Can one find Dirac matrices for any spacetime metric?

Question

For any metric $$g_{μν}$$ is there always a linearly independant spacetime algebra satisfying $$\{\bar{γ}_μ,\bar{γ}_ν\} = 2 g_{μν} I?$$

For a diagonal metric I was able to work out that $$\bar{γ}_μ=\sqrt{n_{μμ}*g_{μμ}}γ_μ$$ satisfied these conditions (the minkowski simply adds negatives to cancel the spacelike gammas). However for metrics which are not diagonalizable at every point in spacetime I've been having trouble.

$$+---$$ is being used here.

Playing around with Tetrads seems like the way to go but I havent had as much luck this far. Thanks in advance to any help!

Answer 1

Yes, tetrads are the way to go. Suppose we have Dirac matrices for Minkowski space, $$ \{ \gamma^a, \gamma^b \} = 2 \eta^{ab} I, $$ and we also have a tetrad/vierbein, $e_a^\mu$, [1] $$ g_{\mu\nu}\ e_a^\mu\ e_b^\nu = \eta_{ab}, \qquad \eta^{ab}\ e_a^\mu\ e_b^\nu = g^{\mu\nu}. $$ Then we can define $\bar{\gamma}^\mu$: $$ \bar{\gamma}^\mu = e_a^\mu\ \gamma^a, \qquad \{ \bar{\gamma}^\mu, \bar{\gamma}^\nu \} = e_a^\mu\ e_b^\nu\ \{ \gamma^a, \gamma^b \} = e_a^\mu\ e_b^\nu\ 2 \eta^{ab} I = 2 g^{\mu\nu} I. $$ When dealing with spinors in curved space, people usually stick to the tetrad formalism.

References:
Carroll, Spacetime and geometry (the last appendix).
Weinberg, The Quantum Theory of Fields: Volume 3 (Section 31.1)

[1] As alluded to by Qmechanic and doetoe in the comments above, there can be topological problems with finding a tetrad or defining spinors. I'd ask them for references if you want to know more about this.