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Gravitational waves propagate through a medium of space-time. Are they traverse waves or longitudinal waves? Or do they propagate without oscillating?

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Gravitational waves are transverse but their possible polarizations are described not by a transverse vector but by a transverse tensor.

Electromagnetic waves moving in the $z$ direction may have two possible polarization vectors $x$ or $y$, or their (complex) linear combinations – vectors perpendicular to the $z$ axis.

Gravitational waves in 3+1 dimensions also have two polarizations that may be described by the components of a tensor $h_{xx}=-h_{yy}$ and by $h_{xy}$. One may again consider complex linear combinations of these polarizations, e.g. circular polarizations not too different from the electromagnetic case.

In the gravitational case, we need a tensor with two indices – that is why we also say that the gravitons have spin $j=2$, unlike photons' $j=1$. The tensor $h_{\mu\nu}$ in general is symmetric so it has 10 components to start with.

However, they have to obey $k^\mu h_{\mu\nu} = 0$ which reduces the number of polarizations to six. This vanishing of the "inner product" is the transverse condition which is why it is right to say that the waves are transverse even though the polarizations are tensor-like.

The polarizations of the form $h_{\mu\nu} = k_\mu\lambda_\nu+\lambda_mu k_\nu$ are "pure gauge", resulting from diffeomorphisms, so they're unphysical. The reduces the 6 candidate polarizations to 3. Finally, there is a traceless condition $h^\mu{}_\mu = 0$ which reduces the number of independent polarizations to 2, just like for the electromagnetic waves.

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