Dirac equation in curved spacetime

In mathematical physics, the Dirac equation in curved spacetime generalizes the original Dirac equation to curved space.

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It can be written by using vierbein fields and the gravitational spin connection. The vierbein defines a local rest frame, allowing the constant Gamma matrices to act at each spacetime point. In this way, Dirac's equation takes the following form in curved spacetime:[1]

${\displaystyle i\gamma ^{a}e_{a}^{\mu }D_{\mu }\Psi -m\Psi =0.}$

Here e_a^μ is the vierbein and D_μ is the covariant derivative for fermionic fields, defined as follows

${\displaystyle D_{\mu }=\partial _{\mu }-{\frac {i}{4}}\omega _{\mu }^{ab}\sigma _{ab},}$

where σ_ab is the commutator of Gamma matrices:

${\displaystyle \sigma _{ab}={\frac {i}{2}}\left[\gamma _{a},\gamma _{b}\right],}$

and ω_μ^ab are the spin connection components.

The modified Klein–Gordon equation obtained by squaring the operator in the Dirac equation, first found by Erwin Schrödinger as cited by Pollock [2] is given by

${\displaystyle \left({\frac {1}{\sqrt {-\det g}}}\,{\cal {D}}_{\mu }\left({\sqrt {-\det g}}\,g^{\mu \nu }{\cal {D}}_{\nu }\right)-{\frac {1}{4}}R+{\frac {ie}{2}}F_{\mu \nu }s^{\mu \nu }-m^{2}\right)\Psi =0.}$

where ${\displaystyle R}$ is the Ricci scalar, and ${\displaystyle F_{\mu \nu }}$ is the field strength of ${\displaystyle A_{\mu }}$. An alternative version of the Dirac equation whose Dirac operator remains the square root of the Laplacian is given by the Dirac–Kähler equation; the price to pay is the loss of Lorentz invariance in curved spacetime.

Note that here Latin indices denote the "Lorentzian" vierbein labels while Greek indices denote manifold coordinate indices.