I was referred here from [this question](https://math.stackexchange.com/questions/3814733/curvature-waves-harmonic-curvature-and-curvature-flow/3875670#3875670) I asked on stackexchange. And now that I'm here, I see that [this other question about geometric wave equations](https://mathoverflow.net/questions/341325/geometric-flow-equations-which-are-second-order-in-time-derivative/341342#341342) is very closely related to mine. But I have a very specific question: I want to know if the d'Alambert equation of the curvature tensor on a Lorentzian manifold


$\Box R\_{\rho\sigma\mu\nu} = 0$


has ever been studied. Section 5 of [this paper](https://arxiv.org/pdf/math/0610256.pdf), linked from the comments of that second question above, comes as close as I have seen, but their version has a whole bunch of extra terms, because their fundamental wave equation is of the metric, not the curvature tensor - it's just a modification of Ricci flow to be 2nd-order in time.


Although I'm not a hard-core math person, I get that lower-order differential equations of the metric are more attractive because they are way more workable. So I don't know, maybe my question is teetering on philosophy, but I'm thinking this "d'Alambert-Riemann" equation is just so elegant. And isn't it possible to say that it's more covariant in some sense? I mean, you can write it with two symbols...



the curvature wave equation
A similar equation that's been used is the Penrose wave equation
\begin{equation}
\square R\_{a b c d} = 2 R\_{a e d f} R{\_b}{^e}{\_c}{^f} - 2 R\_{a e c f} R{\_b}{^e}{\_d}{^f} - R\_{a b e f} R{\_{c d}}{^{e f}} .
\end{equation}
This holds for a vacuum spacetime, i.e. $R\_{a b} = 0$.


I believe that it originates in


R. Penrose, "[A spinor approach to general relativity](https://doi.org/10.1016/0003-4916(60)90021-X)", Annals of Physics 10, 171 (1960),


where it's presented in 2-component spinor notation in equation (3.8) and in the footnote on the following page in a more familiar tensor notation.


For some further references see


M. P. Ryan, "[Teukolsky equation and Penrose wave equation](https://doi.org/10.1103/PhysRevD.10.1736)", Physical Review D 10, 1736 (1974).


