I'm starting to gather information about general relativity and riemannian geometry (as a former future physicist, I'm more interested in ideas and results than in rigor and mathematical proofs) to understand this theory through its formalism. I read in wikipedia that gravitational waves only exist in two tensor polarization modes, and another question on this website alludes to their orthogonality. I would like to know if it is anyhow related to the fact that spacetime is seen as a 4 dimensional pseudoriemannian manifold or if these two features are totally unrelated.
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$\begingroup$ Related: physics.stackexchange.com/q/74307/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Jun 18, 2022 at 4:59
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It is related to space-time having 4 dimensions. In general, the number of polarizations a boson has (and here we take the particle physics perspective of the graviton) is the degeneracy of the representations of its little group. In 4 dimensions, we have, for a massless particle:
$$p_{\mu}=(E,0,0,E)$$
Any rotation in the plane $(0,x,y,0)$ preserves this 4-momentum and is called a little group. In this case, the little group is $E(2)$, and all its representations have degeneracy 2. That is why the photon and the graviton, being massless, have two polarization even when the photon has spin 1 and the graviton spin 2. The intuition behind this is that, irrespective of the spin, you will always only have two helicity states, and if a particle is massless, one cannot boost to a frame of reference in which the helicity changed (thus the helicity is the only physical, Lorentz invariant degree of freedom).
If we have massive particles, on the other hand
$$p_{\mu}=(m,0,0,0)$$
The little group is $SO(3)$ and a spin $j$ representation has degeneracy $2j+1$. So massive spin 1 particles (in the Standard Model they are $W^{\pm}$ and $Z$) have $2+1 = 3$ polarizations.
Observe that the little group, and hence the number of polarizations, depend on spacetime dimension. The same can be derived in the context of General Relativity, where the constraints that leave the gravitational wave with only two polarisations are that it has to be a symmetric tensor, invariant under coordinates changes and invariance under vacuum "gauge" transformations.
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$\begingroup$ Can anyone really explain more than three dimensions? No. Time is not really a dimension. $\endgroup$ Commented Jun 18, 2022 at 0:35
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$\begingroup$ I see you have some qualms with modern physics, but this is not the right place to vent this. $\endgroup$ Commented Jun 18, 2022 at 10:27
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$\begingroup$ It’s not a qualm, it’s just a question that’s never answered. And calling it modern is really milking the cow dry. $\endgroup$ Commented Jun 18, 2022 at 22:45
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$\begingroup$ The idea of dimensionality of vector spaces is a mathematical abstraction. Explaining them goes as far as identifying relevant physical degrees of freedom that propagate in these dimensions. Because you move in three spatial dimensions and age as time passes, one can ascribe to each of these four dimensions some physical significance, and that is all the explaining we could ever ask for. $\endgroup$ Commented Jun 19, 2022 at 12:22