A Generalization of Gravity

TL;DR

This paper explores generalized theories of gravity involving additional symmetric tensors beyond the metric, leading to new geometric frameworks that extend Einstein gravity and could impact cosmology and Lorentz symmetry breaking.

Contribution

It introduces a broad class of diff-invariant gravitational theories built from symmetric tensors and Cayley's hyperdeterminants, extending the metric paradigm and including metric-less models.

Findings

Theories encompass Einstein gravity as a special case.

Possible spontaneous Lorentz symmetry breaking.

Relevance to dark sector cosmology.

Abstract

I consider theories of gravity built not just from the metric and affine connection, but also other (possibly higher rank) symmetric tensor(s). The Lagrangian densities are scalars built from them, and the volume forms are related to Cayley's hyperdeterminants. The resulting diff-invariant actions give rise to geometric theories that go beyond the metric paradigm (even metric-less theories are possible), and contain Einstein gravity as a special case. Examples contain theories with generalizeations of Riemannian geometry. The 0-tensor case is related to dilaton gravity. These theories can give rise to new types of spontaneous Lorentz breaking and might be relevant for "dark" sector cosmology.