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Waves are generally classified as either transverse or longitudinal depending on the they way the propagated quantity is oriented with respect to the direction of propagation. Then what is a gravitational wave? It doesn't make sense to me that a disturbance in the curvature of spacetime has a "direction", so I would say they're neither, much like a wave packet in quantum mechanics.

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Gravitational waves are transverse waves but they are not dipole transverse waves like most electromagnetic waves, they are quadrupole waves. They simultaneously squeeze and stretch matter in two perpendicular directions. Gravitational waves definitely propagate in a given direction but the effect that they have on matter is completely perpendicular to the direction of motion. Below is a picture of what the metric of a passing wave does to space (the wave traveling is perpendicular to the screen). If you imagine a free particle sitting at each grid intersection point, the particle would move sinusoidally right along with the grid:

enter image description here This diagram is from this paper

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I believe this diagram that you have is for $H_+$ (H-plus ? I'm not sure how to pronounce this) gravitational waves, after you take the de Donder gauge $$ \partial^{\mu} \bar{h}_{\mu \nu} = 0 $$ and the longitudinal gauge $$ H_{0 \mu} = 0 $$ the transversility condition $$ H_{\mu 0} + H_{\mu 3} = 0 $$ and finally the traceless condition $$ H^{\mu}_{\; \mu} = 0 $$ yes? Then the other wave is the $H_{\times}$ (again, H-cross ?) wave. Do you have a similiar diagram for this? Your diagram of the metric is far better than the usual one one finds of four test masses oscillating. Thank you. –  Flint72 May 24 at 15:29
    
@Flint72 - if you look at page 7 of the linked paper, you will see the orthogonal wave where the stretching and shrinking occurs at 45 degrees to the figure in the paper. See eftaylor.com/exploringblackholes/GravWaves100707V2.pdf –  FrankH May 24 at 21:05

in the linearized limit of General relativity, as FrankH said, all propagating perturbations of the metric are transverse.

However, it must be noted that the full theory does allow for nonlinear longitudinal modes of propagation. According to the Petrov classification, such regions of longitudinal propagation are region III. There are usually not taken as true propagating modes since they fall of as the inverse cube of distance to the source, so they are interpreted as a kind of evanescent wave

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Is this like in a waveguide for electromagnetic waves in which you can have longitudinal modes despite electromagnetic waves being generally transverse? –  Ignacio Oct 28 '12 at 9:34
    
@Ignacio Since waveguides by definition don't propagate out in all directions, a better analogy would be to the near field of an EM transmitter. This regime has more complicated behavior, including the feedback into the transmitter that makes RFID technology possible. –  Chris White Feb 9 '13 at 23:43

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