Teleparallel theories of gravity as analogue of non-linear electrodynamics

TL;DR

This paper draws an analogy between teleparallel gravity theories and non-linear electrodynamics, showing how various models can be unified under a common framework using constitutive relations, leading to new insights and constraints.

Contribution

It introduces a unified formulation of teleparallel gravity theories using a constitutive relation approach, revealing new constraints and simplifying the field equations.

Findings

Unified formulation of teleparallel gravity models

Identification of a new constraint relating tetrad and spin connection

Simplified, common form of field equations across models

Abstract

The teleparallel formulation of gravity theories reveals close structural analogies to electrodynamics, which are more hidden in their usual formulation in terms of the curvature of spacetime. We show how every locally Lorentz invariant teleparallel theory of gravity with second order field equations can be understood as built from a gravitational field strength and excitation tensor which are related to each other by a constitutive relation, analogous to the axiomatic construction of theories of electrodynamics. We demonstrate how the previously studied models of $f(\mathbb{T})$ and $f(T_\text{ax},T_\text{ten},T_\text{vec})$ gravity as well as teleparallel dark energy can be formulated in this language. The advantage of this approach to gravity is that the field equations for different models all take the same compact form and general results can be obtained. An important new such result we find is a constraint which relates the field equations of the tetrad and the spin connection.