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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
[tex]d s^2 = d t^2 - \left(1+\frac{1}{10}\sin x\right)d x^2 - d y^2 - d z^2 \qquad ,[/tex]
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...?
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
[tex]d s^2 = d t^2 - \left(1+\frac{1}{10}\sin x\right)d x^2 - d y^2 - d z^2 \qquad ,[/tex]
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...?