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Integration and Refinement of Digital Physics,
Unifying Quantum and Classical with a Discrete
Framework:
A Formal Approach to Subparticles, Angular
Momentum,
and Finite-Density Black Holes
Jesse Daniel Brown, PhD1and Anton Fedotov, B.A.2
1Independent Researcher , Email: plasmatoid@gmail.com , ORCID:
0009-0006-3889-534X
2Harbin Institute of Technology , Email: anvifedotov.biz@gmail.com ,
ORCID: 0009-0004-2313-6530
December 22, 2024
Abstract
This paper refines and expands a discrete “digital physics” model of the universe,
in which spacetime is represented by a sequence of discrete frames (or states). We
incorporate:
•Subparticle Metatags: Discrete metadata encoding quantum states, ener-
gies, and momenta of fundamental subparticles in each spacetime pixel.
•Angular Momentum: A local spin or rotation parameter associated with
subparticles or pixel regions, ensuring discrete analogs of rotational effects
and frame-dragging.
•Finite-Density Black Hole Interiors: A synergy between the discrete
frame-based model and a scalar-field approach to avoiding classical singulari-
ties, supported by an anisotropic equation of state (EoS).
By merging these concepts, we aim to unify classical and quantum descriptions
under a single discrete framework, reducing singularities while ensuring compliance
with standard energy conditions. We present both conceptual and mathematical
formalisms, along with references to experimental and computational strategies that
may validate the underlying discrete structure.
Contents
1 Introduction 2
1.1 NewContributions .............................. 2
1
2 Discrete Spacetime Frames: An Overview 3
2.1 Spacetime Pixels and Metatags . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Transition Operator with Rotation . . . . . . . . . . . . . . . . . . . . . 3
3 Refinement: Finite-Density Black Holes 4
3.1 Coupling Λ(ϕ)................................. 4
3.2 Energy Conditions and Metatag Relevance . . . . . . . . . . . . . . . . . 4
4 Quantum Phenomena in the Discrete Model 4
4.1 Quantum Tunneling and Entanglement . . . . . . . . . . . . . . . . . . . 5
4.2 Quantum Scarring and Temporal Crystals . . . . . . . . . . . . . . . . . 5
5 Experimental Validation and Simulations 5
6 Discussion and Outlook 5
6.1 FutureDirections............................... 6
7 Conclusion 6
1 Introduction
Recent advances in both theoretical physics and computational methods motivate the
development of a discrete model of the universe. In such an approach, spacetime is
subdivided into minimal “pixels” or “nodes,” with a frame-based representation capturing
the entire state of the universe at each discrete time step [1–3].
In earlier work, we introduced subparticle metatags, representing fundamental prop-
erties of the smallest constituents in each pixel. This approach draws on insights from
quantum gravity [3], digital physics [4,5], and practical 3D-rendering analogies [6, 7].
1.1 New Contributions
We expand the original discrete framework to include:
1. Angular Momentum in Discrete Frames: Each subparticle (or pixel) is en-
dowed with a spin or rotational parameter, ensuring that total angular momentum
is conserved across frame transitions. This is critical for modeling rotating astro-
physical objects.
2. Finite-Density Black Hole Interiors: We incorporate a scalar field ϕ(r) and
an anisotropic EoS for ultra-dense matter, consistent with the discrete approach.
This approach addresses the singularity problem by imposing finite densities at the
black hole core.
3. Quantum Corrections and Metatags: The coupling Λ(ϕ)=Λ0+λ ϕ2ties the
scalar field to the cosmological constant. This drives nontrivial corrections in the
discrete transition operator, which we relate to subparticle metatags in each frame.
2
2 Discrete Spacetime Frames: An Overview
We denote each discrete global state of the universe by a frame Fn. In this model, time
evolves via an operator:
Fn+1 =T(Fn),
where Tencodes the local and global physical laws dictating the transition from frame
nto frame n+ 1.
2.1 Spacetime Pixels and Metatags
Each frame Fnconsists of “pixels” (or nodes) labeled by integer indices (i, j, k). We write:
Fn=Pn
i,j,k
i, j, k ∈Z,
where Pn
i,j,k is augmented by a metatag:
Mn
i,j,k ={x, y, z, e, p, S, L},
with:
•(x, y, z): Discrete spatial coordinates for the subparticle.
•e: Local energy.
•p: Local momentum (vector or magnitude).
•S: Quantum entanglement or wavefunction parameter.
•L: (New) Angular momentum or spin-like quantity.
The presence of Lallows us to track rotational effects in a discrete manner.
2.2 Transition Operator with Rotation
We revise the transition operator to ensure discrete angular momentum conservation:
Fn+1 =Trot(Fn),
where Trot updates Lin each metatag according to local interactions, while preserving
the total angular momentum
X
i,j,k
Ln
i,j,k =X
i,j,k
Ln+1
i,j,k .
Such a rule parallels standard rotational invariance in continuous physics.
3
3 Refinement: Finite-Density Black Holes
One of the longstanding problems in classical general relativity is the prediction of sin-
gularities at black hole centers. We incorporate recent theoretical progress [8,9] into our
discrete framework by positing:
1. A Scalar Field, ϕ(r):A smooth, bounded function (e.g., ϕ(r) = ϕ0r2
r2+r2
0) that
remains finite at r= 0.
2. An Anisotropic EoS for Ultra-Dense Matter:
ρ(r) = ρ0
1 + r
r0n, Pr(r) = wrρ(r), Pt(r) = wtρ(r).(1)
3. Angular Corrections: For rotating black holes, we couple ϕ(r) to an effective
frame-drag term ω(r) or add a spin metatag for interior fluid cells.
Within the discrete model, each spacetime pixel near the black hole core has a finite
density label ρand a finite value of ϕ. This ensures no pixel can acquire an infinite
curvature, hence avoiding singularities.
3.1 Coupling Λ(ϕ)
Following Refs. [2,10, 11], we adopt:
Λ(ϕ) = Λ0+λϕ2.
In a discrete universe, ϕtakes on pixel-specific values ϕn
i,j,k. Consequently,
Λϕn
i,j,k
varies from pixel to pixel, shaping local geometry at each node. Our transition operator
Tincorporates these local changes in Λ to update the local metatags consistently with
Einstein-like field equations in a discrete setting.
3.2 Energy Conditions and Metatag Relevance
We have shown in separate works [1,8] that carefully chosen ρ(r), Pr(r), Pt(r) can satisfy
the Weak, Dominant, and Strong Energy Conditions throughout the black hole interior.
In this discrete framework:
ρtotal +Pr≥0, ρtotal ±Pr≥0,
etc., are verified at each pixel. The subparticle metatags store the local energy ρand
pressures Pr, Pt, allowing direct numeric checks in a simulation.
4 Quantum Phenomena in the Discrete Model
Digital-physics approaches naturally lend themselves to quantum-like effects, since each
pixel state can be updated by “rule-based” transitions reminiscent of quantum cellular
automata [12, 13]. Below are two phenomena we emphasize:
4
4.1 Quantum Tunneling and Entanglement
By embedding quantum numbers {e, p, S}in each subparticle’s metatag, we can define
discrete tunneling probabilities through potential barriers. Similarly, entangled states
appear as {Sk=Sm}across distant pixels. The synergy with black-hole interiors suggests
subparticle entanglement might be pivotal in horizon-scale phenomena (e.g., black-hole
evaporation models [3]).
4.2 Quantum Scarring and Temporal Crystals
Previously, we introduced the concept of temporal crystals —periodic structures in time
that reflect stable subparticle configurations. With rotation now included, subparticles
can form stable “rotating crystals” that remain robust under discrete frame transitions.
Quantum scarring, or the partial localization of wavefunctions, could also occur if cer-
tain subparticle pathways become self-reinforcing. We model such memory effects by
introducing a small correction term in the transition operator:
∆M=−ϵ Smemory,
where Smemory is a discrete record of prior subparticle states.
5 Experimental Validation and Simulations
Although fully testing a discrete spacetime model is nontrivial, we highlight potential
approaches:
•High-Performance Simulations: Discretize a 3D region around a hypothetical
black hole, assign subparticle metatags, and numerically evolve the system. Check
for compliance with energy conditions, finite curvature invariants, and stable rota-
tion patterns.
•Quantum Optical Analogs: Lab-based “quantum simulators” (e.g., cold atoms,
photonic lattices) can mimic discrete scattering processes. Metatag-like variables
might be realized in the internal states of atoms or photons.
•Cosmological Observations: Look for anomalies in gravitational-wave signals
or cosmic microwave background patterns that might reflect an underlying discrete
structure.
6 Discussion and Outlook
By weaving together:
1. Discrete frames with local metatags,
2. Scalar fields that remain finite at black-hole centers, and
3. Anisotropic EoS ensuring finite density,
we propose a robust, singularity-free framework. The synergy with angular momentum
extends the model’s reach to rotating astrophysical objects—arguably a necessity, since
most observed black holes in nature possess spin.
5
6.1 Future Directions
•Refining Angular Momentum Conservation: One might incorporate a dis-
crete version of the Hartle–Thorne approximation for slow rotation or a near-Kerr
interior model for rapid rotation.
•Higher-Dimensional Discrete Lattices: We can explore embedding these 3D
frames in higher-dimensional “bulk” discretizations reminiscent of brane-world mod-
els.
•Comparison with Loop Quantum Gravity (LQG): While conceptually similar
to the spin network approach, our metatags add an explicit “rendering” analogy
that might lead to new computational methods for quantum gravity.
7 Conclusion
We have integrated angular momentum, finite-density black-hole interiors, and quantum-
corrected scalar fields into a unified discrete-spacetime framework. Subparticle metatags
track position, energy, momentum, spin, and entanglement, while a pixel-by-pixel cos-
mological constant Λ(ϕ) evolves locally. This synergy resolves typical singularities, as
the bounded scalar field and anisotropic pressures ensure finite curvature. Our approach
underscores the feasibility of merging classical and quantum descriptions in a discrete,
computationally friendly manner, laying the groundwork for future simulations and pos-
sible observational checks.
Acknowledgments
We extend our gratitude to colleagues and mentors who have influenced this research,
particularly those working at the intersection of quantum gravity, black-hole physics, and
computational cosmology. Special thanks to John Cumbers and Roman Yampolskiy for
insightful discussions, and Raki Brown for continuous article recommendations.
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