A 256×256 Matrix Form of the Dirac Equation in Curved Spacetime for QED Scattering Without an Explicit Vierbein

Abstract We present a matrix-based formulation of the Dirac equation in curved spacetime that avoids introducing an explicit vierbein and spin connection. The method assembles sixteen two-index gamma matrices into a single 256×256 representation and embeds the metric components directly in the matrix entries. Spacetime remains four dimensional; the number sixteen labels the basis elements of the Dirac algebra. Starting from a Lagrangian identical in form to that of quantum electrodynamics (QED) in flat spacetime, the approach replaces differential-geometric manipulations with matrix products and traces, enabling straightforward symbolic and numerical automation. We develop scattering calculations for Compton, muon–pair production in electron–positron collisions, Møller, and Bhabha processes. Illustrative examples with constant metrics, including off-diagonal components, exhibit characteristic angular modifications of the differential cross sections, whereas in the flat-metric limit the results agree exactly with standard formulas. We delineate the practical scope of the method—constant or slowly varying backgrounds—and identify extensions to coordinate-dependent metrics and loop calculations as priorities for future work, including tests of whether curvature-induced structure can influence high-energy behavior. The formulation thus offers a pragmatic alternative for exploring fermionic processes in gently curved backgrounds while remaining consistent with flat-space quantum electrodynamics.