The mass  remains a scalar in both cases, but it interacts with the geometry in terms of how the Dirac spinor couples to spacetime curvature.
In spinor-tensor theories and certain quantum gravity theories, these objects interact, and the spin connection (derived from the curvature of spacetime) influences the covariant derivative of the spinor field, linking the geometry to the behavior of fermions.
Conclusion:
While you cannot directly transform gamma matrices into the Ricci tensor or energy-momentum tensor, they do interact in the framework of curved spacetime. The vierbein formalism and the spin connection allow gamma matrices (acting in spinor space) to be consistently integrated with the geometry described by the Ricci tensor and the dynamics described by the energy-momentum tensor.
Can quantum gravity by making formalism of transformation among Dirac equation and general relativity equation?
In principle, quantum gravity aims to unify the quantum behavior of matter (as described by quantum field theory, including the Dirac equation for fermions) with the classical description of spacetime geometry provided by general relativity (via Einstein's equations).Â
However, directly transforming the Dirac equation into the Einstein field equations (or vice versa) is not a straightforward task, as these equations describe different types of physical phenomena—matter and spacetime—though they interact closely.
Key Challenges in Unifying Dirac and General Relativity:
1. Different Mathematical Structures:
Dirac equation deals with fermions (spin-1/2 particles) and is written in terms of spinors, which are representations of the Lorentz group in local flat (Minkowski) spacetime.
General relativity describes the geometry of spacetime using the metric tensor and related curvature quantities (Ricci tensor, Einstein tensor, etc.), which operate in a very different mathematical framework (differential geometry and tensors).
Directly transforming one into the other would involve translating spinor fields and their dynamics into a geometrical language, which is not naturally built into general relativity.
