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There are many posts, questions, comments and excellent answers on this website pertaining to cosmology, expansion, universe, spacetime et cetera. Yet, the following question was not asked or answered, so here it goes.

This question deals with - trying to 'force' an analogy between spacetime metrics and the underlying manifold and it's topological properties. (Due to my lack of mathematical knowledge, I will shamelessly abuse some language and notation here.)


We know the following very well:

  1. The dynamics of spacetime metric is excellently described by Einstein's Field eqns: $$G_{\mu\nu} = k\ T_{\mu \nu}$$

  2. The expansion of spacetime metric on a collosal scale is described by: $$ ds^2 = dt^2 - a^2(t) \gamma_{i j} dx^i dx^j$$ The rate of change of this scale factor $a(t)$, in the form of the Hubble parameter, then characterizes expansion of spacetime metric as: $$ H \equiv \frac{\dot{a}}{a}$$ Experiments have measured this parameter and roughly we have [1]: $$H \approx 67.80\pm0.77 \ \frac{Km}{s \ Mpc}$$

Given the above two information and the central role that the metric plays in Riemann Geometry and GR, a colossal amount of information can be extracted about our cosmos. NOTE - metric tensor, like any other tensor on a manifold, is a bi-linear map from a certain space to $\mathbb{R}$. It is also an added physical structure on the Riemannian Manifold, which relates our cherished notions from Geometry to the real world (metric defines physical distances).


There is so much wonderful cosmology that has been done and continues to be studied using the metric. But due to their inherent nature, these studies are limited to the geometrical aspects only. My questions are:

  1. What subject, if any, studies the topological aspect of spacetime (topological cosmology, for example)?
  2. What is our current understanding from such studies? Any references or links will be appreciated.
  3. Since only the physical space-time metric undergoes physical processes, is the topology of the underlying manifold an unchanging static constant?
  4. More specifically: For the last 13.8 billions of years, the physical distance between two objects has increased (from the metric related physics) which is the expansion of universe. Has the coordinate distance between two objects on an appropriate scale changed? Is it even measurable? Does this question make any scientific sense? (appropriate meaning the same scale on which the universe is homogeneous and isotropic)

Finally, any general perspective on such a topic is welcome. Thanks in advance!


[1] arXiv-1303.5062


marked as duplicate by John Rennie cosmology Jun 3 '18 at 8:42

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  • $\begingroup$ This isn't an answer, because it's way beyond my pay grade, but I'm pretty sure the global topology is invariant, any changes to that would be supernatural. ;) But if wormholes or warp bubbles can be created or destroyed that'd change the local topology. I'm not sure how black hole singularities fit with that, I expect that an actual answer will address that issue. $\endgroup$ – PM 2Ring Jun 3 '18 at 8:27
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    $\begingroup$ Q4 doesn't make a lot of sense since there is no such thing as the coordinate distance, GR gives you a lot of freedom of choice of coordinate systems. But perhaps you're asking about comoving coordinates. $\endgroup$ – PM 2Ring Jun 3 '18 at 8:35
  • $\begingroup$ @PM2Ring: Haha, pay grade? I'm scared that you're very certain that 'Global Topology' is invariant. Certainty without evidence, isn't Science. As far as I know, BH singularities are, according to 'expert' opinion, simply point to a lack of complete theory of gravity. You're right, Q4 is hard to define. And no, I don't mean comoving coordinates, I meant exactly what you found as senseless! :) $\endgroup$ – topologically_astounded Jun 3 '18 at 11:49
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    $\begingroup$ No, I'm not certain, but AFAIK under standard GR the global topology cannot change, no matter how the matter and energy evolve in accordance with the EFEs. So if the universe is open it can't turn into a closed universe, or vice versa, or a flat universe can't evolve into a cylindrical one. But I guess you could say that the model simply assumes that invariance. FWIW, I expect that quantum gravity will prevent black hole cores from being mathematical singularities, but that's just my gut feeling, and the universe is under no obligation to conform to my primitive monkey-brain intuitions. ;) $\endgroup$ – PM 2Ring Jun 3 '18 at 15:12
  • $\begingroup$ @PM2Ring: Haha. Agreed. Hope so. and haha good-one. $\endgroup$ – topologically_astounded Jun 3 '18 at 15:29