is relevant, the neo-Newtonian cosmology was developed to fill this gap

Summary

Introduction

General relativity is the usual theory for dealing with gravitational phenomena [1]. Its building blocks, like the equivalence principle and predictions for the trajectories of planets and light in the solar system, have passed the most different tests. With the works of Milne in the 1950s and Harrisson in 1965, pressure effects were correctly incorporated into Newtonian cosmological solutions. The GR solutions for isotropic stars are known as the Tolman–Oppeinheimer– Volkoff (TOV) equations from which we can solve the equilibrium configuration of the stellar interior [9,10]. Hydrostatic equilibrium in stars can be studied with Newtonian mechanics From this approach one obtains the Lane–Emden equation [11] which is basically the Newtonian limit of the TOV system when pressure does not source gravitational effects in the stellar interior. Since the pressure effects, mainly in extreme relativistic stars, are very important for the stellar properties, many systems in nature cannot be described via the Newtonian equations.

Newtonian hydrodynamics in an expanding background: cosmology

Including pressure: the neo-Newtonian cosmology

Numerical results

Conclusions