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Do there exist spacetimes with a timelike and/or null geodesic $\gamma$ with tangent vector $V$ for which $R_{ij}\neq 0$ on the geodesic, but $R_{ij}V^iV^j=0$ on it? If so, are there any general features of such spacetimes; that is any other properties that such spacetimes may possess?

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    $\begingroup$ Are you using $i$ and $j$ as abstract indices, or as spacelike indices according to some slicing? $\endgroup$ Apr 10 at 13:43
  • $\begingroup$ Abstract indices. $\endgroup$
    – Ishan Deo
    Apr 11 at 15:52
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Yes. An example is a FLRW spacetime with equation of state $p=- ρ/3$.

The second Friedmann equation for this equation of state ensures that $\ddot a ≡ 0$, and this means that Ricci tensor component $R_{tt}\sim \ddot a ≡0$. So, with 4-velocity $V^i$ of the comoving frame $R_{ij}V^i V^j≡0$, while spatial components of the Ricci tensor remain nonzero.

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