Quantum-induced revisiting space–time curvature in relativistic regime

General relativity and quantum mechanics are not only fundamentally different theories giving explanations of how nature works but also genuinely incompatible descriptions of reality. The violation of the principles of relativity and equivalence, at quantum scales, belongs to the main conceptual difficulties of reconciling principles of quantum mechanics with general relativity so that their scales of applicability are entirely distinct. When generalized noncommutative Heisenberg algebra accommodating impacts of finite gravitational fields as specified by quantum loop gravity, doubly special relativity, and string theory, for instance, is thoughtfully applied to the eight-dimensional Finsler manifold. In the natural generalization of the pseudo-Riemannian manifold, in which the quadratic restriction on the length measure is relaxed, we have been able to define quantum-induced revisiting fundamental tensor in relativistic regime and thereby extending its applicability to quantum scales. By constructing the affine connections on a four-dimensional pseudo-Riemannian manifold, we have determined the quantum-induced revisiting Riemann curvature tensor and its contractions, the Ricci curvature tensor, and scalar in relativistic regime. Consequently, we have been able to construct the Einstein tensor, in which besides quantization additional geometric structures and curvatures are emerged. As in Einstein’s classical theory of general relativity, we have proved that the covariant derivative of the modified Einstein tensor vanishes, as well. We conclude that the quantum-induced corrections establish quantum properties to the space–time coordinates and momenta. Accordingly, the space–time curvature endows additional curvature and geometrical structure as well as discretization which likely enable sensical predictions of Einstein’s general relativity, at the quantum scale.