Exploring Deformed Metric Tensor through Noncommutative Heisenberg Algebra and Generalized Uncertainty Principle in the Context of Quantum Mechanics and Gravitational Fields

When considering the combination of quantum mechanics (QM) and general relativity (GR), we can extend the fundamental theory of QM by incorporating concepts such as the non-commutative Heisenberg algebra, the generalized uncertainty principle (GUP), and the integration of gravitational fields. This extension leads us to suggest a possible deformation of the metric tensor that combines the effects of QM and GR.
The deformation arises from the non-commutative algebra and the maximal space-like four-acceleration, and it corresponds to curvature in an 8-dimensional spacetime tangent bundle, which is a generalization of Riemannian spacetime. By applying this concept, we can derive a deformed metric tensor that determines how the affine connection on a Riemannian manifold is affected.
In this paper, I explored the symmetric properties of the deformed metric tensor, the affine connection and found out how a parallelly transported tangent vector depends on the spacelike four-acceleration that is given in the units of Length(L) where

\(L=\sqrt{\frac{\hbar\cdot G}{c^ 3}}\)