Gravitational Field Tensor

TL;DR

This paper introduces a new tensorial representation of the gravitational field, which simplifies the Lagrangian formulation and aligns with Newtonian intuition, with potential applications to gravitational wave analysis.

Contribution

It proposes the gravitational field tensor as a tensorial alternative to the affine connection, with a simple scalar Lagrangian and clear physical interpretation.

Findings

Derived the gravitational field tensor for Schwarzschild metric

Presented a scalar Lagrangian density depending only on the metric and its derivatives

Suggested advantages for studying gravitational waves using this tensor

Abstract

We present a tensorial relative of the familiar affine connection and argue that it should be regarded as the gravitational field tensor. Remarkably, the Lagrangian density expressed in terms of this tensor has a simple form, which depends only on the metric and its first derivatives and, moreover, is a true scalar quantity. The geodesic equation, moreover, shows that our tensor plays a role that is strongly reminiscent of the gravitational field in Newtonian mechanics and this, together with other evidence, which we present, leads us to identify it as the gravitational field tensor. We calculate the gravitational field tensor for the Schwarzschild metric. We suggest some of the advantages to be gained from applying our tensor to the study of gravitational waves.