Gravitational dynamics for all tensorial spacetimes carrying predictive, interpretable and quantizable matter

TL;DR

This paper formulates the problem of deriving gravitational dynamics for general tensorial spacetimes that can support predictive, interpretable, and quantizable matter, as a system of linear PDEs, extending beyond Lorentzian metrics.

Contribution

It introduces a mathematical framework to find gravitational actions for a broad class of tensorial geometries satisfying key physical conditions.

Findings

Reformulation of gravitational dynamics as linear PDEs

Extension of spacetime geometries beyond Lorentzian metrics

Mathematical reduction of modified gravity search

Abstract

Only a severely restricted class of tensor fields can provide classical spacetime geometries, namely those that can carry matter field equations that are predictive, interpretable and quantizable. These three conditions on matter translate into three corresponding algebraic conditions on the underlying tensorial geometry, namely to be hyperbolic, time-orientable and energy-distinguishing. Lorentzian metrics, on which general relativity and the standard model of particle physics are built, present just the simplest tensorial spacetime geometry satisfying these conditions. The problem of finding gravitational dynamics---for the general tensorial spacetime geometries satisfying the above minimum requirements---is reformulated in this paper as a system of linear partial differential equations, in the sense that their solutions yield the actions governing the corresponding spacetime geometry. Thus the search for modified gravitational dynamics is reduced to a clear mathematical task.