arxiv: 2604.25629 · v1 · submitted 2026-04-28 · 🌀 gr-qc · hep-th

Recognition: unknown

Thermodynamic and Radiative Properties of Euler-Heisenberg AdS Black Holes Surrounded by Quintessence and Dark Matter with a Cloud of Strings

Faizuddin Ahmed , Edilberto O. Silva

Authors on Pith no claims yet

Pith reviewed 2026-05-07 15:01 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords Euler-Heisenberg black holesAdS black holesquintessencedark mattercloud of stringsblack hole thermodynamicscritical phenomenaHawking radiation

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The pith

Euler-Heisenberg coupling and surrounding matter fields modify the temperature profile, stability structure, and critical point location of AdS black holes.

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

The paper investigates the thermodynamics, criticality, and radiative properties of an Euler-Heisenberg anti-de Sitter black hole surrounded by quintessence, perfect fluid dark matter, and a cloud of strings. Within the extended phase-space formalism the authors derive the thermodynamic quantities and verify the modified first law together with the Smarr relation. They analyze the equation of state and show that the Euler-Heisenberg coupling and the surrounding matter fields change the temperature profile, the stability regions, and the location of the critical point. The work further examines the sparsity of Hawking radiation along with the photon sphere, black hole shadow, and geometric-optics emission rate. A sympathetic reader would care because these changes affect how the black hole behaves in models that include nonlinear electrodynamics and external matter distributions.

Core claim

Within the extended phase-space formalism, the thermodynamic quantities are derived for the Euler-Heisenberg AdS black hole with quintessence, perfect fluid dark matter, and cloud of strings. The modified first law and Smarr relation are verified. The equation of state and critical behavior are analyzed, revealing that the Euler-Heisenberg coupling and surrounding matter fields substantially modify the temperature profile, stability structure, and location of the critical point. Radiative properties including sparsity of Hawking radiation, photon sphere, black hole shadow, and geometric-optics emission rate are also examined.

What carries the argument

The modified metric ansatz that incorporates the Euler-Heisenberg nonlinear electromagnetic term together with the stress-energy tensors of quintessence, perfect fluid dark matter, and a cloud of strings.

Load-bearing premise

The assumed metric ansatz with the Euler-Heisenberg nonlinear electromagnetic term plus the stress-energy tensors of quintessence, perfect fluid dark matter, and cloud of strings is a valid solution to the Einstein field equations.

What would settle it

A direct computation of the surface gravity that yields a Hawking temperature inconsistent with the one obtained from the first law would falsify the thermodynamic analysis.

Figures

Figures reproduced from arXiv: 2604.25629 by Edilberto O. Silva, Faizuddin Ahmed.

**Figure 1.** Figure 1: FIG. 1. Two-panel behavior of the metric function view at source ↗

**Figure 2.** Figure 2: FIG. 2. Two-panel behavior of the Hawking temperature view at source ↗

**Figure 3.** Figure 3: FIG. 3. Two-panel behavior of the specific heat capacity view at source ↗

**Figure 4.** Figure 4: illustrates this behavior for representative isotherms. Below the critical temperature, the system exhibits the familiar oscillatory structure associated with the coexistence of small and large black hole phases; at T = Tc, the inflection point appears; above the critical temperature, the isotherms become monotonic, indicating the disappearance of first-order phase coexistence. A particularly instructive … view at source ↗

**Figure 5.** Figure 5: FIG. 5. Two-panel behavior of the sparsity parameter view at source ↗

**Figure 7.** Figure 7: FIG. 7. Shadow radius view at source ↗

read the original abstract

We investigate the thermodynamics, criticality, and selected radiative and optical properties of an Euler-Heisenberg AdS black hole surrounded by quintessence, perfect fluid dark matter, and a cloud of strings. Within the extended phase-space formalism, we derive the thermodynamic quantities, verify the modified first law and Smarr relation, and analyze the corresponding equation of state and critical behavior. We show that the Euler--Heisenberg coupling and the surrounding matter fields substantially modify the temperature profile, the stability structure, and the location of the critical point. We also examine the sparsity of Hawking radiation, together with the photon sphere, black hole shadow, and the associated geometric-optics emission rate.

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 adds quintessence, dark matter, and string cloud to an Euler-Heisenberg AdS black hole and runs the usual extended-phase-space thermodynamics plus shadow calculations, but the metric ansatz needs explicit confirmation that it solves the Einstein equations with the summed sources.

read the letter

The main takeaway is that this is a straightforward four-field extension of existing Euler-Heisenberg AdS work. The authors write down a metric function that includes the nonlinear electromagnetic correction plus additive terms for each surrounding matter model, derive the thermodynamic quantities in extended phase space, verify the first law and Smarr relation, locate critical points, and then compute the photon sphere, shadow radius, and Hawking radiation sparsity in the geometric-optics limit. The specific combination of fields is new, and the calculations follow the standard template used in dozens of similar papers, so the results are reproducible once the metric is accepted. They show how the extra parameters shift the temperature profile, heat capacity, and critical pressure, which is the expected outcome. The optical sections are likewise routine but cleanly executed. The soft spot is the metric itself. The stress-energy tensors for nonlinear electrodynamics, quintessence, perfect-fluid dark matter, and the string cloud do not automatically superpose; the paper needs to show that the Einstein tensor for the chosen f(r) exactly equals 8π times the total T_μν. If that step is only assumed rather than verified, the thermodynamics and critical behavior are for an approximate or effective model rather than the stated physical system. The large number of free parameters also means all reported shifts are tunable rather than fixed predictions. This work is aimed at researchers already doing black-hole thermodynamics with exotic matter. Someone building similar models would get a useful template and a concrete example of how the parameters move the critical point and shadow size. It is solid enough on the calculational side to deserve a serious referee, provided the metric solution is checked in review.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates the thermodynamics, criticality, and selected radiative/optical properties of Euler-Heisenberg AdS black holes surrounded by quintessence, perfect fluid dark matter, and a cloud of strings. In the extended phase-space formalism, it derives thermodynamic quantities, verifies the modified first law and Smarr relation, analyzes the equation of state and critical behavior, and examines Hawking radiation sparsity, photon sphere, black hole shadow, and geometric-optics emission rate. The central claim is that the Euler-Heisenberg coupling and surrounding matter fields substantially modify the temperature profile, stability structure, and location of the critical point.

Significance. If the metric ansatz is confirmed to be an exact solution, the work would extend studies of AdS black hole thermodynamics by combining nonlinear electrodynamics with multiple matter fields, offering potential insights into modified critical phenomena and observational signatures such as shadows. The explicit verification of the first law and Smarr relation, together with the analysis of radiative properties, would add to the literature on extended phase space thermodynamics.

major comments (1)

[Metric ansatz (likely §2)] The metric ansatz (presented in the setup of the spacetime, likely §2): the assumed form ds² = -f(r)dt² + dr²/f(r) + r² dΩ² with f(r) incorporating the Euler-Heisenberg nonlinear EM correction plus additive contributions from the quintessence, perfect-fluid dark matter, and cloud-of-strings stress-energy tensors is not shown to satisfy G_μν = 8π T_μν^total exactly. Because the Euler-Heisenberg stress-energy is nonlinear, linear superposition with the other tensors may fail, rendering the derived temperature, heat capacity, equation of state, first law, Smarr relation, and critical-point analysis invalid for the stated physical system.

minor comments (2)

[Abstract] The abstract asserts that the matter fields 'substantially modify' the temperature profile and critical point, but no quantitative comparison to the pure Euler-Heisenberg AdS case or to limiting values of the free parameters is provided, leaving the magnitude of the effect unclear.
[Critical behavior analysis] The critical behavior depends on the four free parameters (Euler-Heisenberg coupling, quintessence parameter, dark matter parameter, cloud-of-strings parameter); explicit discussion of their physically allowed ranges or observational constraints would strengthen the analysis of stability and critical points.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment on the metric ansatz. We address this concern directly below and will incorporate the necessary clarification and verification in the revised version.

read point-by-point responses

Referee: [Metric ansatz (likely §2)] The metric ansatz (presented in the setup of the spacetime, likely §2): the assumed form ds² = -f(r)dt² + dr²/f(r) + r² dΩ² with f(r) incorporating the Euler-Heisenberg nonlinear EM correction plus additive contributions from the quintessence, perfect-fluid dark matter, and cloud-of-strings stress-energy tensors is not shown to satisfy G_μν = 8π T_μν^total exactly. Because the Euler-Heisenberg stress-energy is nonlinear, linear superposition with the other tensors may fail, rendering the derived temperature, heat capacity, equation of state, first law, Smarr relation, and critical-point analysis invalid for the stated physical system.

Authors: We appreciate the referee highlighting the need to explicitly confirm that the metric satisfies the Einstein equations with the total stress-energy tensor. In the construction, the individual stress-energy tensors are taken to be minimally coupled with no direct interactions, so T_μν^total = T_μν^EH + T_μν^quint + T_μν^DM + T_μν^strings. For a static spherically symmetric metric, the Einstein equations reduce to first-order differential equations for f(r) that depend only on the sum of the energy densities ρ_total. The nonlinear character of the Euler-Heisenberg sector is fully incorporated when computing its own T_μν^EH from the nonlinear Lagrangian and the Maxwell field strength; once obtained, this contribution adds linearly to the densities of the other fields. Consequently, the integrated f(r) contains additive terms by construction. In the revised manuscript we will add an explicit verification step: we compute the Einstein tensor components G_t^t and G_r^r from the given metric and show that they equal 8π times the summed T_μν^total, thereby confirming that the first law, Smarr relation, equation of state, and critical-point analysis remain valid for the composite system. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows from assumed metric without self-referential reductions

full rationale

The paper assumes a static spherically symmetric metric ansatz incorporating the Euler-Heisenberg nonlinear electrodynamics term plus additive stress-energy contributions from quintessence, perfect-fluid dark matter, and cloud-of-strings. From this ansatz it computes the Hawking temperature via surface gravity, derives the first law and Smarr relation in extended phase space, obtains the equation of state, and locates critical points by standard extremization of the pressure-volume relation. These steps are direct consequences of the input metric and the standard thermodynamic definitions; they do not rename fitted parameters as predictions, invoke self-citations for uniqueness theorems, or smuggle ansatzes via prior work. The reported modifications to temperature profiles and critical locations are explicit functions of the free parameters and therefore constitute model exploration rather than tautological re-derivation. No load-bearing step reduces to its own output by construction.

Axiom & Free-Parameter Ledger

4 free parameters · 3 axioms · 4 invented entities

The central claim rests on a metric ansatz whose validity is assumed, on four phenomenological parameters (Euler-Heisenberg coupling plus three matter densities), and on the standard extended-phase-space dictionary that treats the cosmological constant as pressure. No new entities with independent falsifiable evidence are introduced.

free parameters (4)

Euler-Heisenberg coupling parameter
Controls the strength of nonlinear electromagnetic corrections in the Lagrangian.
Quintessence parameter
Sets the density of the quintessence fluid surrounding the black hole.
Dark matter parameter
Characterizes the perfect-fluid dark-matter halo.
Cloud-of-strings parameter
Sets the density of the string cloud.

axioms (3)

standard math Einstein gravity with negative cosmological constant (AdS) governs the spacetime.
Used for the metric ansatz and thermodynamic pressure identification.
domain assumption The matter fields are described by the standard phenomenological energy-momentum tensors for quintessence, perfect fluid dark matter, and cloud of strings.
Invoked to source the metric solution.
domain assumption Extended phase-space thermodynamics applies, with cosmological constant as pressure.
Central to the equation-of-state and critical-point analysis.

invented entities (4)

Euler-Heisenberg nonlinear electromagnetic field no independent evidence
purpose: Models nonlinear electrodynamics around the black hole.
Phenomenological extension of Maxwell theory; no independent evidence supplied.
Quintessence fluid no independent evidence
purpose: Represents dark-energy-like matter surrounding the black hole.
Standard cosmological model; no new evidence.
Perfect fluid dark matter no independent evidence
purpose: Models dark-matter halo.
Phenomenological; no independent evidence.
Cloud of strings no independent evidence
purpose: Models topological string defects.
Common in some black-hole solutions; no independent evidence.

pith-pipeline@v0.9.0 · 5419 in / 1859 out tokens · 102696 ms · 2026-05-07T15:01:14.124232+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

55 extracted references · 4 canonical work pages · 2 internal anchors

[1]

J. D. Bekenstein, Phys. Rev. D7, 2333 (1973)

1973
[2]

J. M. Bardeen, B. Carter, and S. W. Hawking, Commun. Math. Phys.31, 161 (1973)

1973
[3]

S. W. Hawking, Commun. Math. Phys.43, 199 (1975)

1975
[4]

S. W. Hawking and D. N. Page, Commun. Math. Phys. 87, 577 (1983)

1983
[5]

Kastor, S

D. Kastor, S. Ray, and J. Traschen, Class. Quantum Grav.26, 195011 (2009)

2009
[6]

Kubizˇ n´ ak and R

D. Kubizˇ n´ ak and R. B. Mann, JHEP2012(7), 033
[7]

Gunasekaran, D

S. Gunasekaran, D. Kubizˇ n´ ak, and R. B. Mann, JHEP 2012(11), 110

2012
[8]

Kubizˇ n´ ak and R

D. Kubizˇ n´ ak and R. B. Mann, Can. J. Phys.93, 999 (2015)

2015
[9]

Kubizˇ n´ ak, R

D. Kubizˇ n´ ak, R. B. Mann, and M. Teo, Class. Quantum Grav.34, 063001 (2017)

2017
[10]

Heisenberg and H

W. Heisenberg and H. Euler, Zeitschrift f¨ ur Physik98, 714 (1936)

1936
[11]

G. G. L. Nashed and S. Nojiri, Phys. Rev. D104, 044043 (2021)

2021
[12]

Guerrero and D

M. Guerrero and D. Rubiera-Garcia, Phys. Rev. D102, 024005 (2020)

2020
[13]

Hamil and B

B. Hamil and B. C. L¨ utf¨ uo˘ glu, Fortsc. Phys.73, 2400105 (2024)

2024
[14]

Al-Badawi and F

A. Al-Badawi and F. Ahmed, Chin. J. Phys.94, 185 (2025)

2025
[15]

Ahmed, A

F. Ahmed, A. Al-Badawi, and ˙I. Sakallı, Eur. Phys. J C 85, 554 (2025)

2025
[16]

Ahmed, A

F. Ahmed, A. Al-Badawi, and ˙I. Sakallı, Phys. Dark Univ.49, 101988 (2025)

2025
[17]

Ahmed, A

F. Ahmed, A. Al-Badawi, ˙I. Sakallı, and A. Bouzenada, Nucl. Phys. B1011, 116806 (2025)

2025
[18]

Breton and L

N. Breton and L. A. L´ opez, Phys. Rev. D104, 024064 (2021)

2021
[19]

Magos and N

D. Magos and N. Breton, Phys. Rev. D102, 084011 (2020)

2020
[20]

Salazar I., A

H. Salazar I., A. Garc´ ıa D., and J. F. Pleba´ nski, J Math. Phys28, 2171 (1987)

1987
[21]

V. V. Kiselev, Class. Quantum Grav.20, 1187 (2003)

2003
[22]

Li and K.-C

M.-H. Li and K.-C. Yang, Phys. Rev. D86, 123015 (2012)

2012
[23]

P. S. Letelier, Phys. Rev. D20, 1294 (1979)

1979
[24]

V. B. Bezerra, I. P. Lobo, J. P. Morais Gra¸ ca, and L. C. N. Santos, Eur. Phys. J C79, 949 (2019)

2019
[25]

Abbas and R

G. Abbas and R. H. Ali, Eur. Phys. J C83, 407 (2023)

2023
[26]

A. Sood, M. S. Ali, J. K. Singh, and S. G. Ghosh, Chin. Phys. C48, 065109 (2024)

2024
[27]

Heydari-Fard, S

M. Heydari-Fard, S. Ghassemi Honarvar, and M. Heydari-Fard, MNRAS521, 708 (2023)

2023
[28]

X. Hou, Z. Xu, and J. Wang, JCAP2018(12), 040
[29]

Haroon, M

S. Haroon, M. Jamil, K. Jusufi, K. Lin, and R. B. Mann, Phys. Rev. D99, 044015 (2019)

2019
[30]

M. R. Shahzad, G. Abbas, T. Zhu,et al., Eur. Phys. J. C85, 164 (2025)

2025
[31]

Rehman, G

H. Rehman, G. Abbas, T. Zhu, and B. S. Alkahtani, Phys. Dark Univ.48, 101944 (2025)

2025
[32]

Rehman, G

H. Rehman, G. Abbas, T. Zhu,et al., Eur. Phys. J C83, 856 (2023)

2023
[33]

Abbas and H

G. Abbas and H. Rehman, Fortsc. Phys.71, 2200205 (2023)

2023
[34]

G. D. Acan Yildiz, A. Ditta, A. Ashraf, E. G¨ udekli, Y. M. Alanazi, and A. Reyimberganov, Phys. Dark Univ.46, 101583 (2024)

2024
[35]

Chaudhary, M

S. Chaudhary, M. D. Sultan, A. Malik, Y. M. Alanazi, A. Bin Jumah, A. Ashraf, and A. M. Mubaraki, Int. J Geom. Meth. Mod. Phys.22, 2550129 (2025)

2025
[36]

Sucu and I

E. Sucu and I. Sakalli, Physics of the Dark Universe52, 102295 (2026)

2026
[37]

J Lett.875, L1 (2019)

Event Horizon Telescope Collaboration, Astrophys. J Lett.875, L1 (2019)

2019
[38]

J Lett.875, L4 (2019)

Event Horizon Telescope Collaboration, Astrophys. J Lett.875, L4 (2019)

2019
[39]

J Lett.930, L12 (2022)

Event Horizon Telescope Collaboration, Astrophys. J Lett.930, L12 (2022)

2022
[40]

Perlick and O

V. Perlick and O. Y. Tsupko, Phys. Rep.947, 1 (2022)

2022
[41]

Got¯ o, Prog

T. Got¯ o, Prog. Theor. Phys.46, 1560 (1971)

1971
[42]

Ruffini, Y.-B

R. Ruffini, Y.-B. Wu, and S.-S. Xue, Phys. Rev. D88, 085004 (2013)

2013
[43]

Visser, F

M. Visser, F. Gray, S. Schuster, and A. Van-Brunt, in The Fourteenth Marcel Grossmann Meeting(2017) pp. 1724–1729

2017
[44]

Ahmed, A

F. Ahmed, A. Al-Badawi, and E. O. Silva, Phys. Lett. B 876, 140448 (2026). 12

2026
[45]

Shadow, Sparsity of Radiation and Energy Emission Rate in Skyrmion Black Holes

F. Ahmed, A. Al-Badawi, and I. Sakalli, (2026), arXiv:2604.06259 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[46]

Charged Black Holes in KR-gravity Surrounded by Perfect Fluid Dark Matter

F. Ahmed, M. Fathi, and E. O. Silva, (2026), arXiv:2604.11357 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[47]

Ahmed, S

F. Ahmed, S. Kala, and E. O. Silva, (2026), arXiv:2604.00883 [gr-qc]

work page arXiv 2026
[48]

Ahmed and E

F. Ahmed and E. O. Silva, (2026), arXiv:2603.19034 [gr- qc]

work page arXiv 2026
[49]

F. Gray, S. Schuster, A. Van-Brunt, and M. Visser, Class. Quantum Grav.33, 115003 (2016)

2016
[50]

D. N. Page, Phys. Rev. D13, 198 (1976)

1976
[51]

Ahmed, A

F. Ahmed, A. Al-Badawi, and ˙I. Sakallı, Eur. Phys. J C 85, 984 (2025)

2025
[52]

Wei and Y.-X

S.-W. Wei and Y.-X. Liu, JCAP2013(11), 063
[53]

D´ ecanini, G

Y. D´ ecanini, G. Esposito-Far` ese, and A. Folacci, Phys. Rev. D83, 044032 (2011)

2011
[54]

Mashhoon, Phys

B. Mashhoon, Phys. Rev. D7, 2807 (1973)

1973
[55]

Sanchez, Phys

N. Sanchez, Phys. Rev. D18, 1030 (1978)

1978