A gauge theoretical view of the charge concept in Einstein gravity

TL;DR

This paper explores the analogy between gauge theories and gravity to clarify the concept of charge in Einstein gravity, revealing how mass and other parameters relate to monopole and topological charges within a gauge theoretical framework.

Contribution

It provides a dimensional analysis and strict definitions of charges in gravity, linking gravitational parameters to gauge charges and interpreting classical solutions as monopole or topological charges.

Findings

Mass is an elementary charge of the translation group.

Schwarzschild mass is a quasi-electric monopole charge.

NUT parameter is a quasi-magnetic monopole and topological charge.

Abstract

We will discuss some analogies between internal gauge theories and gravity in order to better understand the charge concept in gravity. A dimensional analysis of gauge theories in general and a strict definition of elementary, monopole, and topological charges are applied to electromagnetism and to teleparallelism, a gauge theoretical formulation of Einstein gravity. As a result we inevitably find that the gravitational coupling constant has dimension $\hbar/l^2$, the mass parameter of a particle dimension $\hbar/l$, and the Schwarzschild mass parameter dimension l (where l means length). These dimensions confirm the meaning of mass as elementary and as monopole charge of the translation group, respectively. In detail, we find that the Schwarzschild mass parameter is a quasi-electric monopole charge of the time translation whereas the NUT parameter is a quasi-magnetic monopole charge of the time translation as well as a topological charge. The Kerr parameter and the electric and magnetic charges are interpreted similarly. We conclude that each elementary charge of a Casimir operator of the gauge group is the source of a (quasi-electric) monopole charge of the respective Killing vector.