Constraints for warped branes

TL;DR

This paper explores the constraints necessary for warped brane solutions in higher dimensions, showing that variable tension leads to additional conditions that restrict possible solutions, with implications for four-dimensional effective theories.

Contribution

It demonstrates that variable tension in warped branes imposes extra constraints, reducing continuous solution families to discrete extrema, and clarifies the inconsistency between higher-dimensional and 4D field equations.

Findings

No static extrema exist for 6D gravity with a cosmological constant.

Variable tension solutions tend to be non-static runaway solutions.

Additional constraints are necessary for solution consistency.

Abstract

We investigate singular geometries which can be associated with warped branes in arbitrary dimensions. If the brane tension is allowed to be variable, the extremum condition for the action requires additional constraints beyond the solution of the field equations. In a higher dimensional world, such constraints arise from variations of the metric which are local in the usual four-dimensional spacetime, without changing the geometry of internal space. As a consequence, continuous families of singular solutions of the field equations, with arbitrary integration constants, are generically reduced to a discrete subset of extrema of the action, similar to regular spaces. As an example, no static extrema of the action with effective four-dimensional gravity exist for six-dimensional gravity with a cosmological constant. These findings explain why the field equations of the reduced four-dimensional theory are not consistent with arbitrary solutions of the higher dimensional field equations - consistency requires the additional constraints. The characteristic solutions for variable tension branes are non-static runaway solutions where the effective four-dimensional cosmological constant vanishes as time goes to infinity.