When Dirac derived his equation he started from the non relativistic time dependent Schrodinger equation and then treated the partial derivative of time as the partial derivative of position and then he had to change the $H$ and the $E$ to make them compatible with SR.However I dont see the problem in writing a equation for particles in curved spacetime the only think we have to additionaly take into account is the metric tensor $g_{\mu\nu} $ since in order to get the equation describing the curvature of spacetime we multiply the metric tensor with the Minkowskian manifold.So why havent we done this?


1 Answer 1


We have. The procedure for moving from special to general relativity is actually pretty easy:

  1. replace all $\eta_{ab}$ inner products with $g_{ab}$ inner products in the Lagrangian
  2. do the same for all partial derivatives, moving $\partial_{a}$ terms into $\nabla_{a}$ terms
  3. Add the Hilbert term $\frac{1}{16\pi G}R$ to the Lagrangian
  4. promote the measure $d^{4}x$ to $\sqrt{|g|}d^{4}x$
  5. treat $g_{ab}$ (or, equivalently, $g^{ab}$) as a variable term when takign the variation

The problem is, that this procedure produces nonlinear classical equations of motion that are very difficult to solve in practice, and when quantized, produces a theory with ultraviolet divergences


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