A note on Pathria's model of the universe as a black hole

TL;DR

This paper investigates Pathria's cosmological model where a universe can be viewed as the interior of a black hole, analyzing the matching conditions between the FRW universe and Schwarzschild spacetime at the event horizon.

Contribution

It examines the matching of the FRW universe with Schwarzschild spacetime through a null hypersurface, revealing the presence of a null shell with surface pressure.

Findings

Matching is not smooth at the event horizon.

The null hypersurface admits a surface pressure.

The event horizon acts as a null shell separating two spacetimes.

Abstract

Pathria has shown that for the certain values of the cosmological constant, a pressureless closed Friedmann-Robertson-Walker universe can be the interior of a Schwarzschild black hole. We examine the Pathria's model from the point of view of the matching of the two different spacetimes through a null hypersurface. We first regard the event horizon of the black hole, which in the Pathria's model is identified with the radius of the universe at the point of its maximum expansion, as a null hypersurface separating the FRW interior from the vacuum Schwarzschild exterior, and then use the null shell formalism for it. It turns out that the matching is not smooth, and in fact, the null hypersurface is the history of a null shell admitting a surface pressure.