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I'm pretty sure Ernst Mach would have said that spacetime cannot exist without matter in it.

But I'm also pretty sure that a black hole can be described as a self-sustaining gravitational field, which is (I guess?) a case of spacetime sustaining itself without the matter meaning much anymore.

So, what is the current theoretical view on this?

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I suppose the only thing you can do is to take Einstein equations and put the stress-energy tensor to 0. You obtain a Ricci-flat universe. – Bzazz Jan 29 at 23:46
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I feel as though this is more of a philosophical question than a physics question. Certainly there are solutions to Einstein's equations for which the stress tensor vanishes identically as @Bzazz points out, but isn't your question more about whether such solutions in some sense "could have" manifested themselves in reality? Perhaps your question would be more usefully stated as "to what extent are such spacetimes relevant to describing our universe?" – joshphysics Jan 30 at 0:32
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Gravitational waves can still exist in an empty universe ($T^{\mu\nu}=0$). In that way the concept of space-time itself has important significance without any reference to either the cosmological constant or inherit matter fields. – Benjamin Horowitz Jan 30 at 2:40
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And yet a scientific positivist of Leibnizian descent would argue that all we can observe are relative displacements and motions of test particles; spacetime is a tool for predicting these but as such it has no independent existence without matter around. – Chris White Jan 30 at 6:28
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@Terry Bollinger, Mach didn't formulate his idea precisely, as you say, but it seems to me that Einstein had it in mind all the time. He does some references to Mach in the Princeton lectures. – Eduardo Guerras Valera Jan 30 at 8:39
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There are two forms or states of "space" 4-momentum space and spacetime. The quantum vacuum is an example of both spaces mutually transforming via the uncertainty relations. In other words at the quantum level, spacetime exists as an ensemble of tiny intervals whose longevity is the same as the virtual particles associated with them. Thus spacetime is generated by the quantum vacuum. This is my opinion. I have worked on Fourier transformation of spacetime.

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user21392, thanks, that's an interesting response and one for which I have immediate empathy, since I've done hobby work myself treating momentum space as just as real as ordinary space, with Fourier linking 4D spaces. Time-energy (or mass) is the really interesting one, yes? But I never remotely thought of connecting the dual $\{tx_i, mp_i\}$ spaces to this question... Hmm. My stuff is just hobby, so I don't quote it for anything. Do you happen to have any peer-reviewed papers on your dual-spaces idea, either by you or someone else? Also, isn't your idea related to space as virtual particles? – Terry Bollinger Feb 26 at 17:46
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Yes. To summarize the comments, there are a lot of physically interesting vacuum solutions to Einstein's field equations, including the eternal Schwarzschild black hole.

If you buy into the idea that gravity is mediated by gravitons, then this idea has even more meaning. You can imagine a bunch of gravitons just floating around. As the quanta of space-time itself, these gravitons could exist independently of any source, just like gravitational waves. So the particular configuration of just space-time itself could contain information.

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The answer is "no": We know that vacuum cannot exist in the classical sense. If you mean "solutions to Einstein equations" rather than "spacetime", then yes (there is the trivial solution...) But this has noting to do with reality, not even with "potentially possible/imaginable reality", assuming that the laws of nature we know for sure to exist, do exist.

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