THE STANDARD "STATIC" SPHERICALLY SYMMETRIC ANSATZ WITH PERFECT FLUID SOURCE REVISITED

Abstract

Considering the standard "static" spherically symmetric ansatz ds2 = −B(r) dt2 + A(r) dr2 + r2 dΩ2 for Einstein's equations with perfect fluid source, we ask how we can interpret solutions where A(r)andB(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) space–times. (ii) Regions with negative A(r) and B(r) of "static" solutions in the literature must be interpreted corresponding to the 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.