Effective spacetime from multi-dimensional gravity

TL;DR

This paper investigates how the effective lower-dimensional spacetimes derived from higher-dimensional Schwarzschild-Tangherlini solutions depend on whether extra dimensions are compact or large, affecting the observed gravitational mass and singularity properties.

Contribution

It provides a detailed analysis of the effective spacetimes from multidimensional gravity, highlighting the impact of extra dimension size on physical properties and potential observational signatures.

Findings

Effective mass is less than Schwarzschild mass for compact extra dimensions.

Effective physics depends on the size of extra dimensions when they are large.

Singularity characteristics in higher dimensions influence lower-dimensional effective theories.

Abstract

We study the effective spacetimes in lower dimensions that can be extracted from a multidimensional generalization of the Schwarzschild-Tangherlini spacetimes derived by Fadeev, Ivashchuk and Melnikov ({\it Phys. Lett,} {\bf A 161} (1991) 98). The higher-dimensional spacetime has $D = (4 + n + m)$ dimensions, where $n$ and $m$ are the number of "internal" and "external" extra dimensions, respectively. We analyze the effective $(4 + n)$ spacetime obtained after dimensional reduction of the $m$ external dimensions. We find that when the $m$ extra dimensions are compact (i) the physics in lower dimensions is independent of $m$ and the character of the singularities in higher dimensions, and (ii) the total gravitational mass $M$ of the effective matter distribution is less than the Schwarzshild mass. In contrast, when the $m$ extra dimensions are large this is not so; the physics in $(4 + n)$ does explicitly depend on $m$, as well as on the nature of the singularities in high dimensions, and the mass of the effective matter distribution (with the exception of wormhole-like distributions) is bigger than the Schwarzshild mass. These results may be relevant to observations for an experimental/observational test of the theory.