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I'm studying General relativity, and I want to clarify the qualitative nature of how gravitational waves propagate.

Simple is best, so I want to imagine a single binary black hole system orbiting in the $x,y$ plane at $(0,0,0)$ forever at the same radius.

Now the quadrupole formula:

$$\overline h_{ij}(\mathbf x,t) = \frac{2G}{|\mathbf x|c^6}\partial_t^2\int y^iy^jT_{00}(\mathbf y, t_r) d^3y$$

$$t_r := t - |\mathbf x|/c$$

$$\overline h_{\mu \nu} := h_{\mu \nu} - \eta_{\mu \nu}\eta^{\sigma \tau}h_{\sigma \tau}$$

tells us that anywhere in space, we have the same perturbation of the Minkowski metric $h_{\mu\nu}$, just scaled by $\frac1{|\mathbf x|}$ and appropriately delayed.

I imagine this situation like the binary system is a small particle oscillating in a block of jelly (gelatin dessert, not jam) with the whole block wobbling in the same plane, and the wobbling diminishing asymptotically.

Where I start to doubt this visualisation though, is when I here that gravitational waves are transverse. Specifically, it seems like gravitational radiation is propagating in the $x$ direction, which conflicts in my head with the fact that the metric is perturbed in this direction.

Is my picture of gravitational waves in some sense accurate? What does it mean to say that gravitational waves are transverse?

Edit: This animation seems to me to conclude that the image I have in my head is wrong, namely, it has a perturbation of the metric in the $z$ direction. I simply cannot reconcile this with the quadrupole formula, which gives no perturbation in the $z$ direction.

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  • $\begingroup$ physics.stackexchange.com/q/41858/150025 Try this. $\endgroup$ Apr 11, 2018 at 10:42
  • $\begingroup$ Hi Yuzuriha, I'm familiar with the diagram given in that answer, which shows distortion when the gravitational wave is passing through the screen. The diagram describes what someone on the $z$ axis would experience. My interpretation is that this would be true (according to the quadrupole formula) for observers on the x axis looking in the $z$ direction. so although I'm convinced the distortion is transverse in the $z$ direction, it doesn't seem so in any other direction. $\endgroup$ Apr 11, 2018 at 10:58

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Gravity waves are transverse quadropole waves, they simultaneously stretch and squeeze spacetime in the x and y angles, where the propagation is in the z direction.

Gravitational waves propagate in the z direction, but the effects they have on spacetime are perpendicular to the direction of motion.

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  • $\begingroup$ Isn't is true that gravitational waves travel in the x-direction as well? I know that they are travelling in the z direction if you stand on the z axis, but if you stand on the x-axis, as far as I can see, either you feel no gravitational effects, or there is gravitational radiation travelling *in the x direction*(by symmetry, it seems there is no other way they would be travelling). Such direction of travel, plus the fact that GWs are transverse demands a perturbation of the metric in the $z$-direction, no? $\endgroup$ May 1, 2018 at 9:00
  • $\begingroup$ It is because we are mixing up here GWs with gravitational effects. They are not the same thing. Let's take the Earth, is it emitting GWs? Well not that we could check with experiments. You need a rotating quadropole moment (binary black holes) to emit GWs. They are emitting GWs in every direction. But let's take just one wavepacket. Make it's direction z. In that case think of it as a traveling mass (it is not true but it is easier to understand). Now lets say the "mass" is traveling in the z direction. $\endgroup$ May 1, 2018 at 9:26
  • $\begingroup$ The "mass" is emitting gravitational effects (not GWs) lets say in this case in the x and y direction, simultaneously. It is stretching and squeezing spacetime in the x and y angles. It is not emitting GWs itself not in the x nor in the y direction. It is like a traveling source of gravity. But in the x and y angle it still stretches and squeezes spacetime without GWs traveling in these directions. In those directions you only have effects of gravity. $\endgroup$ May 1, 2018 at 9:26
  • $\begingroup$ Yes, I understand that the wave travelling in the z direction has effects in the x and y directions, but that is not my question. My question is concerning the apparent contradiction: (a) The quadrupole formula doesn't give any perturbation in the z-direction (b) Some gravitational waves move in a direction other than the z direction. $\endgroup$ May 2, 2018 at 19:58
  • $\begingroup$ Can you please elaborate a little more detailed where you see the contradiction between (a) and (b)? $\endgroup$ May 2, 2018 at 20:06

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