Space time a function of itself, objects in it, or both? Is spacetime a function of itself, objects within it, or both? I am struggling to understand just what is spacetime without objects in it (or theoretical reference points) and thus no frame of reference except itself. It seems that spacetime without objects (on its own) is more of an abstraction and would be hard if not impossible to quantify without someTHING in it by which it could be measured. I feel like I must be missing something.
 A: It is General Relativity that changed our concept of space and time. Even special relativity assumes that time and space exist as coordinates even though it changes the metric with respect to the  Galilean (Newtonian) relativity. 
As physics started to be rigorously mathematically formulated in the eighteenth century , space and time were considered as Euclidean coordinates, conceptually preexisting whether there were any masses or energies in the universe.
With General Relativity, the mathematical formulation:

The space description mathematically is on the left of the equation, 

it is the curvature scalar. It says how much differentially spacetime curves.
On the right is the energy-momentum tensor. If there is no energy or  momentum in the universe then things become more complicated.The solution to the second equation with the right hand side=0 depends on the model of general relativity. If there exists a cosmological constant or it is zero.


Taking the trace of both sides reveals that the constant of proportionality k for Einstein manifolds is related to the scalar curvature R by




Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with k proportional to the cosmological constant. 

In the  case of the cosmological constant being zero one gets zero curvature, and recovers the pseudo-euclidean space of special relativity, which of course is of no descriptive of nature use, since there is no energy or momentum, i.e. observers, to observe this. spacetime.

Is spacetime a function of itself, objects within it, or both?

Spacetime as a term depends on its definition. I have described what the definition is for physics. One can mathematically define coordinates in infinitely many ways. As far as observations go up to now the spacetime nature has chosen is General Relativity's as described above which stops existing if the energy momentum tensor is zero since, for physics, no massless/energyless observers can exist, and physics is about observations. 
