A Memory-Regulated General Relativity: Modifying the Einstein Field Equations to Resolve Spacetime Singularities

The Einstein Field Equations (EFE) of General Relativity predict the formation of spacetime singularities (e.g., at the center of black holes), where curvature diverges to infinity and the theory breaks down. In this study, we propose a modified version of the EFE by introducing a non-Markovian Memory Tensor (M_mu_nu). This memory tensor, inspired by the AMDCF framework, encodes the historical evolution of the mass distribution and acts as an adaptive repulsive force at the Planck scale. The simulation demonstrates that the Memory Tensor successfully regulates the 1/r curvature divergence, producing a 'Regular Black Hole' without a singularity. This framework offers a novel approach to quantum gravity and provides potential explanations for Dark Matter and Dark Energy as memory effects of the early universe. Addendum: On the Non-Markovian Nature of Gravity and the History Operator Following the publication of the modified Einstein Field Equations, an advanced theoretical formulation and corresponding numerical simulations have been developed to rigorously address the nature of the Memory Tensor. The results demonstrate that gravity itself, at its fundamental level, behaves as a non-Markovian memory. We refine the formulation by replacing the simple memory tensor with a History Operator (H_μν[T]), defined as an integral over the past states of the energy-momentum tensor weighted by a memory kernel K(t - t'): H_μν(t) = ∫ K(t - t') T_μν(t') dt' This advanced spacetime geometry, which incorporates historical integration, aligns perfectly with recent findings in quantum gravity (Gravitational Memory Effect). The numerical simulations provide the following proofs: 1. Singularity Avoidance (The Plateau Effect):The gravitational collapse does not proceed to an infinite singularity (1/r divergence). Instead, the non-Markovian memory creates an adaptive repulsive force at the Planck scale, resulting in a mathematically proven "Plateau" of finite curvature. This successfully models a Regular Black Hole. 2. Thermodynamic Stabilization:The closed-loop system achieves absolute thermal equilibrium. The memory kernel prevents the scattering or collapse of energy by dampening the high-frequency fluctuations near the event horizon (Weak Gravity as a Damping Valve). 3. Total Mass Conservation:Unlike classical General Relativity, where mass vanishes into the singularity, the history-dependent spacetime conserves total mass and entropy. This provides a compelling resolution to the Black Hole Information Paradox. This unified framework suggests that Dark Matter and Dark Energy are not exotic particles, but rather the macroscopic manifestation of the universe's gravitational memory of its early expansion and structure formation.