# Can spacetime exist in the absence of matter and energy?

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?

• 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 '13 at 23:46
• 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 '13 at 0:32
• The next interesting thing to consider is a vanishing stress tensor, but a non-zero cosmological constant, the De Sitter universe. It may seem ficticious, but that is where we are going. – Eduardo Guerras Valera Jan 30 '13 at 0:49
• 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 '13 at 2:40
• 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. – user10851 Jan 30 '13 at 6:28

The first problem we run into in answering this question is definitional. What does "matter and energy" mean? The Schwarzschild spacetime has a zero stress-energy tensor everywhere, and yet we characterize it by its mass $m$. A distant observer measures the black hole's gravitational field and says, "Yep, it has mass $m$" (i.e., its Komar, ADM, and Bondi masses are all $m$). Physicists traditionally don't include electromagnetic fields as matter, but relativists call them "matter fields." Gravitational waves don't have a definable contribution to the stress-energy tensor at a given point, but they do have energy if you average them over a wavelength.

A second problem is simultaneity. Suppose for the sake of argument that all matter in our actual universe is eventually going to end up in black holes. (We're pretty sure that this won't happen, but it's not logically impossible, just not likely given what we know about astrophysics and cosmology.) Clearly right "now" our universe contains matter. At some "moment in time" a gazillion years in the future, if we imagine that all that's left is black holes, we could say there will be no matter, in the sense that the stress-energy tensor is identically zero everywhere. If we believed that spacetime could only exist in the presence of matter, then we would have to say that at some "point in time," spacetime would have stopped existing, since the last piece of matter went into a singularity and was no longer present on our spacetime manifold. But the scare quotes around "now," "moment in time," and "point in time" remind us that we can't fundamentally define these times in the sense of universal simultaneity. Relativity doesn't have simultaneity. When matter falls into a black hole, it's valid for a distant observer to say that the matter never makes it past the horizon.

As a final absurd possibility, suppose that Wheeler's geon idea turns out to be valid beyond his wildest dreams. (This is not considered likely anymore, but it's not obviously logically impossible.) We can describe electrons, quarks, etc. as excitations of the gravitational field. We could then say that right now, in the room where I'm writing this, there is no matter whatsoever. The stress-energy tensor is identically zero everywhere, and there are no matter fields, only gravitational waves.

• I had never heard of (or possibly it just didn't register) Wheeler's idea of "geons," but wow, does that ever sound exactly like the kind of idea his remarkable mind would have come up with. It's too bad he didn't work closely for more years with the more conservative Feynman, who was willing to take the kernel of a wild Wheeler idea and make it something solid. (One should also add humble Dyson, who then played a huge role in converting Feynman's ideas into precise equations.) Random: One wonders if a point-like electron is just a black hole that can't evaporate due to charge conservation. – Terry Bollinger Sep 1 '13 at 17:12

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 a bit 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.

• Nice answer, +1, but I don't think anything is added to the answer by talking about gravitons rather than gravitational waves. – user4552 Aug 30 '13 at 3:44

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.

• 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 '13 at 17:46

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.