Metric Measure Space as a Framework for Gravitation
Nafiseh Rahmanpour, Hossein Shojaie

TL;DR
This paper introduces metric measure space as a new framework for gravitation that incorporates conformal invariance, modifies geometric operators, and links to quantum mechanics through a proposed relation with Bohmian trajectories.
Contribution
It develops conformally invariant gravitational field equations within metric measure space and connects this framework to quantum potential and Bohmian mechanics.
Findings
Modified divergence operator in metric measure space.
Generalized conservation of energy-momentum tensor.
Relation between density function and quantum potential.
Abstract
In this manuscript, we show how conformal invariance can be incorporated in a classical theory of gravitation, in the context of metric measure space. Metric measure space involves a geometrical scalar , dubbed as density function, which here appears as a conformal degree of freedom. In this framework, we present conformally invariant field equations, the relevant identities and geodesic equations. In metric measure space, the volume element and accordingly the operators with integral based definitions are modified. For instance, the divergence operator in this space differs from the Riemannian one. As a result, a gravitational theory formulated in this space has a generalized second Bianchi identity and a generalized conservation of energy-momentum tensor. It is shown how, by using the generalized identity for conservation of energy-momentum tensor, one can obtain a conformally…
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