1
$\begingroup$

I was referred here from this question I asked on stackexchange. And now that I'm here, I see that this other question about geometric wave equations 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, 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...

Improve this question
$\endgroup$

1 Answer 1

Reset to default
3
$\begingroup$

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", 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", Physical Review D 10, 1736 (1974).

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .