general relativity - 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?

Answer

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.