Gradient formula for linearly self-interacting branes

TL;DR

This paper derives a simple formula for calculating the regularised gradient of self-interaction potentials in brane models, applicable in higher dimensions with certain codimension conditions, aiding in brane-world scenario analyses.

Contribution

It introduces a universal method to compute regularised gradients of divergent self-interaction potentials for branes with codimension two or more, except in hypermembrane cases.

Findings

Regularised gradient obtained by multiplying potential components by half the trace of the second fundamental tensor.

Method applicable for branes in backgrounds with codimension two or higher.

Fails in hypermembrane case, indicating a limitation of the approach.

Abstract

The computation of long range linear self-interaction forces in string and higher dimensional brane models requires the evaluation of the gradients of regularised values of divergent self-interaction potentials. It is shown that the appropriately regularised gradient in directions orthogonal to the brane surface will always be obtainable simply by multiplying the regularised potential components by just half the trace of the second fundamental tensor, except in the hypermembrane case for which the method fails. Whatever the dimension of the background this result is valid provided the codimension is two (the hyperstring case) or more, so it can be used for investigating brane-world scenarios with more than one extra space dimension.