  Following our work from the previous paper about the study of effective Dirac
algebra and the metric of the simple, special case of relativistic hydrogen
atom, this paper gives the complete metric study defined by the effective Dirac
algebra in the Dirac and Weyl presentation, showing that relativistic
electromagnetic interaction gives the correction of the flat background metric
$\eta_{\mu\nu}$, thus curving spacetime. The curved metric can be nicely broken
down into two parts, the pure correction on the flat spacetime metric and the
projection tensor. We find that the curved metric is independent of the
representation chosen.
