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arxiv: 2605.10992 · v3 · pith:KFE42DCFnew · submitted 2026-05-09 · 🌀 gr-qc

Photon Sphere and Shadow of a Perturbative Black Hole in f(R,mathcal{G}) Gravity

G.G.L. Nashed This is my paper

Pith reviewed 2026-06-30 22:59 UTC · model grok-4.3

classification 🌀 gr-qc
keywords black holephoton sphereshadowf(R,G) gravityperturbative expansionnull geodesicsGauss-Bonnet
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The pith

Higher-curvature corrections in f(R,G) gravity shift the photon-sphere radius of black holes and correct their shadow size.

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

The paper constructs leading-order metric deviations from the Schwarzschild solution in a perturbative f(R,G) theory. It then follows null geodesics to locate the unstable photon orbit and computes the resulting black-hole shadow. Analytic shifts in the photon-sphere radius are derived, showing that the Gauss-Bonnet sector produces larger effects than mixed curvature terms. These geometric changes translate into observable corrections to the shadow radius. The analysis also indicates how the corrections would influence strong-field lensing and quasinormal modes.

Core claim

Within the perturbative expansion, the photon-sphere radius acquires a correction linear in the small coupling constants of the quadratic curvature invariants. The Gauss-Bonnet contribution dominates the shift, and the shadow radius receives an additional direct correction from the metric perturbation itself. These modifications provide a consistent way to estimate deviations from general relativity in strong-field regimes.

What carries the argument

Asymptotic expansion of the field equations around the Schwarzschild metric to obtain leading-order corrections, followed by computation of the effective potential for equatorial null geodesics.

If this is right

  • Analytic expressions for the photon-sphere displacement are obtained as functions of the coupling parameters.
  • The shadow radius correction arises from both the displaced photon sphere and metric perturbations.
  • Strong lensing observables gain sensitivity to higher-curvature effects.
  • Quasinormal modes are expected to receive perturbative frequency shifts.

Where Pith is reading between the lines

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

  • Future very-long-baseline interferometry could place bounds on the quadratic couplings through shadow measurements.
  • The perturbative approach may extend to rotating black holes to study frame-dragging effects.
  • The relative importance of the Gauss-Bonnet term suggests prioritizing constraints on that invariant in observations.

Load-bearing premise

The spacetime stays static and spherically symmetric when the quadratic curvature corrections are treated as small perturbations around the Schwarzschild geometry.

What would settle it

An observed black-hole shadow radius that differs from the general-relativity prediction by an amount inconsistent with the derived linear corrections for any choice of small couplings would rule out the leading-order perturbative model.

Figures

Figures reproduced from arXiv: 2605.10992 by G.G.L. Nashed.

Figure 1
Figure 1. Figure 1: Comparison of the modified metric function, the pho [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Comparison of the modified metric function, the pho [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the black-hole shadow boundary in th [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the black-hole shadow boundary in th [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Deflection angle and its deviation from the Schwarz [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: Deflection angle and its deviation from the Schwarz [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Effective potential Veff (r) in the vicinity of the photon sphere for the Schwarzschild solution and for representative values of the higher-curvature coupling parameters. The plot highlights the enhancement of deviations induced by the f(R, G) corrections in the strong-field region. While the potential coincides with the Schwarzschild case at large distances, noticeable differences arise near the photon s… view at source ↗
Figure 4
Figure 4. Figure 4: Effective potential Veff (r) in the vicinity of the photon sphere for the Schwarzschild spacetime and for several representative choices of the higher-curvature coupling parameters. The plot highlights the enhancement of deviations induced by the f(R, G) corrections in the strong-field region. While the potential coincides with the Schwarzschild case at large distances, noticeable differences arise near th… view at source ↗
read the original abstract

We investigate the impact of higher-curvature corrections on black-hole observables within a perturbative $f(R, G)$ gravity framework. Working in a static, spherically symmetric spacetime, we construct leading-order deviations from the Schwarzschild solution by expanding the field equations in small coupling parameters associated with quadratic curvature invariants. The resulting metric corrections are obtained as asymptotic expansions and used to analyze null geodesics. We derive analytic expressions for the shift in the photon-sphere radius and show that higher-curvature terms modify the location of unstable photon orbits, with the Gauss--Bonnet sector producing a more significant contribution than mixed curvature terms. These modifications propagate to observable quantities, leading to corrections in the black-hole shadow radius. We identify the distinct roles of photon-sphere displacement and direct metric perturbations in determining the shadow size. We further discuss the implications of these corrections for strong gravitational lensing and quasinormal modes, highlighting the enhanced sensitivity of strong-field observables to higher-curvature effects. While the present analysis is based on an asymptotic perturbative treatment, our results provide a consistent framework for estimating leading-order deviations from general relativity and suggest that high-resolution observations, including very-long-baseline interferometry and gravitational-wave measurements, may offer constraints on modified gravity models.

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

1 major / 1 minor

Summary. The manuscript investigates higher-curvature corrections in perturbative f(R,𝒢) gravity for static spherically symmetric black holes. It expands the field equations in small quadratic-curvature couplings around the Schwarzschild background, obtains metric corrections via asymptotic expansions, derives analytic expressions for the resulting shift in the photon-sphere radius (with the Gauss-Bonnet sector dominating), and propagates these corrections to the black-hole shadow radius. Implications for strong lensing and quasinormal modes are also discussed.

Significance. If the perturbative construction and its substitution into the null-geodesic conditions are under control, the work supplies a concrete leading-order framework for estimating deviations in strong-field observables due to quadratic curvature invariants. The explicit separation of photon-sphere displacement from direct metric perturbations and the comparative size of the Gauss-Bonnet versus mixed terms would be useful for future observational constraints with VLBI and gravitational-wave data.

major comments (1)
  1. [Abstract and metric construction section] Abstract and the section describing the metric construction: the metric corrections are obtained as asymptotic expansions (controlled at large r, small M/r). These expansions are then substituted directly into the effective-potential condition that locates the photon sphere at r≈3M. Because M/r∼1/3 at that radius, higher-order terms in the radial expansion can become comparable to the retained correction, rendering the reported analytic shifts uncontrolled. An explicit error estimate, a statement of the truncation order, or a comparison against numerical integration of the field equations at r=3M is required to support the central claim.
minor comments (1)
  1. [Abstract] The abstract states that analytic expressions are derived but does not quote the explicit leading-order metric components or the order at which the expansion is truncated; adding these would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our work's significance and for the detailed comment on the metric construction. We address the concern regarding the asymptotic expansion below.

read point-by-point responses

  1. Referee: [Abstract and metric construction section] Abstract and the section describing the metric construction: the metric corrections are obtained as asymptotic expansions (controlled at large r, small M/r). These expansions are then substituted directly into the effective-potential condition that locates the photon sphere at r≈3M. Because M/r∼1/3 at that radius, higher-order terms in the radial expansion can become comparable to the retained correction, rendering the reported analytic shifts uncontrolled. An explicit error estimate, a statement of the truncation order, or a comparison against numerical integration of the field equations at r=3M is required to support the central claim.

    Authors: We agree that the validity of the asymptotic radial expansion at the photon-sphere location merits explicit discussion. In the revised version, we will add a statement specifying the truncation order of the expansion and provide an order-of-magnitude estimate for the size of the omitted higher-order terms evaluated at r = 3M. This will demonstrate that the leading corrections remain dominant for the small coupling parameters considered. A direct numerical solution of the full field equations is not pursued within the present perturbative treatment, but the requested error analysis will be included to substantiate the analytic results. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is a standard perturbative expansion from field equations

full rationale

The paper derives metric corrections by expanding the modified field equations in small couplings around Schwarzschild, then computes null geodesics and photon-sphere shifts analytically from those corrections. No step reduces by construction to its inputs, no parameters are fitted to the target observables, and no load-bearing uniqueness theorem or ansatz is imported via self-citation. The calculation is self-contained against external benchmarks (the Schwarzschild limit and the perturbative action), so the reported shifts are genuine outputs of the expansion rather than tautological renamings or forced predictions.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on a perturbative expansion whose validity requires the coupling parameters to be small; the abstract does not list explicit numerical values for these parameters or prove that the expansion converges at the photon-sphere radius.

free parameters (1)
  • quadratic curvature couplings
    Small parameters multiplying the R^2, G, and mixed invariants; their magnitudes control the size of the metric corrections but are not given numerical values in the abstract.
axioms (1)
  • domain assumption Existence of a static, spherically symmetric solution that admits an asymptotic expansion around the Schwarzschild metric
    Invoked when the authors state they work in a static spherically symmetric spacetime and expand the field equations in small couplings.

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Works this paper leans on

42 extracted references · 36 canonical work pages · 26 internal anchors

  1. [1]

    we have the derivatives of fR and fG give by Eq. ( 12). On the Schwarzschild background one has R(0) µν = 0, R (0) = 0, A 0(r) = 1 − 2M r , (A9) so that f (0) R = 1 + β G(0), f (0) G = 2γG(0). (A10) 16 Moreover, the Gauss–Bonnet invariant is G(0) = 48M 2 r6 . (A11) Therefore, at first order the higher-curvature sector enters through the effective source ten...

  2. [2]

    Modified Gravity and Cosmology

    T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, Ph ys. Rept. 513, 1 (2012), 1106.2476

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  3. [3]

    Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models

    S. Nojiri and S. D. Odintsov, Phys. Rept. 505, 59 (2011), 1011.0544

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  4. [4]

    Extended Theories of Gravity

    S. Capozziello and M. De Laurentis, Phys. Rept. 509, 167 (2011), 1108.6266

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  5. [5]

    A. A. Ara´ ujo Filho, N. Heidari, I. P. Lobo, and V. B. Bezer ra, JCAP 09, 015 (2025), [Erratum: JCAP 01, E01 (2026)], 2505.08718

    work page arXiv 2025

  6. [6]

    String-inspired Gauss-Bonnet gravity reconstructed from the universe expansion history and yielding the transition from matter dominance to dark energy

    G. Cognola, E. Elizalde, S. Nojiri, S. Odintsov, and S. Ze rbini, Phys. Rev. D 75, 086002 (2007), hep-th/0611198

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  7. [7]

    Introduction to Modified Gravity and Gravitational Alternative for Dark Energy

    S. Nojiri and S. D. Odintsov, eConf C0602061, 06 (2006), hep-th/0601213

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  8. [8]

    Non-singular exponential gravity: a simple theory for early- and late-time accelerated expansion

    E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastiani, an d S. Zerbini, Phys. Rev. D 83, 086006 (2011), 1012.2280

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  9. [9]

    f(R) theories

    A. De Felice and S. Tsujikawa, Living Rev. Rel. 13, 3 (2010), 1002.4928

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  10. [10]

    B. Li, J. D. Barrow, and D. F. Mota, Phys. Rev. D 76, 044027 (2007), 0705.3795

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  11. [11]

    Construction of cosmologically viable f(G) gravity models

    A. De Felice and S. Tsujikawa, Phys. Lett. B 675, 1 (2009), 0810.5712

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  12. [12]

    Vacuum structure for scalar cosmological perturbations in Modified Gravity Models

    A. De Felice and T. Suyama, JCAP 06, 034 (2009), 0904.2092

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  13. [13]

    Gauss-Bonnet dark energy

    S. Nojiri, S. D. Odintsov, and M. Sasaki, Phys. Rev. D 71, 123509 (2005), hep-th/0504052

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  14. [14]

    The geometry of photon surfaces

    C.-M. Claudel, K. S. Virbhadra, and G. F. R. Ellis, J. Mat h. Phys. 42, 818 (2001), gr-qc/0005050

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  15. [15]

    Perlick, Living Rev

    V. Perlick, Living Rev. Rel. 7, 9 (2004)

    2004

  16. [16]

    First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole

    K. Akiyama et al. (Event Horizon Telescope), Astrophys . J. Lett. 875, L1 (2019), 1906.11238

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  17. [17]

    First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way

    K. Akiyama et al. (Event Horizon Telescope), Astrophys . J. Lett. 930, L12 (2022), 2311.08680

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  18. [18]

    A. E. Broderick, T. Johannsen, A. Loeb, and D. Psaltis, A strophys. J. 784, 7 (2014), 1311.5564

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  19. [19]

    Psaltis, L

    D. Psaltis et al. (Event Horizon Telescope), Phys. Rev. Lett. 125, 141104 (2020), 2010.01055

    work page arXiv 2020

  20. [20]

    Battista, Phys

    E. Battista, Phys. Rev. D 109, 026004 (2024), 2312.00450

    work page arXiv 2024

  21. [21]

    Nojiri and S

    S. Nojiri and S. D. Odintsov, Phys. Dark Univ. 47, 101785 (2025), 2412.13775

    work page arXiv 2025

  22. [22]

    Wang and E

    Z.-L. Wang and E. Battista, Eur. Phys. J. C 85, 304 (2025), 2501.14516

    work page arXiv 2025

  23. [23]

    Banerjee, S

    I. Banerjee, S. Chakraborty, and S. SenGupta, Phys. Rev . D 101, 041301 (2020), 1909.09385

    work page arXiv 2020

  24. [24]

    Modified Gauss-Bonnet theory as gravitational alternative for dark energy

    S. Nojiri and S. D. Odintsov, Phys. Lett. B 631, 1 (2005), hep-th/0508049

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  25. [25]

    J. L. Synge, Mon. Not. Roy. Astron. Soc. 131, 463 (1966)

    1966

  26. [26]

    Chandrasekhar, Fundam

    S. Chandrasekhar, Fundam. Theor. Phys. 9, 5 (1984)

    1984

  27. [27]

    A. K. Mishra, S. Chakraborty, and S. Sarkar, Phys. Rev. D 99, 104080 (2019), 1903.06376

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  28. [28]

    Nojiri and S

    S. Nojiri and S. D. Odintsov, Phys. Dark Univ. 46, 101702 (2024), 2410.11530

    work page arXiv 2024

  29. [29]

    Capozziello, S

    S. Capozziello, S. De Bianchi, and E. Battista, Phys. Re v. D 109, 104060 (2024), 2404.17267

    work page arXiv 2024

  30. [30]

    J. M. Bardeen, Proceedings, Ecole d’Et´ e de Physique Th ´ eorique: Les Astres Occlus : Les Houches, France, August, 1 972, 215-240 pp. 215–240 (1973)

    1973

  31. [31]

    K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 62, 084003 (2000), astro-ph/9904193

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  32. [32]

    Gravitational lensing in the strong field limit

    V. Bozza, Phys. Rev. D 66, 103001 (2002), gr-qc/0208075

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  33. [33]

    Strong gravitational lensing --- A probe for extra dimensions and Kalb-Ramond field

    S. Chakraborty and S. SenGupta, JCAP 07, 045 (2017), 1611.06936

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  34. [34]

    S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (John Wiley and Sons, New York, 1972), ISBN 978-0-471-92567-5, 978-0-4 71-92567-5

    1972

  35. [35]

    Capozziello, E

    S. Capozziello, E. Battista, and S. De Bianchi, Phys. Re v. D 112, 044009 (2025), 2507.08431

    work page arXiv 2025

  36. [36]

    K. D. Kokkotas and B. G. Schmidt, Living Rev. Rel. 2, 2 (1999), gr-qc/9909058

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  37. [37]

    Quasinormal modes of black holes and black branes

    E. Berti, V. Cardoso, and A. O. Starinets, Class. Quant. Grav. 26, 163001 (2009), 0905.2975

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  38. [38]

    R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys. 83, 793 (2011), 1102.4014

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  39. [39]

    Geodesic stability, Lyapunov exponents and quasinormal modes

    V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and V. T. Za nchin, Phys. Rev. D 79, 064016 (2009), 0812.1806

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  40. [40]

    I. Z. Stefanov, S. S. Yazadjiev, and G. G. Gyulchev, Phys . Rev. Lett. 104, 251103 (2010), 1003.1609

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  41. [41]

    A. A. Ara´ ujo Filho, N. Heidari, and F. S. N. Lobo, Eur. Ph ys. J. Plus 141, 446 (2026), 2510.25973

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  42. [42]

    Nojiri and S

    S. Nojiri and S. D. Odintsov, Phys. Dark Univ. 46, 101669 (2024), 2408.05668

    work page arXiv 2024