In Carroll's 'Space-time and Geometry', appendix F on congruences, the Raychaudhuri equation is derived.

However, in the process, I seem to miss a calculation step that changes the sign of the Raychaudhuri scalar.

Page 461, Carroll writes:

\begin{align} U^\sigma \nabla_\sigma B_{\mu \nu} &=U^\sigma \nabla_\nu \nabla_\sigma U_\mu + U^\sigma R^\lambda_{\, \, \, \mu \nu \sigma} U_\lambda \\ &= \nabla_\nu (U^\sigma \nabla_\sigma U_\mu) - (\nabla_\nu U^\sigma)(\nabla_\sigma U_\mu) - R_{\lambda \mu \nu \sigma}U^\sigma U^\lambda \end{align}

I can't wrap my head around it. To me, lowering the index on the Riemann tensor and elevating the same dummy index on U would have no influence on the sign whatsoever.

Thanks for your insight!


I think it's a typographical error. He should have changed the sign of the Riemann tensor in the last two lines of his calculation (F.10). Then to obtain the Ricci tensor with the correct sign in the Raychaudhuri equation, you must interchange $\mu$ and $\lambda$, getting the required negative sign. This is because the Ricci tensor is $R^k_{ikj}$ and thus before contracting $\mu$ and $\nu$ you must exchange the positions of $\mu$ and $\lambda$ to get the correct sign.

  • $\begingroup$ Thanks a lot. I should have seen it! $\endgroup$ – Dima Fontaine Jun 13 at 15:33

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