One can consider the connected sum of two smooth n-manifolds $M_{1}$ and $M_{2}$ each with an embedded $\left(n-1\right)$ dimensional submanifold $V$. By deleting the interior of $V$ in both $M$s and identifying $V$ in each $M$ (via an orientation reversing diffeomorphism) one says we have a connected sum $M_{1}\#_{V}M_{2}$ along $V$.

Within the context of General Relativity, one could consider a general space (or spacetime) to be the connected sum of many manifolds.

Consider a prototypical example: two topologically distinct 3-spaces connected via $S^{2}$ (our $V$). For simplicity, one space could be $\mathbb{R}^{3}$ and the other some less stable (from the GR viewpoint) topology within the $V$ boundary.

From a purely qualitative point of view, I would expect the dynamics (as governed by GR vacuum solutions) would consist of $\mathbb{R}^{3}$ “overtaking” the topology interior to $V$ ($S^{2}$ in this case) because it is the more stable of the two. Thus I would expect $V$ to shrink under time evolution until only $\mathbb{R}^{3}$ remains. One can think of $V$ here as a surface where the space is undergoing a topological phase transition My thought was that the majority of dynamics in this (and perhaps the general) case reduces simply to dynamics of V.

To illustrate another example, we might consider two copies of $\mathbb{R}^{3}$ connected via $V$ (still a two-sphere for simplicity).

Now we have an Einstein-Rosin bridge connecting two flat spaces. The dynamics here are known via Wheeler's Causality and multiply connected spacetimes and we know that $V$ will indeed shrink to nothingness (with no sources).Again, $V$ seems to be a surface of topological phase transition. And again, it would seem the dynamics may be described in terms of the boundary of the connected manifolds (the two-sphere).

So my question is: To what extent can we describe an arbitrarily complicated spacetime in terms of the boundary of the connected sums that compose it? Under what conditions can all the dynamics be encapsulated therein? Via Thurston's Geometrization conjecture we might always describe a compact 3-space by such connected sums.

As a final (and admittedly “wild”) thought, suppose we have a spacetime consisting of the connected sum of two 4-manifolds via a 3-manifold $V$. If $M_{1}$ is some less topologically stable (wrt GR and $M_{2}$) spacetime and the $M_{2}$, (the interior of $V$), some more stable space (say $\mathbb{R}^{3,1}$), then one would expect $V$ (lets choose a three sphere for simplicity) to grow overtaking $M_{1}$. In this case, would the dynamics be described mostly/entirely by $V$?

If so, could our actual physical, dynamic space be considered as the (hyper) surface representing a topological phase transition between differing 4 topologies?

If we took the $V$s in this construction at each moment and considered them the foliation of a 4-space, that would coincide with the typical conception of spacetime. I've never seen anything like that mentioned in any GR book, yet the expansion of space appears intriguing from that point of view.

I'd be happy having the 3d case answered, but also find the 4d case intriguing.


1 Answer 1


Great question! I will try to give a relevant comment, from the perspective of quantum gravity, maybe someone else will write classical answer.

Dimensionality and topology is most likely an effective quantity of the universe, as many quantum gravity theories contain for example dimensional reduction (loop quantum gravity, causal sets, dynamical triangulations, group field theories, colored tensor models... etc. ) , where the spacetime seem to be lower dimensional on shorter scales/ higher energies.

This is also observed in an approach called CDT (causal dynamical triangulation). It uses Regge's approach to GR, constructing spacetime from simplices, but with the path integral formalism and the sum over distinct geometries it becomes a quantum theory of geometries (and hopefully of gravity). CDT uses numerical simulations, where the dynamical lattice configuration (connectivity of the simplices) gives rise to geometric properties of a "universe". (its analogous to QCD in its spirit, just it's a lattice formalism of gravity). In 4D CDT there is a globally hyperbolic foliation and each leaf must have the same topology, and during the numerical simulations (monte carlo) the changes to the geometry (moves) leaves both the foliation and the topology intact. Thus it is not that trivial how can one get different topology or dimensionality.

One would expect, that if you glue together 4 dimensional building blocks, then the resulting manifold will be also 4 dimensional, but it's not true. In one of the phases of the model the simplices organise themselves as branched polymers, with Hausdorff dimension = 2 and spectral dimension = 4/3. (spectral dimension is related to the log derivative of the return probability of a random walk and also to the spectrum of the Laplace-Beltrami operator corresponding to the geometry). In the physically relevant phase however (phase C / desitter phase) there is a 4 dimensional universe (hausdorff and long range spectral dimension is 4) that behaves like a desitter universe should (dimensionality, fluctuations, effective action...), but at close range it has spectral dimension = 2. This is the notion of effective dimensionality. Doesn't matter what are the building blocks as in the infinite volume limit the lattice vanishes and one recovers the continuum, effectively a measurable quantity for dimension emerges. The same is true for the topology:

Interestingly, when the spatial topology was chosen to be a 3-torus on every leaf of the foliation, but a non-trivial scalarfield was added, it triggered a phase transition in topology. In the following papers they are explained well:

Matter driven change of Spacetime topology

Scalar fields in CDT quantum gravity

Cosmic voids and filaments from quantum gravity

1 and 3 are letters, so they are short but dense. 2 is quite long but very detailed. (You can find all on arxiv too.)

Numerically the connectivity of the underlying graph still remains toroidal, but the scalar fields, which corresponds to harmonic maps or to deDonder gauge, reveal a spherical distribution (with some highly quantum phenomena) on the toroidal lattice and the "remaining part" is under the cutoff, thus effectively the topology of the spacetime changed. But the Hausdorff dimension of the new structure is 4.

I know that there is no accepted theory of quantum gravity yet. If there will be a theory of quantum gravity through Steven Weinberg's Asymptotic Safety scenario, then most likely cdt will be the lattice approach to quantum gravity.

But the message is, that topology can be an emergent phenomena.

  • $\begingroup$ Thanks! I hadn't even considered the quantum side of things yet! $\endgroup$
    – R. Rankin
    Commented Feb 6, 2023 at 20:23

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