Pith. sign in

REVIEW 4 major objections 4 minor 62 references

This paper shows that AdS compact objects in extended metric-Palatini gravity with a geometric Proca field have standard black hole thermodynamics, including a first law, with stability depending on the AdS length scale.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Thermodynamic quantities for Einstein-Geometric Proca AdS compact objects are derived and plotted, but the first law of thermodynamics is not satisfied for generic parameters.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection Thermodynamics of Einstein-Geometric Proca AdS objects: standard extension, but the first law is not an identity and the q1=0 temperature limit fails. the 4 major comments →

arxiv 2509.09389 v1 pith:ATYZCMT7 submitted 2025-09-11 gr-qc

Thermodynamics of Einstein-Geometric Proca AdS compact objects

Asalkhon Alimova , Elham Ghorani , Beyhan Puli\c{c}e , Farruh Atamurotov , Ahmadjon Abdujabbarov This is my paper
classification gr-qc PACS 04.70.-s04.50.Kd
keywords black hole thermodynamicsmetric-Palatini gravitygeometric Proca fieldAnti-de SitterHawking temperatureheat capacityfirst lawcompact objects
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.

The reading

The paper analyzes the thermodynamics of static spherically symmetric compact objects in a gravity theory that couples general relativity to a geometric Proca field in an Anti-de Sitter background. Using previously constructed solutions, it derives explicit formulas for Hawking temperature, enthalpy, heat capacity, entropy, and Gibbs free energy in terms of the horizon radius and two charge parameters. It finds that temperature decreases with horizon radius and increases with the uniform potential, while heat capacity is negative for small horizons and positive for large ones, indicating a stability transition near the AdS radius. The paper also verifies the first law of black hole thermodynamics along specific contours in parameter space. The upshot is that these geometric-Proca objects behave thermodynamically like ordinary AdS black holes, with the extra fields shifting the stability boundary.

Core claim

The central claim is that the Einstein-Geometric Proca AdS solutions satisfy the standard black hole thermodynamic relations. Using the area-law entropy S = A_h/(4G_N) and the surface-gravity temperature, the authors compute the heat capacity via C = (dM/dr_h)/(dT/dr_h) and find a sign change from negative (unstable) to positive (stable) as the horizon radius grows past roughly the AdS radius. They further verify the first law dM = T dS + Φ₂ dq₂, with the uniform-potential work term dropped, by showing that Δ = dM/dr_h − T dS/dr_h vanishes along the contour plot in their Fig. 8. This establishes, within the model's assumptions, that these geometric-Proca objects have a well-defined thermodyn

What carries the argument

The machinery is the family of static spherically symmetric solutions of extended metric-Palatini gravity: metric components f(r̂) = r̂²/l² + 1 + n₁/r̂^{1−σ} + n₂/r̂ and h(r̂) similarly, built from the Proca field solution φ(r̂) = q₁/r̂^{(1−σ)/2} + q₂/r̂^{(1+σ)/2}. The exponent σ (with 0 ≤ σ < 1 from the Breitenlohner-Freedman bound) encodes the Proca mass relative to the AdS radius. The thermodynamics follows by evaluating surface gravity at the horizon and using the mass/enthalpy expression from the horizon condition, with the parameter pair (q₁, q₂) controlling all deviations from Schwarzschild-AdS behavior.

Load-bearing premise

The analysis assumes that the Bekenstein-Hawking area law S = A/(4G) remains the correct entropy in extended metric-Palatini gravity, and that the work term for the uniform potential q₁ can be omitted from the first law; if either fails, the first-law check and the stability conclusions do not follow.

What would settle it

Compute the first law including the Φ₁ dq₁ term from the full mass formula; if Δ = dM − T dS − Φ₁ dq₁ − Φ₂ dq₂ does not vanish along the same contours, the stated first law is incomplete. Alternatively, derive the entropy from the Euclidean action of the EMPG action and check whether it differs from A/4; any difference would invalidate Eq. (22) and the heat-capacity analysis.

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

If this is right

  • If correct, these compact objects can be treated as thermodynamic systems with a consistent temperature, entropy, and stability analysis, meaning the first law applies in this modified-gravity setting.
  • The heat-capacity sign change implies a phase transition between small unstable and large stable configurations, with the transition scale set by the AdS radius.
  • The q₁ and q₂ parameters provide handles to shift horizon temperature, so astrophysical observations of such objects could constrain these geometric-Proca couplings.
  • The Schwarzschild-AdS limit (q₁ = 0) is recovered, validating the framework against known results.

Where Pith is reading between the lines

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

  • The paper implicitly assumes the Bekenstein-Hawking area law without deriving it from the extended action; a Euclidean-action or Wald-entropy calculation for this theory could test whether geometric-Proca corrections generate additional entropy terms.
  • The dropping of the Φ₁ dq₁ work term is a nontrivial choice; if the correct first law includes it, the Δ = 0 contours would shift, making the verification in Fig. 8 conditional on that choice.
  • The phase transition near r_h ~ l resembles the Hawking-Page transition in AdS; a free-energy comparison between the small and large branches could determine which phase is globally preferred.
  • The monotonicity claims are based on numerical plots for specific parameter values; an analytic proof of the sign of heat capacity would strengthen the stability claim.
Share X Bluesky LinkedIn Reddit HN

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

4 major / 4 minor

Summary. The paper studies the thermodynamics of static, spherically symmetric compact-object solutions in extended metric-Palatini gravity with a geometric Proca field, using the solution previously constructed by the same authors in [29]. It defines Hawking temperature, enthalpy, heat capacity, entropy, and Gibbs free energy as functions of the horizon radius and the parameters q1, q2, and sigma, and attempts to verify the first law dM = T dS + Phi2 dq2. The claimed results are that the temperature decreases with horizon radius and increases with q1, that the heat capacity is negative at small radii and positive at large radii with a phase transition near r_h ~ l, and that the first law holds along the Delta=0 contours of Fig. 8.

Significance. If correct, this would be a useful thermodynamic survey of a nontrivial modified-gravity solution and would provide concrete predictions for the stability of Einstein-Geometric Proca AdS compact objects. The paper makes an explicit attempt at a first-law consistency check and provides analytic formulas and numerical figures. However, the central thermodynamic framework is internally inconsistent: the printed Hawking temperature has the wrong Schwarzschild-AdS limit, the Delta expression in Eq. (27) does not match the temperature from Eq. (18), the heat-capacity expression is not the derivative defined in Eq. (20) and is complex in the stated parameter range, and the first law is verified only on selected contours rather than identically. These are load-bearing failures, so the numerical conclusions about temperature behavior, phase transitions, and stability are not supported by the manuscript as written.

major comments (4)
  1. [Sec. III A, Eq. (18)] In the q1=0 limit the radicand simplifies and gives T_H = (l^2 + 3 r_h^2)/(4 l^2 r_h sqrt(pi)). The surface-gravity formulas (15)-(17) applied to the metric (12) give instead (l^2 + 3 r_h^2)/(4 pi l^2 r_h). The printed expression is therefore sqrt(pi) larger; as l -> infinity it gives 1/(8 sqrt(pi) M), not the stated T0 = 1/(8 pi M). Thus Eq. (18) is not the Hawking temperature of the metric, and every quantity derived from it inherits this error.
  2. [Sec. III F, Eq. (27)] The expression labelled Delta is not dM/dr_h - T dS/dr_h using T from Eq. (18) and S = pi r_h^2. With Eq. (18), T dS/dr_h has an extra 1/sqrt(pi) relative to the second term of Eq. (27); Eq. (27) corresponds to a temperature with denominator 16 sqrt(3) pi l^2 r_h^2, not the printed 16 l^2 r_h^2 sqrt(3 pi). More fundamentally, a genuine first law must give Delta = 0 identically for all r_h at fixed q2; the contour plot in Fig. 8 shows Delta vanishes only on measure-zero curves. Hence M, T, and S are not mutually consistent thermodynamic variables, and the first-law 'verification' is not valid.
  3. [Sec. III C, Eq. (21)] The heat capacity formula contains a factor sqrt(sigma - 3) in the numerator, which is imaginary for the stated physical range 0 <= sigma < 1; the real curves plotted in Figs. 4-5 cannot follow from this expression. Independently, the q1=0 limit of Eq. (21) is pi(l^2+3r^2)/(18 r^3), whereas Eq. (20) applied to Eqs. (19) and (18) gives 2 sqrt(pi) r^2(l^2+3r^2)/(3r^2-l^2) (or 2 pi r^2(...)/(3r^2-l^2) with the corrected temperature). Eq. (21) is therefore not the derivative dM/dT, and the stability and phase-transition conclusions drawn from it are unsupported.
  4. [Sec. III F, Eq. (23)] The first law drops the Phi1 dq1 work term solely because q1 becomes a uniform potential in the Maxwell limit. This is not a justification: M depends on q1 for generic sigma, and variations in q1 must be accompanied by the conjugate work term. If q1 is held fixed, then the Delta=0 contours in Fig. 8, which scan q1 at fixed q2, are not permitted variations and cannot serve as a first-law check. The paper must either include Phi1 dq1 or consistently work at fixed q1 in a way that makes the identity hold for all r_h.
minor comments (4)
  1. [Secs. III B and III E] The paper interchangeably calls M the enthalpy H and then uses M in G = M - T S. This should be clarified, especially because Eq. (19) is derived from f(r_h)=0 and is later identified with the ADM mass.
  2. [Sec. III D, Eq. (22)] The Bekenstein-Hawking area law is assumed without derivation in this extended metric-Palatini theory with a massive Proca field. A Wald or Noether-charge computation would be needed to justify that S=pi r_h^2 remains exact; otherwise the first-law and heat-capacity results could change.
  3. [General] The notation is inconsistent about dimensionless quantities: Eqs. (18)-(27) use r_h and l without the hats defined in Sec. II. Please state the units/conventions used in the thermodynamic formulas.
  4. [Conclusion and captions] There are several typos, e.g., 'bcompact' in the conclusion and 'Fig. (2)' instead of 'Fig. 2'. The caption of Fig. 8 should state which parameters are held fixed in the contour plot.

Circularity Check

1 steps flagged

First-law 'verification' is circular: Δ=0 contours are imposed, not derived; bulk thermodynamic results are non-circular.

specific steps
  1. self definitional [Sec. III F, Eqs. (24)-(27), Fig. 8]
    "To verify the validity of the first law, we plot the contour curves where ∆ = 0. Along each such curve in Fig. 8, the first law of thermodynamics is exactly satisfied."

    The first law, Eq. (24), states dM/dr_h = T dS/dr_h. If M, T, and S are mutually consistent thermodynamic functions, this identity must hold for all r_h and parameters, i.e. Δ ≡ 0 identically. Instead, the paper defines Δ in Eq. (25) as the difference dM/dr_h − T dS/dr_h, then treats the locus Δ=0 as a verification. Those contours are precisely the set of points on which the unproven identity is imposed. The preceding sentence 'we need to satisfy Δ = 0' confirms that the condition is an input, not an outcome. Thus the claim 'the first law is exactly satisfied' along the curves is true by construction: it restates the defining equation of the plotted contours rather than providing an independent consistency check.

full rationale

The paper's main thermodynamic quantities—Hawking temperature (Eq. 18), enthalpy (Eq. 19), heat capacity (Eq. 21), entropy, and Gibbs free energy—are obtained by applying standard black-hole thermodynamic formulas (surface gravity, area-law entropy, ∂M/∂T) to a static spherically symmetric solution imported from the authors' prior work [29]. These derivations are not fitted to the target conclusions and are not circular: the temperature and heat-capacity behaviors follow from the metric functions, and the entropy is the usual horizon-area entropy. The self-citation to [29] is load-bearing only in the normal sense that any paper builds on its earlier solution; it is not a case of an unverified self-citation substituting for derivation. The one genuine circular step is the first-law 'verification' in Sec. III F: instead of demonstrating that dM = T dS holds identically for the derived M, T, and S, the paper defines the discrepancy Δ and then plots curves where Δ=0 as if this were a test. Since Δ=0 is exactly the first law, the verification is equivalent to the input. Separate issues—such as the unargued dropping of Φ1 dq1 and the inaccurate claim that q1=0 recovers T0=1/(8πM) rather than the Schwarzschild-AdS temperature—are correctness concerns, not circularity. Overall, the central thermodynamic predictions are independent, but the first-law validation partially reduces to its own defining condition.

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 0 invented entities

The main input from outside this paper is the EMPG action and the solution from Refs. [26,28,29]. The only new tunable ingredient is the surface-term coefficient γ in the mass/enthalpy. No new entities are introduced here. The dominant external assumptions are the area-law entropy and the omission of the q1 work term.

free parameters (1)
  • γ (surface term coefficient)
    Appears in ADM mass Eq. (14) and enthalpy Eq. (19); its value is not fixed by the theory, and the sentence 'under the normalization M_ADM=1' is unclear. Thermodynamic quantities depend on it.
axioms (4)
  • domain assumption The extended metric-Palatini gravity action (Eq. (2)) and its reduction to GR plus a geometric Proca field (Eq. (5)) are valid.
    Taken from earlier work by the same group (Refs. [26,28,29]); not rederived here.
  • domain assumption The static spherically symmetric solution (10)-(12) from Ref. [29] is the correct solution of the field equations.
    The paper states 'much of the content drawn from our previous work [29]' and uses these metric functions as input.
  • domain assumption The entropy-area law S=A_h/(4G_N) applies to these compact objects in EMPG.
    Eq. (22) is assumed without discussing Wald-type corrections in modified gravity.
  • ad hoc to paper The work term for the uniform potential q1 can be omitted from the first law because in the Maxwell limit q1 is a uniform potential.
    Sec. III F drops Φ1 dq1 without a conjugate charge calculation; this is not a standard thermodynamic argument and is specific to this paper.

reviewed 2026-08-04 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Thermodynamics of Einstein-Geometric Proca AdS compact objects." pith.science (2026). https://pith.science/paper/ATYZCMT7

@misc{pith2026250909389,
  author       = {Pith},
  title        = {Pith review of: Thermodynamics of Einstein-Geometric Proca AdS compact objects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ATYZCMT7}},
  note         = {Machine review of arXiv:2509.09389}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

In this study we explore metric-Palatini gravity extended by the antisymmetric component of the affine curvature. This gravitational theory results in general relativity plus a geometric Proca field. Building on our previous work, where we constructed its static spherically symmetric solutions in the Anti-de Sitter (AdS) background (Eur. Phys. J. C 83(4):318, 2023), we conduct a comprehensive analysis of the system's thermodynamics. We examine the thermodynamic properties of the Einstein-Geometric Proca AdS compact objects, focusing on the Hawking temperature, enthalpy, heat capacity, entropy, and Gibbs free energy. Particular attention is given to the dependence of the Hawking temperature, enthalpy, and heat capacity on the uniform potential $q_{1}$ and the electromagnetic-type charge $q_{2}$. Through numerical analysis we compute the entropy and Gibbs free energy and investigate how these quantities vary with the model parameters.

Figures

Figures reproduced from arXiv: 2509.09389 by Ahmadjon Abdujabbarov, Asalkhon Alimova, Beyhan Puli\c{c}e, Elham Ghorani, Farruh Atamurotov.

Figure 1
Figure 1. Figure 1: illustrates the radial dependence of Hawking tem￾perature on various values of q1 and q2. It is obvious that [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Hawking temperature [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Enthalpy [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Heat capacity [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Heat capacity of Einstein-Geometric Proca AdS compact object as a function of the normalized event [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Entropy with respect [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Gibbs energy with respect to horizon radius [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: The contour plot of equation (27) for ∆ = 0. [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

62 extracted references · 33 canonical work pages · 1 internal anchor

  1. [2]

    Hawking, Communications in Mathematical Physics43(3), 199 (1975)

    S.W. Hawking, Communications in Mathematical Physics43(3), 199 (1975). doi:10.1007/BF02345020

    doi:10.1007/bf02345020 1975

  2. [3]

    Hawking, Nature248(5443), 30 (1974)

    S.W. Hawking, Nature248(5443), 30 (1974). doi: 10.1038/248030a0

    doi:10.1038/248030a0 1974

  3. [4]

    Wald,Quantum field theory in curved spacetime and black hole thermodynamics.(1994)

    R.M. Wald,Quantum field theory in curved spacetime and black hole thermodynamics.(1994)

    1994

  4. [5]

    Cai, Y.S

    R.G. Cai, Y.S. Myung, Y.Z. Zhang, Phys. Rev. D65(8), 084019 (2002). doi:10.1103/PhysRevD.65.084019

    work page doi:10.1103/physrevd.65.084019 2002

  5. [6]

    Nojiri, S.D

    S. Nojiri, S.D. Odintsov, Phys.Rep505(2), 59 (2011). doi:10.1016/j.physrep.2011.04.001

    doi:10.1016/j.physrep.2011.04.001 2011

  6. [7]

    Davies, Reports on Progress in Physics41(8), 1313 (1978)

    P.C.W. Davies, Reports on Progress in Physics41(8), 1313 (1978). doi:10.1088/0034-4885/41/8/004

    doi:10.1088/0034-4885/41/8/004 1978

  7. [8]

    Cai, Y.P

    R.G. Cai, Y.P. Hu, Q.Y. Pan, Y.L. Zhang, Phys. Rev. D 91(2), 024032 (2015). doi:10.1103/PhysRevD.91.024032

    doi:10.1103/physrevd.91.024032 2015

  8. [9]

    Appels, R

    M. Appels, R. Gregory, D. Kubiz ˚A´ ak, Phys. Rev. Lett.117(13), 131303 (2016). doi: 10.1103/PhysRevLett.117.131303

    work page doi:10.1103/physrevlett.117.131303 2016

  9. [10]

    Rakhimova, F

    G. Rakhimova, F. Atamurotov, F. Javed, A. Abdu- jabbarov, G. Mustafa, Nuclear Physics B996, 116363 (2023). doi:10.1016/j.nuclphysb.2023.116363

    arXiv 2023

  10. [11]

    Ditta, X

    A. Ditta, X. Tiecheng, R. Ali, F. Atamurotov, A. Mah- mood, S. Mumtaz, Annals of Physics453, 169326 (2023). doi:10.1016/j.aop.2023.169326

    arXiv 2023

  11. [12]

    Javed, G

    F. Javed, G. Mustafa, S. Mumtaz, F. Atamuro- tov, Nuclear Physics B990, 116180 (2023). doi: 10.1016/j.nuclphysb.2023.116180

    arXiv 2023

  12. [13]

    Ditta, X

    A. Ditta, X. Tiecheng, G. Mustafa, M. Yasir, F. Ata- murotov, Eur. Phys. J. C82(8), 756 (2022). doi: 10.1140/epjc/s10052-022-10708-z

    work page doi:10.1140/epjc/s10052-022-10708-z 2022

  13. [14]

    Yasir, X

    M. Yasir, X. Tiecheng, A. Ditta, R. Ali, F. Ata- murotov, New astronomy105, 102106 (2024). doi: 10.1016/j.newast.2023.102106

    arXiv 2024

  14. [15]

    Sekhmani, S.K

    Y. Sekhmani, S.K. Maurya, M.K. Jasim, ˙I. Sakallı, J. Rayimbaev, I. Ibragimov, Eur. Phys. J. C85(3), 229 (2025). doi:10.1140/epjc/s10052-025-13932-5

    doi:10.1140/epjc/s10052-025-13932-5 2025

  15. [16]

    Rahmatov, M

    B. Rahmatov, M. Zahid, S.U. Khan, J. Rayimbaev, I. Ibragimov, Z. Yuldoshev, A. Dauletov, S. Muminov, Chin. Phys. C49(7), 075105 (2025). doi:10.1088/1674- 1137/adc188

    doi:10.1088/1674- 2025

  16. [17]

    Schr¨ odinger,Space-Time Structure(1985)

    E. Schr¨ odinger,Space-Time Structure(1985)

    1985

  17. [18]

    Hehl, J.D

    F.W. Hehl, J.D. McCrea, E.W. Mielke, Y. Ne’eman, physrep258(1), 1 (1995). doi:10.1016/0370- 1573(94)00111-F

    doi:10.1016/0370- 1995

  18. [19]

    Vitagliano, T.P

    V. Vitagliano, T.P. Sotiriou, S. Liberati, An- nals of Physics326(5), 1259 (2011). doi: 10.1016/j.aop.2011.02.008

    work page doi:10.1016/j.aop.2011.02.008 2011

  19. [20]

    Harko, T.S

    T. Harko, T.S. Koivisto, F.S.N. Lobo, G.J. Olmo, Phys. Rev. D85(8), 084016 (2012). doi: 10.1103/PhysRevD.85.084016 10

    doi:10.1103/physrevd.85.084016 2012

  20. [21]

    Hybrid f(R) theories, local constraints, and cosmic speedup

    S. Capozziello, T. Harko, T.S. Koivisto, F.S.N. Lobo, G.J. Olmo, arXiv e-prints arXiv:1301.2209 (2013). doi: 10.48550/arXiv.1301.2209

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1301.2209 2013

  21. [22]

    Capozziello, T

    S. Capozziello, T. Harko, T.S. Koivisto, F.S.N. Lobo, G.J. Olmo, Universe1(2), 199 (2015). doi: 10.3390/universe1020199

    work page doi:10.3390/universe1020199 2015

  22. [23]

    Capozziello, T

    S. Capozziello, T. Harko, T.S. Koivisto, F.S.N. Lobo, G.J. Olmo, JCAP2013(7), 024 (2013). doi: 10.1088/1475-7516/2013/07/024

    work page doi:10.1088/1475-7516/2013/07/024 2013

  23. [24]

    Capozziello, T

    S. Capozziello, T. Harko, T.S. Koivisto, F.S.N. Lobo, G.J. Olmo, Phys. Rev. D86(12), 127504 (2012). doi: 10.1103/PhysRevD.86.127504

    work page doi:10.1103/physrevd.86.127504 2012

  24. [25]

    Capozziello, T

    S. Capozziello, T. Harko, T.S. Koivisto, F.S.N. Lobo, G.J. Olmo, JCAP2013(4), 011 (2013). doi: 10.1088/1475-7516/2013/04/011

    work page doi:10.1088/1475-7516/2013/04/011 2013

  25. [26]

    Demir, B

    D. Demir, B. Puli¸ ce, JCAP04, 051 (2020). doi: 10.1088/1475-7516/2020/04/051

    work page doi:10.1088/1475-7516/2020/04/051 2020

  26. [27]

    Demir, K

    D. Demir, K. Gabriel, A. Kasem, S. Khalil, Phys. Dark Univ.42, 101336 (2023). doi:10.1016/j.dark.2023.101336

    arXiv 2023

  27. [28]

    and [29], aiming to obtain black hole solutions for Λ = 0 and Λ<0, respectively. In pursuit of a gen- eral static spherically symmetric solution, the following ansatz is employed gµν = diag(−h(r), 1 f(r) , r2, r2 sin2 θ),(8) and the vector field satisfying the equation of motion(7), can be considered as purely time-like ˆYµ = ˆϕ(r)δ0 µ (9) lead to the sol...

    1974

  28. [29]

    Demir, B

    D. Demir, B. Puli¸ ce, Eur. Phys. J. C82(11), 996 (2022). doi:10.1140/epjc/s10052-022-10986-7

    work page doi:10.1140/epjc/s10052-022-10986-7 2022

  29. [30]

    Ghorani, B

    E. Ghorani, B. Puli¸ ce, F. Atamurotov, J. Rayimbaev, A. Abdujabbarov, D. Demir, Eur. Phys. J. C83(4), 318 (2023). doi:10.1140/epjc/s10052-023-11490-2

    doi:10.1140/epjc/s10052-023-11490-2 2023

  30. [31]

    Demir, Adv

    D. Demir, Adv. High Energy Phys.2019, 4652048 (2019). doi:10.1155/2019/4652048

    work page doi:10.1155/2019/4652048 2019

  31. [32]

    Demir, Gen

    D. Demir, Gen. Rel. Grav.53(2), 22 (2021). doi: 10.1007/s10714-021-02797-0

    work page doi:10.1007/s10714-021-02797-0 2021

  32. [33]

    Demir, Phys

    D. Demir, Phys. Rev. D107(10), 105014 (2023). doi: 10.1103/PhysRevD.107.105014

    work page doi:10.1103/physrevd.107.105014 2023

  33. [34]

    Bauer, D.A

    F. Bauer, D.A. Demir, Phys. Lett. B665, 222 (2008). doi:10.1016/j.physletb.2008.06.014

    work page doi:10.1016/j.physletb.2008.06.014 2008

  34. [35]

    Bauer, D.A

    F. Bauer, D.A. Demir, Phys. Lett. B698, 425 (2011). doi:10.1016/j.physletb.2011.03.042

    work page doi:10.1016/j.physletb.2011.03.042 2011

  35. [36]

    Sotiriou, S

    T.P. Sotiriou, S. Liberati, Annals Phys.322, 935 (2007). doi:10.1016/j.aop.2006.06.002

    work page doi:10.1016/j.aop.2006.06.002 2007

  36. [37]

    Koussour, A

    M. Koussour, A. Altaibayeva, S. Bekov, O. Donmez, S. Muminov, J. Rayimbaev, Physics of the Dark Universe 46, 101577 (2024). doi:10.1016/j.dark.2024.101577

    arXiv 2024

  37. [38]

    Koussour, S

    M. Koussour, S. Bekov, A. Syzdykova, S. Muminov, I. Ibragimov, J. Rayimbaev, Phys. Dark Univ.47, 101799 (2025). doi:10.1016/j.dark.2024.101799

    arXiv 2025

  38. [39]

    Vitagliano, Class

    V. Vitagliano, Class. Quant. Grav.31(4), 045006 (2014). doi:10.1088/0264-9381/31/4/045006

    work page doi:10.1088/0264-9381/31/4/045006 2014

  39. [40]

    Vitagliano, T.P

    V. Vitagliano, T.P. Sotiriou, S. Liberati, Phys. Rev. D 82, 084007 (2010). doi:10.1103/PhysRevD.82.084007

    work page doi:10.1103/physrevd.82.084007 2010

  40. [41]

    Buchdahl, J

    H.A. Buchdahl, J. Phys. A12, 1235 (1979). doi: 10.1088/0305-4470/12/8/018

    work page doi:10.1088/0305-4470/12/8/018 1979

  41. [42]

    Tucker, C

    R.W. Tucker, C. Wang, inMathematical Aspects of The- ories of Gravitation(1996)

    1996

  42. [43]

    Obukhov, E.J

    Y.N. Obukhov, E.J. Vlachynsky, W. Esser, F.W. Hehl, Phys. Rev. D56(12), 7769 (1997). doi: 10.1103/PhysRevD.56.7769

    work page doi:10.1103/physrevd.56.7769 1997

  43. [44]

    Ghorani, S

    E. Ghorani, S. Mitra, J. Rayimbaev, B. Puli¸ ce, F. Ata- murotov, A. Abdujabbarov, D. Demir, Eur. Phys. J. C 84(10), 1022 (2024). doi:10.1140/epjc/s10052-024-13373- 6

    work page doi:10.1140/epjc/s10052-024-13373- 2024

  44. [45]

    Tresguerres, Z

    R. Tresguerres, Z. Phys. C65(2), 347 (1995). doi: 10.1007/BF01571892

    work page doi:10.1007/bf01571892 1995

  45. [46]

    Tucker, C

    R.W. Tucker, C. Wang, Class. Quant. Grav.12(10), 2587 (1995). doi:10.1088/0264-9381/12/10/016

    work page doi:10.1088/0264-9381/12/10/016 1995

  46. [47]

    Vlachynsky, R

    E.J. Vlachynsky, R. Tresguerres, Y.N. Obukhov, F.W. Hehl, Class. Quant. Grav.13(12), 3253 (1996). doi: 10.1088/0264-9381/13/12/016

    work page doi:10.1088/0264-9381/13/12/016 1996

  47. [48]

    Mac ´ ıas, J

    A. Mac ´ ıas, J. Socorro, Class. Quant. Grav.16(7), 2323 (1999). doi:10.1088/0264-9381/16/7/312

    work page doi:10.1088/0264-9381/16/7/312 1999

  48. [49]

    Bekenstein, Phys

    J.D. Bekenstein, Phys. Rev. D5(6), 1239 (1972). doi: 10.1103/PhysRevD.5.1239

    doi:10.1103/physrevd.5.1239 1972

  49. [50]

    Bekenstein, Phys

    J.D. Bekenstein, Phys. Rev. D5(10), 2403 (1972). doi: 10.1103/PhysRevD.5.2403

    doi:10.1103/physrevd.5.2403 1972

  50. [51]

    Adler, R.B

    S.L. Adler, R.B. Pearson, Phys. Rev. D18(8), 2798 (1978). doi:10.1103/PhysRevD.18.2798

    work page doi:10.1103/physrevd.18.2798 1978

  51. [52]

    Frolov, Gen

    V.P. Frolov, Gen. Rel. Grav.9(7), 569 (1978). doi: 10.1007/BF00761002

    work page doi:10.1007/bf00761002 1978

  52. [53]

    Gottlieb, R

    D. Gottlieb, R. Hojman, L.H. Rodr ´ ıguez, N. Zamorano, Nuovo Cim. B80(1), 62 (1984). doi:10.1007/BF02899373

    work page doi:10.1007/bf02899373 1984

  53. [54]

    Leaute, B

    B. Leaute, B. Linet, Gen. Rel. Grav.17(8), 783 (1985). doi:10.1007/BF00762627

    work page doi:10.1007/bf00762627 1985

  54. [55]

    Ay´ on-Beato, A

    E. Ay´ on-Beato, A. Garc ´ ıa, A. Mac ´ ıas, H. Quevedo, Phys. Rev. D61(8), 084017 (2000). doi: 10.1103/PhysRevD.61.084017

    work page doi:10.1103/physrevd.61.084017 2000

  55. [56]

    Obukhov, E.J

    Y.N. Obukhov, E.J. Vlachynsky, An- nals Phys.511(6), 497 (1999). doi: 10.1002/andp.1999511060310.1002/(SICI)1521- 3889(199909)8:6¡497::AID-ANDP497¿3.0.CO;2-5

    arXiv 1999

  56. [57]

    Toussaint, Gen

    M. Toussaint, Gen. Rel. Grav.32(9), 1689 (2000). doi: 10.1023/A:1001942420350

    work page doi:10.1023/a:1001942420350 2000

  57. [58]

    H.S. Liu, H. L¨ u, C.N. Pope, JHEP06, 109 (2014). doi: 10.1007/JHEP06(2014)109

    work page doi:10.1007/jhep06(2014)109 2014

  58. [59]

    Wald, (2001)

    R.M. Wald, (2001). Lecture Notes in Physics, vol. 769, pp. 1-37

    2001

  59. [60]

    Unruh, Phys

    W.G. Unruh, Phys. Rev. D14(4), 870 (1976). doi: 10.1103/PhysRevD.14.870

    doi:10.1103/physrevd.14.870 1976

  60. [61]

    Bekenstein, Phys

    J.D. Bekenstein, Phys. Rev. D7, 2333 (1973). doi: 10.1103/PhysRevD.7.2333

    doi:10.1103/physrevd.7.2333 1973

  61. [62]

    Kastor, S

    D. Kastor, S. Ray, J. Traschen, arXiv preprint (2018). URLhttps://arxiv.org/abs/1807.09801

    Pith/arXiv arXiv 2018

  62. [63]

    Mahapatra, I

    S. Mahapatra, I. Banerjee, Physics of the Dark Universe 39, 101172 (2023). doi:10.1016/j.dark.2023.101172. URL https://doi.org/10.1016/j.dark.2023.101172

    arXiv 2023

This paper was first reviewed by deepseek-v4-flash on August 4, 2026.