Kullback-Leibler Divergence as a Measure of Irreversible Information Loss Near Black Hole Horizons

Tatsuaki Tsuruyama

Kullback-Leibler Divergence as a Measure of Irreversible Information Loss Near Black Hole Horizons

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Gravitational time dilation near a black hole drives the Kullback-Leibler divergence between sent and received signals to infinity at a critical radius that approaches the Schwarzschild horizon.

desk verdict A plausible but unverified abstract; the horizon result may simply be time dilation in disguise, but a referee could sort it out. read the letter →

arxiv 2508.04348 v1 pith:YYJTOVXS submitted 2025-08-06 gr-qc math-phmath.MPphysics.space-ph

Tatsuaki Tsuruyama This is my paper · ORCID

classification gr-qcmath-phmath.MPphysics.space-ph PACS 04.70.-s

keywords Kullback-LeiblerdivergenceeventhorizongravitationaltimedilationSchwarzschildblackholeinformationtheorythermodynamicdecodinglimittime-encodedsignalsirreversibleloss

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to show that gravitational time dilation near a black hole makes transmitted information undecodable at a specific radius, and that this radius coincides with the Schwarzschild event horizon in a natural energetic limit. The central tool is the Kullback-Leibler divergence, a standard measure of how much one probability distribution differs from another. The author models a sender emitting time-encoded symbols from a strong gravitational field and a distant receiver; time dilation distorts the received symbol distribution. The paper claims that the divergence between sent and received distributions becomes infinite at a critical radius, so decoding would require infinite thermodynamic work. In the limit where the entropy cost of information is negligible compared with the transmission energy, that critical radius approaches the Schwarzschild radius, giving an information-theoretic reading of the event horizon as a boundary of irreversible information loss.

What carries the argument

The Kullback-Leibler divergence (KLD) — a nonnegative information-theoretic measure of the difference between two probability distributions — is the central object. The paper compares the distribution of time-encoded symbols as transmitted with the distribution as received under gravitational time dilation, and treats the divergence of the KLD as the onset of thermodynamic impossibility. The gravitational time dilation factor from the Schwarzschild metric supplies the radius dependence; the KLD converts that geometric factor into an information-theoretic cost.

What would settle it

Recompute the KLD for the same time-dilation model with a finite symbol interval and additive receiver noise: if the divergence becomes finite as the radius approaches the Schwarzschild horizon, the claimed thermodynamic impossibility disappears. Alternatively, calculate the free energy required for a fixed error probability and check whether it diverges at the paper's critical radius.

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Extended reading notes

Core claim

On its own terms, the paper establishes a quantitative link between time dilation and information loss. It defines the Kullback-Leibler divergence between the probability distributions of transmitted and received symbols in a minimal communication model, and derives a critical radius $r_c$ at which this divergence grows without bound. At $r_c$, no thermodynamic process can decode the received signal, because distinguishing the possible transmitted messages would require infinite free energy. The main result is that $r_c$ approaches the Schwarzschild horizon $r_s$ whenever the information entropy cost per symbol is negligible compared with the energy used to transmit the symbol. The claim is

Load-bearing premise

The argument depends on equating an infinite Kullback-Leibler divergence with thermodynamic impossibility of decoding, and on taking the limit where information entropy cost is negligible relative to transmission energy; neither step is independently derived in the abstract.

Editorial extensions

If this is right

If the result holds, there is a critical radius outside a Schwarzschild black hole at which decoding a time-encoded signal is not merely difficult but thermodynamically forbidden.
The event horizon acquires a new operational characterization: it is the limiting surface at which transmitted and received symbol distributions have zero overlap in the KLD sense.
Finite transmission energy moves the critical radius away from the horizon, implying a thin region outside $r_s$ where communication is still possible but becomes increasingly expensive.
The same reasoning should apply to other time-dilation geometries, so any future horizon (e.g., a cosmological event horizon) may also act as an information-theoretic decoding boundary.
The framework yields quantitative entropic and energetic constraints on communication in strong gravitational fields, potentially testable in relativistic clock networks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the divergence criterion is taken literally, the same KLD argument can be transplanted to accelerated-frame (Rindler) horizons, where the same limiting coincidence should occur; this is a testable application the paper does not spell out.
One could probe the predicted scaling with terrestrial clocks: measure the KLD between time signals sent up and down a tall gravitational potential difference and extrapolate the radius at which it would diverge; the paper's formula makes a definite prediction.
Accounting for measurement noise or finite bandwidth would replace the exact divergence by a large finite KLD, which would shift the critical radius inward or outward; this is an extension the paper does not address.
The approach suggests that information-theoretic divergences, not just geometric singularities, define black hole boundaries; if so, the same quantity may mark the onset of quantum effects where the semiclassical approximation breaks down.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

Based on the abstract alone, this is a paper worth a second look but not yet a convincing result. The genuinely new thing is the idea of using Kullback-Leibler divergence to measure how gravitational time dilation distorts symbol distributions in a communication model, and then asking where decoding becomes impossible. That is a fresh way to frame the old observation that time dilation blows up at the horizon. The minimal model is clear enough at this level, and the ambition to unify information theory, thermodynamics, and GR is real.

The soft spots are obvious even from the abstract. First, the leap from KLD divergence to thermodynamic impossibility is asserted, not argued. In standard information theory, an infinite KLD means the two distributions are perfectly distinguishable—the error exponent goes to infinity, so decoding gets easier, not harder, as sample size grows. To get an impossibility you need an extra step, typically a finite energy budget per symbol and a Landauer-type cost proportional to the KLD. That step might be in the full text, but the abstract does not hint at it.

Second, the limit statement looks tuned. The critical radius approaches the Schwarzschild horizon when the information entropy cost is negligible relative to transmission energy. Since the redshift factor is built into the received symbol distributions, any measure based on it will inherit the divergence at r_s. The limit condition then functions as a knob that tells you which part of the divergence dominates. If the full derivation is just unpacking the metric's time dilation, the horizon result is kinematic, not thermodynamic. If, however, the full text provides a parameter-free derivation where the horizon location emerges from the communication model rather than being put in by hand, the circularity concern evaporates.

Given we only have the abstract, I cannot rule that out. The citation pattern is unknown, so novelty is unverifiable. The paper is significant if true, the idea is worth airing, but the missing derivation is the whole game. I would send it to a referee—someone who knows both info theory and GR—but I would expect heavy revision or a clear negative if the full text does not close the gap. For my own work, I would not cite it until the thermodynamic link is established.

Referee Report

3 major / 3 minor

Summary. The paper proposes a unified framework combining information theory, thermodynamics, and general relativity, using the Kullback-Leibler divergence (KLD) between transmitted and received symbol distributions as a measure of information loss induced by gravitational time dilation. It claims to derive a critical radius at which decoding becomes thermodynamically impossible because the KLD diverges, and shows that this radius approaches the Schwarzschild horizon in the limit where the information entropy cost is negligible relative to the transmission energy. The abstract thus presents the event horizon as a boundary of irreversible information loss governed by thermodynamic principles.

Significance. If fully substantiated, this would be a notable contribution: it would connect a central quantity of information theory (KLD) to the thermodynamically irreversible character of black hole horizons, potentially offering a new entry point for quantum information and gravitational physics. The conceptual premise is plausible—gravitational redshift indeed distorts any time-encoded signal—and using KLD as a quantitative mismatch measure is a reasonable idea. However, as presented in the abstract, the load-bearing links are asserted rather than derived, and no equations, error estimates, or comparisons to known results are provided. The only explicit free parameter is the ratio of information entropy cost to transmission energy, which appears to be a tuning parameter rather than an independently fixed quantity. The strength of the contribution cannot be assessed from the abstract alone.

major comments (3)

[Abstract] The inference from 'divergence of the KLD' to 'decoding becomes thermodynamically impossible' is asserted, not derived. In standard information theory, an infinite KLD between transmitted and received distributions implies perfect asymptotic distinguishability (infinite error exponent), which is the opposite of an impossibility. To reach a thermodynamic conclusion, a further step is required—for example, a Landauer-style erasure cost proportional to KLD and a finite energy budget. Without this derivation, the central claim does not follow. If the full text supplies such a derivation, it must be stated explicitly in equations.
[Abstract] The limit condition 'where the information entropy cost becomes negligible relative to the transmission energy' appears engineered to make the critical radius coincide with the known Schwarzschild horizon. In Schwarzschild geometry, gravitational time dilation diverges as (1 - r_s/r)^(-1/2) at r_s; any KLD built from the redshift factor inherits this divergence. The paper must show that the horizon limit is not a restatement of this kinematic divergence and must justify why the entropy/energy limit is physically realizable (e.g., it does not require infinite energy or zero temperature). A closed-form expression for the critical radius r_c and an error analysis are needed.
[Abstract] The abstract makes quantitative claims ('derive the critical radius,' 'approaches the Schwarzschild horizon') without presenting the communication model, the definitions of the transmitted and received symbol distributions, the transmission energy, or the information entropy cost. These are not presentation details but the mathematical core of the claimed result. The full paper must provide precise definitions and a step-by-step derivation; otherwise the abstract's assertions are unsupported.

minor comments (3)

[Abstract] The phrase 'unified theoretical framework' is vague; specify which elements are unified and how (e.g., via Landauer's principle, quantum measurement theory, or a particular relativistic communication model).
[Abstract] The closing statement that the framework 'may extend to general relativistic and quantum information settings' is speculative. Either state a concrete extension or remove the claim to keep the abstract focused.
[Abstract] No reference is made to known related works on information near black holes (e.g., Bekenstein-Hawking entropy, Hawking radiation, or earlier black-hole communication limits). A brief comparison would help position the claimed novelty.

Circularity Check

2 steps flagged · score 6.0 of 10

Central result reproduces the Schwarzschild time-dilation divergence by constructing KLD from that same divergence.

self definitional [Abstract (introduction of KLD)]
"we introduce the Kullback-Leibler divergence (KLD) as a quantitative measure of the mismatch between the transmitted and received symbol distributions induced by gravitational time dilation."

In Schwarzschild geometry, gravitational time dilation between radius r and infinity is set by sqrt(1 - r_s/r), which becomes singular as r approaches r_s. The paper defines KLD as the measure of the mismatch induced by this time dilation, then uses KLD divergence as the criterion for the critical radius where decoding becomes thermodynamically impossible. Therefore the divergence of KLD at/near the Schwarzschild horizon is not an independent result: it is built into the object's definition from the known metric divergence. The claimed 'critical radius' is thus the same r_s that entered as an input through the time-dilation factor, not a novel information-theoretic consequence.
renaming known result [Abstract (closing claim)]
"This result provides a novel information-theoretic interpretation of the event horizon as a boundary of irreversible information loss governed by universal thermodynamic principles."

The event horizon is already a boundary in the Schwarzschild metric: the redshift/time-dilation factor diverges there and causal signals cannot escape. Re-expressing this known geometric/causal boundary as 'irreversible information loss' via KLD relabels the known horizon behavior in information-theoretic vocabulary. The abstract supplies no additional mechanism connecting thermodynamic principles to the KLD divergence other than the assertion, so the claimed novelty is essentially a renaming of the pre-existing horizon divergence rather than a derived prediction.

full rationale

The abstract-only text makes full verification impossible, but the structure of the central claim is visibly definitional. The KLD is defined as measuring mismatch induced by gravitational time dilation; in Schwarzschild spacetime the same time dilation diverges at r_s. Hence the later statement that the critical radius approaches the Schwarzschild horizon is inherited from the metric input, not derived from an independent information-theoretic mechanism. The limit condition ('information entropy cost becomes negligible relative to transmission energy') is not physically derived in the abstract and acts as a tuning parameter that aligns the divergence point with r_s. No self-citation is present, so the circularity is of the self-definitional and renaming kind rather than a citation-chain problem. There is also a separate correctness issue—the abstract asserts without derivation that KLD divergence makes decoding 'thermodynamically impossible'—but that is an underived premise, not a circular reduction; it does not raise the score further. On the face of the abstract, the main predictive claim reduces by construction to the known Schwarzschild time-dilation divergence, giving a partial-circularity score of 6.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

All entries are extracted from the abstract alone. The free parameter is the entropy-cost to transmission-energy ratio, which tunes the critical radius toward the horizon. The axioms include the Schwarzschild time-dilation background, the use of KLD as a mismatch measure, the asserted identification of KLD divergence with thermodynamic impossibility, and the limiting condition; the last two are ad hoc at the level of the abstract. The full text might reveal more model parameters (symbol rate, bandwidth, detector model), which cannot be audited here. No invented physical entities appear in the abstract.

free parameters (1)

Information entropy cost to transmission energy ratio = epsilon = S_info / E tends to 0
The abstract states the critical radius approaches the Schwarzschild horizon in this limit; the ratio is the tuning knob that sets r_c, and no independent physical origin is given.

assumptions (4)

domain assumption Schwarzschild gravitational time dilation governs symbol clocks
The abstract's framework assumes general relativity and derives the decoding radius near the 'Schwarzschild horizon'; the metric and its time-dilation factor enter as background input.
standard math KLD quantifies the mismatch between transmitted and received symbol distributions
The mathematical definition of KLD as a distribution-divergence measure is standard background; the abstract gives it the semantic role of decoding-failure measure.
ad hoc to paper Divergence of KLD implies decoding is thermodynamically impossible
The abstract asserts decoding becomes 'thermodynamically impossible due to the divergence of the KLD'; this identification is the paper's central modeling postulate, not an established result.
ad hoc to paper The entropy-cost-negligible limit is physically realizable
The horizon result holds 'in the limit where the information entropy cost becomes negligible relative to the transmission energy'; the abstract gives no independent justification for this limit, which appears chosen to produce r_c approaching r_s.

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Cite this review

Pith. "Pith review of Kullback-Leibler Divergence as a Measure of Irreversible Information Loss Near Black Hole Horizons." pith.science (2026). https://pith.science/paper/YYJTOVXS

@misc{pith2026250804348,
  author       = {Pith},
  title        = {Pith review of: Kullback-Leibler Divergence as a Measure of Irreversible Information Loss Near Black Hole Horizons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YYJTOVXS}},
  note         = {Machine review of arXiv:2508.04348}
}

read the original abstract

We present a unified theoretical framework that integrates information theory, thermodynamics, and general relativity to analyze the fundamental limit of decoding time-encoded signals in curved spacetime. In particular, we introduce the Kullback-Leibler divergence (KLD) as a quantitative measure of the mismatch between the transmitted and received symbol distributions induced by gravitational time dilation. Using a minimal communication model, we derive the critical radius at which information decoding becomes thermodynamically impossible due to the divergence of the KLD. We show that this radius approaches the Schwarzschild horizon in the limit where the information entropy cost becomes negligible relative to the transmission energy. This result provides a novel information-theoretic interpretation of the event horizon as a boundary of irreversible information loss governed by universal thermodynamic principles. Our framework offers new insights into the entropic and energetic constraints on communication in strong gravitational fields and may extend to general relativistic and quantum information settings.

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Reviewed August 6, 2026 · model on record in the stance chip above.