Thermodynamic Relativistic Gravity--Classical General Relativity

This paper presents the proposed Curvature Energy Metric Tensor within the Thermodynamic Relativistic Gravity--Classical General Relativity (TRG-CGR) framework. The framework is developed as a hybrid extended-gravity approach that incorporates selected features of General Relativity, classical Newtonian gravity, Modified Newtonian Dynamics, and thermodynamic and entropic interpretations of gravitation. The central formulation begins with a total curvature-energy equation comprising internal gravitational energy, relativistic pressure energy, and entropy-associated energy. By defining the gravitational potential as the total curvature energy per unit mass, a generalized gravitational potential is obtained. This potential is subsequently introduced into the weak-field metric correspondence to construct a generalized static, spherically symmetric spacetime metric. The resulting metric contains a Newtonian contribution, a proposed higher-order curvature correction, and a cosmological curvature contribution. The radial acceleration associated with the total energy is also derived, and its relationship to the proposed metric is examined. Furthermore, the limiting forms of the metric are compared with the Schwarzschild, Schwarzschild--de Sitter, and Reissner--Nordström geometries. A comparison with the Friedmann--Lemaître--Robertson--Walker acceleration equation is included to examine the role of homogeneous pressure and the proposed higher-order correction. The resulting metric is presented as a theoretical ansatz within the TRG-CGR framework, whose complete relativistic consistency requires further investigation through its governing field equations and observational tests.