A Note on the Integral Formulation of Einstein's Equations Induced on a Braneworld

TL;DR

This paper explores the integral formulation of Einstein's equations in braneworld models, emphasizing the subtle role of boundary terms and their implications for Mach's principle and the cosmological constant problem.

Contribution

It revisits the integral approach to Einstein's equations in braneworlds, highlighting the nuanced role of surface terms and their impact on foundational cosmological issues.

Findings

Surface terms in braneworlds are more subtle than in 4D cases.

The integral formulation offers insights into Mach's principle in higher dimensions.

Implications for the cosmological constant problem are discussed.

Abstract

We revisit the integral formulation (or Green's function approach) of Einstein's equations in the context of braneworlds. The integral formulation has been proposed independently by several authors in the past, based on the assumption that it is possible to give a reinterpretation of the local metric field in curved spacetimes as an integral expression involving sources and boundary conditions. This allows one to separate source-generated and source-free contributions to the metric field. As a consequence, an exact meaning to Mach's Principle can be achieved in the sense that only source-generated (matter fields) contributions to the metric are allowed for; universes which do not obey this condition would be non-Machian. In this paper, we revisit this idea concentrating on a Randall-Sundrum-type model with a non-trivial cosmology on the brane. We argue that the role of the surface term (the source-free contribution) in the braneworld scenario may be quite subtler than in the 4D formulation. This may pose, for instance, an interesting issue to the cosmological constant problem.