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I'm sure many here are familiar with the following image showing the 2D representation of how the fabric of spacetime is warped by the presence of mass:-

spacetime fabric

Can this fabric be interpreted as an elasticated sheet?

If so, imagine the following...

2 massive entities (for example black holes, as these are said to cause gravitational waves when they collide) rapidly rotating about their barycentre on this 'sheet' would have a certain displacement, x, between the neutral position of the 'sheet', i.e. no mass placed on it, and the barycentre.

2 neutron stars

When these 2 entities collide, I assume that the displacement, y, of the centre of mass of this new entity on the 'sheet', would be larger than the initial displacement, x, because the 'pressure' exerted on the 'sheet' would be larger owing to the same constant system mass being concentrated on a smaller region of the 'sheet'.

new object

Now, say this collision between the 2 initial massive entities happened 'quickly', this sudden change in displacement would cause the centre of mass, hence the sheet, to oscillate (remember I'm saying that the sheet is elasticated), thus creating ripples that travel across the fabric of spacetime? I.e. gravitational waves.

Is this complete and utter nonsense? If so, can those with wisdom please provide an explanation/analogy of how they are formed? And possibly or if indeed necessary, tell me where my analogy is wrong?

I apologise in advance for the terrible illustrations.

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The rubber-sheet analogy is often used to "explain" the basics of GR to beginners, but actually it has nothing to do with real gravity. It acts much more like a scalar field (the up/down freedom degree) - and there were several attempts to build a scalar gravity. But the correct description turned out to be tensorial and purely geometrical.

GR has 10 potentials instead of one. These potentials aren't physical (like the up/down freedom degree of the rubber sheet) but rather geometrical properties of space-time itself. Moreover, actual solutions are equivalence classes of these potentials with respect to diffeomorphisms, which reflects the gauge nature of GR.

In conclusion: the rubber-sheet analogy is a poor man's attempt to understand gravity. Gravitational waves are just solutions of Einstein's equations.

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  • $\begingroup$ Sorry, that sort of things just happen :) $\endgroup$ – Prof. Legolasov Oct 21 '14 at 3:20
  • $\begingroup$ +1: I'm a little uncomfortable with implying the equivalence classes aren't physical (I mean, the geometry of spacetime is a real, measurable thing - well done for mentioning classes as the "real" solutions), but I don't know how to say it better. But might also be worth mentioning Nordström's theory as a fairly well known scalar one. $\endgroup$ – WetSavannaAnimal Oct 21 '14 at 3:37
  • $\begingroup$ That's cool! What are the 10 potentials? I don't really know much physics (only introductory undergraduate physics) so if there are any analogies to classical mechanics or E/M that'd be really useful. (Say, are they like the magnetic vector potential? Or like the electric or gravitational scalar potential in classical physics?) $\endgroup$ – Mehrdad Oct 21 '14 at 8:27
  • $\begingroup$ The mentioned potentials form the metric tensor $g_{\mu \nu}$ (which is symmetric and thus has 10 independent components in 4-dim space-time). So they basically define the Riemannian geometry of our space time and thus the curvature and the causal structure. You can find more in this wikipedia article en.wikipedia.org/wiki/General_relativity. $\endgroup$ – Prof. Legolasov Oct 21 '14 at 8:40
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To add to Hindsight's great answer: one of the reasons that the analogy fails is the same reason why Nordström's Scalar Theory of Gravitation fails: Waves on rubber sheets are described by linear wave equations; at least in the small amplitude limit.

However, by analogy with Maxwell's equations, waves in gravitation should bear energy. But we are also trying to make the gravitation consistent with our special relativity knowledge. If the waves themselves bear energy, they should themselves be a source of gravitation: they should have an effective gravitational mass of $E/c^2$. Thus, the field equations of gravitation were foreseen to be nonlinear considerably before Einstein published the general theory of relativity in 1916.

There is no simple way to make a rubber sheet wave behave in this way, at least not unless it is made of some highly exotic material which would behave in nonintuitive ways and thus be if little worth as a teaching analogy.

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  • $\begingroup$ I think it's pretty clear that scalar gravity does not work, but why do the waves of a linearized scalar equation not carry energy and mass in a special relativistic framework (just like electromagnetic waves do on top of a flat metric?). Or are you merely concerned about self-consistence if we add the mass-energy in an ad-hoc way? I do agree about the highly exotic character of the vacuum, though... we've been working on that for four centuries... and we may still be at the very beginning of the story. $\endgroup$ – CuriousOne Oct 21 '14 at 4:00
  • $\begingroup$ @CuriousOne I'm talking about self consistency: of course, all waves bear energy, but I'm just saying that in gravitation this energy becomes a "source term" (though not of course, not part of the stress energy tensor). Don't remind me of your last sentence: (I agree wholeheartedly) to be in one's twenties now is a fantastic opportunity! $\endgroup$ – WetSavannaAnimal Oct 21 '14 at 5:38
  • $\begingroup$ Yeah, I don't see it being self consistent either. It's a shoehorn affair with that scalar theory. To be young again? That's about as time-translation invariant as anything... apply Noether to it, and your energy is conserved, too! Or so I wish... $\endgroup$ – CuriousOne Oct 21 '14 at 5:51

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