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From Darkest matter: gravitational waves 2016-06-19 | Espresso

But black-hole pairs orbiting one another emit something else: gravitational waves. These ripples spread outward like sound waves, ever so slightly stretching and squashing spacetime itself as they pass. The first ever to be caught by Earthly detectors, in September, proved Albert Einstein, who predicted their existence a century earlier, was right.

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General relativity is a classical field theory. In such theories, disturbances in the fields propagate as waves. The basic equation of GR is the Einstein field equation, which is a wave equation.

GR replaced Newtonian gravity, which was a theory in which gravity acted instantaneously at a distance. Even as early as 1905, long before GR, it would have been clear that the Newtonian picture was inconsistent with relativity, which doesn't allow cause and effect to propagate at speeds greater than $c$.

There is a funny twist in the story, which is that in 1936, Einstein and Rosen published a paper claiming that gravitational waves did not actually exist. The paper was seen to be incorrect by a referee, but Einstein an Rosen did not initially accept the criticism.

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  • $\begingroup$ I think that describing the EFE as a wave equation is a bit misleading: they are not even linear, as a wave equation certainly should be. My guess is that you're taking about the linearized version, which is indeed a wave equation. $\endgroup$ – Danu Jun 21 '16 at 18:48
  • $\begingroup$ @Danu: No, that's incorrect. A wave equation need not be linear. (You may have in mind the scalar wave equation that some people refer to as "the" wave equation, which is something much more specific.) A good example of something that's not a wave equation would be the heat equation. If you want a good technical definition of a wave equation, probably it's a hyperbolic PDE (i.e., one for which we have existence and uniqueness of solutions to Cauchy problems). Intuitively, it just means a 2nd-order PDE that has oscillatory solutions. $\endgroup$ – Ben Crowell Jun 21 '16 at 20:19
  • $\begingroup$ Do you have oscillatory solutions for the full EFE's? Other than that, I guess our disagreement is just a matter of definitions. $\endgroup$ – Danu Jun 21 '16 at 20:32
  • $\begingroup$ @Danu: As I said in the earlier comment, the best technical definition is probably that it's a hyperbolic PDE; the part about oscillatory solutions is just an intuitive aid. I guess our disagreement is just a matter of definitions Your definition is wrong. You may find it helpful in straightening out your misunderstanding to google on "nonlinear wave equation." $\endgroup$ – Ben Crowell Jun 21 '16 at 21:45
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From Wikipedia

Predicted in 1916[2][3] by Albert Einstein on the basis of his theory of general relativity,[4][5]

Assuming you understand general relativity, why not consult the papers given there?

(2) Einstein, A (June 1916). "Näherungsweise Integration der Feldgleichungen der Gravitation". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin. part 1: 688–696.

(3) Einstein, A (1918). "Über Gravitationswellen". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin. part 1: 154–167.

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