  Conventional relativistic electrodynamics is set on flat Minkowski spacetime,
where all computable quantities are calculated from the flat metric
$\eta_{\mu\nu}$. We can redefine the metric of spacetime from the Dirac
algebra. In this paper, we study how an electrodynamic interaction can alter
the normal gamma matrix to an effective one and result in a shift in the metric
perturbatively. The curvature properties inferred from the curved metric are
also investigated. We also study how the spin operator is changed under the
interaction that contribute to an effective spin operator and how the spin of
an electron will be slightly deviated from $1/2$. Then we perform canonical
quantization of the effective Dirac algebra. Finally we apply our results to
the relativistic hydrogen case and demonstrate how such system curves the
spacetime metric.
