Defining perturbations on submanifolds

TL;DR

This paper redefines how to measure perturbations in the presence of submanifolds, such as branes, by adjusting the standard approach to account for the submanifold's location, and applies this to compute perturbed metrics and curvature tensors.

Contribution

It introduces a modified definition of perturbations on submanifolds and demonstrates its application by calculating perturbed geometric quantities at a brane.

Findings

Derived a new method for defining perturbations on submanifolds.

Computed perturbed metric and extrinsic curvature tensors at the brane.

Provided a gauge-invariant approach for perturbations in braneworld scenarios.

Abstract

We study the definition of perturbations in the presence of a submanifold, like e.g. a brane. In the standard theory of cosmological perturbations, one compares quantities at the same coordinate points in the non-perturbed and the perturbed manifolds, identified via a (non-unique) mapping between the two manifolds. In the presence of a physical submanifold one needs to modify this definition in order to evaluate perturbations of quantities at the submanifold location. As an application, we compute the perturbed metric and the extrinsic curvature tensors at the brane position in a general gauge.