On quantum‐induced revisiting Einstein tensor in the relativistic regime

Abstract

AbstractFor a rigorous approach to reconcile principles of general relativity (GR) and quantum mechanics (QM), we suggest applying the generalized relativistic noncommutative Heisenberg algebra on spacetime coordinates and momenta of a test particle on curved Riemann geometry manifold, which in turn is extended to an eight‐dimensional manifold. The natural generalization of the four‐dimensional Riemann geometry is the Finsler geometry, in which the quadratic restriction on the length measure is relaxed. With the minimum measurable length derived from the relativistic generalized four‐dimensional uncertainty principle, the quantum‐induced corrections to the fundamental tensor could be determined in the relativistic regime. Accordingly, the affine connections could be revisited. With the quantum‐induced revisiting Riemann curvature tensor and its contractions, the Ricci curvature tensor, and then the Ricci scalar, we have been able to construct a quantum‐induced revision of the Einstein tensor, in which besides quantum‐induced corrections, additional geometric structures emerge. On the surface of the 2‐sphere, we compared the quantum‐induced revisiting and non‐revisiting Einstein tensor and concluded that the difference between both versions strongly depends on the minimum measurable length and the local geodesics of the test particle through the additional curvature.