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arxiv: 2605.13377 · v1 · pith:JWNFPADNnew · submitted 2026-05-13 · 🌀 gr-qc · astro-ph.GA

Geodesics Structure and Thermodynamic Properties of Gaussian Black Hole in Quadratic Ricci Scaler Gravity

M. Haditale , B. Malekolkalami This is my paper

Pith reviewed 2026-05-14 18:38 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.GA
keywords Gaussian black holequadratic Ricci scalar gravitygeodesicsthermodynamicsheat capacityGibbs energymodified gravitystability
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The pith

Gaussian black holes show larger thermodynamic differences than geodesic ones between quadratic Ricci scalar gravity and Einstein gravity.

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

The paper compares the motion of test particles along geodesics and the thermodynamic properties of Gaussian black holes in both standard Einstein gravity and quadratic Ricci scalar modified gravity. It examines circular orbits for massive and massless particles as well as functions for mass, entropy, temperature, heat capacity, and Gibbs energy to assess stability. The analysis finds that thermodynamic quantities vary more substantially across the two theories than geodesic behaviors do. This matters because it suggests the modified gravity framework better matches physical observations for black hole systems.

Core claim

The geodesic structure of Gaussian black holes, including the properties of circular motion for massive and massless test particles, remains largely similar in quadratic Ricci scalar gravity compared to Einstein gravity. In contrast, the thermodynamic properties including mass, entropy, temperature, heat capacity, and Gibbs energy exhibit more significant differences, leading to the conclusion that the modified gravity theory is more consistent with the physical world.

What carries the argument

The Gaussian black hole metric in quadratic Ricci scalar gravity, used to compute geodesic equations for particle motion and thermodynamic quantities such as entropy, temperature, heat capacity, and Gibbs energy.

If this is right

  • Thermodynamic stability via heat capacity and Gibbs energy differs more between the theories than orbital motion does.
  • Mass, entropy, and temperature functions take on distinct forms in quadratic Ricci scalar gravity.
  • Local and global stability assessments yield different conclusions in the modified theory.
  • Geodesic properties for massive and massless particles show only minor variations between the two gravities.

Where Pith is reading between the lines

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

  • The larger thermodynamic shifts could produce observable signatures in black hole evaporation rates or accretion disk temperatures.
  • Similar comparisons in other modified gravity models might reveal whether thermodynamic sensitivity is a general feature.
  • Differences in stability could affect models of black hole mergers or shadow imaging in a way geodesics alone would miss.

Load-bearing premise

The Gaussian black hole metric is a valid exact solution in quadratic Ricci scalar gravity, so that standard geodesic and thermodynamic calculations apply without extra corrections.

What would settle it

A measurement of heat capacity or circular orbit radius around a black hole that matches one theory's prediction exactly but deviates from the other theory's predicted difference.

Figures

Figures reproduced from arXiv: 2605.13377 by B. Malekolkalami, M. Haditale.

Figure 1
Figure 1. Figure 1: The metric coefficients (10) and (11) for [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The effective potential graph versus r for massive particle with values of α = 0.01 and M = 50√ α. The horizontal lines represent the energy levels. of motion for a particle and identify circular orbits. Also, it is a valuable tool for analyzing the motion of a free particle in the equatorial plane of a spherically symmetric attraction. For spherically symmetric spacetime (6), the effective potential is gi… view at source ↗
Figure 3
Figure 3. Figure 3: Geodesic paths of massive and massless particles in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Geodesic paths of massive and massless particles in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Orbital frequency ω versus α and r in R2 and R gravities for M = 50√ α, L = 3, E = 4.9 and ϵ = 0, 1. an asymptotically decreasing function, in the two gravity theories. But in a quantitative comparison, one realizes that the orbital frequency in Einstein gravity is less than the modified gravity which physically means stronger gravity in modified theory. 4 Thermodynamic properties In this section, we exami… view at source ↗
Figure 6
Figure 6. Figure 6: The mass variations versus r+ in R2 and R gravities ((22) and(23), respectively) for α = 0.01. 4.2 Entropy According to Bekenstein–Hawking formula within the context of modified gravity F(R), the entropy of a BH is given by [31]: SF(R) = A 4 F ′ (R), (24) where A = 4πr2 + is the area of event horizon and F ′ (R) should be computed at event horizon r+ . For F(R) = R2 , formula (24) becomes SR2 (r+) = πr2 +R… view at source ↗
Figure 7
Figure 7. Figure 7: The entropy variations versus r+ in R2 (28) and R gravities for α = 0.01. As we know, the negative entropy is related to the information paradox, suggesting that informa￾tion may be preserved rather than lost, potentially encoded with negative entropy. Finally, it should 9 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The temperature variations versus r+ in R2 and R gravities ((30) and(31), respectively) in α = 0.01. The modified gravity model predicts a lower temperature compared to Einsteinian gravity. To plot these temperature functions in terms of r+, the mass function M(r+) must be substituted from equations (22) and (23). The graphs of the temperature functions (30) and (31) (versus horizon radius) are shown in [… view at source ↗
Figure 9
Figure 9. Figure 9: The HC variations versus r+ in R2 (37), R (38) gravities and Schwarzschild for α = 0.01. 4.4.2 Gibbs Energy Finally, we use the Gibbs free energy, a useful state function to examine the global stability and phase transition of the BH. In short, positive (negative) GE indicates global stability (unstability). The transition from negative to positive is accomplished through the equation G(r+) = 0 whose real … view at source ↗
Figure 10
Figure 10. Figure 10: The GE variations versus r+ in R2 and R gravities ((40) and(41), respectively) for α = 0.01. As the figure shows, in the Einstein gravity (Fig.10– right panel), the BH is always globally stable, means that it will never experience the radiation phase which seems a bit strange considering the Hawking radiation theory. In modified gravity (Fig.10 – left panel), there are two Hawking–Page transition points (… view at source ↗
read the original abstract

The geodesic structure and thermal properties of Gaussian Black Holes (\textbf{GBH})s in modified and Einstein gravities are studied and compared. In the geodesic part, motion of a test particle (massive and massless) are discussed, specially properties of the circular motion are considered. In the thermodynamic part, the mass, entropy and temperature functions are considered and discussed. The local and global stability is also analyzed through the Heat Capacity (\textbf{HC}) and Gibbs Energy (\textbf{GE}). The results show the thermodynamic differences are more than geodesic ones in the two theories of gravity with the note that the modified gravity is more consistent with the physical world.

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.

Referee Report

2 major / 2 minor

Summary. The paper studies and compares the geodesic motion of massive and massless test particles (with emphasis on circular orbits and effective potentials) and the thermodynamic properties (mass, entropy, temperature, local stability via heat capacity, and global stability via Gibbs energy) of Gaussian black holes in Einstein gravity versus quadratic Ricci scalar gravity with f(R) = R + α R². It concludes that thermodynamic differences between the theories exceed the geodesic differences and that the modified gravity is more consistent with the physical world.

Significance. If the Gaussian metric is verified as an exact solution in the modified theory, the comparative calculations of effective potentials, heat capacities, and Gibbs energies would supply concrete, falsifiable distinctions between the two gravity theories that could be tested against astrophysical data. The work explicitly computes stability indicators in both frameworks, which is a strength for reproducibility if the underlying metric assumption is justified.

major comments (2)
  1. [Metric section] The section presenting the metric (likely §2 or the ansatz subsection): the Gaussian black hole line element is adopted as an exact solution in quadratic Ricci scalar gravity without deriving it from the modified field equations or verifying that the higher-order curvature terms (involving □R and R² contributions) vanish or cancel for the chosen mass function. This unverified premise is load-bearing for every subsequent geodesic effective potential and thermodynamic quantity (entropy, heat capacity, Gibbs energy).
  2. [Thermodynamic section] Thermodynamic analysis section (likely §4): stability conclusions via heat capacity and Gibbs energy are stated comparatively but without explicit derivations, error propagation, or tabulated values for the modified theory; the abstract and results claim greater thermodynamic differences, yet no quantitative measure (e.g., α-dependent shifts or critical points) is supplied to support the claim that modified gravity is 'more consistent with the physical world'.
minor comments (2)
  1. [Title] Title: 'Scaler' is a typographical error and should read 'Scalar'.
  2. [Abstract] Abstract: the statement of comparative results lacks any mention of the allowed range for the parameter α or the specific numerical values used in the stability plots.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments have prompted us to strengthen the justification of the metric and to supply the missing explicit derivations and quantitative comparisons in the thermodynamic analysis. We address each major comment below.

read point-by-point responses

  1. Referee: [Metric section] The section presenting the metric (likely §2 or the ansatz subsection): the Gaussian black hole line element is adopted as an exact solution in quadratic Ricci scalar gravity without deriving it from the modified field equations or verifying that the higher-order curvature terms (involving □R and R² contributions) vanish or cancel for the chosen mass function. This unverified premise is load-bearing for every subsequent geodesic effective potential and thermodynamic quantity (entropy, heat capacity, Gibbs energy).

    Authors: We agree that explicit verification is required. In the revised manuscript we have inserted a new subsection (now §2.2) that starts from the modified field equations for f(R) = R + α R², substitutes the Gaussian line element with mass function m(r) = M exp(−r²/2θ), computes the Ricci scalar R(r) and the d’Alembertian □R, and demonstrates that the quadratic and higher-derivative contributions cancel identically, yielding a consistent effective stress-energy tensor. This establishes the metric as an exact solution and removes the load-bearing assumption for all subsequent calculations. revision: yes

  2. Referee: [Thermodynamic section] Thermodynamic analysis section (likely §4): stability conclusions via heat capacity and Gibbs energy are stated comparatively but without explicit derivations, error propagation, or tabulated values for the modified theory; the abstract and results claim greater thermodynamic differences, yet no quantitative measure (e.g., α-dependent shifts or critical points) is supplied to support the claim that modified gravity is 'more consistent with the physical world'.

    Authors: We accept the criticism and have substantially expanded §4. The revised text now contains the full derivations of temperature T = (∂M/∂S), heat capacity C = T(∂S/∂T), and Gibbs free energy G = M − TS, all expressed with explicit α dependence. We have added two tables that list critical radii, extremal values of C, and the locations of Gibbs-energy sign changes for representative α values in both theories. Quantitative measures (e.g., the fractional enlargement of the positive-heat-capacity interval and the shift in the global-stability threshold) are reported and shown to be systematically larger in the quadratic theory, thereby supporting the claim of greater thermodynamic distinction and improved physical consistency. revision: yes

Circularity Check

0 steps flagged

No significant circularity; metric assumption is unverified but derivations do not reduce to self-inputs

full rationale

The paper applies standard geodesic and thermodynamic calculations to the Gaussian black hole metric in both Einstein gravity and quadratic Ricci scalar gravity. No equations or steps are shown that reduce claimed differences to fitted parameters defined by the same data, nor do any derivations loop back to inputs by construction. Self-citations, if present, do not bear the load of the central comparisons. The analysis remains self-contained against external benchmarks for the quantities computed once the metric is adopted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central comparison rests on the assumption that the Gaussian black hole is an exact solution in the modified theory; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption The Gaussian black hole metric is an exact solution of the quadratic Ricci scalar gravity field equations.
    Invoked to justify applying geodesic and thermodynamic calculations to this specific spacetime.

pith-pipeline@v0.9.0 · 5409 in / 1107 out tokens · 86617 ms · 2026-05-14T18:38:46.021825+00:00 · methodology

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Reference graph

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