Spacetimes where $R_{ij}\neq 0$ but $R_{ij}V^iV^j=0$ on a timelike and/or null geodesic?

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?

• Are you using $i$ and $j$ as abstract indices, or as spacelike indices according to some slicing? Apr 10, 2021 at 13:43
• Abstract indices. Apr 11, 2021 at 15:52

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.