An Algebraic Reformulation of General Relativity via Clifford Algebra of Dirac Gamma Matrices

We present a rigorous reformulation of Einstein’s General Relativity using the real Clifford algebra Cl1,3, constructed from Dirac gamma matrices. In this framework, all geometric and dynamical structures—including the metric, spin connection, curvature, and energy-momentum tensor—are expressed using algebraic operations (symmetrized products, commutators, and traces) of Clifford generators. Rather than invoking the full machinery of differential geometry, we reconstruct the Einstein field equations entirely within an operator algebra framework, while maintaining exact equivalence with the classical theory. The underlying metric structure is assumed through the anticommutation relations defining the Clifford algebra, and is algebraically reconstructed using trace identities. This approach provides a unified representation of both geometry and spinor fields and may offer conceptual and pedagogical advantages in connecting gravity with operator-based formulations. Potential extensions involving bivector sectors and torsion are briefly discussed.