Braneworld Remarks in Riemann-Cartan Manifolds

TL;DR

This paper investigates how torsion in a 5D Riemann-Cartan manifold affects the effective Einstein equations on a 4D brane, revealing that torsion modifies the equations and cosmological constant but not the matching conditions.

Contribution

It extends the Gauss-Codazzi formalism to include torsion effects in braneworld scenarios, showing their impact on the effective Einstein equations and cosmological constant.

Findings

Torsion terms do not alter Israel-Darmois matching conditions.

Torsion significantly changes the effective Einstein equations on the brane.

Contorsion factors drastically modify the effective cosmological constant.

Abstract

We analyze the projected effective Einstein equation in a 4-dimensional arbitrary manifold embedded in a 5-dimensional Riemann-Cartan manifold. The Israel-Darmois matching conditions are investigated, in the context where the torsion discontinuity is orthogonal to the brane. Unexpectedly, the presence of torsion terms in the connection does not modify such conditions whatsoever, despite of the modification in the extrinsic curvature and in the connection. Then, by imposing the Z_2-symmetry, the Einstein equation obtained via Gauss-Codazzi formalism is extended, in order to now encompass the torsion terms. We also show that the factors involving contorsion change drastically the effective Einstein equation on the brane, as well as the effective cosmological constant.