Does Time Require Matter to Exist? I've heard that General Relativity entails matter (or mass) is necessary for time to exist. However, there are vacuum solutions where the universe is empty of matter but still has spacetime.
P.s: I'm not necessarily talking about proper time. I'm referring to time in general.
 A: No, general relativity doesn't make any claim as to whether matter must exist or not. In fact, the simplest of the solutions to the Einstein equations are vacuum solutions. For example, the Kerr-Newman blackholes and their special cases such as the Schwarzschild blackholes and Kerr blackholes. The dimensionality of spacetime is still $4$ in these solutions with one dimension being timelike. While these are all stationary solutions, you can also get non-stationary solutions in the vacuum. For example, gravitational waves. Gravitational waves are purely vacuum solutions and also exhibit non-trivial dynamics unlike the stationary solutions. So, the existence of time isn't contingent upon the existence of matter in general relativity. 
I should clarify that in the case of Kerr-Newman blackholes, there exists electromagnetic fields so they are not truly vacuum solutions but still, they are solutions without the existence of any matter. Also, the special cases of the Kerr-Newman blackholes which are uncharged (i.e. the Schwarzschild blackholes and the Kerr blackholes) are truly vacuum solutions. 
A: In view of general relativity metric $g_{\mu\nu}$ is spacetime. Therefore the solution of Einstein field equations should be global and maximal. The metric of the so-called "vacuum solution" describes only a matter free domain of spacetime. For example, the two famous Schwarzschild solutions are in true just two parts of a single solution with both metrics valid only in their solution domain (interior and exterior).
I think that the trivial solution of Einstein field equations, $g_{\mu\nu}=0$, describes exactly what Einstein meant:
“It was formerly believed that if all material things disappeared out of the universe, time and space would be left. According to relativity theory, however, time and space disappear together with the things.”
https://rloldershaw.medium.com/einstein-without-matter-there-is-no-space-or-time-c2357c75286b
A: safesphere responded to my question by saying that "time and energy are Fourier conjugates (or more generally, spacetime and energy-momentum) and cannot exist in the physical reality without each other. In other words, GR states that spacetime is the field produced by matter just like the electromagnetic field is produced by charges."
My response is the following:
Spacetime and energy-momentum are not Fourier conjugates. In Newton mechanics they are topological duals (flows and generators). 
This connection is different in curved spacetime. In curved spacetime the Fourier transform is not definable. This is the reason in mathematics some differential equations are perfectly understood in a flat space are not nearly understood even in simple curved spaces. 
In GR it is not even possible to formulate the claim that spacetime and energy-momentum are Fourier conjugates. 
There are many more things one could say here. There is a connection between spacetime and energy-momentum (this is the reason why energy is not conserved in GR). But only quantum observables in flat spacetime are connected via some kind Fourier transform. But even this connection is wrong for photons because they have no sharp spacetime observable but an energy-momentum observable.
As I said, I could say many more things here. But bottom line is, that in GR energy-momentum and spacetime are not Fourier conjugates.
