Geometric Evaporation of Primordial Black Holes
from Discrete Vacuum Tension
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
Abstract
Standard semiclassical gravity predicts black hole lifetimes scaling as τ ∝ M
3
via Hawk-
ing radiation. This imposes severe constraints on the abundance of Primordial Black Holes
(PBHs): those with M ∼ 10
15
g should be explosively evaporating today, yet gamma-ray
observations find no such signal. We derive an alternative, faster decay channel from the
Selection-Stitch Model (SSM), where the vacuum is a discrete K = 12 FCC tensor network.
We first prove that a black hole in this framework is a topological vacancy—a region where
gravitational compression has driven the lattice past the metric wall at r
min
= L/
√
3, shat-
tering the network connectivity to K = 0. We show that this vacancy exactly reproduces
the standard Bekenstein-Hawking entropy, the Schwarzschild radius, and the Hawking tem-
perature as emergent lattice quantities. We then derive the quantum boundary dynamics of
this topological void directly from absolute quantum rate theory, obtaining a horizon reces-
sion velocity
˙
R = −(c/2)(l
P
/R
H
) and a lifetime τ
Geo
∝ M
2
. A 10
15
g PBH evaporates in
0.45 ms via this channel, cleanly resolving the gamma-ray constraint. Peierls lattice locking
at L
corr
∼ 1 fm ensures macroscopic black holes remain stable.
Keywords: Primordial Black Holes, Geometric Evaporation, Isometric Tensor Networks,
Topological Vacancy, Bekenstein-Hawking Entropy
1. Introduction
The Hawking radiation mechanism predicts that black holes emit thermal radiation with
a mass loss rate
˙
M
Hawk
∝ M
−2
, leading to a lifetime τ ∝ M
3
[1]. While elegant in the
semiclassical limit, this derivation relies on Quantum Field Theory in Curved Spacetime
(QFTCS) and suffers from the trans-Planckian problem: outgoing Hawking modes originate
from field fluctuations with wavelengths far below the Planck scale, where the continuum
approximation must break down.
This slow decay rate creates a severe constraint on Primordial Black Holes (PBHs). If
PBHs formed during early-universe density fluctuations with masses M ≲ 10
15
g, the τ ∝ M
3
scaling predicts they should be in their final explosive stages today, producing a detectable
gamma-ray background. Constraints from Fermi-LAT and Planck strictly limit such relics
[2].
Email address: raghu@idrive.com (Raghu Kulkarni)
We derive a fundamentally faster decay mode from the discrete vacuum of the Selection-
Stitch Model (SSM) [4]. In this framework, the vacuum is a Face-Centered Cubic (FCC)
tensor network with coordination number K = 12 and lattice spacing L ≈ 1.84 l
P
[3]. A black
hole is not merely a region of extreme curvature—it is a topological vacancy in this network,
where the lattice has been crushed past its geometric exclusion limit. The surrounding intact
lattice exerts mechanical tension on this vacancy, driving a geometric decay that scales as
τ ∝ M
2
.
Interactive 3D visualization. To immediately ground the topological defect healing
and crystallization kinematics discussed in this Letter, readers can explore the mechanism
through an interactive WebGL application. This visualization explicitly illustrates the
K = 0 topological void, the quantum boundary re-stitching, and the resulting mass loss
decay curves:
https://raghu91302.github.io/ssmtheory/ssm_pbh_evaporation.html
2. Black Holes as Topological Vacancies
2.1. From metric wall to lattice shattering
Definition 1 (The Metric Wall). In the SSM, the FCC lattice has an absolute minimum
internode distance:
r
min
=
L
√
3
, (1)
the inscribed sphere radius of the tetrahedral void between four mutually touching spheres of
diameter L. Below r
min
, the tetrahedral voids collapse and the network topology is destroyed
[4].
Definition 2 (Topological Vacancy). A black hole in the SSM is a connected region V where
the local gravitational compression has driven the internode spacing below r
min
. Within V,
the lattice has shattered: the coordination number drops to K = 0, all entanglement bonds are
severed, and no tensor network structure survives. The region V is a topological vacancy—a
hole in the fabric of spacetime.
The mechanism dictates that as gravitational compression forces lattice nodes closer to-
gether, the tetrahedral voids between nodes collapse at r
min
= L/
√
3. The collapse severs
all entanglement bonds passing through the void (see Figure 1a). Without bonds, the nodes
inside the region lose their K = 12 coordination and become disconnected (K = 0). The dis-
connected region is causally isolated from the surrounding lattice. The boundary ∂V between
the intact lattice (K = 12) and the vacancy (K = 0) is the event horizon.
2.2. Recovery of standard black hole properties
Proposition 1 (Bekenstein-Hawking Entropy). The entropy of a topological vacancy of area
A is S = A/(4l
2
P
).
2
Proof. Each bond of the intact K = 12 lattice is a maximally entangled Bell pair con-
tributing ln 2 of entanglement entropy [5]. At the vacancy boundary ∂V, each severed bond
contributes ln 2 to the entanglement entropy between the interior and exterior. On the hexag-
onal boundary sheet, the areal density of bonds is 2/(
√
3 L
2
). The total entropy is calculated
by multiplying this density by the area A and the entanglement entropy per bond ln 2:
S =
2
√
3 L
2
A ln 2 (2)
Substituting the Ryu-Takayanagi relation G
N
=
p
2/3 L
2
/(4 ln 2) [8, 5] explicitly yields:
S =
A
4 G
N
=
A
4 l
2
P
(3)
This mathematically recovers the exact Bekenstein-Hawking entropy [7], derived cleanly from
bond counting.
Proposition 2 (Schwarzschild Radius). The vacancy radius equals the Schwarzschild radius
R
H
= 2GM/c
2
.
Proof. The vacancy forms when the gravitational self-energy density ρ
grav
= M/(4πR
3
/3)
reaches the metric wall density ρ
wall
=
√
2 ℏc/(4L
4
) (the maximum energy density the lattice
can support [3]). Setting ρ
grav
= ρ
wall
determines the onset of shattering:
M
4
3
πR
3
=
√
2ℏc
4L
4
(4)
R
3
=
3
√
2
2π
ML
4
ℏc
(5)
However, once shattering begins at the centre, causal disconnection propagates outward: any
shell whose enclosed mass M(r) satisfies 2GM(r)/(rc
2
) ≥ 1 loses connectivity. This matches
the trapped surface condition of General Relativity. The outermost such shell is exactly
R
H
= 2GM/c
2
.
Proposition 3 (Hawking Temperature). The vacancy radiates at temperature T
H
= ℏc
3
/(8πGMk
B
).
Proof. At the vacancy boundary, the intact lattice terminates abruptly. The nearest-neighbour
bonds that would have connected to interior nodes are dangling maximally entangled half-
pairs. Each dangling bond has a characteristic energy E
bond
= ℏc/(4L) [3]. These dangling
bonds radiate into the vacuum with a thermal spectrum determined by detailed balance.
The effective temperature is set by the surface gravity κ = c
4
/(4GM) via the Unruh relation
T = ℏκ/(2πk
B
c), giving T
H
= ℏc
3
/(8πGMk
B
)—the standard Hawking temperature.
3. Derivation of Geometric Evaporation
Because the event horizon is a discrete boundary of severed high-energy bonds, the vacuum
dynamically attempts to minimize its free energy by re-stitching these bonds and shrinking
the void.
3
Theorem 1 (The Geometric Surface Tension). The discrete surface tension σ of the event
horizon boundary is strictly constrained by the holographic bound, yielding σ = ℏc/4l
3
P
.
Proof. Surface tension is defined as the energy required to create one unit of new surface area.
From Proposition 1, each severed lattice bond on the boundary occupies a fundamental area
of a
0
= 4l
2
P
. The maximum energy associated with a single severed discrete node is the
Planck energy E
P
= ℏc/l
P
. Therefore, the energy per fundamental area—the surface tension
of the topological void—is exactly:
σ =
E
P
a
0
=
ℏc/l
P
4l
2
P
=
ℏc
4l
3
P
(6)
We now derive the dynamic evolution of this boundary directly from the discrete quantum
transition rates of the network, bypassing classical continuum approximations.
Theorem 2 (Quantum Boundary Dynamics). The continuum limit of the discrete quantum
transition amplitude to re-stitch the horizon boundary yields a mean curvature flow defined
by a recession velocity
˙
R = −(c/2)(l
P
/R
H
).
Proof. The topological void heals when a K = 0 node on the boundary undergoes a quantum
transition to a K = 12 intact state (Figure 1b). The drift velocity of the boundary shifting
inward by one discrete lattice step (l
P
) is given by absolute quantum rate theory: v =
(W/ℏ)l
P
, where W is the net thermodynamic work done per quantum jump.
The thermodynamic pressure acting to close a spherical boundary of radius R
H
possessing
surface tension σ is dictated by the variational minimization of area, P = 2σ/R
H
. The work
done over a single Planck volume (l
3
P
) is W = P l
3
P
. Substituting the derived surface tension
(Eq. 6):
W =
2
R
H
ℏc
4l
3
P
l
3
P
=
ℏc
2R
H
(7)
The resulting radial recession velocity of the boundary is therefore directly constrained:
˙
R = −
W
ℏ
l
P
= −
ℏc
2R
H
ℏ
l
P
= −
c
2
l
P
R
H
(8)
In the macroscopic limit, this local energy minimization of the boundary yields a mean cur-
vature flow mathematically isomorphic to the classical Allen-Cahn equation [6], but derived
here strictly from first principles of the discrete quantum lattice.
4. Mass Loss Rate and PBH Lifetimes
This curvature-driven metric healing leads directly to a novel black hole decay scaling.
Converting the spatial recession (Eq. 8) into a mass loss rate using the standard Schwarzschild
4
relation R
H
= 2GM/c
2
=⇒ dR
H
= (2G/c
2
)dM gives:
dM
dt
=
c
2
2G
˙
R (9)
=
c
2
2G
−
c
2
l
P
2GM/c
2
(10)
= −
c
5
l
P
8G
2
M
(11)
Separating variables and integrating from the initial mass M to complete evaporation at
M = 0 over the geometric lifetime τ
Geo
:
Z
0
M
M
′
dM
′
= −
c
5
l
P
8G
2
Z
τ
Geo
0
dt (12)
−
1
2
M
2
= −
c
5
l
P
8G
2
τ
Geo
(13)
τ
Geo
=
4G
2
c
5
l
P
M
2
(14)
Expressing this in terms of the Planck mass m
P
=
p
ℏc/G and Planck time t
P
= l
P
/c, we
substitute G = ℏc/m
2
P
and l
P
= ℏ/(m
P
c):
τ
Geo
=
4
ℏc
m
2
P
2
c
5
ℏ
m
P
c
M
2
(15)
= 4
ℏ
m
P
c
2
M
m
P
2
(16)
= 4 t
P
M
m
P
2
(17)
This τ ∝ M
2
scaling is fundamentally faster than Hawking’s τ ∝ M
3
.
For a PBH at the critical observational threshold M = 10
15
g: Under standard Hawking
radiation, τ
Hawk
≈ 4 × 10
17
s (∼13.8 billion years). These PBHs should be exploding now.
Applying our geometric channel, with M/m
P
≈ 4.6 × 10
19
and t
P
≈ 5.4 × 10
−44
s:
τ
Geo
= 4 × (5.4 × 10
−44
) × (4.6 × 10
19
)
2
≈ 4.5 × 10
−4
s. (18)
A 10
15
g PBH evaporates in 0.45 milliseconds—not 14 billion years (Figure 1c). This cleanly
resolves the gamma-ray abundance constraint: low-mass PBHs do not survive the early
universe.
5. Stability of Macroscopic Black Holes: Peierls Locking
If geometric evaporation applied universally, all black holes would evaporate rapidly. The
Peierls-Nabarro mechanism prevents this.
5
K
= 0
vacancy
K
= 12
intact
severed
bonds
(a) Topological Vacancy
K
= 12
K
= 0
K
= 0
re-stitched
(
K
= 12
)
(b) Geometric Healing
R
= (
c
/2)(
l
P
/
R
H
)
10
10
10
14
10
18
10
22
10
26
10
30
10
34
10
38
10
42
Black Hole Mass
M
(g)
10
50
10
41
10
32
10
23
10
14
10
5
10
4
10
13
10
22
Lifetime (s)
10
15
g
Age of Universe
13.8 Gyr (exploding now!)
0.45 ms
(resolved!)
Peierls
locked
(c) Lifetime Comparison
Geo
M
2
vs
Hawk
M
3
Hawking:
M
3
Geometric:
M
2
Geometric Evaporation: Topological Vacancy Healing in the
K
= 12
Lattice
Figure 1: Geometric Evaporation Mechanism. (a) The black hole as a topological vacancy, where
extreme compression severs bonds to create a K = 0 interior. (b) Curvature-driven metric healing via
discrete quantum boundary re-stitching (
˙
R ∝ −1/R
H
). (c) Lifetime comparison showing the rapid geometric
decay (τ ∝ M
2
) eradicating 10
15
g PBHs in 0.45 ms, while macroscopic black holes become Peierls-locked
and perfectly stable.
In a discrete lattice, a domain wall cannot glide continuously; it must hop between lattice
sites, overcoming a periodic potential barrier (the Peierls stress). For a vacancy of radius
R
H
, the effective mobility is exponentially suppressed:
µ
eff
= µ
0
exp
−
R
H
L
corr
, (19)
where L
corr
is the structural correlation length of the polycrystalline lattice. For geometric
evaporation to remain active for 10
15
g PBHs (R
H
≈ 1.5 fm) but be suppressed for macro-
scopic black holes, we require L
corr
∼ 1 fm. This is the characteristic confinement scale of
the strong nuclear force—the natural correlation length of the QCD vacuum.
Microscopic PBHs (R
H
≪ 1 fm) easily overcome the Peierls barrier and evaporate rapidly.
Stellar-mass and supermassive black holes (R
H
≫ 1 fm) are permanently locked, decaying
only via the standard Hawking channel.
Black Hole M R
H
τ
Geo
PBH (threshold) 10
15
g 1.5 fm 0.45 ms
PBH (Planck mass) 2.2 × 10
−5
g 3.2 × 10
−33
cm 4 t
P
Stellar (Cygnus X-1) 21 M
⊙
62 km Peierls-locked
SMBH (Sgr A*) 4 × 10
6
M
⊙
1.2 × 10
7
km Peierls-locked
Table 1: Geometric evaporation lifetimes. Microscopic PBHs evaporate rapidly; macroscopic BHs are Peierls-
locked and stable.
6. Discussion and Conclusion
The topological vacancy framework recovers all standard black hole properties without re-
lying on continuous background manifolds. The Bekenstein-Hawking entropy, Schwarzschild
6
radius, and Hawking temperature all emerge natively from the severed boundaries of the
shattered K = 0 topological void.
By calculating the transition amplitude for the vacuum to re-stitch these severed entan-
glement bonds, we derived a geometric evaporation mechanism for black holes in the SSM
discrete vacuum. The resulting lifetime τ
Geo
∝ M
2
provides a massive, additive decay chan-
nel (
˙
M
total
=
˙
M
Hawk
+
˙
M
Geo
) that forces a 10
15
g PBH to evaporate in 0.45 ms, definitively
resolving the modern gamma-ray constraint.
This mechanism is strictly falsifiable: the τ ∝ M
2
geometric scaling dictates that the
mass spectrum of evaporating PBHs should show a sharp cutoff at M ∼ 10
15
g with zero
surviving relics, rather than the smooth continuum tail predicted by Hawking evaporation.
References
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