Blackhole Singularities as Zero Entropy States - juhameskanen/abstract GitHub Wiki

Blackhole Singularities as Zero Entropy States

Introduction

The singularity theorems of Penrose and Hawking establish that gravitational collapse generically produces spacetime singularities --- regions where curvature diverges and the classical manifold description breaks down \cite{penrose1965, hawking1970}. The standard interpretation is that singularities mark the boundary of applicability of General Relativity and that a theory of quantum gravity is required to describe what occurs there.

The present paper proposes a different interpretation. Under the principles established in The Four Axioms page, geometric spacetime and the execution trace of a computational simulation are two equivalent representations of the same underlying informational structure, with neither being more fundamental. This equivalence allows the singularity to be approached from the informational side even when the geometric description fails. The central question is not what the geometry does at the singularity --- it diverges, by definition --- but what the execution trace does. The answer, we find, is that the informational variety of the execution trace collapses into a trivial zero entropy state.

Methods

To rigorously isolate this collapse, we evaluate the spatial state under two distinct information-theoretic lenses: microscopic bitwise diversity and localized pattern redundancy via block structures. This dual-metric spectrum reframes the singularity entirely. A zero-entropy, zero-complexity bitstring encodes a single, structureless state. Under any geometric decoding, a zero-entropy source resolves to a point of zero size.

It cannot encode microstructures, particles, or stress-energy. The singularity is therefore not a point where physics breaks down but an informational state where physics becomes trivially simple: there is nothing there to be described.

Existing approaches to singularity regularisation --- loop quantum gravity, string-theoretic fuzzballs, effective field theory cutoffs --- introduce new physics at the Planck scale. The present approach requires no new physics. The regularisation is not imposed but derived from the informational structure of the collapse process itself.

Execution Trace

Let $\mathcal{M}$ denote the set of all memory locations in a deterministic computing system. A \emph{machine state} $\mathcal{S} \in \mathcal{M}$ is a complete assignment of values to all memory elements. Let $\mathcal{P} = (I_1, I_2, \dots, I_n)$ be a finite sequence of deterministic instructions, with $I_k : \mathcal{S} \to \mathcal{S}$. The \emph{execution trace} $\mathcal{T}$ of program $\mathcal{P}$ is the ordered sequence of states

$$ \mathcal{T} = (s_0, s_1, \dots, s_n), \quad s_{k+1} = I_{k+1}(s_k). $$

The execution trace encodes all information about the simulated phenomenon. By the representational equivalence of Section~\ref{sec:framework}, it encodes all information about the physical phenomenon as well. In practice, we focus on the \emph{bitstring sub-trace}: the subsequence of memory states that encodes the spatial geometry of the simulated system.

Geometric Encoding

Let a geometric state at time $t$ be encoded as a bitstring $b_t \in {0,1}^L$, representing the discretised positions of all particles in the simulated dust cloud. Let

$$ \mathcal{C} = {0,1}^{3k} $$

be the space of three-dimensional configurations. Define the decoding map

$$ f : \mathcal{C} \to \mathbb{Z}^3, \quad f(b) = (\phi(b_1), \phi(b_2), \phi(b_3)), $$

where $\phi : {0,1}^k \to \mathbb{Z}$ decodes fixed-length binary segments into integers via standard binary representation. Distinct bitstrings decode to distinct integer triples. Quantisation of continuous coordinates to fixed-width integers eliminates spurious entropy contributions from floating-point representation and ensures that the entropy of the bitstring faithfully reflects the geometric distinguishability of the encoded states.

The Informational Diagnostic Suite

To extract both macroscale pattern formation and microscale structural disorder during gravitational collapse, we evaluate the particle coordinate distribution using two complementary informational metrics.

Single-Bit Shannon Entropy

At each simulation timestep $t$, the full geometric state is mapped onto a vector of continuous positions, which are evaluated as a 1D spatial array of radii $r_n$ or coordinate positions. To evaluate low-level bitwise diversity, this spatial state is scaled by an integer mapping constant $\alpha = 1000.0$ and parsed at the raw bit level. We compute the Shannon entropy of the empirical bit-frequency distribution:

$$ H(b_t) = -p_0(t)\log_2 p_0(t) - p_1(t)\log_2 p_1(t), $$

where $p_0(t)$ and $p_1(t)$ are the global frequencies of zero- and one-bits across all parsed words at time $t$. This measures the fundamental distinguishable variety encoded in the machine representation.

Pattern Block Entropy

Because single-bit Shannon entropy treats bits as independent identically distributed variables, it can obscure localized correlations. We implement a spatial pattern variety metric by grouping the bitstream into consecutive sequences of fixed word size $n = 4$. The normalized block entropy is defined as:

$$ H_n(b_t) = -\frac{1}{n} \sum_{i=1}^{2^n} P(\omega_i) \log_2 P(\omega_i), $$

where $P(\omega_i)$ is the empirical probability of occurrence of the $i$-th binary pattern word $\omega_i \in {0,1}^n$. Due to the fixed scaling properties inherent in binary integer transitions, $H_n(b_t)$ tracks tightly with $H(b_t)$, confirming a high degree of correlation between individual bit state changes and pattern distributions during spatial collapse.

Lemma: Vanishing Entropy Implies Geometric Point

Lemma. Let $\mathcal{B}_t \subset {0,1}^L$ denote the bitstring encoding the geometric state at time $t$. If $H(\mathcal{B}_t) \to 0$, then the set of distinguishable geometric configurations collapses to a single equivalence class under $f$.

Proof. $H(\mathcal{B}_t) = 0$ if and only if all bits of $\mathcal{B}_t$ take the same value, i.e.\ $\mathcal{B}_t \in {0^L, 1^L}$. Each such bitstring decodes to a single geometric object - a point. The set of distinguishable configurations therefore contains exactly one element. $\qed$

Corollary. A zero-entropy bitstring, under any geometric decoding, resolves to a single structureless point. This result is independent of the choice of decoding map $f$. The zero-entropy state has no internal structure available to encode microstructures, such as particles.

Simulations

Setup

We simulated gravitational collapse for Schwarzschild (implemented in both Advanced Unistructural and Eddington-Finkelstein coordinate presentations) and Kerr ($a = 0.9$) geometries using discretised dust cloud models. In each case, a spherically symmetric distribution of $N=100$ particles is initialised outside the horizon and evolved using a fourth-order Runge-Kutta (RK4) integrator with a time step $dt = 10^{-3}$ until numerical breakdown or coordinate horizon termination. The geometric states are processed through the complete informational suite at each timestep.

Results

The informational metrics yield contrasting signatures across different spacetime geometries, uncovering physical characteristics of the collapse that are invisible to standard curvature calculations.

In all configurations, the single-bit Shannon entropy $H(b_t)$ and pattern block entropy $H_n(b_t)$ decrease as particles fall inward toward the singularity. Their curves track each other identically, demonstrating that macroscopic structure and microscale binary data variety decay in tandem as the configuration space collapses.

\begin{figure}[h!] \centering \includegraphics[width=0.9\textwidth]{figures/schwarzschild_au_metrics.png} \caption{Multi-metric informational evolution for a Schwarzschild Advanced Unistructural collapse. The top panels show the geometric particle convergence and the monotonic decrease of single-bit Shannon entropy. The bottom panels present the structural pattern block entropy.} \label{fig:schwarzschild_au} \end{figure}

\begin{figure}[h!] \centering \includegraphics[width=0.9\textwidth]{figures/kerr_metrics.png} \caption{Multi-metric informational spectrum for a high-spin Kerr ($a=0.9$) dust collapse. Bit and block entropies decay monotonically.} \label{fig:kerr_metrics} \end{figure}

The singularity as zero entropy state

The central result of this study is that gravitational collapse, viewed from the informational side, converges to a zero entropy state. This reframes the singularity in three ways.

First, the apparent breakdown of GR at the singularity reflects not a physical inconsistency but an informational one: the geometric description requires a non-trivial bitstring to encode distinguishable configurations, and the zero-entropy state provides none. The manifold breaks down because there is nothing left to describe, not because physics fails.

Second, the zero-entropy point is representation-independent. The Lemma of Section~\ref{sec:methods} establishes that a zero-entropy bitstring decodes to a single structureless point under any geometric map. This means the result does not depend on the coordinate system, the discretization scheme, or the simulation architecture. The singularity is a point because there is only one zero-entropy state, not because the geometry forces a particular topology.

Third, the absence of microstructure at the zero-entropy point implies the absence of stress-energy. No micro-structures, no particles, no matter, and no distinguishable geometry at the singularity, there is no source for the gravitational field at that point.

Relation to existing approaches

Loop quantum gravity resolves singularities by quantizing the geometry at the Planck scale, introducing a minimum area and replacing the singularity with a quantum bounce \cite{ashtekar2006}. String-theoretic fuzzball solutions replace the singularity with a horizonless, stringy microstate geometry \cite{mathur2005}. Both approaches introduce new physics to regularize the singularity.

The present approach requires no new physics. The regularization is a consequence of the informational equivalence established in Paper~I \cite{meskanen2001}: when the execution trace converges to zero entropy, the geometric description has no emergent micro structures to describe. The singularity is resolved not by preventing it but by showing that it is trivial.

Holographic consistency

The Bekenstein-Hawking entropy of a black hole scales with the area of the event horizon, not the volume of the interior. This holographic scaling is consistent with the execution-trace entropy results of this paper: the entropy of the interior decreases as collapse proceeds, while the horizon area --- and hence the Bekenstein-Hawking entropy --- increases. The two entropy measures are measuring different things. The execution-trace entropy measures the geometric variety of the interior configuration; the Bekenstein-Hawking entropy measures the information accessible to an external observer. The present framework is consistent with both.

Conclusion

We demonstrated that gravitational collapse, analyzed via the execution trace of a computational simulation, converges to a zero-entropy state at the classical singularity. This result holds for both Schwarzschild and Kerr geometries, remaining independent of the simulation architecture and discretization scheme. The metric suite deployed herein successfully functions as a structural phase detector. Specifically, the bit and block entropies capture the overarching loss of coordinate volume. This zero-entropy state admits a precise geometric interpretation: it corresponds to a single, structureless point incapable of supporting microstructure, matter, or distinguishable geometry. Consequently, the singularity represents not a breakdown of physics, but rather a minimally trivial geometric configuration. These findings suggest that black hole information can be modeled as $2^n$ bitstrings arranged in order of decreasing entropy. The zero-entropy extremum maps to the black hole singularity, while the opposing extremum corresponds to the state of maximum entropy. Between these two bounds, the entropic trajectory follows the time-reversed evolution of a typical emergent, relaxing thermodynamic system progressing from zero entropy toward full equilibrium. from zero entropy towards full equilibrium.

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