Relaxation Principle in (n+1) Dimensions

TL;DR

This paper generalizes the relaxation principle to arbitrary (n+1) dimensions, explaining why our universe is 3+1 dimensional by analyzing brane interactions and their dimensionalities.

Contribution

It extends the relaxation principle to any number of spatial dimensions and derives the dimensionality constraints of branes using the Friedman equation and scaling solutions.

Findings

The largest interacting brane in n dimensions is always an (n-2)-brane.

The dimensionality constraint explains the universe's 3+1 dimensions.

The relaxation principle can be generalized beyond the original 9-dimensional context.

Abstract

Why do we live in a (3+1) dimensional universe? In this paper, I review the "relaxation principle" first introduced by Karch and Randall (arXiv:hep-th/0506053v2) and generalize its ideas to an arbitrary (n+1) dimensions. This is done by referring to the Friedman equation and the scaling solution to derive the energy densities of the non-interacting and interacting branes, and then formulating their dimensionalities. I also demonstrate that the largest interacting d-brane in n spatial dimensions is always the (n-2)-brane, and that such dimensionality constraint "relaxed" our universe from nine dimensions to three dimensions.