I believe you suspect correctly. Momentum existing along the time axis is very GR, as long as the flow is periodic. Singularity in the universe is the proof of completion of function.

At the Submicroscopic antipode we go in at the present and come out 60 billion years from now, but time movement from the past into the future in the here and now is at right angles geometrically. There are different kinds of time, and time in GR is scaled like space, and for that matter, energy in the universe.

SRT and GR are conceptually almost scary, but the logic is rigid and the experimental verification is awesome.

Look at this. We are made of compacted Photons. "Each" Photon so far as it is concerned, is everywhere in the universe, for time ceases to pass at light "speed". We being made of photons, thus have a geometrically invariant location in the universe, and in a way- fill it! At our frame, we can travel for billions of light years and find worlds unknown, but every frame of reference in the universe, being constructed of photon "stuff" is fixed permanently on the hyperspheres. All frames are invariant because of the universe filling quality of photons...this is where the "geometric universe" comes from.

The vastness of the universe, even from our frame of reference, in GR can be seen to resolve to a singularity, and extend outward to our "light" horizon.

GR is positively iron-clad. No matter which way we turn, we are forced by its logic into a static, eternal cosmos, where "motion" is created by varying energy density at each, photon based frame of reference...and eternal existence is the only kind of existence. If you have ever wondered why the speed of light is measured the same without respect to any motion on the part of the source or the observer, now you know!

Best Wishes, Sam Cox