Mach's Principle selects 4 space-time dimensions

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.