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Quasinormal modes and greybody factors of charge black hole in bumblebee gravity model

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Pith's one-line read For a charged de Sitter black hole in bumblebee gravity, increasing the Lorentz-violation parameter L consistently lowers the perturbation barrier, raises the greybody bound, and slows the quasinormal ringing, giving an observational…

desk verdict A useful but sloppy extension of bumblebee black hole perturbations to charged dS/AdS; the scalar/EM and shadow parts are mostly sound, but internal contradictions and an unverified Dirac reduction sink the current version. read the letter →

arxiv 2506.02508 v1 pith:6JPBXFUP submitted 2025-06-03 hep-ph

Yenshembam Priyobarta Singh , Jayasri Choudhury , Telem Ibungochouba Singh , Dhruba Jyoti Gogoi This is my paper
classification hep-ph MSC 83C5783C4783D05 PACS 04.70.-s11.30.Cp04.62.+v
keywords bumblebeegravityLorentzsymmetryviolationReissner-Nordström-deSitterblackholequasinormalmodesgreybodyfactorsshadowWKBapproximationDiracperturbation
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 extends the perturbation analysis of black holes in bumblebee gravity to charged Reissner–Nordström spacetimes with a cosmological constant, covering scalar, electromagnetic, and Dirac fields. Its central claim is that the Lorentz-violation parameter $L$ and the charge $Q$ systematically control the height of the effective potential barrier between the event and cosmological horizons: for the de Sitter (RNdS) case, raising $L$ lowers the barrier for all three perturbations, while raising $Q$ raises it. Through Visser–Boonserm rigorous bounds, greybody factors inherit this dependence, so a larger $L$ lets more Hawking radiation reach a distant observer; sixth-order WKB with Padé improvement then shows that larger $L$ reduces both the oscillation and damping of quasinormal modes, while larger $Q$ amplifies them. The same parameters also shrink the photon-sphere and shadow radii, offering an observational route to testing Lorentz violation in strong gravity.

What carries the argument

The carrying object is the bumblebee-modified metric (1)–(2), in which the Lorentz-violation parameter $L$ enters both the $g_{rr}$ component (factor $1+L$) and the metric function $A(r)$; the tortoise coordinate $dr_* = \frac{\sqrt{1+L}}{A}dr$ turns all three field equations into one-dimensional Schrödinger form. The effective potentials $V_s$, $V_e$, $V_{d\pm}$ are the load-bearing quantities, since the paper's greybody and QNM statements are read directly off their peak heights and shapes. The greybody part uses the Visser–Boonserm rigorous bound $T\ge \operatorname{sech}^2\!\left(\frac{1}{2\omega}\int \mathcal{P}\,dr_*\right)$ specialized to $h=\omega$ and to two-horizon de Sitter scattering. The QNM part uses the 6th-order WKB formula with Padé improvement, and the time-domain section uses the discretized integration scheme of Gundlach et al. The photon sphere and shadow analysis uses the null geodesic potential and the shadow formula $R_s = r_p/\sqrt{A(r_0) A(r_p)}$.

What would settle it

Take $M=1$, $Q=0.2$, $\Lambda=0.05$, $\ell=1$ and compute the scalar QNM frequencies at $L=0$ and $L=0.6$ with an independent method such as the continued-fraction approach; if $\mathrm{Re}\,\omega$ or $|\mathrm{Im}\,\omega|$ increases rather than decreases with $L$, the claimed monotonicity fails. Likewise, restore the absolute value in the bound and evaluate Eq. (34) with the full effective potential for $m=0.1$; if the bound no longer rises monotonically with $L$, the greybody claim is an artifact of the simplification.

Watch

Extended reading notes

Core claim

The paper establishes, for the RNdS-like bumblebee black hole with metric function $A(r)=1-\frac{2M}{r}+\frac{2(1+L)}{2+L}\frac{Q^2}{r^2}-\frac{1}{3}(1+L)r^2\Lambda$, that the effective potentials for scalar ($V_s$), electromagnetic ($V_e$) and Dirac ($V_d$) perturbations all decrease as the Lorentz-violation parameter $L$ increases, while they all increase as charge $Q$ increases. From the resulting greybody bounds, $T\ge \operatorname{sech}^2\!\left(\frac{\sqrt{1+L}}{2\omega}\int_{r_h}^{r_c} \frac{V}{A}\,dr\right)$, it follows that greybody factors rise with $L$ and fall with $Q$. Quasinormal frequencies computed at 6th-order WKB with Padé approximation show the same monotonicity: larger $L$ lowers both the real frequency and the damping magnitude, larger $Q$ raises them, and the time-domain profiles corroborate this. For the RNAdS-like black hole, the effect of $L$ is perturbation-dependent: it lowers the potential for Dirac perturbations but raises it for scalar and electromagnetic ones. The paper also derives the photon sphere radius $r_p = \frac{3M}{2} + \frac{1}{2}\sqrt{9M^2 - \frac{16(1+L)Q^2}{2+L}}$, independent of $\Lambda$, and shows the shadow radius of the RNdS case decreases with both $L$ and $Q$.

Load-bearing premise

The greybody bound derivation assumes the effective potential is non-negative between the horizons after setting $h=\omega$ and dropping the absolute value in the Visser–Boonserm integrand, but the massive scalar and Dirac plots with $m=0.1$ and $\Lambda=0.05$ use masses below the critical value $m_c=\sqrt{2\Lambda/3}\approx 0.1826$, where the potential is not positive throughout.

Editorial extensions

If this is right

  • If $L$ is positive, the RNdS bumblebee black hole rings at a lower frequency and damps more slowly than its general-relativity counterpart with the same $M$, $Q$, and $\Lambda$, and the effect grows with $L$ across scalar, electromagnetic, and Dirac perturbations.
  • Larger $L$ raises the rigorous greybody bound, meaning a larger fraction of Hawking radiation escapes to infinity and the distant spectrum carries a Lorentz-violation signature.
  • Larger $Q$ does the opposite in every channel, so charge and Lorentz violation can in principle be disentangled in ringdown and greybody observables.
  • The photon-sphere radius is independent of $\Lambda$ but shrinks with $L$ and $Q$, and the RNdS shadow radius shrinks with both, connecting the perturbation observables to black hole shadow measurements.
  • For the RNAdS-like black hole, constraints on $L$ must be field-dependent: scalar and electromagnetic potentials react oppositely to Dirac potentials as $L$ grows.

Reading between the lines

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

  • The paper's pattern — lower barrier implies higher greybody bound and lower QNM frequency — is natural but not proven analytically; one could test the same monotonicity with a continued-fraction or direct-integration code for a single parameter set to check whether the WKB/Padé result is an artifact of the approximation.
  • Since the eikonal (large-$\ell$) limit of QNMs is usually tied to the photon sphere, the reported decrease of $r_p$ with $L$ predicts that the real part of high-$\ell$ QNMs should decrease with $L$; comparing that prediction with the paper's low-$\ell$ tables would provide a consistency test.
  • The rigorous-bound derivation drops the absolute value in the Visser–Boonserm integrand; restoring it, or restricting the analysis to $m>m_c$, could soften the claimed monotonic rise of the greybody factor with $L$ for massive scalar fields.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. This manuscript studies scalar, electromagnetic, and Dirac perturbations of charged, cosmological-constant black holes in bumblebee gravity, using the metric of Eqs. (1)-(2). It derives effective potentials, applies the Visser-Boonserm rigorous bound to obtain greybody factors, computes quasinormal frequencies with 6th-order WKB and Padé approximations, presents time-domain evolutions, and discusses photon-sphere and shadow radii. The central claims are that the Lorentz-violation parameter L and charge Q control the effective potential heights, greybody bounds, and QNM frequencies in a systematic way, with L decreasing the potential for RNdS black holes and Q increasing it according to the body of the paper, although the abstract states the opposite for Q.

Significance. If the calculations were correct, the paper would provide a useful extension of bumblebee-gravity phenomenology by comparing three perturbation channels and relating QNMs, greybody factors, and shadow observables. The manuscript has some tangible strengths: it reports WKB and Padé results with error estimates, includes time-domain profiles, and gives an analytic photon-sphere formula. However, several load-bearing statements are internally contradictory, the Dirac decoupling that underlies the fermion results is not derived, and the greybody-bound applications violate the stated validity conditions. In its current form the manuscript cannot be used as a reliable source for the claimed parameter dependences.

major comments (5)
  1. [Abstract vs. Sections 3-4] The abstract states that increasing charge Q reduces the effective potential in all perturbations, but the body and figures show the opposite. For example, Figs. 4, 6, and 8 show that increasing Q from 0.1 to 0.6 raises the peak of the scalar, electromagnetic, and Dirac potentials for both RNdS and RNAdS black holes, and the text in Sections 3 and 4 states this explicitly. The conclusion also states that the height of the effective potentials increases with Q. The abstract must be corrected to match the actual results, and all summaries need to be checked for consistency.
  2. [Section 7, Table 1] The text in Section 7 says that increasing L increases both the real part and the absolute value of the imaginary part of the QNM frequencies for the RNdS-like black hole, but Table 1 shows the opposite for all three channels. For example, for the scalar field with l=1, n=0, the Padé value of Re(omega) decreases from 0.212355 to 0.179031 to 0.145461 as L goes from 0 to 0.3 to 0.6, and |Im(omega)| decreases from 0.0755182 to 0.0649739 to 0.0532054. The same decreasing trend is visible in the electromagnetic and Dirac rows. The sentence immediately after, which says increasing L 'prevents the rise' of oscillation and damping, contradicts the preceding sentence. This internal contradiction affects one of the central quantitative claims of the paper.
  3. [Section 4, Eqs. (23)-(29)] The reduction of the massive Dirac equation to the decoupled Schrödinger-like equations is not demonstrated. The text jumps from the coupled system (23) to the rotated variables (24), then to Eq. (25) with only 'After some calculations,' and then introduces the frequency-dependent coordinate r_hat* in Eq. (26) and the potential Vdm± in Eq. (29). Because r_hat* depends on omega and r, it is not shown that the QNM boundary conditions are preserved under this transformation, and the derivative dW±/dr_hat* in Eq. (29) is not derived. Since the Dirac rows of Tables 1-2 and the fermion greybody bounds both rely on Vdm±, the Dirac-sector results are unsupported without a complete derivation or a reference to a standard decoupling that applies to this metric.
  4. [Section 5, Eqs. (31)-(38)] The rigorous greybody bound is applied without the absolute value required when h=omega. In Eq. (32), the term (omega^2 - V - h^2)^2 becomes V^2 after setting h=omega, so Eq. (33) should contain |V|/(2omega), not V/(2omega). This is not a minor technicality: Figs. 10 and 13 use m=0.1 with Lambda=0.05, which is below the paper's own critical mass mc=sqrt(2Lambda/3) ~ 0.1826, and Section 3 explicitly says that for m<mc the scalar effective potential has an additional zero and changes sign between the horizons. The bound in Eq. (35) is therefore applied outside its validity regime. In addition, Eq. (38) claims that the integral of |dW/dr*| equals W evaluated at the endpoints, which is only true if dW/dr* does not change sign and W has the same value at both endpoints; the absolute value makes this step invalid in general. The fermion greybody derivation in Section 5.2 needs to be redone with the absolute value and with parameters satisfying the positivity condition.
  5. [Section 5.2.2, Eq. (46)] Equation (46) is dimensionally inconsistent. In natural units, omega, m, and r have inverse-length and length dimensions respectively, while kappa is a dimensionless angular eigenvalue. The argument of the sech function contains the factor (rc-rh) kappa^2/(rc rh), which has dimension 1/length, multiplied by the parenthesis (1 + m^2/(kappa^4 rc rh)), in which the second term has dimension 1/length^4. The overall argument therefore does not reduce to a dimensionless number, so Eq. (46) cannot be a valid expression for a transmission bound. The approximation leading from Eq. (43) to Eq. (46) also needs to be re-examined, since Eq. (43) defines mu_tilde = m/kappa but Eq. (46) contains m^2/kappa^4 in a different combination.
minor comments (5)
  1. [Eq. (36)] The denominator '3Q^2(2+L)' in the last term appears to be a typo: the Q^2 factor should cancel, leaving 3(2+L), since the integral of the charge term is proportional to Q^2 divided by (2+L).
  2. [Eq. (29)] There is a typo in the derivative term where 'k^2_+' appears instead of 'kappa_+^2', and the displayed form of Vdm± would benefit from being typeset in a single equation rather than split with an inline equals sign.
  3. [Eq. (54)] The condition for an unstable circular photon orbit should be V''(r_p) < 0, not V''(r_p) = 0; the text writes all three conditions as equalities, which is the condition for an inflection point rather than a maximum. Only Eq. (55) is used in the subsequent derivation, so this does not affect the photon-sphere radius, but the mathematical statement is incorrect.
  4. [Figures 19-20 captions] The caption of Fig. 19 says 'The parameters are taken as L=0.2' while the figure varies L; this should presumably read Q=0.2. The same issue appears in the caption of Fig. 20, which says L=0.2 while varying Q.
  5. [Section 6, Figure captions] Several greybody comparison figures, such as Figs. 14(b), 15(b), and 16(b), are labeled with Lambda=-0.05 even though they compare SdS-like and RNdS-like black holes that require a positive cosmological constant; the captions and the text should be made consistent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the charged-bumblebee QNM, greybody, and shadow results are computed from an externally imported metric using standard perturbation methods, with no fitted parameter or self-citation used to force the central claims.

full rationale

The central derivation chain is self-contained. The RNdS/RNAdS-like bumblebee metric, Eqs. (1)-(2), is imported from the external reference [52]; the scalar and electromagnetic effective potentials, Eqs. (8)-(10), are obtained by separating the Klein-Gordon and Maxwell equations under the tortoise coordinate (6); the Dirac potential, Eq. (29), is presented as a reduction of the Dirac equation via the Chandrasekhar transformation (24); and the photon-sphere and shadow results, Eqs. (55)-(58), follow algebraically from the null-geodesic conditions. No parameter is fitted to the QNM frequencies, greybody bounds, or shadow radii, and the WKB/Pade and rigorous-bound computations are standard independent methods applied to those potentials. The self-citations [44, 54, 69] appear only as background comparisons (for example, that Schwarzschild/SdS bumblebee photon radii are L-independent) and are not load-bearing for the new charged results. Two rigor concerns are noted but are not circularity: the Dirac decoupling between Eqs. (24) and (29) is asserted with 'After some calculations' and would need independent verification, and several figures use m=0.1 with Lambda=0.05, below the paper's own critical mass m_c = sqrt(2*Lambda/3) ~ 0.183, which can invalidate the positivity assumption in the greybody bound (33)-(34). These are matters of correctness and validity, not reductions of the output to the input by construction.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No numbers are fitted to data; the central claims are parameter scans on an imported metric. The main unprovided inputs are the metric solution itself, the linearity assumption, positivity for the greybody bound, and the unshown Dirac decoupling.

free parameters (1)
  • Time-domain initial wave packet parameters (k1, sigma) = not specified
    The time evolution profiles in Section 8 and Figs. 19-20 depend on the chosen Gaussian initial data width and median, which are not reported, making those profiles non-reproducible as published.
assumptions (4)
  • domain assumption Bumblebee metric (1)-(2) is a valid charged black hole solution with cosmological constant in Einstein-bumblebee gravity.
    This line element is imported from Ref. [52], and all effective potentials, QNMs and shadows are computed on this background.
  • domain assumption Linear test-field perturbation on the fixed background is valid; backreaction is negligible.
    Standard for QNM calculations but never stated explicitly in the paper.
  • domain assumption The Visser-Boonserm bound (31) applies with h=omega and with the absolute value in the integrand omitted when integrating over the two-horizon region.
    The paper uses Eq. (33) directly, but this requires positive V between the horizons; the plotted massive scalar case has m=0.1 below m_c, so the positivity condition is not met.
  • ad hoc to paper The Chandrasekhar-type decoupling (19)-(24) reduces the Dirac equation to the form (28) with the stated frequency-dependent potential.
    The reduction is compressed into 'After some calculations' and is central to the Dirac QNM results; the resulting potential contains omega in the denominator, which is unusual and not discussed.

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Cite this review

Pith. "Pith review of Quasinormal modes and greybody factors of charge black hole in bumblebee gravity model." pith.science (2026). https://pith.science/paper/6JPBXFUP

@misc{pith2026250602508,
  author       = {Pith},
  title        = {Pith review of: Quasinormal modes and greybody factors of charge black hole in bumblebee gravity model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6JPBXFUP}},
  note         = {Machine review of arXiv:2506.02508}
}
abstract

In this paper, we investigate the Dirac field, scalar field and electromagnetic field perturbations of Reissner-Nordstorm-de Sitter (RNdS) and Reissner-Nordstorm-(anti)-de Sitter (RNAdS)-like black holes within the frame work of Einstein-bumblebee gravity. The effective potential, greybody factor and quasinormal modes (QNMs) are also explored by using Dirac equation, Klein-Gordon equation and Maxwell's equation. We find that for RNdS-like black hole increasing the Lorentz violation parameter $L$ consistently leads to decrease in the effective potential for all types of perturbations but for RNAdS case the influence of $L$ varies depending on the types of perturbation. Further for both RNdS and RNAdS-like black holes, increasing charge $Q$ reduces the effective potential in all the perturbations. The greybody factors of all the types of perturbations are also discussed. The results show that the greybody factors depend on the shape of the effective potential: higher (lower) potentials gives lower (higher) greybody factors. The QNMs frequencies of RNdS-like black hole for the massless field perturbations are discussed by using 6th order WKB approximation and Pad\'e approximation. We also analyze the time-domain profiles of the perturbations. The effects of Lorentz violation parameter $L$ and charge $Q$ to the photon sphere radius and shadow radius are also discussed. It is noted that increasing $Q$ and $L$ reduce the rise of shadow radius for RNdS-like black hole.

Figures

Figures reproduced from arXiv: 2506.02508 by the authors.

Figure 1
Figure 1. Variation of metric function for different values of Q with fixed (a) M = 1, Λ = 0.05 and L = 0.2 (b) M = 1, Λ = −0.05 and L = 0.2 . L=0.1 L=0.3 L=0.6 0 1 2 3 4 5 6 7 -0.2 -0.1 0.0 0.1 0.2 0.3 r A (a) L=0.1 L=0.3 L=0.6 0 2 4 6 8 10 -4 -2 0 2 4 6 r A (b) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Variation of metric function for different values of L with fixed (a) M = 1, Λ = 0.05 and Q = 0.2 (b) M = 1, Λ = −0.05 and Q = 0.2 . 3 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Variation of the effective potential for the massive scalar field for RNdS-like black hole for different values of the mass parameter of the field m. The physical parameters are chosen as M = 1, Λ = 0.05, L = 0.2, ℓ = 1 and Q = 0.2. It is noted from the above equation that s = 0 and s = 1 give the effective potential associated with the scalar field perturbation and electromagnetic field perturbation respectively. s… view at source ↗
Figures from the paper (21 more)
Figure 4
Figure 4. Figure 4: Variation of the effective potential for the massive scalar field for different values of Q. The physical parameters are chosen as (a) M = 1, Λ = 0.05, L = 0.2, ℓ = 1 and m = 0.1 (b) M = 1, Λ = −0.05, L = 0.2, ℓ = 1 and m = 0.1. L=0.1 L=0.3 L=0.6 0 2 4 6 8 -0.01 0.00 0…
Figure 5
Figure 5. Figure 5: Variation of the effective potential for the massive scalar field for different values of L. The physical parameters are chosen as (a) M = 1, Λ = 0.05, Q = 0.2, ℓ = 1 and m = 0.1 (b) M = 1, Λ = −0.05, Q = 0.2, ℓ = 1 and m = 0.1. and neutrinos, under extreme gravitation…
Figure 6
Figure 6. Figure 6: Variation of the effective potential for the electromagnetic field for different values of Q. The physical parameters are chosen as (a) M = 1, Λ = 0.05, ℓ = 1 and L = 0.2 (b) M = 1, Λ = −0.05, ℓ = 1 and L = 0.2. L=0.1 L=0.3 L=0.6 0 2 4 6 8 10 -0.01 0.00 0.01 0.02 0.03 …
Figure 7
Figure 7. Figure 7: Variation of the effective potential for the electromagnetic field for different values of L. The physical parameters are chosen as (a) M = 1, Λ = 0.05, ℓ = 1 and Q = 0.2 (b) M = 1, Λ = −0.05, ℓ = 1 and Q = 0.2. where ebν;α = ∂αebν − Γ β αν. We choose the tetrad from E…
Figure 8
Figure 8. Figure 8: Variation of the effective potential for the massive Dirac field for different values of Q. The physical parameters are chosen as (a) M = 1, Λ = 0.05, L = 0.2, ℓ = 1 and m = 0.1 (b) M = 1, Λ = −0.05, L = 0.2, ℓ = 1 and m = 0.1. L=0.1 L=0.3 L=0.6 1 2 3 4 5 6 0.00 0.05 0…
Figure 9
Figure 9. Figure 9: Variation of the effective potential for the massive Dirac field for different values of L. The physical parameters are chosen as (a) M = 1, Λ = 0.05, Q = 0.2, ℓ = 1 and m = 0.1 (b) M = 1, Λ = −0.05, Q = 0.2, ℓ = 1 and m = 0.1. may be calculated by the so-called greybo…
Figure 10
Figure 10. Figure 10: Variation of the Greybody factor for the massive scalar field (a) for di [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: Variation of the Greybody factor for the electromagnetic field (a) for di [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 12
Figure 12. Figure 12: Variation of the Greybody factor for the massless Dirac field (a) for di [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]
Figure 13
Figure 13. Figure 13: Variation of the Greybody factor for the massive Dirac field (a) for di [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]
Figure 14
Figure 14. Figure 14: Variation of (a) effective potential for the scalar field, with fixed M = 1, Λ = 0.05, ℓ = 1 and L = 0.2 (b) greybody factor for the scalar field with fixed M = 1, Λ = −0.05, ℓ = 1 and L = 0.2. S-like SdS-like RN-like RNdS-like 1 2 3 4 5 6 0.00 0.05 0.10 0.15 0.20 r V…
Figure 15
Figure 15. Figure 15: Variation of (a) effective potential for the Dirac field, with fixed M = 1, Λ = 0.05, ℓ = 1 and L = 0.2, (b) greybody factor for the Dirac field, with fixed M = 1, Λ = −0.05, ℓ = 1 and L = 0.2. 7. Quasinormal modes (QNMs) Quasinormal modes are solutions of the wave eq…
Figure 16
Figure 16. Figure 16: Variation of (a) effective potential for the electromagnetic field, with fixed M = 1, Λ = 0.05, ℓ = 1 and L = 0.2, (b) greybody factor for the electromagnetic field with fixed M = 1, Λ = −0.05, ℓ = 1 and L = 0.2. hole incompatible with WKB approximation method. Theref…
Figure 17
Figure 17. Figure 17: Variation of real and imaginary parts of the quasinormal frequencies for scalar, electromagnetic and Dirac perturbations for di [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]
Figure 18
Figure 18. Figure 18: Variation of real and imaginary parts of the quasinormal frequencies for scalar, electromagnetic and Dirac perturbations for di [PITH_FULL_IMAGE:figures/full_fig_p019_18.png]
Figure 19
Figure 19. Figure 19: Time domain profile of (a) scalar field perturbation, (b) electromagnetic field perturbation and (c) Dirac field perturbation for RNdS-like [PITH_FULL_IMAGE:figures/full_fig_p020_19.png]
Figure 20
Figure 20. Figure 20: Time domain profile of (a) scalar field perturbation, (b) electromagnetic field perturbation and (c) Dirac field perturbation for RNdS-like [PITH_FULL_IMAGE:figures/full_fig_p021_20.png]
Figure 21
Figure 21. Figure 21: Plot of effective potential for null geodesic for different values of charge Q (a) with fixed L = 0.2, M = 1 and Λ = 0.05. (b) with fixed L = 0.2, M = 1 and Λ = −0.05. L=0.1 L=0.3 L=0.6 2.0 2.5 3.0 3.5 4.0 4.5 5.0 -0.005 0.000 0.005 0.010 0.015 0.020 r V (a) L=0.1 L=0…
Figure 22
Figure 22. Figure 22: Plot of effective potential for null geodesic for different values of charge L (a) with fixed Q = 0.2, M = 1 and Λ = 0.05. (b) with fixed Q = 0.2, M = 1 and Λ = −0.05. where rp represents the photon sphere radius at r = rp. One may also define the critical impact para…
Figure 23
Figure 23. Figure 23: Plot of the shadow radius varying r0 for (a) RNdS-like black hole (Λ = 0.05) (b) RNAdS-like black hole (Λ = −0.05) with fixed L = 0.3, Q = 0.3 and M = 1. L=0 L=0.3 L=0.6 L=0.9 -4 -2 0 2 4 -4 -2 0 2 4 Y X (a) Q=0 Q=0.3 Q=0.6 Q=0.9 -4 -2 0 2 4 -4 -2 0 2 4 Y X (b) [PITH…
Figure 24
Figure 24. Figure 24: Plot of the shadow radius of RNdS-like black hole for di [PITH_FULL_IMAGE:figures/full_fig_p024_24.png]

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