BTZ black hole from (3+1) gravity

TL;DR

This paper presents a method to derive (2+1) dimensional BTZ black hole solutions from (3+1) gravity with a cosmological constant, using a specific reduction mechanism based on Weyl tensor conditions and foliation techniques.

Contribution

It introduces a novel reduction approach from (3+1) to (2+1) gravity that produces BTZ black hole solutions without matter content.

Findings

Successfully derives BTZ solutions from (3+1) gravity

Establishes criteria involving Weyl tensor for dimensional reduction

Provides a new perspective on connecting higher and lower-dimensional gravity solutions

Abstract

We propose an approach for constructing spatial slices of (3+1) spacetimes with cosmological constant but without a matter content, which yields (2+1) vacuum with $\Lambda$ solutions. The reduction mechanism from (3+1) to (2+1) gravity is supported on a criterion in which the Weyl tensor components are required to vanish together with a dimensional reduction via an appropriate foliation. By using an adequate reduction mechanism from the Pleba\'nski-Carter[A] solution in (3+1) gravity, the (2+1) BTZ solution can be obtained.