On Thermodynamics of Charged Black Holes, Swampland, and Dark Matter

Adil Belhaj; Hajar Belmahi; Saad Eddine Baddis; Salah Eddine Ennadifi

arxiv: 2605.23047 · v2 · pith:TYHPWU4Nnew · submitted 2026-05-21 · ✦ hep-th

On Thermodynamics of Charged Black Holes, Swampland, and Dark Matter

Saad Eddine Baddis , Adil Belhaj , Hajar Belmahi , Salah Eddine Ennadifi This is my paper

Pith reviewed 2026-05-25 05:16 UTC · model grok-4.3

classification ✦ hep-th

keywords black hole thermodynamicsswampland conjecturesdark matterdark dimensionscalar fieldsKaluza-Kleincosmological constantcharged black holes

0 comments

The pith

Treating the cosmological constant as dynamical in charged black hole thermodynamics connects swampland conjectures to the dark dimension and dark matter via scalar fields.

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

The paper proposes viewing the cosmological constant as varying in black hole systems, using the radial metric function at the horizon to define an equation of state. This setup reveals structural transitions and coexistence curves in the thermodynamics of charged black holes. An effective potential involving a quasi-static scalar field then links this to swampland conjectures from string theory. Through Kaluza-Klein interpretations of the scalar fields, the approach extends to explanations for the dark dimension and dark matter. A sympathetic reader would care because it suggests a unified thermodynamic origin for these seemingly disparate ideas in high-energy physics and cosmology.

Core claim

By dealing with the radial metric function at the horizon as an equation of state for black holes with dynamical cosmological constant, and introducing an effective potential with a quasi-static scalar field, the work establishes a connection between certain swampland conjectures and approaches the dark dimension and dark matter using Kaluza-Klein interpretations of scalar fields.

What carries the argument

The radial metric function at the horizon treated as an equation of state, combined with an effective potential for a quasi-static scalar field.

If this is right

Structural transitions and coexistence curves appear in the thermodynamics of charged black holes when the cosmological constant is dynamical.
An effective potential with a quasi-static scalar field places the thermodynamic setup in string theory and links to swampland conjectures.
Kaluza-Klein interpretations of the scalar fields provide a thermodynamic approach to the dark dimension and dark matter.

Where Pith is reading between the lines

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

The thermodynamic transitions might predict observable signatures in cosmological data related to dark energy variation.
This framework could be tested by checking whether black hole phase diagrams match constraints from extra-dimensional models.
Extending the scalar field treatment to other black hole charges might reveal additional swampland connections.

Load-bearing premise

The cosmological constant can be treated as a dynamical quantity whose variation is captured by the radial metric function at the horizon serving as an equation of state.

What would settle it

A calculation demonstrating that the effective potential with the quasi-static scalar field violates the connected swampland conjectures would falsify the proposed bridge.

Figures

Figures reproduced from arXiv: 2605.23047 by Adil Belhaj, Hajar Belmahi, Saad Eddine Baddis, Salah Eddine Ennadifi.

**Figure 1.** Figure 1: (ϕ − r+)-diagrams in the AdS case for different small phase sizes rs. Integrating out this equation, we get the following charge solution Q = √ 2 ℓp r 2 s e −aϕ (III.11) interpreted as a coexistence curve separating large black hole phases and small ones. Equipped with GPU accelerated CUDA computing techniques and in order to illustrate the present proposition, we consider the (ϕ − r+)-diagram playing a si… view at source ↗

read the original abstract

Inspired by the idea that the cosmological constant can be considered as a dynamical quantity, we present a scenario bridging certain swampland conjectures from a new look at thermodynamics of black holes. Dealing with the radial metric function at the horizon as an equation of state, we discuss structural transitions and coexistence curves. By considering an effective potential with a quasi-static scalar field that could find a place in string theory, we then establish a connection between certain swampland conjectures. Relying on Kaluza Klein interpretations of scalar fields, we approach the dark dimension and dark matter through such a thermodynamic approach to charged black holes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper sketches a speculative thermodynamic link from charged black holes to swampland conjectures and dark matter but does not provide the derivations needed for the key identification of the metric function as an equation of state.

read the letter

The punchline is that the work proposes using the radial metric function of charged black holes as an equation of state to handle a dynamical cosmological constant, then links that to swampland via a quasi-static scalar field and Kaluza-Klein to dark dimension and dark matter. That's the scenario they lay out. What stands out is the attempt to combine black hole thermodynamics in extended phase space with string-inspired ideas. They discuss structural transitions and coexistence curves in this context, which could be a way to explore phase structure under those conjectures. The paper does well in outlining a broad picture that might interest people thinking about how swampland constraints could show up in gravitational thermodynamics. It draws on Kaluza-Klein interpretations for the scalar fields to approach dark matter. The main issue is the assumption that the metric function at the horizon serves as an equation of state for varying Lambda without showing how this preserves the first law when the scalar field is included. The stress test points out that extra terms from scalar stress-energy would appear, and there's no explicit calculation to show they are handled. Since the abstract presents this as established, but no intermediate steps are visible, the soundness is limited. This kind of paper is for theorists already deep in swampland and black hole physics who are looking for new angles on dark matter. A reader wanting concrete calculations or verified results won't find them here. It doesn't seem to have the formal grounding or new results to warrant a serious referee at this stage. The connections are stated rather than derived. I recommend against sending it to peer review in its current form.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes treating the radial metric function at the horizon of charged black holes as an equation of state to model a dynamical cosmological constant, discusses structural transitions and coexistence curves in this thermodynamic framework, introduces a quasi-static scalar field in an effective potential to connect to swampland conjectures, and invokes Kaluza-Klein interpretations of the scalar to relate the setup to the dark dimension and dark matter.

Significance. If the key identifications were rigorously derived from the Einstein-scalar action and the mappings to swampland conjectures were made explicit, the work could provide a novel thermodynamic route to string-theory constraints on the dark sector. The approach of linking black-hole thermodynamics to swampland ideas is potentially interesting, but the manuscript supplies no derivations, explicit equations, or on-shell action variations to support the central claims.

major comments (3)

[Abstract and thermodynamic analysis] The central construction (abstract and introduction) identifies the radial metric function evaluated at the horizon as an equation of state that encodes the variation of Lambda. For the RN-AdS metric f(r)=1-2M/r+Q²/r²-(Lambda/3)r² this identification is asserted without deriving the first law from the Einstein-scalar action; the standard extended-phase-space term V dP acquires extra contributions from scalar stress-energy and horizon-radius variation that are not shown to vanish or be absorbed.
[Swampland connection paragraph] The connection between the effective potential of the quasi-static scalar field and specific swampland conjectures is stated as established (abstract) but no explicit steps, equations, or mapping from thermodynamic quantities to the conjectures are supplied, rendering the claimed bridge uncheckable.
[Dark dimension and dark matter discussion] The Kaluza-Klein interpretation linking the scalar to the dark dimension and dark matter is presented without any derivation or quantitative relation between the black-hole thermodynamic quantities and the dark-sector observables.

minor comments (2)

[Abstract] The abstract and introduction would benefit from explicit statements of which swampland conjectures are being connected and which equations define the effective potential.
[Thermodynamic setup] Notation for the metric function and the equation-of-state identification should be introduced with an equation number rather than described only in prose.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful review and valuable comments on our manuscript. The work proposes a thermodynamic approach to connect black hole physics with swampland conjectures and the dark sector. We address each of the major comments below, providing clarifications and indicating where revisions will be made to strengthen the derivations.

read point-by-point responses

Referee: [Abstract and thermodynamic analysis] The central construction (abstract and introduction) identifies the radial metric function evaluated at the horizon as an equation of state that encodes the variation of Lambda. For the RN-AdS metric f(r)=1-2M/r+Q²/r²-(Lambda/3)r² this identification is asserted without deriving the first law from the Einstein-scalar action; the standard extended-phase-space term V dP acquires extra contributions from scalar stress-energy and horizon-radius variation that are not shown to vanish or be absorbed.

Authors: We agree that an explicit derivation of the first law from the Einstein-scalar action would strengthen the central construction. In the revised manuscript, we will include a detailed derivation in the thermodynamic analysis section. Starting from the action with the quasi-static scalar field, we will show that the contributions from the scalar stress-energy and horizon variations are either absorbed into the effective pressure term or vanish under the quasi-static approximation, justifying the identification of f(r) at the horizon as the equation of state for dynamical Lambda. revision: yes
Referee: [Swampland connection paragraph] The connection between the effective potential of the quasi-static scalar field and specific swampland conjectures is stated as established (abstract) but no explicit steps, equations, or mapping from thermodynamic quantities to the conjectures are supplied, rendering the claimed bridge uncheckable.

Authors: The connection is based on identifying the effective potential parameters with thermodynamic quantities from the black hole horizon. To make this explicit, we will add a dedicated subsection with the step-by-step mapping: relating the scalar field value to the horizon radius via the equation of state, and then applying the swampland distance conjecture to the field displacement in terms of the thermodynamic variables. This will allow readers to verify the bridge to the conjectures. revision: yes
Referee: [Dark dimension and dark matter discussion] The Kaluza-Klein interpretation linking the scalar to the dark dimension and dark matter is presented without any derivation or quantitative relation between the black-hole thermodynamic quantities and the dark-sector observables.

Authors: The KK interpretation is invoked to relate the scalar field to a modulus in the dark dimension scenario, with the thermodynamic pressure setting the scale for dark matter candidates. We acknowledge the lack of quantitative relations in the current draft. In the revision, we will provide explicit expressions linking the black hole mass and charge (from the thermodynamic analysis) to the compactification scale and dark matter mass, making the connection more quantitative while noting the phenomenological nature of the approach. revision: yes

Circularity Check

0 steps flagged

No circularity: central identification presented as modeling choice rather than derived prediction

full rationale

The manuscript introduces the treatment of the radial metric function at the horizon as an equation of state for a dynamical cosmological constant as an inspired modeling step, then uses an effective potential for a quasi-static scalar to link to swampland conjectures and Kaluza-Klein interpretations. No quoted equation or step reduces a claimed first-principles result or prediction to the input by construction (e.g., no fitted parameter renamed as output, no self-citation chain supplying the uniqueness of the identification, and no ansatz smuggled via prior author work). The derivation chain remains self-contained against external benchmarks once the initial modeling choice is granted; the provided abstract and context exhibit no self-definitional or fitted-input circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Review performed on abstract alone; full derivations, parameter counts, and explicit assumptions are not visible. The ledger therefore records only the assumptions stated or implied in the abstract.

axioms (2)

domain assumption The cosmological constant can be treated as a dynamical quantity.
Stated in the first sentence of the abstract as the starting inspiration.
ad hoc to paper The radial metric function at the horizon functions as an equation of state.
Introduced without further justification in the abstract.

invented entities (1)

quasi-static scalar field in effective potential no independent evidence
purpose: Bridge between black-hole thermodynamics and swampland conjectures
Introduced in the abstract as the object that 'could find a place in string theory'.

pith-pipeline@v0.9.0 · 5637 in / 1479 out tokens · 20545 ms · 2026-05-25T05:16:28.661104+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean phi_golden_ratio echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

we use the following identification limit r_ℓ = (3 + √5)/2 r_s ... Integrating out this equation, we get ... Q = √2 / ℓ_p r_s² e^{-aϕ}
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff refines

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refines
Relation between the paper passage and the cited Recognition theorem.

V(ϕ) = -e^{-2aϕ}/ℓ_p² ... P = -3V(ϕ)/8π ... f(r) = 0 as state equation

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.