A vacuum-like configuration in General Relativity as a manifestation of a Lorentz-invariant mode of five-dimensional gravity

TL;DR

This paper constructs a Lorentz-invariant cosmological model within five-dimensional gravity, demonstrating that vacuum configurations in general relativity can originate from a Lorentz-invariant mode of five-dimensional space, linking higher-dimensional theories to GR vacuum states.

Contribution

It proves a five-dimensional analog of the Birkhoff theorem, showing how vacuum-like configurations in GR emerge from a Lorentz-invariant mode of 5D gravity, with implications for Kaluza-Klein models.

Findings

A Lorentz-invariant mode of 5D gravity corresponds to vacuum configurations in GR.

The model demonstrates a dynamically realized Kaluza-Klein scenario with a scalar field.

Vacuum configurations in GR can be interpreted as zero modes of 5D space.

Abstract

A Lorentz-invariant cosmological model is constructed within the framework of five-dimensional gravity. The five-dimensional theorem which is analogical to the generalized Birkhoff theorem is proved, that corresponds to the Kaluza's ``cylinder condition''. The five-dimensional vacuum Einstein equations have an integral of motion corresponding to this symmetry, the integral of motion is similar to the mass function in general relativity (GR). Space closure with respect to the extra dimensionality follows from the requirement of the absence of a conical singularity. Thus, the Kaluza-Klein (KK) model is realized dynamically as a Lorentz-invariant mode of five-dimensional general relativity. After the dimensional reduction and conformal mapping the model is reduced to the GR configuration. It contains a scalar field with a vanishing conformally invariant energy-momentum tensor on the flat space-time background. This zero mode can be interpreted as a vacuum configuration in GR. As a result the vacuum-like configuration in GR can be considered as a manifestation of the Lorentz-invariant empty five-dimensional space.