On Kottler's path: origin and evolution of the premetric program in gravity and in electrodynamics

TL;DR

This paper explores Kottler's 1922 program to formulate gravity and electrodynamics without relying on spacetime metrics, highlighting the success in electrodynamics and potential approaches for gravity.

Contribution

It analyzes the premetric formulation of electrodynamics and gravity, demonstrating the viability of metric-independent descriptions and proposing future directions using teleparallelism.

Findings

Electrodynamics admits a consistent premetric formulation.

Kottler's approach to gravity yields quasi-Maxwellian equations.

Potential of teleparallelism to develop premetric gravity theories.

Abstract

In 1922, Kottler put forward the program to remove the gravitational potential, the metric of spacetime, from the fundamental equations in physics as far as possible. He successfully applied this idea to Newton's gravitostatics and to Maxwell's electrodynamics, where Kottler recast the field equations in premetric form and specified a metric-dependent constitutive law. We will discuss the basics of the premetric approach and some of its beautiful consequences, like the division of universal constants into two classes. We show that classical electrodynamics can be developed without a metric quite straightforwardly: the Maxwell equations, together with a local and linear response law for electromagnetic media, admit a consistent premetric formulation. Kottler's program succeeds here without provisos. In Kottler's approach to gravity, making the theory relativistic, two premetric quasi-Maxwellian field equations arise, but their field variables, if interpreted in terms of general relativity, do depend on the metric. However, one can hope to bring the Kottler idea to work by using the teleparallelism equivalent of general relativity, where the gravitational potential, the coframe, can be chosen in a premetric way.