All Static Circularly Symmetric Perfect Fluid Solutions of 2+1 Gravity

TL;DR

This paper systematically derives all static circularly symmetric perfect fluid solutions in 2+1 gravity with a cosmological constant, revealing their structure, specific cases, and similarities with 3+1 gravity solutions.

Contribution

It provides a complete classification of static circularly symmetric perfect fluid solutions in 2+1 gravity, including explicit solutions for various equations of state and demonstrating their uniqueness.

Findings

Explicit solutions for linear and polytropic fluids

Demonstration of the uniqueness of constant energy density solutions

Establishment of similarities between 2+1 and 3+1 gravity solutions

Abstract

Via a straightforward integration of the Einstein equations with cosmological constant, all static circularly symmetric perfect fluid 2+1 solutions are derived. The structural functions of the metric depend on the energy density, which remains in general arbitrary. Spacetimes for fluids fulfilling linear and polytropic state equations are explicitly derived; they describe, among others, stiff matter, monatomic and diatomic ideal gases, nonrelativistic degenerate fermions, incoherent and pure radiation. As a by--product, we demonstrate the uniqueness of the constant energy density perfect fluid within the studied class of metrics. A full similarity of the perfect fluid solutions with constant energy density of the 2+1 and 3+1 gravities is established.