Why does spacetime propagate gravitational waves?

Indeed it is assumed that in the absence of a source of gravitation spacetime will be uniform. In that sense we can say that spacetime curvature must be regarded as a form of elastic deformation. As in: when the source of deformation goes away, the deformation goes away.

Other than that the nature of spacetime is unknown. This elastic property must be granted in order to formulate a theory at all.

More generally:
In terms of GR very little is known about the nature of spacetime. In order to formulate GR at all some assumptions must be made. For sure: the fact that such a succesful theory has been constructed is strong supporting evidence for these assumptions. Our concept of the nature of spacetime is limited to the content of those assumptions.

In response to the comment by Ben Crowell:

I will first present this response, then I will explain why I judged this lengthy response to be necessary.

In the original question Mads asked: "Why doesn't spacetime preserve deformations after the object that caused the has moved away?"

I infer that Mads is asking: "Why does spacetime undergo elastic deformation rather than plastic deformation?"

Of course, plastic deformation of spacetime is such a bizarre concept that it just never occurs to a physicist to consider it, it doesn't enter the mind. But of course we can interpret the question in a more general sense: "what is the underlying physics that makes spacetime behave in the way that it does?"

In the following the background is the distinction between 'implicit' and 'explicit'

As we know: Einstein used Einstein's principle of equivalence and the demand that in non-relativistic conditions GR must reproduce Newton's inverse square law of gravity as things that his new theory would have to satisfy in order to be considered viable. By and large these two contraints were sufficient to narrow down the possibilities to the 1915 GR.

Implicitly the principle of equivalence has vast consequences.
If you grant these two:
- the principle of equivalence leads to GR
- GR implies the existence of gravitational waves
Then it follows that you must grant the combination: principle of equivalence ultimately implies the existence of gravitational waves.

While the logical implications are far reaching, the principle of equivalence does not describe spacetime explicitly. In that sense the principle of equivalence is very cautious. The principle of equivalence is minimal; it states the very minimum that is needed in order to narrow down the GR search sufficiently.

Let me make a comparison: historically the second law of thermodynamics was first an empirical law. Later on the science of statistical mechanics moved the description of entropy to a deeper level. Before the development of statistical mechanics the pre-statistical thermodynamics was a completely succesful theory. Still, at a later time a framework was developed that established a foundation underneath its predecessor.

We cannot exclude the possibility that at some point in the future a theory of spacetime will be developeded that is to GR what statistical mechanics was to pre-statistical thermodynamics. If such a theory of spacetime is developed it may describe on a fundamental level a physical mechanism allows spacetime to curve, and that causes it to return to uncurved state when the source of gravitation moves away again.

To Mads, who asked the original question:
by conscious decision the assumptions that are granted in order to formulate GR are minimal. We don't know why spacetime has the properties that it has. What we know is what we need in order to be able to formulate GR; the principle of equivalence. We grant that. Because of the success of GR we judge this to be a good call.

Now: why such a lengthy response, far longer than my first reply to Mads.
To Ben Crowell:

1) I urge to you always apply historical awareness.
Every new theory of physics is formulated by starting from one or more assumptions that must be granted in order to formulate the theory at all. Example: on publication of the Principia many of Newton's contemporaries objected that since no physical mechanism for the inverse square law was described the theory should be rejected. Newton argued that the very success of the inverse square law of gravitation was sufficient evidence in itself. Newton was right of course.

In the history of physics it is never the case that the reigning theory is the exhaustive theory that will not have a successor. A new theory replaces a predecessor if it moves the description of the physics taking place to a deeper level of description, thus deepening the understanding.

Yes, of course GR is a thoroughly successful theory of motion and gravitation. But like all theories of physics in order to be able to formulate it at all some assumptions are granted. Granting these assumptions is justified by the success of the theory.

2) I urge you to be always aware of far reaching implications of starting assumptions. Yes, of course GR does not make explicit statements about elastic properties of spacetime. But if you grant that propagation of gravitational waves is possible you have implicitly granted the elastic property that enables that.

Here on stackexchange comments are limited to 600 characters for a good reason; the stackexchange designers want to give some room for comment, but everybody is urged to refrain from getting sucked into protracted one-on-one debate. I support that policy wholeheartedly.

So yeah, writing this protracted response-to-a-comment feels very awkward.

In general, if you feel an answer that I wrote is off, then please cast your response in the form of a additional answer to the question, an answer where you try to lead the reader in the direction that you deem better. Let the reader decide what answer is the best fit for the question.

Gravitational waves apparently travel through space, so in my primitive understanding of things, space (or spacetime?) must have a neutral "shape" that it is compelled to resume after being deformed. Is this true or am I misunderstanding?

I would say this is not a good way of thinking about it. For example, after a gravitational wave passes over the earth, the spacetime around the earth doesn't go to a preferred flat shape, it just goes back to the original state of curvature that it had before because of the earth's field.

A good analogy is what happens if I broadcast a radio beep. The radio signal is an electromagnetic field, and the earth also has its own static magnetic field. While the radio signal is propagating, its electric and magnetic fields just add onto the earth's magnetic field. When the radio wave is gone, the earth's magnetic field is back to what it had been.

This is called superposition, which is just a fancy way of saying addition. Another classic example of superposition is two sets of ripples on a pond spreading through each other without interacting.

Now it's not actually true that general relativity obeys a law of superposition, but it is an extremely good approximation for a small-amplitude gravitational wave passing through the static curvature of an object like the earth.

This is an answer to update 1 of your question.

I should point out: the design of stackexchange is that a sufficiently distinct follow-up question should be asked separately, as a question on its own.

The question:
Is there any hand-wavy path from the principle of equivalence to the existence of gravitational waves?

The case of gravitational waves is very different from electromagnetic waves. For more about that I suggest you read the article Aberration and the speed of gravity by Steven Carlip.

Despite the differences I will still use electromagnetism as an analogy.
Around the time of Faraday the concepts of 'electric field' and 'magnetic field' were introduced, and Faraday introduced a concept of 'field lines'. The field was thought of as a mediator of the interactions, electrostatic interaction and magnetic interaction. It wasn't known whether these fields actually existed. This hypothesized field is not an observable; what we measure is how in the presence of such a hypothesized field trajectories of particles are affected. (For example the trajectories as particles move through a cloud chamber.) Whatever the case, the field concept is at least a good instruments to guide thinking about the problems. To aid his thinking, Maxwell used a physical model of the ether. This model was designed to embody known properties of electricity and magnetism. Some of the logical implications of the model proved to be true physical properties of the real world, that is how Maxwell's model was worthwile.

A necessary condition for wave propagation is capability of oscillation in whatever form. Maxwell noticed that in his model conditions for oscillation were present. There is an unstrained state, a force can stress the state of the model away from the unstrained state, and on rebound the changing state will overshoot. That is, once the state is at a rate of change, it tends to continue at that rate of change, analogous to inertia. That is what you need for oscillation. Maxwell showed that if you granted the existence of an hypothetical (but plausible) phenomenon that he named 'displacement current', then according to his theory electromagnetic waves should exist, and he could calculate the speed of propagation, and it was to within known accuracy of measurement the same as the speed of light.

The very existance of electromagnetic waves also implied that the electromagnetic field is a true physical entity, not just a mental construct. An electromagnetic wave continues to propagate, regardless of whether the source that emitted still exists.

Maxwell's prediction of the speed of electromagnetic waves is somewhat analogous to the way that Isaac Newton calculated the speed of sound propagation from first principles. Air meets the requirements for oscillation:
- air has elasticity
- air has inertial mass, so on rebound it will overshoot
In the Principia Newton presented a calculation that used these properties: given the elasticity of air and its mass per unit of volume the speed of sound can be derived.

In the case of electromagnetism the restoring force that acts to restore the state back in the direction of unstressed state is very, very strong. Thus the speed of propagation of electromagnetic waves is very, very fast.

Principle of equivalence and gravitational waves

As mentioned elsewhere, the principle of equivalence has vast implications. As electromagnetism grew from strength to strengh physicsts began to explore the possibility of the existence of a gravitational field, a field that acts as the mediator of gravitational interaction. The principle of equivalence implies the supposition that spacetime itself is the very gravitational field. According to GR spacetime itself is acting as the mediator of gravitational interaction.

If you grant the principle of equivalence you have implicity granted that spacetime itself has physical properties, to be described by a quantitive theory. There is an unstrained state, which is referred to as 'geometrically flat spacetime', and any source of gravitation causes a strained state of the field, that is referred to as 'curvature of spacetime'.

My knowledge of GR is insufficient to go beyond this stage of discussion.

Let me point out some stark differences with the history of Maxwell's conclusion that electromagnetic waves must exist. It was possible for Maxwell to calculate the propagation speed of electromagnetic waves from first principles. However, in the case of gravitational waves it is assumed that they propagate at the same speed as light. It is clearly very plausible that gravitational waves propagate at lightspeed, but logically there is room for alternative theories of gravitation according to which gravitational waves propagate slower. (A recent LIGO observation with gravitational wave and light from a cataclysmic event arriving effectively simultaneously is strong evidence that gravitational waves indeed propagate at lightspeed.)

According to the wikipedia page about gravitational waves Einstein went back and forth on the question as to whether gravitational waves exist, so evidently this is among the most opaque aspects of GR.