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

(posted October 27, 2013

last modified June 20, 2014)

Abstract

We use Thurston's Geometrization Conjecture and related results from the mathematics of 3-manifolds to model geometry at the Planck scale. Spacelike hypersurfaces are treated as ensembles of hyperbolic 3-manifolds connected to spacetime by thin regions containing incompressible surfaces. Assuming that the probability associated with any given manifold is a function only of its volume, we find the partition function of the manifolds in the ensemble. Applying this to the Schwarzschild Black Hole, we find the entropy to be proportional to the 3/2 power of the black hole mass. This result has appropriate "second law behavior": the entropy of a black hole into which matter has fallen is larger than the sum of the prior entropies of the black hole and the matter. We also apply the model to the Friedmann-Robertson-Walker cosmology and discuss the evolution of the universe from initial conditions described by such an ensemble.

The Schwarzschild vacuum extends, via the Kruskal extension, to either the curvature singularity at the center of the horizon, or to
whatever non-vacuum region (described by Quantum Gravity) that lies in the interior. Let us call the neighborhood of the singularity N_{s}. Since the
Schwarzschild singularity is spacelike [4], it is in principle possible to foliate N_{s} into a family of
spacelike surfaces orthogonal to some time coordinate. Following the works of Thurston and of Perelman [5], we propose that N_{s}
is an ensemble of 3-manifolds, connected to the vacuum via thin regions containing incompressible surfaces.
Adapting results from Maher [17] and Dunfield and Thurston [18] to characterize the ensemble, we are able to compute the
entropy of a black hole from a fundamentally statistical mechanical point of view. Our results do not agree with those of Bekenstein
and others, but if N_{s} admits geometric structure, we feel that our approach has merit.

The rest of the paper is organized as follows: section 2 contains a brief review of the mathematics of 3-manifolds relevant to our
purpose, and describes the resulting model for N_{s}. In section 3 we compute the partition function, the average volume
of the manifolds in N_{s}, and the entropy. We show that the entropy obeys a "second-law" inequality.
In section 4 we discuss some implications of the model, and some limitations of our approach. Finally, in section 5 we apply the model to the
Friedmann-Robertson-Walker cosmology and discuss possible solutions to Einstein's Equations.

M = X / Γhas an (X,G) geometric structure: a diffeomorphism-invariant class of charts on M whose transition functions are elements of G. For the moment, we will assume that M is prime (every separating S

Thurston's Geometrization Conjecture [6] states that every oriented closed prime 3-manifold has a torus decomposition such that the resulting components have geometric structures in which X is one of the following eight model geometries:

This set is complete, and is similar to Bianchi's classification [19] (for instance, Nil is Bianchi type II and Sol is Bianchi type VI.)

- E
^{3}(Euclidean 3-space)- S
^{3}- H
^{3}(hyperbolic space)- H
^{2}⊗ R- S
^{2}⊗ R- the universal cover of SL(2,R)
- Nil (the 3 -dimensional Heisenberg Group)
- Sol (a Lie Group with topology R ⊗ R
^{2})

There are no geometries which continuously interpolate between any of the eight; the only way they can be connected is along incompressible tori, which are obstructions to the existence of geometrical structure [12] (a torus is incompressible if its embedding into M injects its fundamental group into that of M.) The result of the decomposition is a finite collection of compact 3-manifolds whose boundaries are tori.

Perelman's proof of the Geometrization Conjecture [7-9] utilizes Ricci Flow [12] with surgery to isolate and excise the singularities occurring at the incompressible tori. The flow separates the manifold into "thick" and "thin" parts [13]; most have structures based on the model geometries, but the singular tori (or spheres, in the non-prime case) are contained within thin parts, called ε-necks. Being diffeomorphism invariant, the Ricci Flow preserves isometries, so the geometric structures of the component manifolds are valid arbitrarily close to the singularities. The essence of Perelman's proof is to show that only a finite number of surgeries are required to complete the decomposition.

Our model takes N_{s} to be an ensemble of 3-manifolds after the thick-thin decomposition has been carried out, but
before the singular incompressible surfaces have been excised. The result is that instead of one singularity, there are many:
we picture N_{s} as a region where every ε-neighborhood is the entry to a 3-manifold based on one of the model
geometries, connected by an ε-neck containing an incompressible sphere or torus. In the sequel, we will refer to these
3-manifolds connected to N_{s} as the "ensemble manifolds."

Every 3-manifold can be obtained by gluing together the boundaries of two handlebodies (solid tori) of some genus [18];
this is called a Heegaard splitting. By considering two handlebodies of arbitrary genus and taking random walks
in the mapping class group of the torus
boundaries, Maher [17] has shown that the probability of getting a hyperbolic manifold (with X = H^{3})
from random Heegaard splittings is asymptotically 1. Within that definition of randomness, almost all 3-manifolds are hyperbolic manifolds.
As a result, the number of non-hyperbolic manifolds in the ensemble will be relatively small.

Our model assumes that all of the ensemble manifolds are hyperbolic.
It is important to note that although we have ignored non-hyperbolic manifolds in our computations, they presumably occur,
albeit with low frequency. Since S^{3} and S^{2} ⊗ R are possible geometries, it follows that some
topology change is inherent in this approach.

All 3-manifolds can also be obtained by gluing together tetrahedra [10]; if the gluing results in a hyperbolic manifold,
the volume of
the resulting 3-manifold is the sum of the Lobachevsky functions of the dihedral angles of the tetrahedra [20].
By the Mostov Rigidity Theorem,
volume is a topological invariant of a hyperbolic 3-manifold [11]. Further, the set of volumes is a discrete set,
and there is a manifold
with the minimum volume: it is called the Week's manifold, and is formed by a (2,1) Dehn fill on the "figure 8 sibling"
(census manifold m003; Dehn filling is the process of gluing a solid torus to a hyperbolic manifold with torus boundary; in
a (2,1) fill, the meridian of the solid torus is wrapped twice around that of the boundary, and the longitude
is mapped once [21].) Let us number the volumes V_{i} in ascending order, with the volume of the Week's manifold
taken as V_{0}.

Dunfield and Thurston [18] have conjectured that the volumes of random Heegaard splittings asymptotically become linear with respect to the lengths of the random walks. Our model extrapolates that conjecture to a linear dependence of volume on the "volume number". We normalize the volumes so that the volume of the Week's manifold is 1 Planck volume. We therefore have

V_{i} = (μ * i + 1) * V_{P},

where we denote the Planck volume as V_{P}. We expect μ to be small; in the 166,049 volumes from the SnapPy [22] orientable
cusped and closed censuses, along with the census of knot and link exteriors up to 14 crossings, the normalized volumes range from
1 to only 32.0252.

P_{i} ∝ e^{-β Vi},

and that all manifolds with the same volume are indistinguishable (here β has dimensions of inverse volume.) Since the ensemble manifolds are
all geometrically separated from each other by singular incompressible surfaces, they must be taken to be completely independent. Likewise,
the N_{i} are independent and not constrained. It is also important to note that ε-necks can branch; *any* number of ensemble
manifolds (even *all* of them) can connect to the region N_{s} through a single ε-neck.
Hence the ensemble manifolds must behave as Bose-Einstein objects.

With this ansatz the partition function is

Z = Π_{i,0}^{∞} (1 - e^{- β VP (μ i + 1)})^{-1}.

Taking the log of both sides, we have

ln Z = - Σ_{i,0}^{∞} ln (1 - e^{- β VP (μ i + 1)}).

We can expand the log as

- Σ_{j,1}^{∞} e^{- β VP j (μ i + 1)} / j.

Interchanging the sums, we have

ln Z = Σ_{j,1}^{∞} Σ_{i,0}^{∞} e^{- β VP j (μ i + 1)} / j

= Σ_{j,1}^{∞} e^{- β VP j} / ((1 - e^{- β VP j μ}) j).

Since V_{P} << 1, we approximate the denominator as β V_{P} μ j^{2} and the sum evaluates to give us

ln Z = Li_{2} (e^{- β VP}) / (β V_{P} μ).

(Li_{n} (z) is the polylogarithm function.)

Taking the derivative with respect to β we find the average total volume occupied by the ensemble manifolds to be

V = Li_{2} (e^{- β} V_{P}) / (β^{2} V_{P} μ) - ln (1 - e^{- β VP}) / (β μ).

The dominant term is π^{2} / (6 β^{2} V_{P} μ). But how does this volume relate to the parameters of the black hole?

During the collapse of a supernova remnant, if degeneracy pressures are sufficient to halt the collapse at a radius

R > R_{hor} = 2 M G / c^{2},

a trapped surface will not form and the remnant becomes either a white dwarf or a neutron star. But if degeneracy pressures are insufficient,
the collapsing matter will fall behind R_{hor}. At the instant when the collapsing remnant falls behind the horizon, the entire volume
contained within the horizon is filled with matter. General Relativity predicts that all of that matter falls "into" the singularity
at the center of the horizon. But our model essentially "hides" the matter from the Schwarzschild vacuum behind singular incompressible
surfaces. Just as a positive cosmological constant corresponds to negative pressure, positive energy density corresponds to hyperbolic
curvature: the curvature induced by the matter is now induced by the curvature of the (mostly hyperbolic) ensemble manifolds. This leads us to interpret the model
as "hiding" the volume inside the horizon in the ensemble manifolds. The logical result
is to equate V with the only natural volume associated with a Schwarzschild black hole, the volume contained within the horizon:

4/3 π (2 M G / c^{2})^{3}

Doing so, we obtain

β = c^{3}/8 √(π / (μ V_{P} G^{3} M^{3})).

The entropy is

S = 2 k Li_{2} (e^{- β VP}) / (β V_{P} μ) - k ln (1 - e^{- β VP}) / μ.

The leading term is

k π^{2} / (3 β V_{P} μ)

= (8 k / (3 c^{3})) √(π^{3} G^{3} M^{3} / (μ V_{P})).

This result obviously differs from the accepted dependence on area, which for a Schwarzschild black hole is
proportional to M^{2}.
But what other powers of mass might have "second law behavior?" That is, if black holes of masses M and d*M collide, for what values of b will

(M + d M)^{b} > M^{b} + (d M)^{b} ?

This inequality obviously holds for any integer b > 1, so we expand for b near 1 to find

(b - 1) M ((d + 1) ln (d + 1) - d ln (d)) > 0,

which clearly holds for d ≥ 1 and b > 1. If we consider d to be small, we have

(b - 1) d M (1 - ln (d)) > 0,

which still holds for b > 1. Now for ordinary particles or objects, mass is not entropy, so is this limit relevant? In [2], Bekenstein considered the smallest increase in entropy to be 1 bit. However, within the context of a theory of Quantum Gravity, one might reasonably expect the entropy of a black hole to increase more if a tau particle entered the horizon than if an electron entered. As the only obviously extensive quantum number, mass might reasonably be taken as a proxy for entropy. While this is not a proof of the validity of the inequality for small masses, we think it merits consideration.

(e^{β (μ i + 1) VP} - 1)^{-1}.

Summing over all i, we find the total average number of manifolds to be

N = 1/m -½ - (ln (e^{β μ VP} - 1) + ψ_{q} (1/μ))
/ (β μ V_{P})

where q is e^{β μ VP} (ψ_{q} (z) is the q-digamma function.)

The thermodynamic temperature of the ensemble, obtained from

dE/dM = T dS/dM,

is

(c^{5} / (4 k)) √(μ V_{P} / (M G^{3} π^{3}))

Using an order of magnitude estimate for μ of 0.0001, for a 3 Solar Mass black hole we were able to approximate:

N ≈ 6.63 * 10^{61}

T ≈ 1.52 * 10^{10} K

S ≈ 2.36 * 10^{37} J/K

N_{0} ≈ 5.16 * 10^{55}, and the N_{i} do not drop below 1 until i ≈ 10^{59}.
With these values, the model seems to suggest that there are a vast number of ensemble manifolds in a very small volume at high temperature.
This implies that the region N_{s} may well be akin to Wheeler's [24] spacetime foam.
We note that the boundary of N_{s} is a compact trapped
surface; it would seem difficult to recover information lost into such an environment.

We also note that the presence of non-hyperbolic manifolds in the ensemble, though negligible in our model, indicates that the ensemble could contain sources of negative energy density.

We conjecture that this analysis holds for the Kerr-Newman black hole with the mass M replaced by the "irreducible mass" [25]

M_{irr} = ½ √(a^{2} + (M + √(M^{2} - a^{2} - e^{2}))^{2}).

It can be shown that M_{irr}^{3/2} also has "second law behavior" for both the merger of similar black holes (where the masses,
angular momenta and charges all differ by the same factor), and for sufficiently small increases in mass.
It is not clear that the model is applicable to the Reissner-Nordstrom black hole, which has a timelike singularity [4]. We believe it unlikely that
any such black holes are realized in nature, however.

Finally, we must keep in mind that any model for physical behavior inside a compact trapped surface such as a black hole horizon is not subject to direct experimental verification. At this stage it seems likely that there is no experimental method available to distinguish this model from any other.

Because they are manifolds of constant sectional curvature, the ensemble manifolds are Einstein Manifolds [26]. Because
each manifold is separated from every other manifold, as well as the "outside" universe, by a curvature singularity, Einstein's Equations must
be solved separately in each manifold, using a coordinate patch consistent with its geometry. Let g_{ij} denote the metric on
any of the manifolds, and g_{μν} denote the 4-metric formed as

g_{μν} = a^{2}(t) g_{ij} - dt^{2}.

Near t=0, Electroweak symmetry is unbroken (energies are far above the Higgs mass) [27], so the only possible curvature sources are radiation and the cosmological constant. For Einstein spaces, the spatial components of Einstein's Equations reduce to

12 a(t) a''(t) + 6 a'^{2}(t) + a^{2}(t) (2 α ρ(t) - 6 Λ) + R = 0

and the time-time component becomes

6 a'^{2}(t) - 2 a^{2}(t) (α ρ(t) + Λ) + R = 0

In these equations, R is the constant curvature of the Einstein space, the speed of light is taken to be 1,
and α has units of 1 / (energy density * length^{2}).

In the outside universe, R = zero, Λ > 0, and a(0) = 0. The familiar solution is given in this context by

ρ(t) = ρ_{now} / a^{4}(t)

a(t) = e^{- t √ (Λ / 3)} (α ρ_{now} / Λ)^{¼} √
((e^{4 t √ (Λ / 3)} - 1) / 2)

But the ensemble manifolds are vacuum manifolds of constant volume, so for them ρ = 0, a(0) = 1, and of course R < 0. There are three cases to consider:

- Λ = 0: a(t) = 1 ± √ ( |R| / 6 ) t.
- Λ > 0: there are a set of related solutions,
a
_{±}(t) = e^{- t √ (Λ / 3)}(e^{2 t √ (Λ / 3)}(η_{±}- (± R)) ± R) / η_{±}where η

_{±}= ± 4 Λ + 2 √ (2 Λ (2 Λ - R)), anda_{±}(t) = e^{- t √ (Λ / 3)}(± (e^{2 t √ (Λ / 3)}- 1) R + η_{±}) / η_{±}.In these solutions, either all of the ± are +, or all are -. Note that η

_{±}is always positive. As t gets large, the solutions become proportional to ± e^{t √ (Λ / 3)}. - Λ < 0: In this case, the only real solution occurs when R = 2 Λ, and a(t) = cos (t √ (|Λ| / 3)).

In the first two cases, the ensemble manifolds grow as the outside universe grows (albeit more slowly in the Λ = 0 case). The ensemble manifolds are still separated from the universe by the incompressible tori singularities, but the ε-necks would become thick. If the negative curvature isolated by the singularities is to be interpreted as the energy density associated with ρ in the outside universe, it is unclear how these models could become what we observe today.

However, we find the last case intriguing. In it, the ensemble manifolds are rapid breathers, but never grow past their original size. The ε-necks remain small, and the relationship we posit between the hidden negative curvature and the external radiation could be consistent. It also suggests a possible reason for the size of the observed cosmological constant: the total energy density of the universe (including the ensemble manifolds) could be zero, and the small positive Λ we observe could be offset by the negative Λ in the ensemble manifolds.

This is of course purely speculative.

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