The Physical Interpretation of the Lanczos Tensor

TL;DR

This paper explores the analogy between the Lanczos tensor in general relativity and the vector potential in electromagnetism, proposing that the Lanczos tensor may have a significant quantum role similar to the Aharonov-Bohm effect.

Contribution

It introduces a novel interpretation of the Lanczos tensor's physical significance and suggests its potential importance in quantum gravity based on electromagnetic analogies.

Findings

Lanczos tensor can be viewed as analogous to the electromagnetic vector potential.

Quantum significance of the Lanczos tensor is proposed based on electromagnetic analogy.

Potential pathway for quantizing gravity via the Lanczos tensor is discussed.

Abstract

The field equations of general relativity can be written as first order differential equations in the Weyl tensor, the Weyl tensor in turn can be written as a first order differential equation in a three index tensor called the Lanczos tensor. The Lanczos tensor plays a similar role in general relativity to that of the vector potential in electro-magnetic theory. The Aharonov-Bohm effect shows that when quantum mechanics is applied to electro-magnetic theory the vector potential is dynamically significant, even when the electro-magnetic field tensor $F_{ab}$ vanishes. Here it is assumed that in the quantum realm the Lanczos tensor is dynamically significant, and this leads to an attempt to quantize the gravitational field by pursuing the analogy between the vector field and the Lanczos tensor.