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I'm reviewing an exam, and I can't figure this one out. I know the covariant derivative, but I'm not seeing how it necessarily has a mass dimension.

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A spatial (temporal) covariant derivative $\nabla_{\mu}$ has dimension $[\nabla_{\mu}]$ of inverse length (time), respectively. In natural units where $\hbar=1=c$, that is also dimension of mass.

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  • $\begingroup$ However, in geometric units ($G=c=1$) units of inverse length would be inverse mass as well. I guess this depends on the context of the question. $\endgroup$ – mmeent Dec 3 '17 at 22:09
  • $\begingroup$ Dang it. It's one of those things that in retrospect is obvious. Thanks @Qmechanic $\endgroup$ – Zack Vacanti-Mitchell Dec 3 '17 at 23:22
  • $\begingroup$ @mmeent: Right. Moreover, in Planck units $c=\hbar=G=1$, length, time and mass are all dimensionless. $\endgroup$ – Qmechanic Dec 4 '17 at 19:06

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