Normalization conventions for Newton's constant and the Planck scale in arbitrary spacetime dimension

TL;DR

This paper derives the relationship between the Einstein-Hilbert action coefficient and Newton's constant in arbitrary spacetime dimensions, clarifying normalization conventions and their implications for the Planck scale and brane world models.

Contribution

It provides a general formula connecting Einstein-Hilbert coefficient and Newton's constant in any dimension, extending the well-known four-dimensional case.

Findings

Derived the formula K^2=2[(d-2)/(d-3)]Vol(S^[d-2])G_N for arbitrary dimensions

Clarified the normalization conventions for the Planck scale in different dimensions

Discussed implications for brane world models and Planck mass definitions

Abstract

We calculate, in d spacetime dimensions, the relationship between the coefficient 1/K^2 of the Einstein-Hilbert term in the action of general relativity and the coefficient G_N of the force law that results from the Newtonian limit of general relativity. The result is K^2=2[(d-2)/(d-3)]Vol(S^[d-2])G_N, where Vol(S^n) is the volume of the unit n-sphere. While the d=4 case is an elementary calculation in any general relativity text, the arbitrary case presented here is slightly less well known. We discuss the relevance of this result for the definition of the so-called "reduced Planck mass" and comment very briefly on the implications for brane world models. [abstract abridged]