There are many ways for spacetime to warp, which can be put into two categories. The simpler kind, Ricci curvature, is the only kind of curvature in <4 dimensions and is produced by mass-energy, momentum, pressure, and shear stress, according to general relativity. The other kind, Weyl curvature, only exists in 4 or more spacetime dimensions and can exist in a vacuum.
Gravitational waves are Weyl-curvature distortions of spacetime that propagate in a vacuum according to general relativity.
(Also, gravitational waves do carry a little bit of energy, so they cause a small amount of Ricci curvature, but this is a secondary effect.)
I'm not sure why you want to draw attention to non-Lorentzian spacetimes in this context.
> ... that propagate ...
"propagate".
That requires a decomposition of spacetime into space+time, and of course the decomposition of the Riemann curvature tensor, the setting of a background value for the Weyl tensor, and the use of perturbation theory.
But if you're going down that path, why not use the metric tensor? g_munu = eta_munu + h_munu + h.o.t. is standard in post-Newtonian expansion approximations, and in particular https://en.wikipedia.org/wiki/Linearized_gravity (which doesn't track the higher-order terms).
The Weyl curvature tensor C_abcd is useful in understanding that in a spherical region of space (not spacetime, so really we're in the land of extracting 3-Cotton-York C_ab) where a GW is incident suffers not from a volume deficit but from an ellipsoidal stretch-squash. But conceptual understanding of and calculation are... well, not really on speaking terms. Even theorists who take the full covariant theory seriously will decompose further, into e.g. an electrical and magnetic part, and add further structure to match the worldlines to the Raychaudhuri equation in shear and vorticity.
> they cause a small amount of Ricci curvature
???
If nothing else, I think you'd need to choose between explaining this or explaining why "Ricci curvature is produced by [matter but] Weyl curvature ... can exist in a vacuum" (or choose neither).
The sticky bead apparatus is a breaking of the T_munu = 0 vacuum condition.
I was trying to give a conceptual understanding on the level of the person I was responding to, not instructions for doing actual calculations.
I didn't mean to suggest euclidean metric, I just thought "3+1" would be extra jargon.
The person asking questions was confused because they'd heard that "mass causes curvature" in GR, and that gravitational waves involve curvature. I figured it would help to explain that these are different kinds of curvature.
I think I was totally wrong re: Ricci curvature. I was thinking "oh, GW carry energy and there's nonlinear evolution..." and got carried away, whoops.
If you want to give a more accurate explanation (that makes sense to someone who's never taken a GR class) please do!
There is no question that this stuff is hard (and has been cutting edge for decades) and that it is easy to make mistakes.
> give a more accurate explanation (that makese sense to someone who's never taken a GR class)
It's hard to know what level of understanding to aim for on HN. There are non-relativist working physicists here who somewhat casually read other areas of physics (and mathematics) here, for example, rather than e.g. physics SE or looking through literature reviews.
I'm guessing that you have done GR but probably not much with approximations like GEM, nor looked into the history of gravitational waves (e.g. the 1950s-60s work by Bondi with collaborators like Pirani) before the wide availability of powerful computers and observational support for the linearized theory. My goal here is not to nitpick you, but rather to offer a couple of references that might interest you or anyone who is quietly reading along.
> I think I was totally wrong re: Ricci curvature.
R_munu = 0 for one (uncharged) BH, and also for two. Charged BHs are different (Reissner-Nordström's Ricci tensor is R_munu = +- g_munu r_{Q}^2 / r^4 where r_{Q}^2 = \frac{Q^2G}{4 \pi \varepsilon_0 c^4} and ε_0 is the electric constant, all thanks to the electromagnetic stress-energy tensor; the Ricci scalar remains 0).
Superposing two Schwarzschild or Kerr solutions is messy [Krivan & Price 1998 Phys. Rev. D 58, 104003 was a nice overview arxiv html5[*] <https://ar5iv.labs.arxiv.org/html/gr-qc/9806017>, 'two locally Kerr holes, no matter how close they are, will not superpose into a single Kerr hole. Though [this] "failure" of the close limit is physically correct, it is inconvenient [for doing perturbation theory]'] but doesn't change Rmunu = 0.
A charged binary is beyond the scope of this comment.
> I was thinking "oh, GW carry energy ..."
I can recommend two really good papers written at very different times.
Weber & Wheeler 1957 <https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.29....> (it's also on sci-hub) where they struggle with the question of the sense in which GWs are real ("the only well defined way there is to express the influence of [gravitational] radiation [is] in terms of its effect upon invariant space time intervals betwee two test bodies"). I enjoyed seeing the rare \dotequalsdot in eqns 28,30, "geometrically equal to". Also, they lead right into the next suggestion with, "Similarties between gravitational and elecgromagnetic waves thus make it simple to draw a number of reasonable inferences. The significance of these inferences has a much more subtle character in the gravitational case than in the electromagnetic case. Neither field densities nor test particle motions have a meaning independent of the choice of coordinate systems. The simple observable consequences of wave action are instead changes in the separation of nearby test particles -- changes that are related to the covariant components of the curvature tensor R_ijkl."
Goswami & Ellis 2021 <https://iopscience.iop.org/article/10.1088/1361-6382/abdaf3> arxiv html5[*] <https://ar5iv.labs.arxiv.org/html/1912.00591> with the somewhat stentorian title "Tidal forces are gravitational waves". It is really about the Weyl curvature in binaries. The authors explicity consider (for orbital motions of two massive bodies) a decomposition of the Weyl tensor C_abcd into an electric and magnetic part (for which see §III; and eqn 6 repeats the point on R_ab in vaccum), the former compared to Newtonian gravitation and the latter being general-relativistic and encoding the gravitational waves that do not arise in the Newtonian theory. Their focus is on the magnetic part of the Weyl tensor, since (they argue) that is where all the interesting stuff is encoded, and how energy gets from the orbital system to an observer and between the binary partners themselves.
One highlight is a short paragraph summarizing some ~1960 work by Bondi and MacCrae [**], "Through a series of shape changing operations ... one sees that after each rotation, Tweedledum is gaining internal energy as the external tidal force is doing work on him, while Tweedledee is losing energy as she is doing work against the external force. This is an excellent example of how the internal energy of a system can be transferred to another system via gravitational induction, in Newtonian gravity." [emphasis mine].
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[*] lost way downpage and heavily downvoted is a HN user pointing out that the page of cartoons linked at the top is not accessibilty-friendly. I accept the point, and make up for it a little with the html5 links here. One can straightforwardly paste the trailing element of the ar5iv into a search engine or whatnot in order to reach a PDF version of each arxived paper above.
[**] The actual Bondi & McCrae paper(s) is(are) proving a bit elusive, but this gives a good overview: §§3-4, Tweedledum and Tweedledee and Energy Transfer (the latter starts with an excellent quote from Bondi in 1957), https://link.springer.com/article/10.1007/s10701-022-00660-z (open access). So this quickly became three papers, and I'll stop now, because I'm fairly sure you can find your way through them if you really want, and figure out how to get help if you get stuck.
A simple point that might clear up the confusion is: note that the electromagnetic field warps in the presence of charge, yet no charge is transmitted in an electromagnetic wave either.
Explanation: the wave is the warping itself. If a charge or mass exists, the field is (permanently) warped to contain it, in a way that doesn't propagate (although it can be mathematically described as an exchange of particles). But if the charge or mass accelerates, then the warping changes, and the information about that acceleration propagates away. Basically the wave is other charges/masses 'finding out' about a distant change in the velocity of a charge.
Water & sound waves don't transmit any mass, either, but just energy. In fact, that's pretty much the definition of a wave: An oscillation in space and time that transports energy.
Necessary disclaimer: Gravitational energy / energy in General Relativity is notoriously difficult to define and the story of energy conservation is even worse. In general, you can no longer expect energy to be conserved globally. Locally yes, due to the 4-divergence of the energy-momentum tensor vanishing, but globally you can have effects such as cosmic expansion that causes all photons to lose energy as they travel through the universe and, as far as we know, that energy is simply lost.
The "carrier" is the metric tensor. Roughly, at every infinitesimal point in a spacetime like ours there is a tensor value which encodes the distance from that point to its neighbours along all four orthogonal dimensions. "The metric" of a spacetime is usually taken to mean an integration of the metric tensor values at each point along some set of points.
Let's draw a supermassive black hole binary. In the rough <https://en.wikipedia.org/wiki/Minkowski_diagram> schematics below, each equal-mass black hole is an O and we show an observer ("y" for "you") who sees the binary's circular orbit edge-on. The y axis is time, oriented with the future towards the start of this comment. The x axis is one spatial dimension. The diagram-angle from the binary to the observer is meant to be 45 degrees, representing a null aka lightlike separation.
Let's consider two snapshots of the binary's mutual orbit:
1.
y
OO
2.
y
8
The "OO" vs "8" is an abuse of notation; read the OO as the orientation where "y" sees one eclipsing the other, and the 8 as the orientation where y sees the two distinctly.
"y" is at a spatial remove. But "y" is drawn a couple of lines up because of the finite propagation speed c. The orientation OO...y or 8...y is not felt by y at the time, but later. A null curve connects the binary "at the time" with the "but it's felt later at" observer.
We then integrate all the infinitesimal points along the null curve between y and the centre of mass ("COM") of the binary and find that the null curve in diagram 1 is shorter* than the null curve in diagram 2, because there is more mass between y and the COM in the first diagram.
Above, "y" is pointlike and y would free-fall towards the centre of mass. In y's proper time the free-fall would be faster during orientation 1 than during orientation 2. Since the binary is in a mutual orbit, there is a speeding-up and slowing-down of the free-fall measured by y in y's proper time.
Now we consider some extended-body stuff. Things above and below the SMBHB orbital plane will tend to fall towards the plane. If "y" has some height and the binary isn't treated as practically pointlike, the various parts of now line-like y want to free-fall to the black hole closest to them. In orientation OO the top and bottom of the y will squash inwards. In orientation 8 the top and bottom of the y will stretch up and down. Interpolate through the orientations between OO and 8. The "height" here is along one axis in the plane perpendicular to the binary's orbital plane; a "y" with "width" would be extended in that plane in the orthogonal axis. In orientation 8 there is a stretch along that axis compared to OO.
The magic is all in the orientation of the observer/gravitational-wave-detector to the two mutually-orbiting bodies, and the integration of the metric tensor between every part of the observer and every part of the binary. This is ... generally analyitically intractable, so one does numerical methods ("NR", numerical relativity) or one works in the approximation known as linearized gravity. Most non-specialists, and even many members of the LIGO/Virgo/KAGRA et al collaborations who have encountered the mathematics of gravitational waves did so in the context of linearized gravity, and only sometimes encounter NR and (assuming they aren't writing the NR codes) even then don't think too hard about how the "block universe" spacetime described by the field equations are split into 1+3 dimensions.
So, in summary, the crucial thing is the length of the null curves connecting every part of the y with every part of the binary. The periodic rotation of the binary causes the lengths and angles of those curves to oscillate, shortening and spreading or lengthening and narrowing. Looking closely at a large number of such null curves at once, one can successfully model them in bulk as obeying the massless wave equation, even though nothing actually propagates (the whole spacetime-filling metric tensor field is solved all at once; it only "evolves" or "propagates" when we think in terms of the initial values formalism or a 1+3 formalism).
Finally, here is an pulsar-timing-array astrophysicist (<https://www.aoc.nrao.edu/~tcohen/>) doing an intepretive dance (after a bit of explanation):
(there are further links in the video description).
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* someone is bound to pounce on "shorter" given that the interval of a null curve is always 0 (thus the name and from that flows crazy wrong ideas about "photons experience no time"), but we can use the very much timelike worldlines of the (components of the) binary and observer and retarded time to construct a notion of the length of the null curves connecting them. Eric Poisson has gory technical details for anyone actually interested, in the context of the self-force formalism which is more suitable than linearized gravity for high-mass-ratio binaries, especially as the binary hardens.
PS. "Fabric of spacetime" is a cliché that causes me to grit my teeth. I grit harder when someone who has worked with the Einstein Field Equations writes or says it. It's not only overused, it's misleading to non-experts (and sometimes even to experts). Spacetime is not a substance. It does not "distort", a solution to the EFEs is what it is; "distortions" imply comparing the solution to some alternative that one likes better (e.g. the spacetime without any matter, despite any non-zero stress-energy in the solution). It is at best a mathematical container for coincidences ("events" where two objects at the same point in spacetime can have e.g. their velocities usefully compared) and interactions like collisions, scatterings or the formation of molecular bonds.
Unfortunately, colloquially or informally it's very hard to talk about gravitational waves without talking about the stretching and squashing of some region of space (not spacetime), because people (even experts) internalize "fabric" and similar metaphors, because formalisms and approximations to general relativity used in studying inspiralling-binary sources of gravitational waves split spacetime into space + time, and don't invite the consideration of solutions of the geodesic equation (for tractability reasons).
The confusing thing here is that spacetime warps in the presence of mass.
But no mass is transmitted in a gravity wave.
So how do the waves travel?