Spherically symmetric brane spacetime with bulk $f(\mathcal{R})$ gravity

TL;DR

This paper derives and analyzes spherically symmetric solutions in a brane-world model with five-dimensional $f(\,\mathcal{R})$ gravity, revealing black hole solutions and phase transitions influenced by bulk curvature effects.

Contribution

It introduces a method to obtain brane solutions with $f(\mathcal{R})$ gravity in the bulk, highlighting the impact of dark radiation and pressure on these solutions.

Findings

Existence of black hole solutions with parameters from the bulk.

Dark pressure and dark radiation significantly influence brane geometries.

Identification of second order phase transitions in the thermodynamics of these solutions.

Abstract

Introducing $f(\mathcal{R})$ term in the five-dimensional bulk action we derive effective Einstein's equation on the brane using Gauss-Codazzi equation. This effective equation is then solved for different conditions on dark radiation and dark pressure to obtain various spherically symmetric solutions. Some of these static spherically symmetric solutions correspond to black hole solutions, with parameters induced from the bulk. Specially, the dark pressure and dark radiation terms (electric part of Weyl curvature) affect the brane spherically symmetric solutions significantly. We have solved for one parameter group of conformal motions where the dark radiation and dark pressure terms are exactly obtained exploiting the corresponding Lie symmetry. Various thermodynamic features of these spherically symmetric space-times are studied, showing existence of second order phase transition. This phenomenon has its origin in the higher curvature term with $f(\mathcal{R})$ gravity in the bulk.