Effective field equation on m-brane embedded in n-dimensional bulk of Einstein and f(R) gravity

TL;DR

This paper derives effective gravitational equations on lower-dimensional branes embedded in higher-dimensional Einstein and $f(R)$ gravity bulk spacetimes, extending the formalism to multiple dimensions and exploring solutions in vacuum and cosmology.

Contribution

It generalizes the derivation of effective field equations for branes of arbitrary codimension in Einstein and $f(R)$ gravity, including applications to static and cosmological solutions.

Findings

Derived effective equations for branes in Einstein and $f(R)$ gravity.

Obtained vacuum static spherically symmetric solutions.

Explored cosmological solutions and implications.

Abstract

We have derived effective gravitational field equations on a lower dimensional hypersurface (known as a brane), placed in a higher dimensional bulk spacetime for both Einstein and $f(\mathcal{R})$ gravity theories. We have started our analysis on $n$-dimensional bulk from which the effective field equations on a $(n-1)$-dimensional brane has been obtained by imposing $Z_{2}$ symmetry. Subsequently, we have arrived at the effective equations in $(n-2)$-dimensions starting from the effective equations for $(n-1)$ dimensional brane. This analysis has been carried forward and is used to obtain the effective field equations in $(n-m)$-dimensional brane, embedded in a $n$-dimensional bulk. Having obtained the effective field equations in Einstein gravity, we have subsequently generalized the effective field equation in $(n-m)$-dimensional brane which is embedded the $n$-dimensional bulk spacetime endowed with $f(\mathcal{R})$ gravity. We have also presented applications of our results in the context of Einstein and $f(\mathcal{R})$ gravity. In both the cases we have discussed vacuum static spherically symmetric solutions as well as solutions in cosmological context. Implications are also discussed.