1/r potential in higher dimensions

TL;DR

This paper explores how gravitational potentials behave in higher dimensions within Einstein and Lovelock gravity, revealing conditions under which the potential mimics the familiar 1/r form and discussing implications for gravitational wave properties.

Contribution

It identifies specific dimensional conditions where Lovelock gravity reproduces the 1/r potential, linking it to Einstein gravity in four dimensions and analyzing gravitational wave degrees of freedom.

Findings

In Einstein gravity, potential scales as 1/r^{d-3} in d dimensions.

Pure Lovelock gravity yields a 1/r potential in dimensions d=3m+1.

Gravitational wave polarizations differ between four dimensions and higher-dimensional theories.

Abstract

In Einstein gravity, gravitational potential goes as $1/r^{d-3}$ in $d$ non-compactified spacetime dimensions, which assumes the familiar $1/r$ form in four dimensions. On the other hand, it goes as $1/r^{\alpha}$, with $\alpha=(d-2m-1)/m$, in pure Lovelock gravity involving only one $m$th order term of the Lovelock polynomial in the gravitational action. The latter offers a novel possibility of having $1/r$ potential for the non-compactified dimension spectrum given by $d=3m+1$. Thus it turns out that in the two prototype gravitational settings of isolated objects, like black holes and the universe as a whole --- cosmological models, the Einstein gravity in four and $m$th order pure Lovelock gravity in $3m+1$ dimensions behave in a similar fashion as far as gravitational interactions are considered. However propagation of gravitational waves (or the number of degrees of freedom) does indeed serve as a discriminator because it has two polarizations only in four dimensions.