Geodesics, Scalar Fields, and GUP-Corrected Thermodynamics of Charged BTZ-like Black Holes in Bopp-Podolsky Electrodynamics

Abdelmalek Bouzenada; Ahmad Al-Badawi; Erdem Sucu; Faizuddin Ahmed; \.Izzet Sakall{\i}

REVIEW 5 major objections 4 minor 187 references

Geodesics, Scalar Fields, and GUP-Corrected Thermodynamics of Charged BTZ-like Black Holes in Bopp-Podolsky Electrodynamics

T0 review · 5 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Adding a minimum-length correction to the Hawking temperature suppresses thermal radiation and can leave a stable remnant of this charged BTZ-like black hole.

desk verdict A routine geodesic/scalar-field catalogue for a BP-modified BTZ metric is undermined by a GUP-thermodynamics section whose central derivation assumes a Killing horizon that does not exist. read the letter →

arxiv 2507.10874 v1 pith:DUZFURUD submitted 2025-07-15 gr-qc hep-th

Faizuddin Ahmed , Ahmad Al-Badawi , Abdelmalek Bouzenada , Erdem Sucu , İzzet Sakall{i} This is my paper · ORCID

classification gr-qchep-th PACS 04.70.-s04.60.-m

keywords BTZblackholeBopp-PodolskyelectrodynamicsgeneralizeduncertaintyprincipleHawkingtemperatureremnantnullgeodesicsdisclinationKeplerianfrequencies

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 studies a circularly symmetric charged BTZ-like black hole pierced by disclinations, a conical-defect parameter $\beta=1-4\lambda$, in Bopp-Podolsky electrodynamics, a nonlinear electrodynamics whose nonminimal coupling makes the two metric functions $f(r)$ and $g(r)$ unequal. It derives the effective potential for null geodesics, circular photon orbits, the critical impact parameter and shadow radius, and the Keplerian orbital frequency, showing how the Bopp-Podolsky coupling $b^2$, electric charge $Q$, cosmological constant $\Lambda$, and disclination parameter shift these observables. Its central new claim is a GUP-corrected Hawking temperature: tunneling computed with a minimum-length uncertainty relation gives a negative correction proportional to $\beta_{\rm GUP} E^2$, so quantum gravity suppresses Hawking radiation most strongly for small horizons, which the authors interpret as evidence for stable black hole remnants. If the paper is right, this charged BTZ-like black hole would not evaporate completely, and the predicted shadow sizes and orbital frequencies give concrete signatures to search for in observations.

What carries the argument

The argument runs on three pieces. The spacetime is fixed by the metric functions $f(r)=-M-\Lambda r^2-4b^2Q^2M/r^2-2Q^2\ln(r/r_0)$ and $g(r)=-M-\Lambda r^2+8b^2Q^2\Lambda-4b^2Q^2M/r^2-2Q^2\ln(r/r_0)$, whose inequality $f\neq g$ encodes the Bopp-Podolsky nonminimal coupling. The quantum step is the GUP-modified commutation relation, which deforms the Klein-Gordon equation; the WKB ansatz then gives, after a near-horizon expansion and residue evaluation, an imaginary action $\mathrm{Im}\,W_+=\pi E/\sqrt{f'(r_h)g'(r_h)}\,\bigl(1+\beta_{\rm GUP}E^2/f'(r_h)\bigr)$. Feeding this into the tunneling probability $\Gamma\sim\exp(-2\,\mathrm{Im}\,S)=\exp(-E/T_{\rm GUP})$ produces the closed-form GUP-corrected temperature that carries the remnant claim.

What would settle it

Take the Table 5 parameters $M=1$, $Q=1$, $\Lambda=-0.1$, $b=0.1$, $r_0=1.0$: the event horizon defined as the largest positive root of $g(r)=0$ is a larger root, not the $0.10767$ used in the table, so computing the corrected temperature at both roots settles whether the suppression pattern survives. A second check is to evaluate $f(r_h)$ at the tabulated roots; if $f(r_h)$ is not numerically zero, $\sqrt{f'(r_h)g'(r_h)}/(4\pi)$ is not the standard surface gravity, and a calculation using the proper Killing-horizon relation gives a different temperature.

Watch

Extended reading notes

Core claim

The paper's central claim is that GUP corrections change the thermal life of a charged BTZ-like black hole in Bopp-Podolsky electrodynamics. Starting from the modified commutation relation $[x_i,p_j]=i\hbar\delta_{ij}(1+\beta_{\rm GUP}p^2)$, the authors deform the Klein-Gordon equation, apply the WKB tunneling ansatz $S=-Et+W(r)+j\phi$, and evaluate the imaginary part of the action at the horizon to obtain a corrected temperature whose $\beta_{\rm GUP}\to 0$ limit is $T_H=\sqrt{f'(r_h)g'(r_h)}/(4\pi)$. The correction is negative and grows with the emitted particle energy $E^2$, and the paper's Table 5 and Figure 7 show the corrected temperature decreasing as $\beta_{\rm GUP}$ increases, most strongly for small horizon radii. The authors interpret this as suppression of Hawking radiation that can stabilize the black hole into a remnant, and they connect the same parameter set to observable quantities: the shadow radius grows with charge, and the Keplerian frequency $\Omega_\phi=(1/\beta)\sqrt{-\Lambda+Q^2/r^2-4b^2Q^2M/r^4}$ steepens with the Bopp-Podolsky coupling $b^2$.

Load-bearing premise

The remnant conclusion assumes the Hawking temperature is correctly given by $\sqrt{f'(r_h)g'(r_h)}/(4\pi)$ evaluated at the chosen root $r_h$ of $g(r)=0$, even though $f(r_h)$ is not zero there, so the horizon is not a Killing horizon for the time translation and the standard surface-gravity formula may not apply.

Editorial extensions

If this is right

As $\beta_{\rm GUP}$ grows, the corrected temperature falls below the classical Hawking temperature for every horizon radius, with the suppression strongest for small $r_h$, so micro black holes would evaporate more slowly and could end as stable remnants.
The $\beta_{\rm GUP}\to 0$ limit reproduces the semiclassical temperature, so the GUP result is presented as a consistent deformation of the standard tunneling temperature.
The heat capacity develops divergences and sign changes for nonzero Bopp-Podolsky coupling $b$, signaling phase transitions in the thermodynamic stability of these black holes.
The shadow radius and the Keplerian frequency $\Omega_\phi$ both grow with charge and with $b^2$, steepening the frequency gradient near the hole and giving parameter-dependent signatures for timing and imaging searches.

Reading between the lines

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

The temperature table's remnant pattern is computed at the smallest positive root of $g(r)$ (for example $r_h\approx 0.10767$ for $r_0=1$), whereas the paper defines the event horizon as the largest positive root; re-evaluating the corrected temperature at that larger root could erase the claimed suppression, so the remnant conclusion should be checked against the paper's own horizon definition.
The closed-form corrected temperature contains no $\beta$ (disclination parameter), even though the text says the topological defect modifies the thermodynamic quantities; the remnant mechanism as written does not depend on the defect, so any claim about its interplay with quantum gravity needs a separate derivation.
Because the suppression scales with $E^2$, different particle species emitted at different energies would experience different effective temperatures, changing the shape of the Hawking spectrum rather than merely lowering the luminosity; this is a testable consequence distinct from a uniform temperature drop.

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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

Good to have a clear look at this one. The paper is the usual 'take a known metric, run standard methods' exercise: geodesics, circular photon orbits, shadows, scalar field via Klein-Gordon, GUP tunneling, Keplerian frequencies, all for the Bopp-Podolsky BTZ-like solution from Ref [1] with disclinations. The geodesic and scalar-field parts are routine but mostly sound. The equations of motion, the photon-orbit condition (24), the shadow tables, and the Schrödinger-like reduction all hang together algebraically. If you want a catalogue of orbital and wave properties for this specific background, the first half of the paper delivers that.

The trouble is the GUP-thermodynamics section, which carries the abstract's headline claim about remnants. The problems are load-bearing. From the metric (2), g(r) = f(r) + 8 b^2 Q^2 Λ. For Λ<0 and b≠0, any root of g(r)=0 has f(r) = 8 b^2 Q^2 |Λ| > 0. So ∂_t is not null at r_h; the horizon is not a Killing horizon. The near-horizon expansion f(r) ≈ f'(r_h)(r-r_h) used in the derivation behind Eq. (41) is therefore invalid, and the tunneling integral does not give the simple pole that produces the temperature. The formula T_H = sqrt(f' g')/(4π) in Eq. (43) presupposes a Killing horizon. On top of that, Eq. (42) does not actually follow from Eq. (41), and the numbers in Table 5 do not match between Eqs. (43) and (44) for several rows. The table also evaluates temperatures at the smallest positive root of g in some rows even though Section 2 defines the event horizon as the largest positive root. Any one of these would be a serious issue; together they sink the remnant conclusion.

This is not a matter of a typo. The thermodynamic section is built on a misidentification of the horizon. The first half of the paper could be salvaged, but the abstract's main physical claim is unsupported. The paper also inherits a solution from Ref [1] that is explicitly perturbative and valid far from the horizon, which raises an additional question about applying it at the horizon at all.

For peer review: I would desk reject this version. The geodesic and scalar-field material is fine but not novel enough to carry a paper with a broken central claim. The authors could fix the thermodynamic section or, better, drop the GUP/remnant claim and publish a clean 'geodesics and scalar fields' paper. That version would deserve referee time.

Referee Report

5 major / 4 minor

Summary. The manuscript analyzes geodesics, scalar-field perturbations, GUP-corrected thermodynamics, and Keplerian frequencies for a charged BTZ-like spacetime in Bopp-Podolsky electrodynamics with disclinations, using the metric of Ref. [1] (Eqs. (1)-(2)). It derives effective potentials for null and timelike geodesics, circular photon orbits and shadow estimates, a Klein-Gordon reduction to a Schrödinger-like equation, a GUP-modified Hawking temperature via the tunneling method, and orbital frequencies. The advertised central result is that GUP corrections suppress Hawking radiation and may lead to stable black hole remnants.

Significance. The geodesic and circular photon orbit analysis in Section 3 is standard and the algebraic check of Eq. (24) is sound; the scalar-field reduction in Section 4 follows a familiar route. If the thermodynamic claim were valid, the paper would add a lower-dimensional example of GUP-induced remnants in a nonlinearly charged geometry. However, the thermodynamic section contains a fundamental error: at every zero of g(r) used as the horizon, f(r) is nonzero, so the surface is not a Killing horizon and the near-horizon tunneling calculation is inapplicable. The numerical tables are also inconsistent with the displayed formulas. These problems invalidate the paper's main advertised claim, and they are not local presentation issues.

major comments (5)

[§5, Eqs. (2), (40), and (41)] The tunneling calculation is performed at a surface that is not a Killing horizon. Since g(r) = f(r) + 8 b^2 Q^2 Λ, any root r_h of g(r_h)=0 has f(r_h) = 8 b^2 Q^2 |Λ| > 0 for b, Q ≠ 0 and Λ < 0, so the Killing vector ∂_t is not null at r_h. The near-horizon expansions f(r) ≈ f'(r_h)(r-r_h) and g(r) ≈ g'(r_h)(r-r_h) used before Eq. (41) are therefore not both valid; near r_h the integrand of Eq. (40) behaves as 1/√(r-r_h), not as a simple pole. Equation (41) is not obtained, and Eqs. (42)-(44) and the remnant conclusion are unsupported. The same problem makes Eq. (10) vanish, since that expression contains √g(r_h)=0.
[§5, Eq. (42)] Equation (42) does not follow from Eq. (41). With the tunneling probability Γ = exp(-4 Im W+) = exp(-E/T_GUP), Eq. (41) gives T_GUP ≈ (1/4π)√(f'g') [1 - β_GUP E²/f'(r_h)] to first order in β_GUP. The β_GUP-dependent term in Eq. (42) is instead -E²π²β_GUP r_h^6 added to the classical numerator. For the small-horizon rows of Table 5 (b=0.10, r0=1.00, r_h=0.10767) the denominator of Eq. (42) is negative, so that term raises TGUP above TH, exactly opposite to the suppression claimed in the text and shown in Table 5.
[§5, Eqs. (43)-(44) and Table 5] Equation (44) is not the explicit form of Eq. (43). Because g'=f', Eq. (43) is |f'(r_h)|/(4π). Equation (44), with the numerator factored as (Λr_h^4 + Q^2 r_h^2 - 4 M b^2 Q^2)^2, reduces to (Λr_h^4 + Q^2 r_h^2 - 4 M b^2 Q^2)/(2π r_h^3), which is -f'(r_h)/(4π), not its absolute value. For the first small-horizon row of Table 5 this gives a negative value (about -3.62), while the table lists 3.62417848. Thus Table 5 is not generated by Eq. (44), and the 'explicitly becomes' claim is incorrect.
[Table 5 and Sec. 2] The horizon radii used in Table 5 do not follow the paper's own horizon convention. Section 2 defines the event horizon as the largest positive root of g(r)=0; for b=0.10, r0=1.00, Λ=-0.1, M=Q=1, the largest positive root of Eq. (2) is approximately 7.0, while Table 5 lists r_h=0.10767, the smallest positive root. All temperature values in the r0=1.00 and r0=1.50 rows are therefore evaluated at the wrong root even under the paper's stated convention. Combined with the Killing-horizon problem, this removes the numerical support for the remnant claim.
[§3.4, Tables 3 and 4] The shadow radii are computed at a fixed rcpo=3.5 that is not a solution of the circular photon orbit condition Eq. (24) for the stated parameters. For example, with M=1, r0=1, Q=0.5, b=0.2, Eq. (24) has no root at r=3.5 (the left-hand side is positive for all r>0 in this parameter set), and with Q=0.1, b=0.1 it likewise is not satisfied at r=3.5. Consequently the Rs entries in Tables 3 and 4 are not shadow radii of the geometry described by Eq. (24), and the monotonic trends reported for Rs in the text are not supported by the displayed equations.

minor comments (4)

[§4, Eq. (34)] Equation (34) mixes R and ψ: after the transformation (33), the equation should be R''(r) + (ω² - Veff) R(r) = 0, not R''(r) + (ω² - Veff) ψ(r) = 0.
[Fig. 2 caption] The caption repeats 'the scale parameter r0 = 1' twice; one occurrence should be removed.
[§6, Eq. (52)] The conversion factor νφ = c³/(2πGM) Ωφ is dimensionally unclear in (2+1) dimensions, where Newton's constant has units of inverse mass; please state the units of G and the intended normalization.
[General] There are typos such as 'chracteristic' in Section 4 and 'Author declare(s)' in the conflict-of-interest statement; these should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the metric and GUP machinery are imported from external references, and no fitted parameter is renamed as a prediction.

full rationale

The paper's central derivation chain is not circular. The metric functions in Eq. (2) are taken from Ref. [1], an external author group, so the geodesic, scalar-field, shadow, and Keplerian-frequency sections are applications of an externally fixed background rather than reconstructions of the input. The GUP-corrected Hawking temperature is obtained from the standard GUP-modified Klein-Gordon equation cited from the external literature (Eq. (36), Refs. [65,125,129]) and is an algebraic consequence of that assumed deformation; no parameter is fitted to a subset of data and then presented as a prediction. The suppression direction when beta_GUP is increased is indeed hardwired into the assumed negative correction, but that is model application rather than circularity. The self-citations [36], [67], and [72] appear only in background literature lists and are not load-bearing for the central claim. The reviewer-raised issues that f(r_h) is nonzero at roots of g(r)=0, that the horizon root selection in Table 5 differs from the largest-root definition, and that the near-horizon expansion may therefore be invalid are potential correctness problems, but they are not circularity: they do not make any derived quantity equal to its own input by construction.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The central results rest on the metric from Ref [1], the Bopp-Podolsky field equations, the GUP modified commutation relation, and the standard WKB tunneling framework. The free parameters are model inputs chosen by hand in the plots and tables; none are fit to data. The main questionable axiom is the use of TH = sqrt(f'g')/(4 pi) when f at the horizon is nonzero, and the choice of which root of g(r)=0 is the thermodynamic horizon.

free parameters (8)

M (black hole mass) = 1 in all figures and tables
Chosen by hand as a model input; central quantities depend on it.
Q (electric charge) = 0.1 to 3.0 across tables
Chosen by hand as a model input; explored for its influence on horizons, orbits, and temperature.
Lambda (cosmological constant) = -0.1, -0.2, -0.3 in different sections
Chosen by hand as a model input; sign and magnitude determine AdS behavior.
b (Bopp-Podolsky coupling) = 0 to 1.8 across tables
Chosen by hand as a model input; drives the f vs g asymmetry and horizon structure.
r0 (scale parameter) = 0.5, 1.0, 1.5 in Table 5
Chosen by hand; sets the logarithmic scale in the metric functions.
beta (disclination parameter, beta = 1 - 4 lambda) = Set to 1 in most plots; otherwise varied
Chosen by hand; encodes the conical defect strength.
beta_GUP (GUP deformation parameter) = 0, 0.01, 0.05, 0.1, 0.2 in Table 5
Chosen by hand; controls the magnitude of quantum gravity corrections.
E (particle energy in tunneling) = 1 in Table 5
Chosen by hand; enters the GUP correction term quadratically.

assumptions (6)

domain assumption The metric in Eq. (2) from Ref [1] is a valid solution of (2+1)-dimensional Bopp-Podolsky electrodynamics coupled to gravity.
The entire analysis starts from this background without re-deriving the field equations.
domain assumption The Bopp-Podolsky energy-momentum tensor is given by Eqs. (6)-(7).
This defines the nonminimal coupling responsible for f(r) != g(r).
domain assumption The GUP modified commutation relation [x_i, p_j] = i hbar delta_ij (1 + beta_GUP p^2) underpins the modified Klein-Gordon equation (36).
The GUP deformation is assumed as the model for quantum gravity corrections rather than derived.
domain assumption The Hawking temperature formula TH = sqrt(f'(rh) g'(rh))/(4 pi) applies even though f(rh) is nonzero.
The horizon at g(rh)=0 is not a Killing horizon for the timelike Killing vector because f(rh) != 0; the paper does not justify this formula in the non-Killing-horizon setting.
standard math WKB tunneling with the residue theorem at the simple pole near the horizon gives the imaginary part of the action.
This is the standard semiclassical tunneling method, applied without modification.
domain assumption The disclination parameter beta = 1 - 4 lambda models cosmic strings as a redefinition of the angular coordinate.
The conical defect interpretation is taken from the cosmic string literature and applied to this 3D black hole.

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Cite this review

Pith. "Pith review of Geodesics, Scalar Fields, and GUP-Corrected Thermodynamics of Charged BTZ-like Black Holes in Bopp-Podolsky Electrodynamics." pith.science (2026). https://pith.science/paper/DUZFURUD

@misc{pith2026250710874,
  author       = {Pith},
  title        = {Pith review of: Geodesics, Scalar Fields, and GUP-Corrected Thermodynamics of Charged BTZ-like Black Holes in Bopp-Podolsky Electrodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DUZFURUD}},
  note         = {Machine review of arXiv:2507.10874}
}

abstract

In Ref. [1], the spacetime geometry generated by compact objects in $(2+1)$-dimensional Bopp-Podolsky electrodynamics is derived. Using a perturbative approach, the authors derived a charged BTZ-like black hole solution and computed corrections up to second order in a perturbative expansion valid far from the horizon. In this work, we investigate the same circularly symmetric three-dimensional charged BH solution pierced by disclinations. We begin by analyzing the motion of photons within this spacetime, focusing on how the geometric parameters influence the effective potential governing null geodesic motion. The corresponding equations of motion are derived, and the resulting orbital dynamics are explored through graphical methods to show the influence of key parameters including the BH mass $M$, electric charge $Q$, cosmological constant $\Lambda$, and BP coupling parameter $b^2$. Extending our analysis to wave dynamics, we examine the propagation of massless scalar fields in the BH solution by solving the Klein-Gordon equation. Through suitable coordinate transformations, we derive a Schr\"odinger-like equation with an effective potential that encodes the influence of the topological defect and electromagnetic corrections. Additionally, we investigate quantum gravitational effects by applying the Generalized Uncertainty Principle (GUP) to derive a GUP-corrected Hawking temperature, revealing systematic suppression of thermal radiation that could lead to stable BH remnants. Finally, we compute Keplerian frequencies for circular orbits, demonstrating how the interplay between charge, nonlinear electrodynamics, and disclination parameters creates distinctive observational signatures that could potentially test modified gravity theories in strong-field regimes.

Figures

Figures reproduced from arXiv: 2507.10874 by the authors.

**Figure 2.** Comparison of the effective potential Veff(r) for null geodesics under three different scenarios: (i) uncharged BTZ BH without coupling (purple curve), (ii) charged BTZ BH without coupling (red curve), and (iii) charged BTZ BH with non-minimal coupling (blue curve). The parameters are fixed as the BH mass M = 1, the cosmological constant Λ = −0.3, r0 = 1, the scale parameter r0 = 1, and the angular momentum L = 1, a… view at source ↗

**Figure 3.** provides a clear visualization of the influence of both the BP coupling and charge parameters on the circular photon orbit. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 4.5 5.0 5.5 b rcpo 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2 3 4 5 Q rcpo [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Plot of impact parameter γc for different values of BP coupling parameter b with fixed Q = 2 (left panel) and for different values of charge parameter Q with fixed b = 0.6 (right panel). Here, we set M = 1 and r0 = 1. 3.3 Stability Analysis and Photon Forces To study t…

**Figure 5.** Figure 5: Illustration of the force on photon particles as a function of [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: Comparison of the force Fph(r) on the photon particles under three different scenarios: (i) uncharged BTZ BH without coupling (purple curve), (ii) charged BTZ BH without coupling (red curve), and (iii) charged BTZ BH with BP coupling (blue curve). The parameters are fi…

**Figure 7.** Figure 7: Plot of TGUP versus rh for different values of βGUP. The plots are governed by [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: Heat capacity C as a function of horizon radius rh for the charged BTZ-like BH with parameters Λ = −1, Q = 1, M = 1, and β = 1. The plot reveals the thermodynamic stability regions and potential phase transitions in the system. 14 [PITH_FULL_IMAGE:figures/full_fig_p01…

**Figure 9.** Figure 9: Variation of the Keplerian frequency Ωϕ as a function of radial coordinate r for a charged BTZ-like BH with Λ = −0.1, Q = 1, M = 1, and β = 1. The plot highlights the impact of the BP electrodynamics and cosmological constant on the orbital dynamics near the BH, showin…

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