Gravitational Energy-Momentum in the Tetrad and Quadratic Spinor Representations of General Relativity

TL;DR

This paper explores gravitational energy-momentum in different representations of General Relativity, showing how local Lorentz invariance affects tensor properties and proposing a tensorial energy-momentum density using spinor degrees of freedom.

Contribution

It demonstrates the equivalence of Lorentz freedom in tetrad formalism with spinor degrees of freedom in quadratic spinor representation, enabling a tensorial energy-momentum density.

Findings

Moller energy-momentum is a tensor under coordinate transformations but not under Lorentz rotations.

Spinor degrees of freedom correspond to local Lorentz invariance in the quadratic spinor representation.

A new local energy-momentum density tensor is proposed that is invariant under both coordinate and Lorentz transformations.

Abstract

In the Tetrad Representation of General Relativity, the energy-momentum expression, found by Moller in 1961, is a tensor wrt coordinate transformations but is not a tensor wrt local Lorentz frame rotations. This local Lorentz freedom is shown to be the same as the six parameter normalized spinor degrees of freedom in the Quadratic Spinor Representation of General Relativity. From the viewpoint of a gravitational field theory in flat space-time, these extra spinor degrees of freedom allow us to obtain a local energy-momentum density which is a true tensor over both coordinate and local Lorentz frame rotations.