Gravity on parallelizable manifold

TL;DR

This paper proposes a new gravitational field theory on parallelizable manifolds, deriving field equations from an action principle, and finds solutions close to Schwarzschild black holes with distinct scalar curvature properties.

Contribution

It introduces a gravitational model on parallelizable manifolds with a quadratic Lagrangian, leading to unique solutions like the Rosen metric, differing from classical general relativity.

Findings

Derivation of field equations as a second order Laplacian-type system.

Existence of a unique static, spherically symmetric solution close to Schwarzschild.

The model admits black hole solutions with positive scalar curvature.

Abstract

General relativity postulates that the gravity field is defined on a Riemannian manifold. The field equations are $R^\mu_\nu = 0$ i.e. Ricci's curvature tensor vanishes. The field equations have to be augmented by natural physical requirements like orientability, time orientability and existence of a spinorial structure. Moreover, it is impossible to define the energy of the gravity field only by the metric tensor. We suggest to impose an additional structure and consider parallelizable manifold i.e. manifolds for which a smooth field of frames exists. The derivation of the field equations is by an action principle. A Lagrangian, which is quadratic in the differentials, is defined. The minimum of the action is achieved at a $16 \times 16$ second order quasi linear system. It is of Laplacian type. The system admits a unique exact solution for a centrally symmetric, static and assimptotically flat field. The resulting metric is the celebrated Rosen metric, which is very close to the Schwarzchild metric. The two are intrinsicly different since the scalar curvature of the Schuarzschild metric is zero, in contrast to Rosen's which is positive. The suggested field admits black holes which are briefly discussed.