Why gravitational waves have 2 possible polarizations?

Question

I was under the impression that light waves have 2 polarizations because even if the photon is spin 1, the photon is massless which imposes additional constraints. In the case of gravity, it is associated with a tensor (or a hypothetical spin 2), so I would have expected to have 3 polarizations (4-1 for travelling at $c$). What is the actual way to think about this?

Answer 1

The deep group theory explanation for this fact comes from the Wigner classification of unitary representations of the Poincaire group. There are many resources which cover this material, such as Volume 1, Chapter 2 of Weinberg's QFT textbook. I will not attempt to reproduce the argument. The result is that, in $3+1$ spacetime dimensions:

• A massive spin-$s$ particle has $2s+1$ degrees of freedom. You can understand this relatively easily by going to the rest frame of the particle, and then applying the usual arguments from non-relativistic quantum mechanics that give you the eigenvalues of $S_z$ once $S^2$ is known.
• A massless spin-$0$ particle has $1$ degree of freedom. A massless spin-$s$ particle has $2$ degrees of freedom when $s>0$. (This is at least true for bosons, it's possible there are extra subtleties for fermions; I am not 100% sure and it's not relevant for this question so I don't want to comment on that case). The argument here is more subtle. Essentially you can't go into the rest frame of a massive particle, but you can go into a frame where the particle is traveling along, say, the $z$ axis. In this frame, it becomes clear that the degrees of freedom are determined by representations of the group of isometries of the Euclidean plane (ie, rotation and translation in 2 spatial dimensions), and following this line of reasoning leads you to concluding there are two degrees of freedom.

In the case of general relativity, we can at least understand why there are two independent degrees of freedom by carefully counting the equations and gauge symmetries (coordinate transformations).

As a warmup, consider Maxwell's equations. When written in terms of the gauge potential $A_\mu$, you can combine Faraday's law and Ampere's law to get 4 second order wave equations for the four components $A_\mu$. However, there is one gauge redundancy (sometimes called a gauge symmetry) $A_\mu \rightarrow A_\mu + \partial_\mu \lambda$. This is a redundancy because $A_\mu$ and $A_\mu+\partial_\mu \lambda$ represent the same physical state. The gauge redundancy means that one of the components of $A_\mu$ is not really physical -- for example, we can use our freedom to choose $\lambda$ to set one of the components of $A_\mu$ to zero. Additionally, Gauss's law of electromagnetism is a constraint -- that is, an equation without time derivatives, that restricts the space of allowed initial conditions, and which effectively makes one of the three remaining components of $A$ dependent on the others. Combining this information, $4$ second order equations - $1$ gauge redundancy - $1$ constraint equation leads to $2$ truly independent components of $A$, corresponding to the two polarizations of the photon.

In the case of GR, Einstein's equations form 10 second order differential equations for the 10 components of the metric tensor, $g_{\mu\nu}$. However, there are 4 gauge symmetries, corresponding to the freedom to choose four coordinate functions $x^\mu$. Furthermore, Einstein's equations are not really independent; there are four constraint equations that come about because of the contracted Bianchi identities. These four constraints restrict the space of allowed initial conditions for the metric. Combining this information, $10$ second order equations - $4$ gauge redundancies - $4$ constraints = $2$ independent degrees of freedom, which we can identify with the $+$ and $\times$ polarizations of gravitational waves.