Dynamics of Hyperbolic Manifolds

K. Koehler

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

(posted October 31, 2014,
last modified November 14, 2014)


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.

1. Introduction

Consider a region of spacetime which can be foliated into spacelike hypersurfaces orthogonal to some time coordinate. If such a region admits geometric structure, then Thurston's Geometrizaton Conjecture [1], proved by Perelman [2-4], completely describes the geometries of the spacelike hypersurfaces.

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.

2. Quantum Numbers and Interactions of Hyperbolic 3-manifolds

Hyperbolic 3-manifolds may have any number of cusps: open regions with toral cross sections in which the area of the torus decreases exponentially along the length of the cusp. [8] In order to attach a cusp to the external spacetime, we must glue a solid torus to the boundary of the cusp. This solid torus is then embedded in the external spacetime, with the effect that from the point of view of the spacetime, all that is "seen" is the solid torus. In effect, the closed hyperbolic manifold is "inside" the solid torus.

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

Λinternal = -3 / LPlanck2.

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

E = c4 V Λinternal / 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

qnew / pnew = q1 / p1 + q2 / p2

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,

volume and the Dehn invariants are in general insufficient to uniquely identify any given manifold.

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

quanta can exist in identical multiplets,

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.

3. What can we learn from the SnapPy Orientable Closed Census?

At this time it is not possible to classify all hyperbolic 3-manifolds. Therefore we must test the viability of our model with a "sufficiently general" set. We have chosen the SnapPy [7] Orientable Closed Census of 11,031 closed hyperbolic manifolds for this purpose. In one respect it is not at all general: every manifold in the census is the completion of a manifold with only one cusp. Also, there are two caveats:

  1. 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


  2. 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.
This decision implies that for a different choice of precision the multiplet structures will be different from what we present here. We are confident, however, that the qualitative aspects of our results are robust.

The census manifolds feature:

Given such a diverse set of parameters (modulo the number of cusps in the manifolds they were surgered from), particularly in the volumes, it is remarkable that any 3-vertex interactions satisfying the required conservation principles exist at all.

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:

4. Discussion

Until a complete classification of hyperbolic 3-manifolds is constructed, this work represents at best a proof of concept. One might naively hope that once the classification is completed, some relationship between (possibly the low-energy portion of) this model might be related to the Standard Model with restored electroweak symmetry [12]. Note that above the Higgs scale, all particles are massless, rendering families indistinguishable.

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

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 M3 ⊗ time, while it also features multiple species of quanta which can interact in a fashion very similar to that of ordinary quantum field theory.


We would like to thank D. Freeman and C. Vaz for discussions regarding this work.


  1. Thurston, W. P., "Three Dimensional Manifolds, Kleinian Groups and Hyperbolic Geometry", Bulletin AMS 6, 357-381 (1982)
  2. Perelman, G., "The entropy formula for the Ricci flow and its geometric applications", arXiv:math/0211159 (2002)
  3. Perelman, G., "Ricci flow with surgery on three-manifolds", arXiv:math/0303109 (2003)
  4. Perelman, G., "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds", arXiv:math/0307245 (2003)
  5. Koehler, K. R., "Geometrization at the Planck Scale" (2014)
  6. Dunfield, N. M. and Thurston, W. P., "Finite covers of random 3-manifolds", Invent. Math. 166, 457-521 (2006)
  7. Culler, M. and Dunfield, N., a Python interface to SnapPea, by Weeks, J., at www.math.uic.edu/t3m/SnapPy/doc/
  8. Weeks, J., "Computation of Hyperbolic Structures in Knot Theory", in Handbook of Knot Theory, Elsevier (2005)
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  10. Thurston, W. P., "The Geometry and Topology of Three-manifolds", www.msri.org/publications/books/gt3m/ (2002)
  11. Thurston, W. P., How to See 3-manifolds, Classical and Quantum Gravity 15 pp. 2545-2571 (1998)
  12. Kolb, E. and Turner, M., The Early Universe, Westview (1994)

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