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Within relativity (both special and general), changes of reference frames can change both the notions of space and of time, with one depending on the other as well. As a consequence, it is necessary to treat both concepts in a unified manner. Hence the term spacetime.
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Spinor space and curved spacetime
I am currently thinking about the Dirac equation in curved (1+1)-dimensional spacetime. … I always thought about spacetime and spinor space as two different spaces. …
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Spin connection raise and lower flat indices
The spin connection $\omega^a_{b\nu}$ is used to define the covariant derivative of a spinor in curved spacetime. … If I try to explicitly compute $\Omega_\nu$, I just get $\Omega_0 = \Omega_1 = 0$, which doesn't make sense in a curved spacetime. …