The Dirac equation in curved spacetimes using coordinate-free notation

TL;DR

This paper formulates the Dirac equation in curved spacetimes using coordinate-free notation, demonstrating invariance under local coframe rotations, and derives related physical quantities like the stress-energy tensor.

Contribution

It introduces a coordinate-free formulation of the Dirac equation in curved spacetimes, highlighting invariance properties and providing explicit expressions for physical tensors.

Findings

The Dirac equation is invariant under local coframe rotations.

A conserved current independent of local coframe orientation is established.

The stress-energy tensor for the Dirac field is explicitly calculated.

Abstract

The Dirac equation in curved spacetimes is formulated using coordinate-free notation. A Lagrangean density which corresponds to the subject equation is presented. The subject equation is invariant under a local rotation of the coframe. The current is independent of the local orientation of the coframe and it is conserved. The subject equation has an equivalent formulation which uses the Christoffel gamma. The stress-energy tensor is calculated.