Structure of a gravitational wave?

[ClamShell]

Questions concerning GW's are very common on PF,
but I think I have a new question on the concept.

Maxwell explains that EM waves(photons) propagate
in space due to a relationship between the sinusoidal
representations of the electric and magnetic fields
of the wave. Namely that these two sinusoids are
orthogonal and 90 degrees out of phase (if memory
serves). Kind of a push-pull relationship. IE, if these
components are not present there is no propagation
of the EM wave into space.

My question is that if EM waves are analogous to
to Gravitational Waves, what would be the orthogonal
components that would force the GW's to radiate
according to Maxwell's explanation of EM waves?

Without orthogonal components would not the GW
be stationary?

 

[arkajad]

You may like to read about http://arxiv.org/abs/0908.1326" - in section 2 it is explained how for weak fields gravity can be treated, in some respect, in a similar way as electromagnetism.

 

[bcrowell]

Namely that these two sinusoids are
orthogonal and 90 degrees out of phase (if memory
serves).

They're in phase.

My question is that if EM waves are analogous to
to Gravitational Waves, what would be the orthogonal
components that would force the GW's to radiate
according to Maxwell's explanation of EM waves?

Without orthogonal components would not the GW
be stationary?

The analogy isn't as close as you're thinking; in a gravitational wave there are not two different fields that are orthogonal to one another. The wave equation of GR is the Einstein field equations, which don't have the same structure as Maxwell's equations. There is something somewhat similar to the "push-pull" in GR, which is that contraction along one transverse axis requires expansion along the other transverse axis axis; this is because the Einstein field equations in vacuum are basically statements of conservation of the volume of a cloud of test particles.

One thing that is similar about the two is that in both cases, the vibrations are transverse. (In the case of gravitational waves, this is only strictly true in the far-field limit.) In the GR case, this is because a purely longitudinal plane wave propagating along the x-axis would be equivalent to a change of coordinates $x \rightarrow x'(x,t)$, which can't mean anything in GR because GR is completely invariant under any smooth change of coordinates.

One way to see that GR can't be strictly analogous to EM is that in the EM case you have two planes of polarization, like | and -, whereas in the GR case you have polarizations that look like + and x.

[EDIT] Arkajad makes a good point about the analogy in the weak-field case.

 

[ClamShell]

The analogy isn't as close as you're thinking; in a gravitational wave there are not two different fields that are orthogonal to one another. The wave equation of GR is the Einstein field equations, which don't have the same structure as Maxwell's equations. There is something somewhat similar to the "push-pull" in GR, which is that contraction along one transverse axis requires expansion along the other transverse axis axis; this is because the Einstein field equations in vacuum are basically statements of conservation of the volume of a cloud of test particles.

Thanks for pointing out that E & B are in phase for EM waves. What I had hoped,
was that for GW's the orthogonal fields would be gravity and momentum. Thanks
for the eye-opener.