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Supermassive black hole in NGC 4649 (M60) with a dark matter halo: Impact on shadow measurements and thermodynamic properties

T0 review · 6 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The dark matter halo of NGC 4649 changes its black hole's shadow and thermodynamic behavior.

desk verdict Two worked DM-halo black hole metrics for NGC 4649 with a thorough thermodynamic package, but the Model-1 photon-sphere equation has a factor-3/2 error that puts the photon sphere on the horizon, and the Model-2 surface gravity is off by two orders of magnitude. read the letter →

arxiv 2505.03661 v1 pith:TXWWXEAI submitted 2025-05-06 gr-qc astro-ph.HEhep-th

Francisco S. N. Lobo , Jorde A. A. Ramos , Manuel E. Rodrigues This is my paper · ORCID
classification gr-qcastro-ph.HEhep-th MSC 83C5783C20 PACS 04.70.Bw95.35.+d
keywords blackholeshadowdarkmatterhaloNGC4649SchwarzschildSmarrformulathermodynamicsphotonspheregeneralrelativity
topics Dark Matter
open problems Dark Matter
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

The paper claims that a supermassive black hole sitting inside the dark matter halo of NGC 4649 cannot be treated as an isolated Schwarzschild hole: the halo changes where the event horizon sits, how strong the curvature is near the hole, how large the observed shadow is, and how mass, entropy, temperature, and surface gravity are related. Two static, spherically symmetric metrics are built from the same observationally fitted halo density profile, and both reduce to Schwarzschild when the halo is switched off. The stakes are concrete: the second model predicts a shadow radius per unit mass that grows with the halo's circular velocity, so a sharp enough shadow measurement could become a probe of dark matter in the strong-field regime. The paper also extracts Smarr-type mass formulas and first laws from both solutions, treating the halo parameters as genuine thermodynamic variables.

What carries the argument

The load-bearing object is the cored dark matter density profile of Eq. (2), which carries the observed halo parameters (core radius $a$ and circular velocity $V_c$) into both metrics; everything else hangs on how that profile is fed into the field equations. For the first model, the machinery is the construction of [39], which adds a correction $F_1(r)$ to the Schwarzschild metric function and fixes it through the mass function $m(r)=V_c^2 r^3/(a^2+r^2)$, yielding $F_1(r)=\gamma(a^2+r^2)^{V_c^2}-2M/r$. For the second model, the machinery is the approach of [49], which adds the perfect-fluid dark matter term $2V_c^2 r^2/(a^2+r^2)$ to $1-2M/r$. The shadow calculation is carried by the photon-sphere condition $d(r^2/f(r))/dr=0$ and the observer rescaling $r_{\rm sh}=r_{\rm ph}\sqrt{F(r_O)/F(r_{\rm ph})}$, while the Smarr formulas come from treating the mass as a homogeneous function of entropy and the halo variables.

What would settle it

Measure the shadow of the NGC 4649 black hole (or another supermassive black hole with a well-fitted halo) with horizon-scale interferometry and compare $r_{\rm sh}/M$ to Eqs. (38) and (65): a shadow that decreases with $V_c$, or that lies outside the predicted $1\sigma/2\sigma$ band for the quoted halo parameters, would falsify the halo-induced growth in the second model. Even without an image, an independent probe of the dark matter density at $r\sim 10^{12}$-$10^{13}$ m, for instance from stellar or pulsar orbits, that disagrees with the extrapolated cored profile would remove the foundation of both metrics.

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Extended reading notes

Core claim

The central claim is that the halo profile $\rho_{\rm DM}(r)=\frac{V_c^2}{4\pi G}\frac{3a^2+r^2}{(a^2+r^2)^2}$, fitted by the cited authors to Hubble Space Telescope, stellar-dynamical, and globular-cluster data for NGC 4649, can be substituted directly into Einstein's equations to produce two exact Schwarzschild-type metrics: $F_1(r)=\gamma(a^2+r^2)^{V_c^2}-2M/r$ for the first solution and $F_2(r)=1-2M/r+2V_c^2 r^2/(a^2+r^2)$ for the second. In the regime $r\ll a$ and $V_c\ll 1$, the paper derives approximate analytic expressions for the photon sphere, event horizon, and shadow radius; the shadow-radius-to-mass ratio remains inside the $1\sigma$ and $2\sigma$ constraints from Sagittarius A* observations for both data sets. The second model's shadow ratio increases monotonically with $V_c$, which the authors read as a possible observational handle on the halo. On the thermodynamic side, the paper derives mass-entropy-temperature relations, two new halo potentials in each model ($A_\gamma$ and $A_a$ in the first; $A_{V_c}$ and $A_a$ in the second), and Smarr formulas $M=2TS-2V_c^2\gamma A_\gamma+aA_a$ and $M=2TS+aA_a$, together with the corresponding first laws.

Load-bearing premise

The density profile of Eq. (2) is measured at galactic scales (core radius $a$ roughly 10 kpc), but the paper uses its $r\ll a$ limit to build the metric at the horizon and photon sphere, about ten million times closer in, where the dark matter distribution is not constrained by the cited data.

Editorial extensions

If this is right

  • In the second model the shadow-radius-to-mass ratio $r_{\rm sh}/M$ increases monotonically with the halo circular velocity $V_c$, so a horizon-scale image of a black hole with an independently known mass and halo would constrain $V_c$.
  • Both solutions pass the Sgr A* shadow constraints at $1\sigma$ and $2\sigma$ for the quoted parameter intervals, meaning a dark halo of this type is not ruled out by current shadow data.
  • The first model introduces $\gamma$ as a third model parameter coupled to the observer distance; distant observers see a slightly larger shadow than near-horizon observers because $\gamma_O\approx r_O^{-2V_c^2}$ while $\gamma_c\approx a^{-2V_c^2}$.
  • As entropy grows, the halo-corrected mass and temperature both approach $(1+2V_c^2)$ times their Schwarzschild values, so the halo's thermodynamic imprint does not vanish at large horizon areas.
  • Both models violate the classical third law in the limit $M\to 0$ because the surface gravity vanishes, and the paper attributes this to the absence of quantum and Hawking-evaporation corrections.

Reading between the lines

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

  • If the same construction is applied to other galaxies with measured halo parameters, the ratio $r_{\rm sh}/M$ should carry a specific, $V_c$-dependent shift; comparing two or more supermassive black holes would separate halo effects from deviations in the gravity theory itself.
  • The use of the kpc-scale cored profile at horizon scales is an extrapolation; a dark matter spike, annihilation, or baryonic inflow near the hole would alter $F(r)$ there and could change the predicted shadow more than $V_c$ does. Testing the inner profile with pulsar timing or lensing would sharpen the prediction.
  • The Smarr formulas suggest that halo parameters ($\gamma$, $a$, $V_c$) behave like thermodynamic state variables; extending the mass function to rotation and charge could produce phase-transition structure analogous to anti-de Sitter black holes, but that extension is not 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.

Referee Report

6 major / 5 minor

Summary. The paper constructs two static, spherically symmetric Schwarzschild-type metrics intended to describe a black hole embedded in the dark-matter halo of NGC 4649, using the density profile of Eq. (2) from Shen and Gebhardt (2010) and two representative parameter sets, Data I and Data II. For each metric the authors compute the event horizon, Kretschmann scalar, photon-sphere radius, shadow radius, and a set of thermodynamic quantities including entropy, temperature, Smarr-type relations, and surface gravity. The shadow predictions are compared with the Sgr A* constraints of Vagnozzi et al. (2023), and the paper concludes that the halo alters both the geometry and the thermodynamics of the black hole, with the second model showing a monotonic increase of r_sh/M with the halo velocity V_c.

Significance. If the calculations were correct, the paper would provide a concrete example of how a galactic dark-matter halo could leave an imprint on black-hole shadow and thermodynamic observables. The paper is clearly organized and contains explicit derivations of the entropy-mass relations and of the Smarr formulas via homogeneity arguments. However, the central shadow calculation contains an algebraic factor error, the printed shadow formula does not reduce to the Schwarzschild limit, the second model's photon-sphere equation is not the condition it claims to be, and the reported surface-gravity values for the second model are inconsistent with direct evaluation. These are not local typos: they affect the quantitative predictions advertised as the main results. The manuscript is therefore not publishable in its present form.

major comments (6)
  1. [IV.A.3, Eq. (35)] Substituting Eq. (16) into the photon-sphere condition Eq. (32) gives gamma[(a^2+r^2)^{V_c^2} - V_c^2 r^2 (a^2+r^2)^{V_c^2-1}] = 3M/r, not 2M/(gamma r). The missing factor 3/2 propagates: with the r<<a approximation, Eq. (36) yields r_ph = 2M a^{2V_c^2}/gamma, which is exactly the horizon radius of Eq. (37), contradicting the stated requirement r_h < r_ph. The numerical test at the end of Section IV.A.3 compares Eq. (36) with a numerical solution of the same erroneous Eq. (35), so it cannot validate the approximation. With the corrected factor, r_ph = 3M a^{2V_c^2}/gamma; for Data I and gamma0=1.1 this gives r_ph/M ~ 2.73 and r_sh/M ~ 4.5, outside the quoted 1-sigma interval. Eq. (38), Fig. 3, and the claimed halo-induced change of the shadow therefore rest on an algebraic error.
  2. [IV.A.3, Eq. (38)] Eq. (38) does not reduce to the Schwarzschild shadow radius in the limit claimed in the text. Setting V_c=0 and gamma=1 gives r_sh = 3M * sqrt(1/3) = sqrt(3) M, whereas the text states that this limit yields 3 sqrt(3) M. In addition, the right-hand side contains powers of a and M that do not combine into a quantity with well-defined length units. The expression is therefore not a reliable shadow formula even after the factor error in Eq. (35) is corrected.
  3. [IV.B.3, Eq. (63)] Eq. (63) is not the photon-sphere condition (32) for the metric in Eq. (23); it is the horizon condition F(r)=0. The correct photon-sphere condition is 1 - 3M/r + 2 V_c^2 r^4/(a^2+r^2)^2 = 0. The subsequent statement that 'considering r << a, we have r_ph ~ 3M' does not follow from the printed Eq. (63), which would give r ~ 2M at leading order. Although the value r_ph ~ 3M happens to be the result of the correct equation in the r << a limit, the derivation as written is internally inconsistent and needs to be corrected.
  4. [IV.B.4.b, Eq. (75)] The surface-gravity values obtained from Eq. (75) are inconsistent with a direct evaluation of kappa = (1/2) F'(r_h) using the metric function of Eq. (59). For Data I, with r_h = 1.034 x 10^13 m and M_I = 5.17 x 10^12 m, the direct evaluation gives kappa ~ 4.8 x 10^-14, close to the Schwarzschild value 1/(4M_I), whereas the paper reports kappa_I = 3.79 x 10^-16, roughly a factor of 100 smaller. Eq. (75) and Fig. 16 therefore do not represent the surface gravity of the second solution, and the discussion of halo effects on kappa is unsupported.
  5. [II and IV] The density profile of Eq. (2) is constrained by stellar and globular-cluster kinematics on scales of order a ~ 10 kpc. The strong-field quantities computed here, such as the horizon, photon sphere, and shadow at r ~ 10^13 m ~ 10^-7 pc, require extrapolating this cored profile many orders of magnitude inward, where the cited data provide no constraint. Processes such as a dark-matter spike, annihilation, or baryonic contamination could completely change the metric in this region. The paper should either justify this extrapolation or present the near-horizon results as conditional on an assumed extrapolation.
  6. [IV.A.3 and Fig. 3] In the first model, gamma is a free integration constant with no independent determination. The text uses the Sgr A* shadow bounds to define allowed gamma intervals and then treats agreement with those same bounds as evidence for the model. This is circular: the observation is used both to constrain gamma and to validate the shadow prediction. Because the shadow ratio depends sensitively on the chosen gamma0=1.1, the claim of a halo-induced shadow modification in the first model is not a parameter-free prediction.
minor comments (5)
  1. [Abstract] The sentence 'The second incorporates halo effects through V_c=0 and a=0' appears to have an inverted condition; it should presumably read 'through V_c and a'.
  2. [IV.A.3, Eqs. (36)-(38)] The factors a^{-2V_c^2} involve non-integer powers of a length, so gamma carries a V_c-dependent dimension; the paper nevertheless sets gamma0=1.1 as if it were dimensionless. The unit convention for gamma should be stated explicitly.
  3. [IV.A.3] The reported deviations 1 - r_ph/r_ph,n are given without error bars, and they compare with a numerical solution of Eq. (35); they should not be described as supporting the physical approximation when the base equation is itself incorrect.
  4. [IV.B.3 and Fig. 11] The inset label 'mBH' is not defined; the black-hole mass should be denoted M consistently throughout the figure and text.
  5. [Throughout] There are typographical inconsistencies, e.g., 'Schwarszchild' in the caption of Fig. 4; these should be corrected in a revision.

Circularity Check

1 steps flagged · score 6.0 of 10

First-model shadow validation scans free parameter γ against Sgr A* bounds; otherwise the construction and the second-model prediction are open.

  1. fitted input called prediction [Section IV A 3, Black Hole Shadow Radius (Eqs. (38)-(42) and Fig. 3)]
    "The behavior of the shadow radius, as described in Eq. (38), is analyzed as a function of the values of the constant γ, as shown in Fig. 3. For both pairs of values, there is an interval of γ within the valid range (blue areas) that includes γ0. This shows that γ can be treated as a new parameter in addition to Vc and a. The γ intervals obtained for r_sh/M are 0.934408≤γ_I≤1.23404 and 0.934282≤γ_II≤1.23387 for Data I and II, respectively. In fact, the model agrees with the 1σ and 2σ intervals, which supports its validity and the approximations made for r_ph."

    Eq. (38) gives r_sh/M as a function of γ. Eqs. (39)-(42) impose the observed Sgr A* bounds on r_sh/M, so solving Eq. (38) for γ defines the 'valid range'; Fig. 3 is then plotted for γ values already restricted to that range. Reporting this as 'the model agrees with the 1σ and 2σ intervals' is an inversion of the same formula: the data are used to select γ, and the agreement is therefore built into the construction. The parameter γ is not independently predicted: γ0=1.1 is set by hand, while γ_c and γ_O are only asymptotic consistency choices. Hence this shadow 'validation' reduces to a free-parameter consistency check rather than a parameter-free prediction.

full rationale

The only load-bearing self-reference in the paper is the first-model shadow comparison. Writing Eq. (38) as r_sh/M(γ), the Sgr A* bounds in Eqs. (39)-(42) are substituted to solve for the allowed γ intervals; those intervals are then drawn as the 'valid range' in Fig. 3 and the overlap with the hand-picked γ0=1.1 is announced as the model's agreement with the 1σ and 2σ intervals. This is an inversion of the same formula, so that particular validation is by construction; it is effectively a free parameter fitted to the shadow bound and then called support for the model. The rest of the derivation chain is not circular: the density profile (2) comes from an external kinematic fit, the metric constructions in Eqs. (16) and (23) are explicit, the Smarr/first-law relations are Euler-theorem identities on the mass functions, and the second-model shadow increase with Vc (Eq. (65), Fig. 11) is an independent consequence of the externally fitted halo parameters. Self-citations are not load-bearing. The separate factor-3/2 mismatch between Eq. (32) and Eq. (35) would be a correctness defect, not a circularity. Overall: partial circularity confined to one 'prediction', hence 6.

Assumptions & free parameters 8 free parameters · 7 assumptions · 0 invented entities

Everything the central claim rests on that the reader did not pay for upstream. The metric construction starts from an empirical halo profile fitted in [47]; the free parameters V_c, a, M are inputs from that fit. The first metric adds an integration constant gamma that the paper renormalizes by hand (gamma_0=1.1) and later constrains by Sgr A* shadow data. No new entities are introduced; the load-bearing extras are the strong-field extrapolation of the profile, the f=g metric ansatz, the applicability of the cited construction formalisms, and the transfer of Sgr A* bounds to an NGC 4649 model.

free parameters (8)
  • gamma_0 (first solution) = 1.1
    Hand-chosen constant used in Figs. 1-7; not derived from data or from the limit gamma->gamma_c. It sets the horizon shift and is later allowed to vary over [0.934,1.234] to match Sgr A* shadow bounds.
  • gamma (shadow-constrained) = Data I: [0.934408,1.23404]; Data II: [0.934282,1.23387]
    Integration constant in the first metric treated as a free parameter; its allowed range is set by requiring r_sh/M to lie inside the Sgr A* Keck+VLTI 1-sigma and 2-sigma intervals (Eqs. 39-42).
  • V_c (Data I) = 13.68e-4 (geometrized)
    Critical tangential velocity of the DM halo fitted to HST, stellar dynamics, and globular cluster data of NGC 4649 in [47]; input parameter for both metrics.
  • V_c (Data II) = 18.35e-4 (geometrized)
    Second data set value from [47] for the critical velocity.
  • a (Data I) = 30.86e19 m
    Core radius of the DM halo from [47]; entering the density profile and both metrics.
  • a (Data II) = 46.29e19 m
    Core radius from [47] for the second data set.
  • M (Data I) = 5.17e12 m
    Black hole mass from [47] for Data I.
  • M (Data II) = 6.65e12 m
    Black hole mass from [47] for Data II.
assumptions (7)
  • domain assumption The DM density profile from Shen and Gebhardt (2010), rho(r)=V_c^2(3a^2+r^2)/(4*pi*G*(a^2+r^2)^2), holds at all radii down to the horizon.
    Used in Section III for both metrics; the profile is fitted to HST, stellar and globular cluster data at kpc scales, and extrapolating it into the strong-field region is an unconstrained assumption.
  • domain assumption The metric is static and spherically symmetric with f(r)=g(r), i.e., ds^2=-F(r)dt^2+F(r)^{-1}dr^2+r^2dOmega^2.
    Stated in Section II: 'throughout this work, we assume f(r)=g(r)'. This restricts the class of solutions and is not derived from the field equations.
  • standard math The construction recipes of Xu et al. (2018) and Ma et al. (2024) produce valid Einstein equations with the DM energy-momentum tensor as source.
    The paper relies on these cited formalisms without re-derivation; errors or hidden assumptions in those methods propagate into the present results.
  • ad hoc to paper Sgr A* shadow bounds from Vagnozzi et al. (2023) can be transferred to a model whose halo parameters are those of NGC 4649.
    Used in Sections IV A 3 and IV B 3; no argument is given that the two systems share the same halo-modified spacetime law, and the bounds are far broader than the predicted effect.
  • domain assumption Euler homogeneity with exponents c_1=1, c_2=-V_c^2, c_3=1/2 gives a physically meaningful Smarr relation and first law for these spacetimes.
    The Smarr formulas (53) and (73) follow from formal scaling of M(S,gamma,a) without derivation from asymptotic charges or boundary terms; the interpretation of A_gamma and A_a as thermodynamic potentials is assumed.
  • standard math The Kretschmann scalar formula K=f''^2+4f'^2/r^2+4(f-1)^2/r^4 is correct for f(r)=g(r).
    Used in Eq. (3) as the basis for the curvature analysis; it is a standard textbook result.
  • domain assumption The r<<a and V_c<<1 limits are valid for the near-horizon and photon-sphere calculations, and the density profile's main effect in that regime is a constant gamma*a^{2V_c^2}.
    Imposed in Eqs. (35)-(37) for the shadow analysis; the resulting approximation r_ph=r_h indicates the approximation or equation is mishandled.

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Pith. "Pith review of Supermassive black hole in NGC 4649 (M60) with a dark matter halo: Impact on shadow measurements and thermodynamic properties." pith.science (2026). https://pith.science/paper/TXWWXEAI

@misc{pith2026250503661,
  author       = {Pith},
  title        = {Pith review of: Supermassive black hole in NGC 4649 (M60) with a dark matter halo: Impact on shadow measurements and thermodynamic properties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TXWWXEAI}},
  note         = {Machine review of arXiv:2505.03661}
}
abstract

We investigate black hole (BH) solutions embedded in a dark matter (DM) halo, modeled as extensions of the Schwarzschild metric. The DM density profile is constrained by Hubble Space Telescope data, stellar dynamics, and globular cluster (GC) measurements of the elliptical galaxy NGC 4649 (M60). Using this profile, we construct two distinct spacetime solutions characterized by the black hole mass ($M$), critical velocity ($V_c$), and core radius ($a$), all reducing to the Schwarzschild case in the limit $V_c=0$ and $a=0$. Our results show that the DM halo modifies essential BH features, such as the event horizon radius and spacetime curvature, as reflected in the Kretschmann scalar. We also derive an approximate analytical expression for the BH shadow radius, which increases slightly due to the halo's influence. Comparisons with two observational datasets further validate the analysis. Thermodynamic properties are examined across the two models. In the first, a generalized Smarr formula is obtained via two additional variables, $\gamma$ and $a$. The second incorporates halo effects through $V_c=0$ and $a=0$. These results underscore the role of DM in shaping both geometric and thermodynamic aspects of BHs.

Figures

Figures reproduced from arXiv: 2505.03661 by the authors.

Figure 1
Figure 1. Graphical representation of F(r), Eq. (24), for the parameter sets: (VcI , aI , MI , γ0) given as Data I; (VcII , aII , MII , γ0) given as Data II; and the solution of the Schwarzschild-type without halo (0, 0, M0, 1). rhII = 1.209×1013 m, and r (sch) h = 1.27×1013 m, respec￾tively for the data sets Data I, Data II and for solutions of Schwarzschild-type, i.e. without a halo [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 3
Figure 3. Graphical representation of the ratio rsh/M from Eq. (38) for values of γ within the measurement inter￾vals corresponding to 1σ and 2σ. For the parameter sets: (VcI , aI , MI , γ), given by Data I, (VcII , aII , MII , γ), given by Data II, and the solution of type SS without halo (0, 0, M0, 1). The subfigure of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. shows the behavior of mass in relation to en￾tropy. Although in general M2 ∝ S is conserved due to the low velocity, the intensity of the entropy decreases with the presence of the halo at the same mass value. This reflects the response to the reduction of rh since S ∝ r 2 h . However, the mass is changed by the presence of the halo [47]. These relationships indicate that the BH with halo reacts with a lower entropy… view at source ↗
Figures from the paper (9 more)
Figure 5
Figure 5. Figure 5: Graphical representation of the temperature [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Graphical representation of Aγ(S) − Aγ|0,0 from Eq. (48), where Aγ|0,0 refers to the values Vc = 0 and a = 0. For the values: (VcI , aI ) given by Data I and (VcII , aII ) given by Data II. Solutions Data I Data II 2× 1042 4× 1042 6× 1042 8× 1042 1× 1043 5.0× 10-7 1.0×…
Figure 7
Figure 7. Figure 7: Graphical representation of Aa(S) from Eq. (49). For the values: (VcI , aI , γ0) given by Data I and (VcII , aII , γ0) given by Data II. transformation M(S, γ, a) → M(q c1 S, qc2 γ, qc3 a), i.e M = q c2+c3+2c3V 2 c γ √ S 2 √ π  a 2 + S π V 2 c , (50) where c1 = 2c3. …
Figure 9
Figure 9. Figure 9: Graphical representation of F(r) from Eq. (59), for the following values: (VcI , aI , MI ) given by Data I, (VcII , aII , MII ) given by Data II, and for the Schwarzschild solution without halo (0,0,M0). as defined with respect to the metric in Eq. (23). The correspond…
Figure 11
Figure 11. Figure 11: shows the ratio rsh/M depending on the speed range described by [47], see Tab. I. The slight variation of the shadow is a reaction to the high values of a and the low values of Vc. The subfigure in [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]
Figure 12
Figure 12. Figure 12: Graphical representation of M(S)/M(sch)−1, from Eq. (66). For the parameter sets: (VcI , aI ), given by Data I, (VcII , aII ), given by Data II, and the Schwarzschild solution without a halo (0,0). To obtain the temperature, we use T = ∂M ∂S , from Eq. (66). Thus, T(S…
Figure 13
Figure 13. Figure 13: Graphical representation of T(S)/T (sch) − 1, from Eq. (68). For the values: (VcI , aI ) given by Data I, and (VcII , aII ) given by Data II, and for the Schwarzschild so￾lution without a halo (0,0). a. Smarr Formula: To define the Smarr formula, we rewrite the mass f…
Figure 14
Figure 14. Figure 14: Graphical representation of AVc (S), from Eq. (70). For values: (VcI , aI ) given by Data I and (VcII , aII ) given by Data II. Solutions Data I Data II 2× 1042 4× 1042 6× 1042 8× 1042 1× 1043 -2.0× 10-6 -1.5× 10-6 -1.0× 10-6 -5.0× 10-7 0 S A a [PITH_FULL_IMAGE:figur…
Figure 15
Figure 15. Figure 15: Graphical representation of Aa(S), from Eq. (71). For values: (VcI , aI ) given by Data I and (VcII , aII ) given by Data II. where the contributions of the halo to the mass are em￾phasized by its velocity and its distance from the BH center. b. Surface Gravity: As in…

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Works this paper leans on

67 extracted references · 21 canonical work pages · cited by 1 Pith paper

  1. [1]

    With the first solution (16) F(r) =γ a2 +r 2 V 2 c − 2M r .(24) we have considerable difficulty in obtaining an analyti- cal solution

    Event Horizon The condition for determining the event horizonr h is the solution of the equationF(r h) = 0. With the first solution (16) F(r) =γ a2 +r 2 V 2 c − 2M r .(24) we have considerable difficulty in obtaining an analyti- cal solution. However, we may represent the behavior ofF(r) with respect torand calculater h using defined values for the parame...

    work page

  2. [2]

    (3) K=F ′′(r)2 + 4F′(r)2 r2 + 4[F(r)−1] 2 r4 .(26) Substitute Eq

    Kretschmann Scalar Assuming thatf(r) =g(r), so thatF(r) =G(r), the Kretschmann scalarKis obtained from Eq. (3) K=F ′′(r)2 + 4F′(r)2 r2 + 4[F(r)−1] 2 r4 .(26) Substitute Eq. (16), yeilds the following expression K= 4 r2 2γrV 2 c a2 +r 2 V 2 c−1 + 2M r2 2 + 4 r4 γ a2 +r 2 V 2 c − 2M r −1 2 + h 4γr 2V 2 c V 2 c −1 a2 +r 2 V 2 c−2 +2γV 2 c a2 +r 2 V 2 c−1 − 4...

    work page

  3. [3]

    Using the metric coefficient from Eq

    Black Hole Shadow Radius The existence of a region outside the BH containing a photon sphere allows an observer at distancerO to detect the shadow of this BH [48]. Using the metric coefficient from Eq. (1), and following the approaches outlined in [48, 53], we define a functionh(r) as h(r) = s r2 f(r).(30) The radius of the photon spherer ph can therefore...

    work page

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    (16) and using Eq

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