Why can't the speed of gravitational waves be greater than the speed of light if the universe can expand faster than the speed of light?

Since the expansion speed of the universe can be greater than the speed of light, why can't gravitational waves, which also uses space as the medium, travel faster than the speed of light?

• I am not expert on this, but I am guessing that while superluminal expansion of the universe does not allow superluminal transmission of information, superluminal propagation of gravitational waves would allow, and thus is forbidden by causality? Commented Jul 29, 2021 at 12:40
• @aystack That's not a bad intuition to have. The way I would say it is that in GR, we can only really define the relative velocity of two objects locally. In more detail, we need to be able to find a patch of spacetime containing both objects where the curvature in this patch is small enough that we can ignore it. In any local frame, GWs travel at the speed of light. Conversely, since the expansion of the Universe is due to spacetime curvature, we can't talk about it in a local frame. "The universe expanding faster than light" is something people like to say, but doesn't mean much. Commented Jul 29, 2021 at 13:22

Gravitational waves are solutions to the linearized field equations $$\Box h_{\mu \nu} = 0,$$ where $$\Box$$ is the d'Alembertian differential operator.

They are of the form \begin{align} h_{\mu \nu}= \text{Re}[H_{\mu \nu} e^{i k_{\rho} x^{\rho}}] , \end{align}

where $$H_{\mu \nu}$$ is just a complex symmetric matrix setting the polarization of the wave and $$k_{\rho}$$ is its wave-vector.

For the above ansatz to satisfy the equations, we plug it into the equations to get the condition $$k_{\mu} k^{\mu} =0$$. This is just the statement that the wave-vector must be null, meaning the wave propagates at the speed of light.

Gravitational waves propagate at the same speed as light and in a similar manner. The point here is that the speed $$c$$ is only measured locally. The speed of objects with mass is also only limited to be $$\lt c$$ locally. The relative speed of two objects separated by great distance is not well-defined. See this post for a detailed explanation.

The expansion of the universe does not give objects any "speed" because the universe does not lie in a higher-dimensional space. For the same reason, there is no "central point" in the expansion. However, that being said, gravitational waves do experience cosmological redshift just like light.

While gravitational waves and cosmic expansion are both gravitational phenomena, they are not the same.

The universe "expanding faster than the speed of light" is not a local issue. The Hubble parameter describes the expansion rate now $$H_0 \sim 70$$ km/s/Mpc (kilometers per second per megaparsec). So for every megaparsec of empty space between two points the expansion makes them appear to recede from each other at a speed of $$70$$ km/s. For reference the Milky Way has a radius of about $$30$$ kiloparsecs ($$0.03$$ Mpc, but it's not really empty).

This is apparent motion. The two points don't actually move. A person at each point would perceive themselves to be at rest. Each point is stationary with respect to its local spacetime. The addition of new space between the points makes the distance between them get bigger, but nothing is actually moving.

We can calculate the current separation required for the apparent recessional velocity to be the speed of light $$c$$ $$d_\mathrm{horizon} = \frac{c}{H_0} = \frac{300\,000 \,\,\mathrm{km/s}}{70\,\,\mathrm{km/s/Mpc}} = 4300\,\,\mathrm{Mpc}$$

Something $$4400$$ Mpc away appears to move faster than the speed of light away from us. The light it emits will never reach us. But its speed relative to the local spacetime will always be less than the speed of light.

(We can see things farther away than this distance, because they were closer when they emitted the light that we see, but that's a different question)

Gravitational waves (GWs) are another aspect of spacetime. The waves move across spacetime, but they don't carry spacetime with them. The wave speed of GWs is a consequence of general relativity and is a fundamental property of spacetime itself (like the wave speed on a string depends on the properties of the string like its density and tension).