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The paper claims that finite-time state transitions of the BTZ black hole are optimal thermodynamic paths: geodesics of the Weinhold or Ruppeiner metric, whose length sets the probability and duration of the process.

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2026-08-03 18:03 UTC pith:64CNV6UK

load-bearing objection A clean geometric workout for BTZ that inherits, rather than establishes, the physical optimality claim. the 3 major comments →

arxiv 2512.06931 v3 pith:64CNV6UK submitted 2025-12-07 gr-qc

Optimal Control Theory of the (2+1)-Dimensional BTZ Black Hole

M. Radomirov , R. C. Rashkov , G. S. Stoilov , T. Vetsov This is my paper
classification gr-qc MSC 83C57 PACS 04.70.Dy
keywords BTZ black holethermodynamic geometryoptimal controlgeodesicsWeinhold metricRuppeiner metricfinite-time thermodynamicsblack hole evaporation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to establish that the (2+1)-dimensional BTZ black hole changes state along geodesics of a Hessian thermodynamic metric on its space of equilibrium states. Depending on the representation, those geodesics extremize energy dissipation (Weinhold metric, Hessian of energy) or entropy production (Ruppeiner metric, Hessian of entropy), and the square of the thermodynamic length gives the minimal energy or minimal entropy cost of the transition, with smaller length meaning higher probability. For the rotating hole in the energy representation, every numerically studied optimal process ends at a static, non-rotating configuration with nonzero mass, and complete evaporation never occurs; in the entropy representation, outcomes split into three classes: asymptotic approach to near-extremal states (unattainable in finite time, as the third law requires), approach to a fixed nonzero spin, and finite-time termination at a static state. For the static hole, analytic solutions show evaporation length proportional to initial entropy in the energy representation and to the fourth root of initial energy in the entropy representation, implying larger holes evaporate more slowly and less probably. If right, this is the first geometric optimal-control formulation for the BTZ black hole and offers a fluctuation mechanism distinct from blackbody Hawking evaporation in a holographically relevant setting.

Core claim

The paper's central claim is that finite-time transitions between BTZ thermodynamic states are geodesics of the information metric, so that optimal processes extremize entropy production or energy dissipation depending on representation. Solving the geodesic equations with initial data (S0=5, J0=1) in the energy representation, the authors find that every process terminates at a static BTZ configuration: accretion-type paths can increase final energy by factors up to ~100 and briefly bring the specific spin close to extremality (a≈0.994), while evaporation-type paths reduce energy by up to ~99%, with the strongest evaporation near initial angle 259°; complete evaporation is never reached. In

What carries the argument

The central machinery is the Hessian thermodynamic metric and its geodesic flow. Weinhold's metric is the Hessian of E(S,J); Ruppeiner's metric is the Hessian of S(E,J), and a sign parameter ϵ controls whether the information geometry is elliptic (R>0) or hyperbolic (R<0). Geodesic equations, built from the Christoffel symbols of these metrics, define optimal profiles S(t),J(t) or E(t),J(t) that extremize thermodynamic length L=∫√(g_ab ẋ^a ẋ^b)dt. L² is interpreted as the minimal energy (energy representation) or minimal entropy (entropy representation) required for the process, so smaller L implies higher probability, and the affine time gives the duration. In the static case the geodesic e

Load-bearing premise

The load-bearing premise is that geodesics of the Hessian thermodynamic metrics describe physically realizable optimal processes on the BTZ horizon, with L² governing probability and duration; the paper adopts this mapping from the thermogeometric optimization approach rather than deriving a specific mechanism, and Section 4.4.1 explicitly states that the algorithm does not encode or refer to any specific physical mechanism.

What would settle it

A microscopic or holographic calculation of transition rates between two BTZ states that does not select geodesic paths — for example, a static hole whose spontaneous fluctuation probability does not fall with L, or whose entropy does not decay linearly with L∝S0 — would falsify the claim. Equivalently, if near-horizon fluctuations are shown to obey equations of motion different from the geodesic equations (or a probability law not governed by the thermodynamic length), the optimal-control description fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • In the energy representation, all optimal processes from a rotating BTZ state reach a static configuration with nonzero energy and entropy; complete evaporation of the rotating hole never happens, so the third law is respected.
  • In the entropy representation, the late-time state is controlled by the initial direction: near-extremal approach (a→1), fixed non-extremal spin (0<a<1), or static configuration (a=0) reached in finite time.
  • Static BTZ optimal evaporation scales are fixed: thermodynamic length grows linearly with initial entropy in the energy representation and as the fourth root of initial energy in the entropy representation, so larger black holes take longer and are less likely to fluctuate.
  • Thermodynamic length squared carries physical units and its minimal value fixes the metric scale ϵ (1/2 for the energy representation, −1/4 for entropy), giving the framework predictive specificity rather than free parameters.
  • Optimal evaporation paths are not constrained by Stefan–Boltzmann or blackbody power laws, so the model makes different, testable time profiles from Hawking evaporation for the same initial state.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the L²-probability link is right, the static scaling law L∝S0 is a concrete target: it predicts that the likelihood of spontaneous evaporation falls exponentially with initial entropy, a statement that could be tested against CFT₂ or near-horizon fluctuation calculations.
  • The entropy-representation attractor values (e.g., a→0.966 and 0.725) suggest a check: whether these fixed spins are universal functions of initial data or of the metric alone; if universal, they would be a verifiable signature of the optimization principle.
  • Because the paper explicitly says the algorithm encodes no physical mechanism, the natural next step is to derive the same geodesic equations from a microscopic horizon model; until then the 'optimal processes' are best read as mathematical geodesics rather than demonstrated black hole control.
  • The linear-in-time entropy decay (energy representation) and quartic-in-time energy decay (entropy representation) differ sharply from the time profiles of blackbody Hawking evaporation; comparing these profiles in a 2+1-dimensional setting could distinguish optimal from non-optimal evolution observationally or numerically.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper applies the thermogeometric optimization (TGO) framework of [31] to the (2+1)-dimensional BTZ black hole. It computes Weinhold and Ruppeiner metrics on the two-dimensional thermodynamic state space, solves the resulting geodesic equations analytically for the static case and numerically for the rotating case, and interprets the geodesics as optimal evaporation/accretion protocols. In the energy representation all numerically studied processes terminate at a static BTZ configuration (Table 1), while in the entropy representation the endpoints split into asymptotic near-extremal states, fixed non-extremal spin, and finite-time static configurations (Table 2). The paper also derives static analytic profiles, fixes the metric scale ϵ by identifying L^2 with the initial energy/entropy, and contrasts optimal evaporation with blackbody Hawking evaporation.

Significance. If the TGO identification is accepted, the paper is the first optimal-control/thermodynamic-length analysis of BTZ and provides a useful classification of endpoint behavior. The analytic static profiles and explicit expressions for the metric and geodesic equations are strengths, and the numerics appear reproducible from the stated initial data. The main limitation is that the physical optimality interpretation is inherited from the authors' own prior framework and is not derived for BTZ; the probability/duration reading of L^2 is an assumption. The paper's value is therefore primarily as a mathematical geodesic study, with the physical conclusions requiring either additional support or explicit qualification.

major comments (3)
  1. [§2.2, Eq. (2.4)] The two fundamental relations in Eq. (2.4) are not inverses as printed. For J=0, E=ζ^2 S^4/(2λ) implies S=(2λE)^{1/4}/ζ^{1/2}, whereas the stated S(E,0)=√(2λE)/ζ. All later formulas—T in Eq. (2.6), the Weinhold metric in Eq. (4.1), and the entropy-representation geodesics in Sec. 5—are consistent with E=ζ^2 S^2/(2λ)+λ J^2/(2S^2), not with the printed E(S,J). The exponent in the first line of Eq. (2.4) should be corrected; as it stands the fundamental equation contradicts the remainder of the paper.
  2. [§5.2, Eqs. (5.7)–(5.10)] Eq. (5.8) does not follow from Eq. (5.7). With E(t)=E0(1-|\dot E0|t/(4E0))^4 and g_EE = -ϵ(2λ)^{1/2}/(4ζ) E^{-3/2} (for J=0), the integrand in Eq. (3.3) is √(-ϵ(2λ)^{1/2}/(4ζ)) |\dot E0| E0^{-3/4}, independent of t. Hence the line-integral length is proportional to |\dot E0| E0^{-3/4} τ, not to |\dot E0| E0^{3/4} τ/√λ as printed. This also makes the equality between (5.8) and (5.9) used to obtain (5.10) problematic. Eq. (5.9) and the resulting E0^{1/4} scaling in the conclusion are correct, but (5.8) must be recomputed.
  3. [§3; §4.4.1; Tables 1–2] Sec. 4.4.1 explicitly states that 'the algorithm does not encode or refer to any specific physical mechanism,' and the identification of geodesics as optimal evaporation/accretion protocols is imported from [31] without derivation for BTZ. The equations solved in Secs. 4–5 are geodesics of Hessian metrics; they become optimal-control statements only if an underlying dynamical model exists whose dissipation functional is the thermodynamic length. No such model is given, and Sec. 2.3 rules out Hawking evaporation while invoking unspecified fluctuations. Consequently the probability statements in Tables 1–2 ('smaller L indicates higher probability') and the abstract's claim that state transitions are 'described by paths that extremize entropy production or energy dissipation' are unsupported. Please either provide a concrete derivation/test (e.g., a stochastic horizon-fluctuation model whos
minor comments (5)
  1. [Table 1, row 315°] Eτ=50 is inconsistent with δE=+103% and E0=1.46 under the definition in Eq. (4.16); if δE is as defined, Eτ should be about 3.0. Please also reconcile the text values in Sec. 4.4.2 (δE=+10226%, δS=+992%) with the table entries (+103, +104 are listed in different conventions).
  2. [§4.4.1 footnote 12] The sentence ends with 'traditionally called ).' — incomplete; please fix.
  3. [§4.4.1] The numerical integration is not described. Please specify the solver, tolerances, and the stopping criterion used to define τ (especially when J→0 or when a→0).
  4. [§5.2 notation] The notation '4√2λ' is ambiguous; please use \sqrt[4]{2λ} consistently.
  5. [§2.1, Eq. (2.2)] The paper interchanges E and M without explicitly stating E=Mc^2 until a footnote. Please state this once in the main text and check the factors of c in the entropy relation.

Circularity Check

0 steps flagged

No significant circularity: the BTZ geodesic computations are derived from the standard BTZ fundamental relations; the only caveat is that the physical-optimality interpretation is inherited from the authors' own TGO framework [31], but this is an unsupported assumption rather than a circular reduction.

full rationale

The paper's load-bearing mathematical content is not circular. The geodesic equations in Secs. 4.2 and 5.1 are derived from the standard BTZ fundamental relations (2.4) via the Hessian Weinhold and Ruppeiner metrics (3.1)-(3.2); the final states in Tables 1-2 are numerical solutions of these geodesic equations, not fits to the claimed outcomes. The epsilon values, epsilon=1/2 in the energy representation and epsilon=-1/4 in the entropy representation, are fixed by normalizing L^2 to the initial energy or entropy (Secs. 4.3 and 5.2), not by fitting the predicted transitions. The physical interpretation, however, is imported from the authors' own prior work [31]: the claim that thermodynamic geodesics represent optimal evaporation/accretion protocols and that smaller L implies higher probability is not re-derived for BTZ. Indeed, Sec. 4.4.1 states that 'the algorithm does not encode or refer to any specific physical mechanism.' That is a foundational assumption and a correctness risk for the central physical claim, but it is not a circularity in the derivation chain: the BTZ geodesic results do not reduce by construction to the framework's conclusion, and [31] was an independent application to the Kerr black hole rather than a parameter fitted to the present paper's data. Therefore the appropriate finding is no significant circularity, with a small penalty for the load-bearing self-citation in the interpretive step.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

The central calculation uses standard BTZ thermodynamics plus the TGO assumption imported from [31]. No new particles, forces, or dimensions are introduced. The hand-set numbers are the metric scale epsilon and the initial rate directions; the latter control the classification results, while the former is a normalization choice.

free parameters (2)
  • epsilon (metric scale factor) = 1/2 (energy representation), -1/4 (entropy representation)
    Introduced in TGO [31]; fixed here by requiring L^2 to be at least the initial energy (Sec. 4.3) or initial entropy (Sec. 5.2). It does not change the geodesic equations but scales thermodynamic length and the probability interpretation.
  • initial rate vector (u, phi) = u=0.01 in Planck units; phi scanned over {0,45,60,90,135,180,225,240,259,270,315} degrees
    Boundary conditions chosen by hand. The final-state classification (static, near-extremal, fixed spin, accretion vs evaporation) depends directly on these initial rates, so they are inputs rather than derived quantities.
axioms (4)
  • domain assumption BTZ thermodynamic fundamental relations in Eqs. (2.2)-(2.4) are correct and give the energy and entropy as functions of S,J or E,J.
    Standard BTZ thermodynamics from refs. [36,40]; used to construct both thermodynamic metrics.
  • domain assumption Hessians of energy (Weinhold) and entropy (Ruppeiner) provide valid thermodynamic information metrics whose geodesics are physically meaningful.
    Standard thermodynamic geometry [11-13]; assumed without re-derivation.
  • domain assumption Geodesics of thermodynamic length are optimal finite-time protocols, and smaller thermodynamic length L implies higher process probability.
    Core TGO premise from the authors' prior work [31]; not derived for BTZ and noted in Sec. 4.4.1 to encode no specific physical mechanism.
  • domain assumption Extremal states (a=1, T=0) are excluded and cannot be reached in finite classical time.
    Used to restrict the state space and interpret asymptotic trajectories; stated in Eqs. (2.8)-(2.9) as a form of the third law.

pith-pipeline@v1.3.0-alltime-deepseek · 17920 in / 29438 out tokens · 264118 ms · 2026-08-03T18:03:37.811578+00:00 · methodology

0 comments
read the original abstract

We apply a finite-time geometric optimization framework to investigate thermal fluctuations and (non)equilibrium optimal processes in the $(2+1)$-dimensional BTZ black hole. Employing Hessian thermodynamic information metrics, we construct geodesic trajectories that define optimal protocols connecting distinct thermodynamic configurations. Finite-time state transitions are described by paths that extremize entropy production or energy dissipation, depending on the chosen thermodynamic representation. { We compare our optimization framework with a non-optimal blackbody Hawking evaporation model, revealing substantial differences between the two descriptions. Finally, we quantify the intrinsic efficiency of both types of processes in terms of the extractable rotational energy stored in the black hole configurations.} This work presents the first formulation of a geometric optimal control theory for the BTZ black hole.

Figures

Figures reproduced from arXiv: 2512.06931 by G. S. Stoilov, M. Radomirov, R. C. Rashkov, T. Vetsov.

Figure 1
Figure 1. Figure 1: The Weinhold thermodynamic curvature R is presented as a function of S and J in Planck units with ℓ = 1. An increase in the interaction strength among the horizon degrees of freedom is ob￾served near the extremal BTZ curve (dashed red), where a phase transition takes place. In the asymp￾totic regime, far from extremality, the thermodynamic geometry of the BTZ black hole approaches flatness, indicating weak… view at source ↗
Figure 2
Figure 2. Figure 2: Profiles for ϕ = 0◦ . (a) Time evolution of E (orange), S (green), and J (blue). (b) The dependence of E, S, and J on the specific spin a. The final configuration corresponds to a static BTZ black hole with Eτ = 151 and Sτ = 55 in Planck units. (c) Time evolution of the specific spin a(t). It peaks at apeak ≈ 0.994 indicating the state closest to extremality, and then gradually decreases to zero. (d) The g… view at source ↗
Figure 3
Figure 3. Figure 3: Profiles for ϕ = 240◦ . (a) Time evolution of E (orange), S (green), and J (blue). (b) The dependence of E, S, and J on the specific spin a. The final configuration corresponds to a static BTZ black hole with Eτ = 0.4 and Sτ = 2.9 in Planck units. (c) The evolution of a(t) is strictly monotonic, tending towards zero. (d) The geodesic trajectory of states (green curve) in (S, J) space, with a starting point… view at source ↗
Figure 4
Figure 4. Figure 4: Geodesics paths of states in (S, J) space. (a) The thermodynamic curvature R is represented as a colored background. The initial state (S0 = 5, J0 = 1) lies in a relatively strong curved region (yellow) with high interactions, though it remains sufficiently distant from the extremal boundary (the dashed red curve). (b) A magnified view near the initial point. Polar circles help visualize the initial angles… view at source ↗
Figure 5
Figure 5. Figure 5: Profiles for ϕ = 0◦ . (a) Time evolution of E (orange), S (green), and J (blue). (b) The dependence of E, S, and J on the specific spin a. The BTZ black hole does not settle to a final state. It fluctuates between near-extremal states forever asymptotically approaching extremality. (c) Time evolution of the specific spin a(t). The spin asymptotically increases towards the extremal value. (d) The geodesic t… view at source ↗
Figure 6
Figure 6. Figure 6: Profiles for ϕ = 240◦ . (a) Time evolution of E (orange), S (green), and J (blue). (b) The dependence of E, S, and J on the specific spin a. The BTZ black hole settles to a final static state with energy Eτ = 3 and Sτ = 8 in Planck units. (c) Time evolution of the specific spin a(t). The spin monotonically decreases to zero. (d) The geodesic trajectory of states (orange curve) in the (E, J) space starts at… view at source ↗
Figure 7
Figure 7. Figure 7: Geodesic trajectories of states in the (E, J) space. Since the thermodynamic curvature is R = 0, the manifold is Ricci-flat. The initial state (E0 = 5, J0 = 1) is far from the extremal boundary (dashed red curve). The figure presents a magnified view around this initial point. Concentric polar circles indicate the initial angles ϕ, which set the direction of the corresponding geodesics. Each trajectory evo… view at source ↗

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