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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.

Christian Fieldhouse
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  • https://physics.stackexchange.com/q/41858/150025 Try this. – Yuzuriha Inori Apr 11 '18 at 10:42
  • 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. – Christian Fieldhouse Apr 11 '18 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.

Árpád Szendrei
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  • 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? – Christian Fieldhouse May 01 '18 at 09:00
  • 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. – Árpád Szendrei May 01 '18 at 09:26
  • 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. – Árpád Szendrei May 01 '18 at 09:26
  • 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. – Christian Fieldhouse May 02 '18 at 19:58
  • Can you please elaborate a little more detailed where you see the contradiction between (a) and (b)? – Árpád Szendrei May 02 '18 at 20:06
  • Specifically, I would expect a transverse wave moving on the x-direction to have some z-component perturbation. – Christian Fieldhouse May 02 '18 at 20:07
  • Oh ok, so you mean that there is a GW moving in the z direction, and another GW moving in the x direction. Now the two are both stretching and squeezing perpendicular to their own direction, so the two effects would meet, and disturb each other? So one GW would disturb another squeezing and vica versa? – Árpád Szendrei May 02 '18 at 20:28
  • No, sorry, I mean that gravitational waves are propagating radially from the binary (right?) in particular, there is a gravitational wave moving along the x-axis, and that should (because it's some combination of x/+ polarised waves) have some z-direction perturbation. – Christian Fieldhouse May 02 '18 at 21:07
  • i believe you are thinking that GW propagate radially. They in fact propagate spherically. The binary holes rotate not only on a plane axis, but their rotation axis also rotates. Otherwise we would only see the GWs from the holes if we would happen to be in the plane. You are right that these GWs and their effects disturb each other somewhat. But we see from Earth their net effect that arrives here. – Árpád Szendrei May 03 '18 at 09:20
  • I apologise because it seems I have not communicated what I'm confused about. I am not thinking about the gravitational waves disturbing each other at all. Can you answer two simple questions: a) Is there or is there not Gravitational radiation travelling along the x-axis? b) If there is gravitational radiation travelling along the x-axis, would this imply a metric perturbation in the z-direction? – Christian Fieldhouse May 04 '18 at 09:47
  • Yes for both questions. I think I like this as favorite question. If you like the answer please upvote. – Árpád Szendrei May 11 '18 at 15:14
  • Do you also agree that the quadruple formula gives no perturbation in the z-direction? If so, this is an obvious contradiction of a) and b), right? – Christian Fieldhouse May 12 '18 at 06:18
  • I agree. But i would like to comment that this is a static view of the binary masses. They are in reality rotating and that is why you see perturbation, the rotation. – Árpád Szendrei May 12 '18 at 08:37
  • I'm not sure what you mean by a static view. The Quadrupole formula relies on the second derivative (with respect to time) of an integral. When you plug in a parametrisation of two rotating bodies in the y-z plane you get no x-perturbation. The bodies aren't in-spiralling, but they are moving. Do you mean that there is no radiation in the z-direction without in-spiral? – Christian Fieldhouse May 13 '18 at 06:03
  • Correct. I believe that is the answer. – Árpád Szendrei May 13 '18 at 06:06