Mach's Principle selects 4 space-time dimensions

TL;DR

This paper investigates how Mach's principle influences the dimensionality of space-time by analyzing the ghost-free conditions of a Mach operator derived from Einstein's equations, finding that 4D space-time uniquely avoids ghosts.

Contribution

It introduces a novel approach linking Mach's principle to space-time dimensionality through ghost analysis of the Mach operator in various backgrounds.

Findings

Ghosts appear in the Mach operator on most backgrounds.

Only in 4-dimensional space-time does the Mach operator lack tensor ghosts.

The ghost-free condition constrains the possible dimensions of certain background spaces.

Abstract

Bi-tensor kernel in integral form of Einstein equations realizing Mach's idea of non-existence of empty space-times is taken as an inverse of differential operator ("Mach operator") defined conventionally as a second variation of Einstein's gravity Action over contravariant components of metric tensor. The choice of transverse gauge condition used in this definition does not influence results of the paper since only transverse and traceless tensor modes written on different background space-times are studied. Presence of ghosts among modes of Mach operator invalidates the integral formulation of Einstein equations. And the demand of absence of these ghosts proves to be a selection rule for dimensionality of the background space-time. In particular Mach operator written on De Sitter background or on the background of so called "Einstein Universe" does not possess tensor ghosts only in 4-dimensions. The similar demand gives non-trivial formula for dimensionalities of subspaces of the Freund-Rubin background.