Embeddings for 4D Einstein equations with a cosmological constant

TL;DR

This paper explores various methods of embedding 4D Einstein solutions with a cosmological constant into higher-dimensional Ricci-flat or anti-de Sitter manifolds, revealing different induced physics and consistent definitions of effective mass in 4D.

Contribution

It extends previous work by demonstrating multiple embedding approaches and analyzing their physical implications, especially regarding the effective mass of test particles.

Findings

Different embeddings produce distinct 4D cosmological terms.

Effective mass definitions are equivalent across multiple theoretical approaches.

Effective mass remains consistent for null and non-null 5D motions.

Abstract

There are many ways of embedding a 4D spacetime in a given higher-dimensional manifold while, satisfying the field equations. In this work we extend and generalize a recent paper by Mashhoon and Wesson ({\it Gen. Rel. Gravit.} {\bf 39}, 1403(2007)) by showing different ways of embedding a solution of the 4D Einstein equations, in vacuum with a cosmological constant, in a Ricci-flat, as well as in an anti-de Sitter, 5D manifold. These embeddings lead to different physics in 4D. In particular, to non-equivalent cosmological terms as functions of the extra coordinate. We study the motion of test particles for different embeddings and show that there is a complete equivalence between several definitions for the effective mass of test particles measured in 4D, obtained from different theoretical approaches like the Hamilton-Jacobi formalism and the principle of least action. For the case under consideration, we find that the effective mass observed in 4D is the same regardless of whether we consider null or non-null motion in 5D.