Applications of the Kretschmann Scalar and Raychaudhuri Equation in f(R, Σ,T) Gravity

This study presents a comprehensive geometric analysis of the Schwarzschild spacetime within the framework of Formula: see text gravity, a theory founded on Parameterized Absolute Parallelism (PAP) geometry. The core of our investigation employs two fundamental tools: the Kretschmann scalar, a curvature invariant, and the Raychaudhuri equation, which governs the evolution of gravitational focusing. We first derive the exact expression for the modified Kretschmann scalar, Formula: see text, which incorporates contributions from both spacetime curvature and torsion, the latter being controlled by the dimensionless PAP parameter Formula: see text. Our analysis confirms that the well-known General Relativity (GR) result, is recovered in the limit Formula: see text, validating our approach. Crucially, we demonstrate that Formula: see text acts as a dial for gravitational strength: values like Formula: see text intensify curvature and tidal forces everywhere, while values like Formula: see text dramatically suppress them, mimicking a repulsive anti-gravity effect. However, the central singularity at Formula: see text persists for all values of Formula: see text. Furthermore, we derive the full Raychaudhuri equation within PAP geometry, revealing new kinematic terms that couple torsion to the expansion, shear, and vorticity of a congruence. This shows that the fundamental theorem of geodesic focusing is altered; torsion can either enhance or oppose the tendency of gravity to cause convergence, implying that the standard energy conditions of GR are no longer sufficient to predict singularity formation. By applying this equation to a radial timelike congruence, we compute its expansion scalar Formula: see text and its rate of change, confirming consistent kinematic behavior within the modified theory. Our work establishes that Formula: see text gravity offers a rich and viable extension of GR, providing a continuous bridge between Riemannian and teleparallel geometries through the parameter Formula: see text, with significant implications for understanding gravitational strength, singularities, and cosmic acceleration.