Exact Solutions of the Field Equations for Empty Space in the Nash Gravitational Theory

TL;DR

This paper derives exact solutions to Nash's gravitational field equations for empty space, revealing connections to known spacetimes like Schwarzschild-de Sitter and de Sitter, and explores implications for dark energy and matter inclusion.

Contribution

It provides the first exact solutions of Nash's field equations for empty space, linking them to classical spacetimes and suggesting a potential explanation for dark energy.

Findings

Static case yields Schwarzschild-de Sitter spacetime.

Cosmological case results in de Sitter universe.

Empty space corresponds to Lorentz Invariant Vacuum Energy.

Abstract

John Nash has proposed a new theory of gravity. We define a Nash-tensor equal to the curvature tensor appearing in the Nash field equations for empty space, and calculate its components for two cases. 1. A static, spherically symmetric space, and 2. The expanding, homogeneous and isotropic space of the Friedmann-Lemaitre-Robertson-Walker (FLRW) universe models. We find the general, exact solution of the Nash field equations for empty space in the static case. The line element turns out to represent the Schwarzschild-de Sitter spacetime. Also we find the simplest non-trivial solution of the field equations in the cosmological case, which gives the scale factor corresponding to the de Sitter spacetime. Hence empty space in the Nash theory corresponds to a space with Lorentz Invariant Vacuum Energy (LIVE) in the Einstein theory. This suggests that dark energy may be superfluous according to the Nash theory. We also consider a radiation filled universe model in an effort to find out how energy and matter may be incorporated into the Nash theory. A tentative interpretation of the Nash theory as a unified theory of gravity and electromagnetism leads to a very simple form of the field equations in the presence of matter. It should be noted, however, that the Nash theory is still unfinished. A satifying way of including energy-momentum into the theory has yet to be found.