On possible attenuation of space singularity in modified general relativity with quantum-deformed conformal metric

With a geometric quantization ansatz that combines phase-space Finsler/Hamilton geometry and the Relativistic Generalized Uncertainty Principle (RGUP), the metric tensor in general relativity (GR) is approximately conformally reformulated. Thereby, different quantum-mechanical operators can be properly reconciled with the principles of GR. The phase-space Hessian associated with the squared Finsler/Hamilton structure [Formula: see text] results in a phase-space metric tensor that can be transformed into a four-dimensional metric tensor which is approximately conformally revisited. The conformal coefficient [Formula: see text] depends on space–time, momentum space and the maximal proper force. By means of analytical and numerical timelike geodesic congruence associated with the Schwarzschild metric, the spherically symmetric vacuum solution of the Einstein field equations, it was concluded that the sign and magnitude of [Formula: see text] play a crucial role in governing the Raychaudhuri expansion: specifically, [Formula: see text] tends to reduce focusing and thereby reduce singular behavior, while [Formula: see text] tends to increase focusing and thereby enhance singular behavior. Therefore, the space singularity can be partially controlled or entirely regulated depending on the approximate values assigned to [Formula: see text]. We investigate the assumptions used to treat [Formula: see text] as an effective Hamiltonian and to truncate the quantized metric tensor, as well as their relevance to the maximal proper force. Although there are considerable limitations to the proposed formulation due to various approximations, such as the estimation of quantum operators, we conclude that rigorous operator approaches open up numerous research possibilities.