I'm following the derivation of Fermi coordinates in MTW, section 13.6. Equation 13.60 states(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\mathbf{\nabla_ue}_{\hat{\alpha}} = -\mathbf{\Omega\cdot e}_{\hat{\alpha}}[/tex]

where [itex]\Omega^{\mu\nu}[/itex] is antisymmetric (and [itex]\hat{\alpha}[/itex] is the tetrad label). My question is, over which index is the contraction of Omega with e performed? We have

[tex]\Omega^{\mu\nu}e_\mu = - \Omega^{\nu\mu}e_\mu[/tex]

so the result changes sign depending on the index chosen, but I can't find the authors' definition of the dot product between anything besides two vectors. Also, just want to check that my interpretation of the rest of the equation is correct... It's equivalent to

[tex]u^\beta\left(\partial_\beta e^\mu_{\hat{\alpha}} - \Gamma^\mu_{\beta\gamma}e^\gamma_\hat{\alpha}\right) = -g_{\beta\gamma}\Omega^{\mu\beta}e^\gamma_\hat{\alpha}[/tex]

maybe with the mu and beta swapped on the Omega, right?

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# Fermi normal coordinates

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