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I have problem with understading how Riemann tensor in orthonormal frame transforms using Lorentz transformation of frames. I was reading Morris Thorne paper from 1988 (American Journal of Physics 56, 395 (1988); https://doi.org/10.1119/1.15620). Authors have used metric: $g_{\mu\nu}=diag[-e^{2\phi(r)},1/(1-b(r)/r),r^{2},(rsin\theta)^{2}]$ to create orthonormal (static observer) frame: enter image description here

and orthonormal moving observer frame:enter image description here

Then, they've calculated riemann tensor components for static observer case - nothing horrifying. Afther that they transformed Riemann tensor from static frame to moving one (special relativity transformation). I'm not sure how to do it -i'm guessing that i have to use lorentz transformation matrices:

$\Lambda^{\mu}_{\nu}= $$ \begin{pmatrix} \gamma & ^-_+\gamma(v/c) &0 &0\\ \gamma(v/c) & ^-_+\gamma & 0 &0\\ 0 & 0 & 1 &0\\ 0& 0 & 0 &1 \\ \end{pmatrix} $

Riemann tensor is (1,3) rank tensor so it will be 3 inverse lorentz matrices ($\Lambda^{\nu}_{\mu}$) and one standard $R^a_{bcd}=\Lambda^{a}_{\mu}\Lambda^{\nu}_{b}\Lambda^{\sigma}_{c}\Lambda^{\zeta}_{d}R^\mu_{\nu\sigma\zeta}$

Is this right way to do it?

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