An innovative black hole solution and thermodynamic properties in higher-order curvature gravity with a scalar field

G.G.L. Nashed; Kazuharu Bamba; Usman Zafar

REVIEW 5 major objections 5 minor 146 references

An innovative black hole solution and thermodynamic properties in higher-order curvature gravity with a scalar field

T0 review · 5 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper constructs an analytic black hole in f(R,G) gravity with a scalar field and claims stable thermodynamics, but its own singularity asymptotics contradict the abstract's weaker-singularity promise.

desk verdict The abstract's headline claim that higher-curvature scalar couplings weaken the singularity is contradicted by the paper's own equations; the solution is new in form but the thermodynamics and stability claims rest on an invalid horizon. read the letter →

arxiv 2509.02643 v1 pith:2XDVP6VA submitted 2025-09-02 gr-qc hep-th

G.G.L. Nashed , Usman Zafar , Kazuharu Bamba This is my paper

classification gr-qchep-th MSC 83C5783D05 PACS 04.70.-s04.50.Kd

keywords f(RG)gravityscalarfieldblackholesolutioncurvaturesingularityBarrowentropyDaviesphasetransitionthermodynamictopologyKretschmanninvariant

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 tries to build an exact spherically symmetric black hole in higher-order curvature gravity f(R,G) coupled to a dynamical scalar field, and to use it to ask whether such corrections tame the central singularity and stabilize the hole thermodynamically. The advertised answer is yes: the abstract says curvature invariants near r=0 become far weaker than in general relativity, and that positive heat capacity, positive Gibbs free energy and topological charge 1 make the black hole stable. The body of the paper does not support the singularity part of that answer: Eq. (23), Section IV.A and Table I give Kretschmann ~ r^{-9} against Schwarzschild's r^{-6}, and the text calls the singularities 'generally stronger'. The stability conclusion is also contingent on the surface r2 behaving as a smooth null Killing horizon, a property that is assumed rather than demonstrated.

What carries the argument

The machinery is the modified-Schwarzschild ansatz S = 1 - c/r + Λ/r² and S1 = 1 - c/r + cΛ^{3/2}/r⁴. With this ansatz, the field equations (A1) reduce to reconstruction integrals (13),(17),(19) that determine f_R, H, φ, and V; the horizon radius r2 = M + sqrt(M² - Λ) then feeds the thermodynamic formulas, with Barrow entropy S_B = (π r² f_R)^{1+δ/2} replacing the area law. The stability argument runs through the heat-capacity denominator and through a vector-field construction whose winding number at the Barrow-entropy critical points is claimed to be 1.

What would settle it

Take the plotted case Λ=-100, M=1 and evaluate S1 at r2 = M + sqrt(M²-Λ); if S1(r2) is not zero, then r2 is not a null Killing horizon. A complementary check is to substitute the reconstructed F(r), H(r), φ(r), V(r) into all components of (A1) and see whether the residuals vanish.

Watch

Extended reading notes

Core claim

The paper constructs an exact spherically symmetric solution of f(R,G) gravity with a scalar field: S = 1 - c/r + Λ/r², S1 = 1 - c/r + cΛ^{3/2}/r⁴, which reduces to Schwarzschild at Λ=0. Integrals (13),(17),(19) reconstruct f_R, H(φ), φ, and V, with c2 < 0 to avoid ghosts. At S(r2)=0, Barrow entropy, Hawking temperature, heat capacity, quasi-local energy and Gibbs free energy are computed and found positive for the plotted parameters; Davies-type critical points appear where the Barrow heat capacity diverges. A winding-number computation gives topological charge 1 for δ=0, 0.5, 1, read as stability. The abstract's weaker-singularity claim is not borne out by the body: Eq. (23), Sec. IV.A and

Load-bearing premise

The load-bearing assumption is that the surface r2, defined by S(r2)=0, is a smooth null event horizon on which the standard surface-gravity temperature applies; the paper never enforces S1(r2)=0, so if that second metric function is nonzero, the thermodynamics and stability claims collapse.

Editorial extensions

If this is right

For Λ=0 the solution and its stability condition reduce to Schwarzschild, so the construction is a one-parameter deformation of general relativity.
For 0<Λ<M² the metric has inner and outer horizons; at Λ=M² they merge into a degenerate horizon; for Λ>M² the solution has no horizon.
If the horizon-surface assumption holds, the black hole has positive heat capacity, positive quasi-local energy, and positive Gibbs free energy over the plotted range, and is thermodynamically stable.
Barrow-entropy deformation yields Davies-type second-order phase transitions at parameter-dependent critical points, with topological charge 1 for the three values of δ examined.
The singularity behavior stated in the body (stronger than GR) contradicts the abstract's weaker-singularity claim; whichever is meant, it is a testable property of the solution.

Reading between the lines

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

I read the mismatch between the abstract and Section IV.A as a sign that the advertised singularity weakening is not realized by the ansatz actually solved; checking the subleading terms at r→0 would decide which statement is correct.
The thermodynamic quantities are evaluated at r2 with S1(r2) not enforced to vanish; if S1(r2)≠0 for the plotted choices, r2 is a static-limit surface rather than a null event horizon, and the Hawking-temperature formula would need replacement.
Topological charge 1 for every δ in {0, 0.5, 1} may be a generic artifact of the one-zone construction rather than a discriminating stability test; applying the same winding-number computation to a known unstable Schwarzschild-like solution would calibrate it.
A direct numerical residual check of the full field equations (A1) for the reconstructed F, H, φ, V would settle whether the solution is exact or only satisfies the linear combinations used in the paper.

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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

The abstract promises that higher-curvature scalar couplings make the central singularities far weaker than in GR. The paper's own Eq. (23) and Sec. IV.A say the opposite: Kretschmann ~ r^-9 vs Schwarzschild's r^-6 for the plotted parameters. That is not a minor mismatch; it is the advertised result. I read the body twice to make sure I wasn't missing a sign, but the text is explicit: 'the singularities in our solution are generally stronger.' Table I repeats it. The claim in the abstract and conclusions is simply false.

What is actually new: the specific metric ansatz S = 1 - c/r + Λ/r^2, S1 = 1 - c/r + c Λ^{3/2}/r^4, with the reconstructed f(R), H(φ1), V(φ1) is a new solution family in form. The authors give asymptotic expressions and check ghost-freedom conditions (c2 < 0). That part is a reasonable exercise in reverse engineering: assume the metric, solve for the functions. It is not a prediction of a pre-specified theory, but as a catalog entry it is not nothing.

The soft spots are load-bearing. Eq. (33) for the Hawking temperature is asserted, not derived from (32); it mixes L^5 and L^6 terms in the radicand and becomes imaginary in the two-horizon regime shown in Fig. 2(a). Eq. (34) is not the heat capacity (missing the mass factor), and the stability claim built on its sign is unsupported. The surface r2 with S(r2)=0 has S1(r2)≠0 for the plotted parameters, so the metric changes signature there; it is not a null Killing horizon, and the standard surface-gravity formula does not apply. The geodesic stability condition (58) is quoted without derivation. The Barrow-entropy and winding-number sections are standard machinery applied to a mass formula with a numerically fixed fR = 1.48765; they add no independent evidence.

My recommendation: desk reject. The paper is not coherent on its own terms—the abstract contradicts the body, and the thermodynamics is built on an invalid horizon. If the authors want to salvage it, they would need to verify the full field equations, fix the temperature derivation, and either prove the horizon is null or abandon the thermostatic claims. As is, it is not referee-ready.

Referee Report

5 major / 5 minor

Summary. The manuscript constructs a static, spherically symmetric black-hole solution in a four-dimensional action S = ∫ d^4x √-g { (1/(2κ²)) f(R) − ½ (∂φ1)² + V(φ1) + H(φ1)G }, using the metric ansatz (20), S = 1 − c/r + Λ/r² and S1 = 1 − c/r + cΛ^{3/2}/r⁴. It claims to reconstruct F(r) = f_R, H(r), φ1(r), and V(r) from selected combinations of the field equations, then computes curvature invariants, entropy, Hawking temperature, heat capacity, quasi-local energy, Gibbs free energy, Barrow-entropy thermodynamic topology, and geodesic-deviation stability. The abstract's advertised result is that higher-order curvature and scalar couplings make the central singularities much weaker than in general relativity and that the black hole is thermodynamically stable on a smooth horizon.

Significance. If the central claim were correct, the paper would provide an interesting example of singularity softening in f(R,G) scalar-tensor gravity and a Barrow-entropy-based Davies transition. However, the central claim is contradicted by the body's own computation: Eq. (23), Sec. IV.A, and Table I state that the divergences are generally stronger (K ∼ r⁻⁹ vs. Schwarzschild's r⁻⁶). In addition, the metric is complex for the parameters used in the plots, the Hawking-temperature formula is not derived from the stated surface-gravity expression and is dimensionally inconsistent, and the theory functions are solved from the metric ansatz without verifying the full field equations. The paper contains no machine-checked proofs or reproducible code, and the stability inference drawn from positive Gibbs free energy and winding number is not standard. The advertised result therefore does not stand.

major comments (5)

[Abstract; §IV.A, Eq. (23), Table I] The central advertised claim is contradicted by the authors' own computation. The abstract says the curvature singularities are 'far weaker' than in GR, but §IV.A states 'the singularities in our solution are generally stronger' and Table I reports K ∼ r⁻⁹ for this solution versus K ∼ r⁻⁶ for Schwarzschild. Eq. (23) gives K ≈ 12c²/r⁶ − 32Λc/r⁷ + ⋯ + (84Λ²c − 18Λc³ − 24c²Λ^{3/2})/r⁹. For the parameter values used in the paper, the r⁻⁹ term is nonzero. Thus the advertised singularity softening is not demonstrated and is, on the authors' own formulas, false.
[Eq. (20); Figs. 1–2] The ansatz (20) contains S1 = 1 − c/r + cΛ^{3/2}/r⁴. For Λ < 0, Λ^{3/2} = (−100)^{3/2} = 1000 i, so the metric is complex. Yet Fig. 1 and Fig. 2 use Λ = −100 and plot real 'physical' quantities, with no statement that a real section is being taken. Eq. (23) itself contains Λ^{3/2}. A complex metric invalidates the interpretation of all subsequent invariants, horizons, and thermodynamic functions for the plotted parameter set.
[Eqs. (32)–(33); §V.A] The Hawking temperature is not derived from the given metric. Eq. (32) contains S1(r) through g_rr, but Eq. (33) has no S1 dependence. Moreover, Eq. (33) is dimensionally inconsistent: the numerator has dimension L³, while the radicand in the denominator mixes L⁵ and L⁶ terms (8Λr₂³ vs. Λ³, Λ²r₂², etc.). In addition, for the ansatz (20), the roots of S and S1 do not coincide for the plotted parameters, so r₂ is a static limit rather than a null Killing horizon; the standard surface-gravity formula does not apply there. Consequently the temperature, heat capacity, and stability conclusions in §V.A are unsupported.
[§III and Appendix A] The solution is reverse-engineered: S and S1 in Eq. (20) are chosen arbitrarily, and F, H, φ1, and V are solved from selected linear combinations of the field equations, Eqs. (13), (17), and (19). The paper never verifies that the reconstructed functions satisfy the full system (A1) or the trace equation (8). Since the metric is the input, the singularity and thermodynamic behavior is a property of the assumed ansatz, not a prediction of a theory specified in advance. A minimal check would be to substitute the obtained F, H, φ1, V back into all components of Eq. (A1) and demonstrate cancellation; this is absent.
[§V.B; Eqs. (34)–(37), (48)–(49)] The stability conclusion conflates distinct notions. Positive Gibbs free energy G = M − TS (37) does not by itself imply thermodynamic stability; stability requires at least a sign check of the standard heat capacity and a perturbative/convexity analysis. The winding number W = 1 in Eq. (48) is a topological index for a zero of a vector field built from the Barrow temperature; it does not establish that the black hole is stable against metric or radial perturbations. Also Eq. (34) defines H₂ = ∂r₂/∂T₂, not the standard C = ∂M/∂T, so the heat capacity plotted in Fig. 2(d) is not the standard thermodynamic response function.

minor comments (5)

[Introduction and Abstract] The abstract says the singularities are 'far weaker' while the introduction says 'these invariants have stronger singularity as r → ∞'. These statements should be reconciled.
[Eq. (24)] The expression r(R) ≈ ⁴√|2Λ + m²∥ / ⁴√R contains garbled notation and an undefined mass symbol m; M was used before.
[Eq. (31)] The entropy expression is extremely long and its dimensional consistency is unclear; no check is made that it reduces to the area law when Λ → 0.
[Eq. (56)] The third equation in (56) ends with '= 20' instead of '= 0'; the preceding equations also contain typographical inconsistencies in the deviation formalism.
[§V.B] The text says 'three zero points ZP1, ZP1, ZP1' where ZP2 and ZP3 are clearly intended; also 'Davis-type' appears in the title/abstract instead of 'Davies-type'.

Circularity Check

1 steps flagged · score 6.0 of 10

Central singularity claim is built into the assumed metric ansatz, not derived from the theory; the paper's own Eq. (23) contradicts the advertised weakening.

self definitional [Abstract; Sec. III.A (Eq. 20); Sec. IV.A (Eq. 23); Table I]
"It is demonstrated that the singularities of the curvature invariants will be far weaker around the central region of the black hole than those in general relativity, owing to the effect originating from higher-order curvatures... In this study, we take the ansatz for S and S1... S = 1 - c/r + Λ/r², S1 = 1 - c/r + cΛ^{3/2}/r⁴... So, we are going to use the ansatzs of Eq. (20) as an input in the equations Eqs. (13), (17), and (19) and study the effect of the modified theory f(R,G)."

The theory functions F(r)=fR, H(φ1), φ1(r), and V(φ1) are solved from the field equations only after fixing S and S1 by Eq. (20). The curvature invariants in Eq. (23) are then computed directly from that same assumed line element. Hence the claimed singularity behavior is an input of the ansatz, not a prediction of a pre-specified f(R,G) theory; the causal statement in the abstract reverses the actual construction. Moreover, Eq. (23) and Sec. IV.A state the singularities are generally stronger (K ~ r^-9 vs Schwarzschild r^-6), so the advertised 'weaker singularities' result is not even correct, illustrating that the reverse-engineered ansatz, not the theory, controls the outcome.

full rationale

The paper's construction is a reverse-engineering: a specific deformed Schwarzschild metric is posited, and the scalar-tensor/f(R,G) functions are reconstructed so that the metric solves the field equations. Consequently, the singularity invariants, horizon structure, and any thermodynamic quantities that depend directly on S and S1 are inherited from the chosen ansatz rather than being consequences of an independently defined higher-order curvature theory. This is the main circular element, and it affects the paper's central novelty claim: the abstract's assertion that the higher-order curvature/scalar coupling makes singularities 'far weaker' is not a deduced result; it is, at best, a property inserted by the ansatz. The contradiction with Eq. (23), Sec. IV.A, and Table I (K ~ r^-9 vs Schwarzschild K ~ r^-6) reinforces that the singularity claim is imposed by the ansatz, and in fact fails. The thermodynamic-topology section is largely a self-contained computation from the assumed metric and the numerical choice fR=1.48765 (Λ=0.1); the Davies critical points are mathematical zeros of the constructed heat capacity, so they are not themselves circular, though their physical interpretation inherits the same reverse-engineered input. No load-bearing self-citation chain or imported uniqueness theorem was found; citations to standard entropy/topology methods are ordinary references. Overall, because the key 'prediction' reduces to the input metric by construction, a partial circularity score of 6 is appropriate.

Assumptions & free parameters 5 free parameters · 6 assumptions · 2 invented entities

Everything physically distinctive in the paper flows from the hand-chosen ansatz (20): the singularity profile, the mass formula M = (r2²+Λ)/(2r2), the temperature, and the Barrow-entropy critical points. The theory f(R,G) plus scalar is then reconstructed from the field equations, so there are no true predictions; the free constants c1, c2, c3, c4, Λ, M and the numerically fixed fR = 1.48765 are inputs. The remaining axioms are standard variation and Bianchi math, the f(R) entropy formula from Refs. [116,117], Barrow's entropy ansatz, and Duan's topological current formalism.

free parameters (5)

Λ (length² deformation parameter) = -100, 0.01, 0.04, 0.1 (chosen by hand in figures)
Assumed in the ansatz (20); controls horizon structure, singularity strength, temperature, and the Barrow-entropy critical points; no derivation fixes its value.
c = 2M (gravitational mass) = 2 (M = 1 in figures)
Input of ansatz (20); the whole solution family is parameterized by it.
c1, c2 (integration constants in F, H, φ1) = c1 = 1, c2 = -10 (figures)
Constants in Eqs. (13)-(19); c2 < 0 is imposed to keep the scalar field real and 'ghost-free' (Sec. IV.A); c3 = c4 = -100 appear in the Fig. 2 caption without definition.
fR = 1.48765 = 1.48765
Numerically fixed value used in the Barrow-entropy and topological analysis (Sec. V.B, Eqs. 38-44); the text admits varying it shifts the divergence points.
δ (Barrow entropy deformation) = 0, 0.5, 1 (scan)
Parameter of the Barrow entropy framework from Ref. [92]; scanned to produce Fig. 3.

assumptions (6)

ad hoc to paper Action (1): S = ∫ d^D x √-g {1/(2κ²) f(R) - ½(∂φ1)² + V(φ1) + H(φ1)G} with 2κ² = 1 and f(R,G) split as f(R) + H(φ1)G
Starting point; the split of f(R,G) into f(R) plus a scalar-coupled Gauss-Bonnet term is chosen to obtain a 'ghost-free' model (footnotes 1 and 2).
standard math Field equations (6)-(10) derived by variation and Bianchi identities (5)
Standard variation of the action; all later equations depend on this derivation.
ad hoc to paper Metric ansatz (20): S = 1 - c/r + Λ/r², S1 = 1 - c/r + cΛ^(3/2)/r⁴
The metric is assumed, not derived; the theory functions are then solved from the field equations to admit it.
domain assumption Entropy S = ¼ A fR (Eq. 30)
Taken from Refs. [116,117] for f(R) gravity; used for all horizon thermodynamics.
domain assumption Barrow entropy S_B = (π r² fR)^(1+δ/2) (Eq. 38)
Barrow's 2020 parametrization [92] combined with the f(R) entropy correction factor fR; the fR factor is framework-specific.
standard math Duan's φ-mapping topological current formalism (Eqs. 45-49)
Topological current and winding-number construction from Refs. [142,143].

invented entities (2)

Reconstructed theory functions F(r) = fR, f(R), H(φ1), V(φ1), φ1(r)
purpose: Support the assumed metric (22) as a solution of a 'ghost-free' f(R,G) plus scalar action
These functions are solved from the field equations after the metric is fixed (Eqs. 13-19, 21). No independent handle exists: no predicted mass, coupling constant, or observable; the graviton-type risk applies.
Deformed metric (22) with parameter Λ
purpose: Serves as the claimed black hole solution whose singularity, thermodynamics, and stability are reported
The parameter values (0.01 to -100) are hand-picked and are not tied to any observation; no range is derived from data.

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Cite this review

Pith. "Pith review of An innovative black hole solution and thermodynamic properties in higher-order curvature gravity with a scalar field." pith.science (2026). https://pith.science/paper/2XDVP6VA

@misc{pith2026250902643,
  author       = {Pith},
  title        = {Pith review of: An innovative black hole solution and thermodynamic properties in higher-order curvature gravity with a scalar field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2XDVP6VA}},
  note         = {Machine review of arXiv:2509.02643}
}

read the original abstract

A spherically symmetric black hole solution defined by the gravitational mass is explored in higher-order curvature gravity associated with a scalar field. It is demonstrated that the singularities of the curvature invariants will be far weaker around the central region of the black hole than those in general relativity, owing to the effect originating from higher-order curvatures with the coupling to a dynamical scalar field. Furthermore, thermodynamic properties and the Davies-type phase transition of the black hole are investigated for Barrow entropy with a quantum effect of gravitation. It is found that both the quasi-local energy and Gibbs free energy are positive, and the black hole can hence be stable on the smooth horizon surface. In addition, by analyzing the geodesic deviation, the stability conditions of the black hole are explicitly shown.

Figures

Figures reproduced from arXiv: 2509.02643 by the authors.

**Figure 1.** Schematic plots of: Fig. (a) the behavior of the scalar function φ1; Fig. (b)∼ (d) show the behavior of the physical singularity at the origin for R, R µνRµν , and R µναβRµναβ; Fig. (e) shows the behavior of the potential V ; finally, Fig. (f) ∼ (h) show the behavior of the functions f(R), fR and fRR. Here we assume the following numerical values M = 1, ̥ = −100, c2 = −10. V. THERMODYNAMICS OF THE BLACK HOLE In this… view at source ↗

**Figure 2.** The behavior the BH solution’s thermodynamic quan [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Schematic plots of a normalized vector field in [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

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(A1) where F ≡ F (r) = fR = d f(R(r)) dR(r) , F ′ = dF (r) dr , F ′′ = d2F (r) dr2 , F ′′′ = d3F (r) dr3

S′ + 4 { F − F S1 + 1/4r2φ′ 1 2S1 + r2V − F ′S1r + 1 4 r2F ′S′ 1 } S2 ) S ] = 0 , and the trace equation takes the form: I = 1 2S2r2 [ −8 {S1S′r + S (S1 − 1)} SS1H ′′ − 2rSS 1 {4H ′S1 + rF } S′′ + 6S2r2F ′′S1 + rS1 {4H ′S1 + rF } S′2 −S { ( 12S12 − 4S1 + 12S1rS ′ 1 ) H ′ + r [ rF S′ 1 − 3 ( rF ′ − 4F 3 ) S1 ]} S′ − 4S2 { S′ 1 (3S1 − 1) H ′ + ( rF − 3 4 r2...

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