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

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.

Gravitational properties of light—the gravitational field of a laser pulse

“…). In the following, we call the first three independent components in equation (21) longitudinal curvature components referring to the majority of their indices being 0 or z while the tangent vector to the trajectory of our pulse has non-zero components 0 and z. In contrast, we will call the last three components in equation (21) transversal.…”

Gravitational closure of matter field equations

“…Employing the technology developed in the previous section, we now significantly improve and extend the results of [5]. The crucial technical advance is the identification of the geometric phase space with the non-tensorial configuration variables and canonically conjugate momentum densities, whose transformation behavior already captures the non-linear constraints on the canonical geometry that was left to be implemented only afterwards in previous treatments.…”

“…Thirdly, in section IV, we now convert the entire constraint algebra for the gravitational dynamics into a countable set of linear homogeneous partial differential equations, for whose solution powerful methods are available [6]. Unlike the construction in [5], this reveals one single and immutable set of equations for the gravitational Lagrangian. Only the coefficient functions appearing in these partial differential equations vary with the choice of matter dynamics and can now be constructed swiftly according to simple rules.…”

Etherington’s Distance Duality with Birefringence

“…Indeed, there is no need to artificially construct a notion of a covariant derivative, neither in the standard theory nor here, since all equations follow without such theoretical overhead. Having identified the appropriate notion of energy-momentum for our spacetime, we can now compute it from the action (11). The corresponding equations of motion are a set of modified Maxwell's equations,…”

“…where k a = −∂ a S is the wave covector as usual. Then, the energy-momentum can be found from (11), and, upon averaging over the period of oscillation, one obtains…”