The standard "static" spherically symmetric ansatz with perfect fluid source revisited

TL;DR

This paper revisits the static spherically symmetric ansatz with perfect fluid sources in Einstein's equations, exploring interpretations of solutions with non-positive metric functions and deriving related equations, concluding no new simple black hole solutions exist.

Contribution

It introduces a novel interpretation of solutions with negative metric functions as tachyonic sources and derives corresponding equations, expanding understanding of such spacetimes.

Findings

Regions with negative A(r) and B(r) correspond to tachyonic sources.

Derived Oppenheimer-Volkoff-like equations for various source interpretations.

No new simple black hole solutions supported by perfect fluid found.

Abstract

Considering the standard "static" spherically symmetric ansatz ds2 = -B(r) dt2 + A(r) dr2 + r2 dOmega2 for Einstein's Equations with perfect fluid source, we ask how we can interpret solutions where A(r) and B(r) are not positive, as they must be for the static matter source interpretation to be valid. Noting that the requirement of Lorentzian signature implies A(r) B(r) >0, we find two possible interpretations: (i) The nonzero component of the source four-velocity does not have to be u0. This provides a connection from the above ansatz to the Kantowski-Sachs (KS) spacetimes. (ii) Regions with negative A(r) and B(r) of "static" solutions in the literature must be interpreted as corresponding to tachyonic source. The combinations of source type and four-velocity direction result in four possible cases. One is the standard case, one is identical to the KS case, and two are tachyonic. The dynamic tachyonic case was anticipated in the literature, but the static tachyonic case seems to be new. We derive Oppenheimer-Volkoff-like equations for each case, and find some simple solutions. We conclude that new "simple" black hole solutions of the above form, supported by a perfect fluid, do not exist.