Energy nonconservation: f (R, T) gravity and noncommutative quantum theory

The f (R, T ) gravity theory was proposed as an extension of the f(R) theories, for which besides geometrical correction terms, proportional to the Ricci scalar R, one has also material correction terms, proportional to the trace of the energy-momentum tensor T. Those material extra terms prevent the energy-momentum tensor of the theory to be conserved. On the other hand, in the context of noncommutative quantum mechanics, the presence of compact dimensions whose coordinates do not commute with time imply that time evolution is discretized, a feature which induces violations of energy conservation. In the present work we propose a connection between these two effects, so that the energy nonconservation observed in the 4-dimensional f(R, T ) gravity can be understood as a macroscopic effect of nonconservative quantum transitions involving the compact extra dimension. It turns out that the energy flows between the ordinary (commutative) 4-dimensional spacetime and the compact extra dimension.