Gravity on a parallelizable manifold. Exact solutions

TL;DR

This paper explores a wave-type field equation for gravity on a parallelizable manifold, deriving exact static and cosmological solutions that connect classical Newtonian gravity with a relativistic framework.

Contribution

It introduces a novel wave-type field equation for gravity, linking classical solutions to relativistic models via a scalar harmonic function, and provides exact static and cosmological solutions.

Findings

The wave equation admits a unique static, spherical, asymptotically-flat solution.

The pseudo-conformal frame satisfies the wave equation when the conformal factor relates to a harmonic function.

A new class of exact static and cosmological solutions is derived.

Abstract

The wave type field equation $\square \vt^a=\la \vt^a$, where $\vt^a$ is a coframe field on a space-time, was recently proposed to describe the gravity field. This equation has a unique static, spherical-symmetric, asymptotically-flat solution, which leads to the viable Yilmaz-Rosen metric. We show that the wave type field equation is satisfied by the pseudo-conformal frame if the conformal factor is determined by a scalar 3D-harmonic function. This function can be related to the Newtonian potential of classical gravity. So we obtain a direct relation between the non-relativistic gravity and the relativistic model: every classical exact solution leads to a solution of the field equation. With this result we obtain a wide class of exact, static metrics. We show that the theory of Yilmaz relates to the pseudo-conformal sector of our construction. We derive also a unique cosmological (time dependent) solution of the described type.