Stellar models with Schwarzschild and non-Schwarzschild vacuum exteriors

TL;DR

This paper investigates how different vacuum exterior solutions in stellar models affect gravitational bounds and redshift, revealing that these exteriors depend on internal star parameters and can lead to observable differences from general relativity.

Contribution

It demonstrates that vacuum exteriors in braneworld scenarios are determined by star parameters, altering gravitational bounds and redshift limits compared to classical models.

Findings

Maximum $M/R$ ratio varies with exterior type, reaching up to 1.

Surface redshift can be unbounded for certain exteriors.

Exterior solutions depend on internal star structure, not just mass.

Abstract

A striking characteristic of non-Schwarzschild vacuum exteriors is that they contain not only the total gravitational mass of the source, but also an {\it arbitrary} constant. In this work, we show that the constants appearing in the "temporal Schwarzschild", "spatial Schwarzschild" and "Reissner-Nordstr{\"o}m-like" exteriors are not arbitrary but are completely determined by star's parameters, like the equation of state and the gravitational potential. Consequently, in the braneworld scenario the gravitational field outside of a star is no longer determined by the total mass alone, but also depends on the details of the internal structure of the source. We show that the general relativistic upper bound on the gravitational potential $M/R < 4/9$, for perfect fluid stars, is significantly increased in these exteriors. Namely, $M/R < 1/2$, $M/R < 2/3$ and $M/R < 1$ for the temporal Schwarzschild, spatial Schwarzschild and Reissner-Nordstr{\"o}m-like exteriors, respectively. Regarding the surface gravitational redshift, we find that the general relativistic Schwarzschild exterior as well as the braneworld spatial Schwarzschild exterior lead to the same upper bound, viz., $Z < 2$. However, when the external spacetime is the temporal Schwarzschild metric or the Reissner-Nordstr{\"o}m-like exterior there is no such constraint: $Z < \infty$. This infinite difference in the limiting value of $Z$ is because for these exteriors the effective pressure at the surface is negative. The results of our work are potentially observable and can be used to test the theory.