Four-momentum and Dirac equation in curved spacetime

Question

Norm of four momentum in Minkowski spacetime is proportional to the square of rest mass as

\begin{equation}|P|^2= P^\alpha \eta_{\alpha\beta}P^\beta= (E/c)^2 - p^2 = (mc)^2 \end{equation}

While in a curved spacetime the norm is given as

\begin{equation}|P|^2= P^\alpha g_{\alpha\beta}P^\beta= g_{00}P^0P^0+g_{0i}P^0P^i+g_{ij}P^iP^j=(mc)^2 \end{equation}

Since Dirac equation in Minkowski Spacetime is obtained from first equation, it seems natural to assume that Dirac equation in curved spacetime is obtained from the second. Is it a correct assumption? If yes then can someone please help me with derivation of Dirac equation in curved spacetime starting from the second equation? If this is not a correct assumption, then why not?

This post imported from StackExchange Physics at 2015-10-13 09:53 (UTC), posted by SE-user amateurRebel

Comments

I found some reference arxiv.org/abs/0911.0014 where the equation is derived from the second equation as I had expected but I am not sure if this is valid or not

This post imported from StackExchange Physics at 2015-10-13 09:53 (UTC), posted by SE-user amateurRebel

Answer 1

A better way to think about this is to think about the Dirac equation being obtained from the action of a fermionic field in curved spacetime. Then a lot of the concepts generalize from the Minkowski metric in a more straightforward manner. In flat spacetime we have

\[\int d^4x ~\bar{\Psi} (i \gamma^{\mu} \partial_{\mu} - m) \Psi\]

But in a curved spacetime we move to

\[\int d^4x ~\sqrt{-g} ~\bar{\Psi} (i \gamma^a e^{\mu}_a\nabla_{\mu}-m) \Psi\]

where the \(e_a^{\mu}\) are the vielbein, which allow us to establish a locally Minkowski frame where the standard Dirac matrices \(\gamma^{a}\) satisfy the Clifford algebra, and \(\nabla_{\mu}\)is the covariant derivative.