Tests for Radiative Character of Spacetimes

[bcrowell]

Given a metric, is there any general technique that can be used in order to determine whether or not it contains gravitational waves?

The equivalence principle says that the gravitational field isn't a tensor, so we can't write an expression for the local energy density in terms of the gravitational field, as we could with the electromagnetic fields. For the same reason, I assume it is not possible to create anything in GR that would play the role of the Poynting vector.

If someone gives me a metric like
d s^2 = d t^2 - \left(1+\frac{1}{10}\sin x\right)d x^2 - d y^2 - d z^2 \qquad ,
I can pretty easily tell that it can't be a real gravitational wave, because it can be eliminated by a change of coordinates. But that relies on the simple form of the example. I'm guessing that if there are any general techniques for recognizing this, they aren't completely elementary, since as late as the 1930's, Einstein was still flip-flopping on whether gravitational waves were real.

I guess one thing you can check is whether the Weyl tensor vanishes. If it does, then the solution is definitely not a gravitational wave. But plenty of non-radiative solutions have nonvanishing Weyl tensors. In particular, any non-flat vacuum solution is going to have a nonvanishing Weyl tensor, whether it's radiative or not.

I'm not even sure whether it's possible to make a totally well-defined distinction between radiative and non-radiative...? Is the distinction only a well-posed one in, say, asymptotically flat spacetimes? In the Petrov classification scheme, types III, N, and II are radiative, and types D and O aren't. But the Petrov scheme is, as far as I understand it (which is not very much!), basically a method for classifying exact solutions, not all spacetimes in general, and in any case I get the impression that it maybe nontrivial to find the Petrov type of a given spacetime...?

 

[Naty1]

I understand your question only vaguely, so the following may be off...

There may some hints or even an answer in Roger Penrose's THE ROAD TO REALITY,beginning around 14.7 "What a metric can do for you" in the Calculus on Manifolds chapter.

I had to give up detailed reading about a hundred pages earlier as the math got too advanced and I could not make sense of the many,many,many notations used.

Here Penrose makes several mathematical statements relating to a positive definite g_ab..references the Levi-Civita connection,killing vectors, and Kronecker delta (yes, I realize these are tensor related) and I think the gist of these pages are tests for various types of connections relationing to types of smoothness (my term) and types of geodesics along with "hoop notation"...

I just checked the prior section.."Lie Derivatives".. and while I don't get them either, Penrose is again looking at manifold continuity,geometric interpretations of lie group operations and questions of curvature...

If no one else has an answer, the above text section is worth a look...

 

[Naty1]

In the above text, section 14.8 Sympletic Manifolds here is a quote that will give you a possible idea what he's discussing:

It should be remarked that there are not many local tensor structures that define a unique connection, so we are fortunate that metrics (or pseudometrics) are often things that are given to us physically. ...

He then notes sympletic

...The local structure of a sympletic manifold has no notion of curvature...which might serve to distinguish one from another...

and he contrasts these with Riemannian manifolds...

So it again seems he is talking gravitational curvature considerations, but I don't know how that relates to gravitational waves..(a speculation: no defined manifold curvature, then no gravitational wave??)