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Charged black hole solutions in $f(R,T)$ gravity coupled to nonlinear electrodynamics

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Pith's one-line read This paper claims that Eq. (22) is an exact static spherically symmetric charged black hole solution of f(R,T)=R+βT gravity coupled to the NLED Lagrangian L=f0+F+αF^p.

desk verdict Central exact-solution claim is unverified and fails a direct substitution as printed; otherwise a routine f(R,T)+NLED exercise with weak shadow constraints. read the letter →

arxiv 2412.00582 v2 pith:XWBUJDDY submitted 2024-11-30 gr-qc astro-ph.HEhep-th

Gabriel I. Róis , José Tarciso S. S. Junior , Francisco S. N. Lobo , Manuel E. Rodrigues , Tiberiu Harko This is my paper · ORCID
classification gr-qcastro-ph.HEhep-th MSC 83C5783D0583C50 PACS 04.70.-s04.50.Kd04.40.Nr
keywords f(RT)gravitynonlinearelectrodynamicschargedblackholesstaticsphericalsymmetryholeshadowSagittariusA*effectivecosmologicalconstanthorizonstructure
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 claims to construct a new family of exact, static, spherically symmetric charged black hole solutions in f(R,T) gravity with \(f(R,T)=R+\$\beta$ T\), sourced by nonlinear electrodynamics with Lagrangian \(L=f_0+F+\$\alpha$ F^p\). The central object is the metric function \(A(r)=1-\frac{2M}{r}+\frac{$q^{2}$}{$r^{2}$}-\frac{2}{3}(2\$\beta$+1)f_0 $r^{2}$+\frac{$2^{{1-p}}$}{3-4p}\$\alpha$[2\$\beta$(p-1)-1]$q^{{2p}}$$r^{{2-4p}}$\), which the authors present as an exact solution for a purely magnetic charge \(F_{23}=q\sin\$\theta$\). The solution carries an effective cosmological constant \(\Lambda_{\rm eff}=2(2\$\beta$+1)f_0\), so the spacetime approaches a de Sitter-like geometry at large distances while remaining singular at the origin. For \(p=2,4,6\) the authors give explicit metrics, analyze the horizon structure (including critical masses and charges with up to four horizons), and reconstruct the NLED Lagrangian in terms of \(F\), finding that the nonlinearity persists even when \(\$\alpha$=0\) because the f(R,T) coupling \(\$\beta$\) itself generates a nonlinear term. The paper also uses the observed shadow size of Sagittarius A* to set allowed ranges on \(\$\beta$\), \(\$\alpha$\), and \(p\).

What carries the argument

The engine of the argument is the power-law NLED Lagrangian \(L=f_0+F+\$\alpha$ F^p\) combined with the linear f(R,T) choice \(f(R,T)=R+\$\beta$ T\) and the magnetic ansatz \(F_{23}=q\sin\$\theta$\); substituting these into the field equations (13)-(15) under the assumption \(B(r)=A(r)\) yields the metric function (22). The consistency relations \(L_F=(\partial L/\partial r)(\partial F/\partial r)^{-1}\) and \(L_{FF}=(\partial L_F/\partial r)(\partial F/\partial r)^{-1}\) are then used to reconstruct the Lagrangian, and the effective metric \(g_{\mu\nu}^{\rm eff}=L_F g_{\mu\nu}-L_{FF}F_{\mu\$\sigma$}F^\$\sigma${}_\nu\) carries the shadow calculation.

What would settle it

Solve the full field equations (13)-(15) without imposing \(B(r)=A(r)\), for example by integrating the difference of the \(tt\) and \(rr\) components for a general \(B(r)\); if the claimed metric survives only under the imposed equality, or if a distinct branch with \(B(r)\neq A(r)\) changes the horizon counts or shadow radius, the central claim is settled.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the system consisting of \(f(R,T)=R+\$\beta$ T\) gravity and magnetic nonlinear electrodynamics has an exact spherically symmetric black hole solution with metric function (22), obtained under the symmetry \(B(r)=A(r)\). The solution reduces to a Reissner-Nordström type solution with an effective cosmological constant when \(\$\alpha$=0\), and to Schwarzschild when \(q=0\), \(\Lambda_{\rm eff}=0\), and \(\$\alpha$=0\). For \(p=2,4,6\), the Kretschmann scalar diverges as \(r\to 0\) and tends to \(8\Lambda_{\rm eff}^2/3\) at infinity, and the horizon equations produce critical masses and charges separating configurations with one, two, three, or four horizons. A second, independent route fixes the first and second derivatives \(L_F(r)\) and \(L_{FF}(r)\) from the field equations and integrates them, yielding the same Lagrangian \(L_{\rm NLED}(F)=f_0+F+\$\alpha$ F^p+$2^{{1/(2\beta+1)}}$f_1 $F^{{1/(2\beta+1)}}$$q^{{-1/\beta-2}}$\), so the f(R,T) coupling can sustain nonlinear electrodynamics even with \(\$\alpha$=0\). Finally, the shadow radius computed from the effective metric (55) is compared with the observed Sagittarius A* shadow, constraining the model parameters.

Load-bearing premise

The load-bearing assumption is that the metric functions are equal, \(B(r)=A(r)\), before solving; the exact solution, horizon analysis, and shadow constraints all rely on it, and no theorem is given that forces this equality in f(R,T) gravity with these matter sources.

Editorial extensions

If this is right

  • For \(\beta=0\), the metric function reduces to the GR plus NLED charged black hole, so the family contains the standard limit and any deviations from GR are controlled by \(\beta\).
  • With \(p=2,4,6\), the Kretschmann scalar diverges at \(r=0\) and approaches \(8\Lambda_{\rm eff}^2/3\) as \(r\to\infty\), so the solutions are singular cores embedded in a de Sitter-like asymptotic geometry.
  • The horizon analysis predicts up to four horizons depending on \(M\) and \(q\), with critical values such as \(M_c=5.682\) for \(\alpha=0.5\), \(\beta=0.002\), \(f_0=0.001\), \(q=5.75\).
  • The shadow-radius comparison with Sagittarius A* permits \(0\le\beta\lesssim1.6\times10^4\) for \(p=2\), \(0\le\beta\lesssim3\times10^{10}\) for \(p=4\), and \(0\le\beta\lesssim7.8\times10^{16}\) for \(p=6\), with corresponding bounds on \(\alpha\) and \(p\).
  • The reconstructed Lagrangian (50) stays nonlinear when \(\alpha=0\) because the \(\beta\) term contributes \(F^{1/(2\beta+1)}\), so f(R,T) itself acts as a source of nonlinear electrodynamics.

Reading between the lines

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

  • Because the paper integrates the equations only under \(B(r)=A(r)\), solving the full system without this ansatz is a natural next step; a distinct two-function branch would change the horizon structure and the quoted shadow bounds.
  • The allowed ranges for \(\beta\) grow from \(10^4\) to \(10^{16}\) as \(p\) increases, so the shadow measurement mostly pins down combinations of \(p\), \(\beta\), and the effective cosmological constant rather than \(\beta\) alone; tighter future shadow measurements could break this degeneracy.
  • The equality of the two reconstruction strategies hints at an integrability condition in this theory; if it holds, any metric function satisfying it could be paired with an explicit NLED Lagrangian, generating further exact solutions without new assumptions.
  • Applying quasinormal-mode, thermodynamic, and accretion-disk tests to the shadow-compatible parameter ranges would clarify whether the multi-horizon configurations are dynamically stable and observationally distinguishable from general-relativistic black holes.
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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. The paper studies static, spherically symmetric black hole solutions in f(R,T)=R+βT gravity coupled to nonlinear electrodynamics with a power-law Lagrangian L=f0+F+αF^p and a purely magnetic charge. It claims to derive a one-parameter family of exact solutions, Eq. (22), specializes to p=2,4,6, computes Kretschmann scalars, discusses horizon structure, reconstructs the NLED Lagrangian from the field equations, and uses EHT Sgr A* shadow observations to constrain β, α, and p. The abstract and introduction state that the solutions are derived and that regularity is verified, although the printed Kretschmann scalars diverge at small r.

Significance. If Eq. (22) were a genuine exact solution, the paper would provide a new family of charged black holes in f(R,T) gravity with an effective cosmological constant and p-dependent corrections, and the shadow constraints could be a useful observational test of the model. Those are potentially valuable contributions. However, the manuscript does not show the derivation of Eq. (22), the direct substitution check against the independent field equation appears to fail, the reconstruction in Sec. III.B has dimensional and circularity problems, and the shadow constraints constrain a combination of f0 and β rather than β alone. These issues affect the central claims of the paper. I also note the absence of any machine-checkable derivation or reproducibility artifact to support the exactness statements.

major comments (5)
  1. [Sec. III.A, Eq. (22)] The central exact solution is not verified. With B(r)=A(r) and f(R,T)=R+βT, f_R=1, so Eqs. (13) and (14) are identical and Eq. (15) is an independent constraint. The paper states that Eq. (22) follows from the field equations but does not show the substitution. I performed a direct substitution for p=2 using F=q^2/(2r^4), L_F=1+αpF^(p-1), and L_FF=α(p-1)pF^(p-2) into Eq. (15) as printed: after clearing denominators the residual contains powers r^0, r^-4, and r^-8, whose coefficients vanish only for incompatible values β=-4/7, β=0, and β=4/13 unless α=f0=0. This indicates that Eq. (22) is not a solution of the field equations as written, or that the printed equations carry an unstated normalization/sign convention. Because every subsequent result uses Eq. (22), the authors must provide a full derivation or a computer-algebra verification, and must reconcile the normalization of the field equations with the action (1).
  2. [Sec. II.A and Eqs. (13)-(15)] The normalization of the field equations is ambiguous. The action (1) contains 2κ^2 L_NLED with κ^2=8π, but the field equation (3) and its components (13)-(15) have no κ^2 on the matter right-hand side. If κ^2=8π is intended, the right-hand sides of Eqs. (13)-(15) should be proportional to κ^2 L_NLED; if instead a different convention is used, it must be stated explicitly. This ambiguity propagates into Eq. (22) and into the reconstruction formulas (46)-(47), and it may be the source of the residual terms found in the substitution check.
  3. [Sec. III.B, Eqs. (46)-(47)] The reconstructed derivatives L_F(r) and L_FF(r) appear to be dimensionally inconsistent. In Eq. (46), for example, the numerator contains r[A'(r)+2 L_NLED(r)(2βr+r)] + A(r) - 1. The term r A'(r) is dimensionless, while r multiplied by 2 L_NLED(r)(2βr+r) has dimension of inverse length squared, so the two terms cannot be added unless a highly nonstandard unit convention is in force. The same issue appears in Eq. (47). A derivation of these formulas and a statement of the units used (including the role of κ^2) are needed before they can be used.
  4. [Sec. III.B, Eqs. (48)-(50)] The Lagrangian reconstruction is circular in an important sense. Eq. (48) is evaluated on the metric function (22), which was derived by assuming L=f0+F+αF^p. The recovered Lagrangian (50) therefore necessarily contains the input terms, and the claim that 'even if α=0, the Lagrangian remains nonlinear due to β' is not an independent result. To establish that β alone can generate a nonlinear electromagnetic Lagrangian, one would need a metric that is not constructed from a nonlinear Lagrangian input, or an explicit statement that the reconstruction is only a self-consistency check rather than a new derivation.
  5. [Sec. IV.B, Figs. 5-7 and Subsec. IV.B.1] The reported constraints on β are effectively constraints on the combination f0(1+2β), not on β itself. In the metric (22), the f0 term enters as Λ_eff r^2/3 with Λ_eff=2(2β+1)f0, and for p=4 and p=6 the α-dependent terms decay as r^-14 and r^-22, making them negligible at the photon-sphere and shadow scales. With f0 fixed to Λ/2=5×10^-42, the values β_max≈1.6×10^4, 3×10^10, and 7.8×10^16 correspond simply to different effective cosmological constants. The paper states that β is constrained and that Λ_eff 'increases with β,' but no independent constraint on β is obtained. A joint constraint in terms of physical dimensionless parameters (e.g., f0 M^2 and β) is required to support the observational conclusions.
minor comments (5)
  1. [Sec. IV.B and figures] The text states that the observer is at rO ∼ 8 Mpc, but Sagittarius A* is at ∼8 kpc and the figure captions say kpc. This should be corrected.
  2. [Sec. III.A.2, Eq. (38)] The large-F asymptotic expression for the p=4 Lagrangian is written as F+αF^2, but Eq. (36) has αF^4; Eq. (38) should be F+αF^4.
  3. [Abstract and Sec. V] The manuscript says the regularity of the solutions is verified, but the Kretschmann scalars in Eqs. (33), (39), and (45) diverge at small r. I recommend rephrasing to 'analyze the regularity' or 'check for curvature singularities' to avoid the impression that regular black holes are obtained.
  4. [Sec. III.A, p=6 text] There is a typo: 'the linear case is recieveed' should read 'the linear case is recovered.'
  5. [Sec. IV.B.3 and Fig. 9] The text says the shadow radius is analyzed 'within the range p∈[2,6],' but the manuscript presents only p=2,4,6. Please clarify whether p is treated as continuous or as three discrete values.

Circularity Check

1 steps flagged · score 6.0 of 10

The Lagrangian reconstructed in Sec. III.B is the input Lagrangian (19) recovered by inverting the same metric that was built from it; the exact-solution and shadow parts are otherwise self-contained.

  1. self definitional [Section III.B.1 (Strategy one), Eqs. (46)-(50); input Lagrangian Eq. (19) and metric Eq. (22)]
    "By considering the form of the function (18), along with the expressions (19)–(21) in the equations of motion, and assuming the symmetry B(r) = A(r), we derive the following metric function from the components of the field equations (13-15) ... By imposing f1 = 0, we recover the form of the Lagrangian given by (19)."

    Eq. (22) is obtained by solving the field equations with L = f0 + F + αF^p fixed as input. In Sec. III.B the same metric (22) is then substituted into the field equations to 'derive' L_F(r) and L_FF(r) and integrate them into L_NLED(F), Eq. (50). With f1 = 0, Eq. (50) returns exactly the input Lagrangian (19), term by term; the only new piece is the homogeneous integration term proportional to f1. Thus the reconstructed Lagrangian is a self-consistency check on the original ansatz, not an independent derivation. The claim that the nonlinearity is controlled by both α and β follows from the f1-term, but the f0 + F + αF^p part is recycled from the input.

full rationale

The paper's central exact-solution claim, Eq. (22), is not itself circular: it is proposed as a solution of the stated field equations for a fixed matter Lagrangian, and the shadow analysis in Sec. IV is an honest parameter constraint against external EHT observations rather than a prediction. The citation of the foundational f(R,T) paper [11], which includes two co-authors, is a normal reference to an independent external result and is not load-bearing circularity. The circular element is confined to Sec. III.B: the metric (22) was constructed from the assumed Lagrangian (19)-(21), and the reconstruction procedure then feeds this same metric back into the field equations to recover L_NLED(F). The recovered expression (50) reduces to the input Lagrangian (19) when the integration constant f1 is set to zero, so the 'derivation' of the general Lagrangian is a self-consistency check, not an independent result. This does not invalidate the solution (22), but it does mean the Lagrangian-reconstruction claim and the associated 'nonlinearity governed by β' conclusion are partially circular. Score 6 reflects one central 'prediction' reducing by construction, while the rest of the derivation chain is self-contained.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new fundamental entities. Its results depend on the f(R,T)=R+βT gravity action, the power-law NLED Lagrangian, the B=A metric ansatz, and the effective metric for photon propagation. The free parameters β, α, p, f0, f1, q, and M are either chosen by hand or fitted to the Sgr A* shadow data, with the shadow constraints being so loose that they barely limit the model.

free parameters (7)
  • β = 0 ≤ β ≲ 1.6e4 (p=2), ≲3e10 (p=4), ≲7.8e16 (p=6)
    Coupling constant in f(R,T)=R+βT; constrained from Sgr A* shadow radius, effectively fitted to the observed band.
  • α = 0 ≤ α ≲ 7 (p=2), ≲1.4 (p=4), ≲3.8 (p=6)
    Coupling constant in the NLED Lagrangian L=f0+F+αF^p; constrained from the same shadow comparison.
  • p = 2, 4, 6 (chosen)
    Power-law index in the NLED Lagrangian; selected a priori and then shown to be compatible with shadow data.
  • f0 = Λ/2 with Λ=1e-41
    Constant term in the NLED Lagrangian that produces the effective cosmological constant; chosen to set the cosmological scale.
  • f1 = 1e-60
    Integration constant in the reconstructed Lagrangian; set to an extremely small value in the shadow calculations without physical justification.
  • q (magnetic charge) = 0.5M (shadow analysis), 5.75 (horizon analysis)
    Magnetic charge parameter in the solution; chosen for illustrative plots and shadow computations.
  • M (mass) = 1 (shadow analysis)
    Mass parameter; normalized to unity for the shadow radius comparison.
assumptions (5)
  • domain assumption The f(R,T) field equations (3) with the coupling f_T(T_μν+Θ_μν) are assumed correct.
    The entire solution construction rests on this specific form of the modified Einstein equations from Harko et al. [11].
  • ad hoc to paper The metric ansatz B(r)=A(r) is imposed.
    This simplifying equality is not derived from a Birkhoff theorem; all solutions and shadows depend on it.
  • standard math The magnetic field ansatz F_{23}=q sinθ satisfies the modified Maxwell equation (2).
    For a purely magnetic field in spherical symmetry, the divergence of F^{μν} vanishes identically, so the equation is automatically satisfied.
  • domain assumption The effective metric (55) governs photon trajectories in nonlinear electrodynamics.
    The shadow calculation uses the effective metric from Novello et al. [114], an established result for NLED but not derived here.
  • ad hoc to paper The power-law Lagrangian L=f0+F+αF^p is assumed as the matter source.
    This particular functional form is chosen for tractability; it is not derived from any deeper principle.

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Cite this review

Pith. "Pith review of Charged black hole solutions in $f(R,T)$ gravity coupled to nonlinear electrodynamics." pith.science (2026). https://pith.science/paper/XWBUJDDY

@misc{pith2026241200582,
  author       = {Pith},
  title        = {Pith review of: Charged black hole solutions in $f(R,T)$ gravity coupled to nonlinear electrodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XWBUJDDY}},
  note         = {Machine review of arXiv:2412.00582}
}
abstract

In this work, we investigate static and spherically symmetric black hole solutions in $f(R,T)$ gravity, where $R$ is the curvature scalar and $T$ is the trace of the energy-momentum tensor, coupled to nonlinear electrodynamics (NLED). To construct our solutions, we adopt a linear functional form, $f(R,T) = R + \beta T$. In the limit $\beta = 0$, the theory reduces to General Relativity (GR), recovering $f(R,T) \approx R$. We propose a power-law Lagrangian of the form $\mathcal{L} = f_0 + F + \alpha F^p$, where $\alpha =f_0= 0$ corresponds to the linear electrodynamics case. Using this setup, we derive the metric functions and determine an effective cosmological constant. Our analysis focuses on specific cases with $p = 2$, $p = 4$, and $p = 6$, where we formulate analytic expressions for the matter fields supporting these solutions in terms of the Lagrangian as a function of $F$. Additionally, we verify the regularity of the solutions and study the structure of the event horizons. Furthermore, we examine a more specific scenario by determining the free forms of the first and second derivatives $\mathcal{L}_F(r)$ and $\mathcal{L}_{FF}(r)$ of the Lagrangean of the nonlinear electromagnetic field. From these relations, we derive the general form of $\mathcal{L}_{\text{NLED}}(r)$ using consistency relations. This Lagrangian exhibits an intrinsic nonlinearity due to the influence of two constants, $\alpha$ and $\beta$. Specifically, $\alpha$ originates from the power-law term in the proposed Lagrangian, while $\beta$ arises from the assumed linear function $f(R,T)$. The interplay of these constants ensures that the nonlinearity of the Lagrangian is governed by both $\alpha$ and $\beta$, rather than $\alpha$ alone.

Figures

Figures reproduced from arXiv: 2412.00582 by the authors.

Figure 2
Figure 2. The plot of the metric function A(r), described by Eq. (40), for the values α = 0.5, β = 0.002, f0 = 0.001 and q = 5.75. ditional solution for the critical mass. In this analysis, we obtained a critical mass value of Mc = 5.680, consid￾ering the constants α = −1.5, β = 0.02, f0 = 0.001, and q = 5.75. The behavior of the metric function (40) with respect to the radial coordinate r is illustrated in [PITH_FULL_IMAGE:… view at source ↗
Figure 3
Figure 3. The plot of the metric function A(r), described by Eq. (40), for the values α = −1.5, β = 0.02, f0 = 0.001 and q = 5.75. 5 10 15 20 25 30 -0.2 -0.1 0.0 0.1 0.2 0.3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. The plot of the metric function A(r), described by Eq. (40), for the values α = 0.45, β = 0.0019, f0 = 0.0009, and M = 10.0. (i) qe > qc: When the electric charge exceeds the crit￾ical value (qe > qc), the metric function reveals the pres￾ence of a single horizon. (ii) qe = qc: At the critical charge (qe = qc), the solu￾tion exhibits two event horizons. (iii) qe < qc: For electric charges below the critical threshol… view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: The shadow radius rsh for our black hole model (solid red curve), described by the metric function given in Eq.(22) with power p = 2, was calculated according to Eq. (59) for an observer located at rO ∼ 8000 kpc. The shaded regions in blue and light blue represent the …
Figure 6
Figure 6. Figure 6: The shadow radius rsh for our black hole model (solid red curve), described by the metric function given in Eq.(22) with power p = 4, was calculated according to Eq.(59) for an observer located at rO ∼ 8000 kpc. The shaded regions in blue and light blue represent the E…
Figure 8
Figure 8. Figure 8: For p = 4 and β = 1010, the shadow radius starts to deviate from the constraints imposed by the estimated value for Sgr A* when α ∼ 1.4. However, for values of α smaller than this, the shadow radius of our model remains consistent with the EHT observational limits for …
Figure 9
Figure 9. Figure 9: The shadow radius rsh for our black hole model (solid red, yellow and blue curve), described by the metric function given in Eq.(22), was calculated according to Eq.(59) for an observer located at rO ∼ 8000 kpc. The shaded regions in blue and light blue represent the E…

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Reference graph

Works this paper leans on

127 extracted references · 24 canonical work pages · cited by 2 Pith papers

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    (28) The Lagrangian in terms of the radial coordinate, Eq

    Solution with p = 2 For this solution, we consider p = 2 in the metric func- tion given by (24), resulting in the following form: A (r) = 1 − 2M r + q2 r2 − Λeff r2 3 − α (2β − 1) q4 10r6 . (28) The Lagrangian in terms of the radial coordinate, Eq. (25), for this case becomes LNLED (r) = f0 + q2 2r4 + αq4 4r8 . (29) 6 In principle, we can rewrite r(F ) fr...

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    With this imposition, the metric function (24) takes the following form: A(r) = 1 − 2M r + q2 r2 − Λef f 3 r2 − α [6β − 1] q8 104r14

    Solution with p = 4 We now consider the solution for p = 4. With this imposition, the metric function (24) takes the following form: A(r) = 1 − 2M r + q2 r2 − Λef f 3 r2 − α [6β − 1] q8 104r14 . (34) Thus, the Lagrangian in terms of the radial coordinate is LNLED (r) = f0 + q2 2r4 + αq8 16r16 . (35) By writing r(F ), we find that the Lagrangian is now des...

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    In this case, the metric function (24) takes the following form: A(r) = 1 − 2M r + q2 r2 − Λeff 3 r2 − α(10β − 1)q12 672r22

    Solution with p = 6 Finally, we consider p = 6. In this case, the metric function (24) takes the following form: A(r) = 1 − 2M r + q2 r2 − Λeff 3 r2 − α(10β − 1)q12 672r22 . (40) which provides the following Lagrangian LNLED (r) = f0 + q2 2r4 + αq12 16r24 . (41) If we write r(F ), we find that the Lagrangian is now described by: LNLED (F ) = Λeff 4β + 2 +...

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    Below, we illustrate the event horizon representations obtained by solving Eqs

    Horizons Next, we present the horizon graphs, focusing exclu- sively on the metric function with the power p = 6, as the graphical behavior for the other powers is analogous. Below, we illustrate the event horizon representations obtained by solving Eqs. (16) and (17) simultaneously, applied to the metric function given in Eq. (40). This approach allows u...

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    (46), into the consis- tency relation given by Eq

    Strategy one Subsequently, by substituting the derivative of the La- grangian density, as described in Eq. (46), into the consis- tency relation given by Eq. (11), we derived an expression that enabled us to determine the Lagrangian density in the following form: LNLED(r) = − r− 2 β −4 Z r β+2 β (rA′(r) + A(r) − 1) β dr + f1r− 2 β −4. (48) Substituting th...

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    M ” that appears in our metric to the mass. For this rea- son, we specify the shadow radius in units of the local mass “ M

    Strategy two In this second approach, we determine the Lagrangian in terms of the electromagnetic scalar, LNLED(F ), by em- ploying the second consistency relation, Eq. (12). The process is analogous to that described in the previous section, with the key difference being the initial use of Eq. (12) instead of Eq. (11). Specifically, by substitut- ing the...

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    5 we analyze the behavior of the shadow radius rsh of our model for the metric function (22) with power p = 2, i.e

    Constraining the β parameter In Fig. 5 we analyze the behavior of the shadow radius rsh of our model for the metric function (22) with power p = 2, i.e. Eq. (28). We consider the shadow size of the black hole Sgr A* as a function of the parameter β and follow the uncertainties given in Eqs. (53) and (54). The assumed values of the constants are: M = 1, q ...

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    Constraining the α parameter Similar to the procedure developed in the previous sec- tion, we now constrain the parameter α based on the data obtained by the Event Horizon Telescope (EHT) for Sgr A*, as illustrated in Fig. 8. We analyze the be- havior of the shadow radius rsh for the metric function of our model described by Eq. (22), considering the same...

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