(e-mail address: kenneth.koehler@uc.edu)

(posted October 31, 2014,

last modified November 14, 2014)

Abstract

We construct a toy model for the dynamics of hyperbolic 3-manifolds orthogonal to some time coordinate in spacetime. We identify externally measurable quantum numbers associated with those manifolds, and describe interactions allowed by natural conservation principles associated with those quantum numbers. Using the SnapPy Orientable Closed Census of hyperbolic 3-manifolds, we illustrate the viability of the model.

In [5], we proposed a model in which the quanta of quantum gravity are hyperbolic 3-manifolds connected to the "external" spacetime by ε-necks. These ε-necks contain incompressible tori singularities separating incompatible geometries. By extrapolating a conjecture of Dunfield and Thurston [6], and arguing that the hyperbolic 3-manifolds must behave as Bose-Einstein objects, we were able to compute the entropy of a Schwarzschild black hole, and to show that there exist solutions to Einstein's equations in which the time evolution of the hyperbolic manifolds makes them "rapid breathers".

In this paper we will attempt to identify the externally measurable quantum numbers of the manifolds, and using the SnapPy [7] Orientable Closed Census as a test bed, examine the interactions allowed by the natural conservation principles associated with those quantum numbers. In section 2, we will address the quantum numbers, and describe possible interactions. In section 3, we will describe the SnapPy Orientable Closed Census, justify its use as a test bed and categorize the possible interactions within that set of manifolds. Finally, in section 4, we will summarize what we have learned.

The torus, however, must be incompressible: its embedding into the spacetime must inject its fundamental group into that of the spacetime. [9] For that reason, it cannot appear as a doughnut, whose fundamental group is rendered trivial by the embedding. Rather it must appear as a cylinder with the end caps identified. For any given torus, there are an infinite number of ways to make the identification, described by integers p and q: the number of times the longitude (meridian) wraps around the torus. Provided p and q are relatively prime (and q ≠ 0), and except in finitely many cases, the 3-manifold will remain hyperbolic. [10] These integers are called the Dehn surgery invariants; (p,q) Dehn surgery corresponds to a relative end cap twist of 2πq/p. If the one of the invariants is negative, the meridian rotation will be clockwise, and if both are negative, the resulting manifold is equivalent to one in which both are positive ((-3,-1) is the same as (3,1).) Note that the twisted torus is symmetrical with respect to both end caps: a 2πq/p twist at one end is equivalent to a 2πq/p twist at the other end.

If a Maxwellian daemon were to send a test particle into one of the end-caps of such a torus, it would immediately exit the other end-cap, but with a translation corresponding to the Dehn invariants. Hence the Dehn invariants are candidates for observable quantum numbers associated with the hyperbolic 3-manifold quanta in our model. Note, however, that to be considered as candidate quanta for a theory of quantum gravity, these tori must have length scales of order the Planck length.

If we consider the breather solutions from [5],
the scalar curvature of the hyperbolic manifold must be twice the "internal" cosmological constant.
But the scalar curvature for *all* hyperbolic 3-manifolds is

R = -6 / (length scale)^{2},

so assuming that the scale of all of the manifolds is the same, for all hyperbolic manifolds in this model

Λ_{internal} = -3 / L_{Planck}^{2}.

By the Mostov Rigidity Theorem,
the volume V of a hyperbolic manifold is a topological invariant. [10] Since
we chose Λ_{internal}
to be the same for all hyperbolic 3-manifolds (again, considering only the breather solutions), the energy

E = c^{4} V Λ_{internal} / G

is also a topological invariant, and is hence another candidate for an observable quantum number associated with the hyperbolic 3-manifold quanta in our model.

All cusped hyperbolic 3-manifolds can be constructed by the identification of the sides of one or more polyhedra. [10] Their volumes are determined by the dihedral angles of the parent polyhedra, as are the shapes of the toral cross sections of each cusp. When Dehn surgery is performed on the cusps, both the torus shapes and the volume are modified; the volume of the closed manifold is strictly less than the volume of the corresponding cusped manifold. [10] So the volumes, and therefore the energies, of all hyperbolic 3-manifolds in our model, form a discrete set. As the Dehn invariants approach ∞, the energies will become an approximately continuous set. Note that in general, every cusp of every parent manifold will have a different limit set.

Intrinsically, hyperbolic 3-manifolds also possess at least one other topological invariant: the Chern-Simons number. It is unclear how one might measure this from the external spacetime, however. If the incompressible tori have kinetic motion with respect to each other, momentum and angular momentum will of course be conserved. But aside from the usual suspects, the quantum observables intrinsic to the hyperbolic 3-manifolds themselves seem to be three: E, p and q.

Suppose that two incompressible tori were to meet and join at one pair of end caps. Considering the end cap
identifications, one might naively think that the result would be a pair of hyperbolic 3-manifolds hidden
behind a solid torus of genus 2. However, a genus 2 torus cannot be an incompressible surface separating
incompatible geometries. [11] Therefore we must consider the resulting object as a *single*
incompressible torus, "inside" of which is a new hyperbolic manifold.

We expect that the energy of the new manifold must be the sum of the energies of the component manifolds. But what of the Dehn invariants? Since the twist angles must add, we have

q_{new} / p_{new} = q_{1} / p_{1} + q_{2} / p_{2}

We will call a joining which satisfies those conservation conditions a "3-vertex." Because of the nature of the 3-vertex, all interactions involving 4 or more quanta can be constructed by combining 3-vertex interactions.

As we shall see below, from the point of view of the hyperbolic 3-manifolds,

This means that from the point of view of our model,

that is, multiplets of quanta which are different yet physically indistinguishable. They would, however, affect the probability amplitudes associated with any given vertex in a Feynman diagrammatic context.

We will therefore also define a "flavor-changing" interaction, in which one member of a multiplet can spontaneously change into another member of the same multiplet. Flavor multiplets will also affect the "landscape" of possible interactions, as we will see below.

- 11 manifolds could not
be completely evaluated by SnapPy 2.2 because of a software problem
(m015(-6,1), m285(1,2), s298(5,1), s445(-3,2), s502(-2,3), s698(2,3), s802(-1,3),
s849(3,4), s855(-2,3), v2297(-1,3) and v2833(-1,3), where our notation here and below is interpreted as
*census_name_of_parent_manifold*(p,q)); - more significantly, since all numerical computations incur representation and convergence error,
we must decide in advance how much precision to demand. This will affect decisions of numerical
equality for volumes, etc. Because the high precision manifold class involved prohibitively long
execution times, we chose to use the normal manifold class, for which the minimum precision of
all the calculations performed on the census was nine significant digits. Because the last digit can
incur representation error, we elected to round to eight digits and express all volumes as 32-bit integers
* 10
^{-8}.

The census manifolds feature:

- 9,218 different volumes (at our choice of significance);
- 37 different Dehn invariants (-9 ≤ p ≤ 9, 1 ≤ q ≤ 4)
- 47 manifolds which have zero Chern-Simons number;
- 479 nontrivial homology groups, ranging from
/423 to 2*Z*, the most common being*Z*/2 (228 manifolds);*Z* - 21 nontrivial symmetry groups, including 10 which are dihedral or have a dihedral factor; the
most common symmetry group is
/2 (7,025 manifolds;*Z*is second most common with 3,474 manifolds);*Z* - 37 manifolds which are amphicheiral (possessing an orientation-reversing symmetry);
- 1,159 manifolds which form 498 multiplets, distributed as follows:
- 397 doublets
- 71 triplets
- 14 quadruplets
- 10 quintuplets
- 2 each of sextets and septets
- 1 each of an octet and a 12-plet

In fact, modulo flavor-changing within multiplets, there are 46 such interactions possible in the census we used, involving 85 different manifolds.

The interactions group into the following disjoint graphs:

- one involving 18 interactions among 29 manifolds with 20 different volumes; 24 different homology
groups are represented, but only five symmetry groups (D
_{4}, D_{6},, 3**Z**/2 and**Z**/2 + (at least**Z**)/2); none of the manifolds are amphicheiral:**Z**- m003(-3,4) + m262(-3,2) ↔ s944(-1,2)
- m003(-3,4) + v2101(1,2) ↔ v3145(3,2)
- m004(6,1) + m004(6,1) ↔ m223(3,1)
- m004(6,1) + m039(6,1) ↔ s554(3,1)
- m006(3,1) + m006(3,1) ↔ m206(3,2)
- m006(3,1) + m140(-6,1) ↔ s823(6,1)
- m007(3,1) + m007(3,1) ↔ m019(3,2)
- m007(3,1) + s772(-3,2) ↔ v3213(-3,1)
- m011(-3,1) + m011(-3,1) ↔ s772(-3,2)
- m011(-3,1) + m136(6,1) ↔ v2274(-6,1)
- m015(6,1) + m303(-3,1) ↔ v2274(-6,1)
- m039(6,1) + m039(6,1) ↔ v3469(3,1)
- m039(6,1) + m303(-3,1) ↔ v3066(-6,1)
- m130(-3,1) + m136(6,1) ↔ v3066(-6,1)
- m147(3,1) + m147(3,1) ↔ v3145(3,2)
- m223(3,1) + m223(3,1) ↔ v2168(3,2)
- m223(3,1) + m147(3,1) ↔ v3066(3,2)
- m223(3,1) + m140(-6,1) ↔ v3066(6,1)

Note that

The structure of these interactions is somewhat complex, but graphically the most interesting feature is a cycle connecting m136(6,1), m303(-3,1), v2274(-6,1) and v3066(-6,1), with most of the remaining manifolds connected to the latter two via acyclic trees (one with five manifolds and one with 18 manifolds.)- m147(3,1) and v3469(3,1) are each members of a different doublet;
- m136(6,1), s772(-3,2) and v3145(3,2) are each members of a different triplet; and
- m303(-3,1) is a member of a quartet.

So, for instance, production of a v3066(-6,1) via the m303(-3,1) channel will be 33 1/3 % more likely than through the m136(6,1) channel.

- one graph involving six interactions among 10 manifolds with seven different volumes; nine different homology
groups are represented, but only three symmetry groups (
,**Z**/4 and 3**Z**/2); one of the manifolds is amphicheiral (m303(-1,3)):**Z**- m010(-4,3) + m010(-4,3) ↔ m395(-2,3)
- m010(-2,3) + m010(-2,3) ↔ m303(-1,3)
- m010(-2,3) + m016(-2,3) ↔ m350(-1,3)
- m010(-2,3) + m239(-2,3) ↔ v3213(-1,3)
- m016(-2,3) + m016(-2,3) ↔ m371(-1,3)
- m016(-2,3) + m395(-2,3) ↔ v3491(-1,3)

The structure of these interactions is much simpler; m010(-2,3), m016(-2,3) and m395(-2,3) are the only manifolds with more than one edge in the graph. m395(-2,3), v3213(-1,3) and v3491(-1,3) are all members of different doublets, m371(-1,3) is a member of a triplet, and m303(-1,3) is a member of a quartet.

- one graph involving three interactions among five manifolds with four different volumes; the
manifolds all have different homology groups, but all have symmetry group
except m017(-1,3), which has D**Z**_{6}symmetry; none of the manifolds in this group nor in either of the following two groups are amphicheiral:- m003(-2,3) + m003(-2,3) ↔ m017(-1,3)
- m003(-2,3) + m339(-2,3) ↔ v2573(-1,3)
- m006(2,3) + m017(-1,3) ↔ m339(-2,3)

This graph too has only three manifolds with more than one edge in the graph: m003(-2,3), m017(-1,3) and m339(-2,3).

- one involving two interactions among 4 manifolds with three different volumes;
all have different homology groups, but all have symmetry group
except m220(5,2), which has D**Z**_{6}symmetry:- m015(5,1) + m015(5,1) ↔ m220(5,2)
- m015(5,1) + m221(-5,2) ↔ v3184(-5,1)

The only manifold in this graph with more than one edge is m015(5,1).

- and 17 isolated interactions, three of which involve three manifolds and 14 of which involve two; nearly
all of the volumes and homology groups are different, but all of the manifolds have either
D
_{4}, D_{6},, 3**Z**/2 or**Z**/2 symmetry:**Z**- m006(1,3) + m135(-2,3) ↔ v1859(2,3)
- m026(1,4) + m140(-1,2) ↔ v3066(1,2)
- m036(-1,3) + m030(2,3) ↔ v2296(-2,3)
- m003(-4,3) + m003(-4,3) ↔ m033(-2,3)
- m003(-3,1) + m003(-3,1) ↔ m016(-3,2)
- m015(-6,1) + m015(-6,1) ↔ v2202(-3,1)
- m015(8,1) + m015(8,1) ↔ s781(4,1)
- m022(4,3) + m022(4,3) ↔ s772(2,3)
- m035(-6,1) + m035(-6,1) ↔ v3076(-3,1)
- m036(-4,3) + m036(-4,3) ↔ v2668(-2,3)
- m036(-2,3) + m036(-2,3) ↔ s916(-1,3)
- m037(2,3) + m037(2,3) ↔ v2626(1,3)
- m081(2,3) + m081(2,3) ↔ v2719(1,3)
- m116(-2,3) + m116(-2,3) ↔ v3185(-1,3)
- m116(2,3) + m116(2,3) ↔ v3504(1,3)
- m168(3,1) + m168(3,1) ↔ v2831(3,2)
- m206(1,2) + m206(1,2) ↔ v3043(1,4)

Of these, m006(1,3), m016(-3,2), m026(1,4), m036(-4,3), m035(-6,1), v2296(-2,3), v2626(1,3), v3066(1,2) and v3185(-1,3) are each members of separate doublets; s781(4,1) and v3076(-3,1) are each members of a different triplet; s916(-1,3) and v2668(-2,3) are each members of a different quartet; and s772(2,3) is a member of a quintet.

Even at this juncture, however, it has many tantalizing features:

- several well-defined discrete quantum observables;
- a well-defined set of classical limits (as p and q → ∞);
- a potentially complex interaction landscape;
- if one of the interaction subgroups were to make contact with known particle physics, the remaining manifolds would constitute a dark sector;
- the presence of multiplets; should the classification allow further observables to be defined, multiplet splitting and flavor-changing interactions could be modeled;
- the vast majority of the manifolds appear to be chiral, so it may be possible to implement handedness into the model; and
- the preponderance of
/2 and**Z**symmetry groups could fit into emergent continuous symmetries (**Z**is the discrete center of U(1) and**Z**/2 is the discrete center of SU(2), all of the orthogonal and symplectic groups, and E**Z**_{7}.)

We also note that our model is something of a hybrid: it requires that in every coordinate patch, the
spacetime geometry must have the form M_{3} ⊗ time, while it also features multiple
species of quanta which can interact in a fashion very similar to that of ordinary quantum field theory.

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©2014, Kenneth R. Koehler. All Rights Reserved.