Upper bound on the GUP parameter using the black hole shadow
Juliano C. S. Neves






nevesjcs@if.usp.br
Centro de Cincias Naturais e Humanas, Universidade Federal do ABC,
 Avenida dos Estados 5001, Santo Andr, 09210-580 So Paulo, Brazil






An upper bound on the parameter that provides a generalized uncertainty principle  (GUP) is obtained from the black hole shadow. With the aid of a recent constraint between regular black holes and the GUP parameter, it is indicated a relation between this parameter and the deviation from circularity of the black hole shadow. In the case of the recent announcement of the M87* results from the Event Horizon Telescope collaboration, a deviation from circularity ≲ 10% imposes a GUP parameter β_0 <10^90.     


04.70.Bw,04.20.Dw,04.70.Dy

Upper bound on the GUP parameter using the black hole shadow
    Juliano C. S. Neves
    June 27, 2019
============================================================





§ INTRODUCTION



The Event Horizon Telescope collaboration announced the first black hole image ever captured <cit.>. By
 using interferometry, the collaboration built the shadow of the supermassive black hole (M87*) at the center of the Messier 87 galaxy. As pointed out by the collaboration, the black hole shadow is well-described by general relativity by  adopting the Kerr metric in order to interpret the phenomenon. Moreover, the supermassive M87*—whose mass is  M=(6.5 ± 0.7 )× 10^9M_⊙ and is r_obs= (16.8 ± 0.8)Mpc distant from us—presents an
     almost circular shadow. Accordingly, the shadow of M87* indicates a deviation ≲ 10% from circularity. Even
      with the good agreement with the Kerr metric, options are not totally ruled out. Due to the uncertainty on both the
       rotation  parameter and observation angle, options to Kerr metric are still candidates for M87* shadow <cit.>. This article explores that avenue and applies the M87* parameters to Kerr-like objects, like rotating regular black holes.   

Shadows of black holes have been studied since the pioneer work of Synge <cit.>, in which the shadow of the
 Schwarz-schild  black hole was obtained. Bardeen <cit.> built the first shadow for a rotating black hole,
  namely,  for the Kerr metric. Since then, whether in the general relativity context or beyond, shadows of different 
  black holes have been published, like shadows for the Reissner-Nordstrm
    black hole <cit.>, the Kerr-Newman black hole <cit.>, black holes with a cosmological constant
     <cit.>, Kerr-Newman-NUT black hole <cit.>, Kerr-Newman-Kasuya black hole <cit.>, braneworld black holes <cit.>,
      rotating wormholes <cit.> and for regular black holes <cit.>, which are focused
       on this article. It is worth emphasizing that the results exposed here do not consider an accretion disk around the black
        hole. Accretion disks modify the shadow's silhouette, and models that consider such influences are being developed
         <cit.>. The results presented in this article can be used in future researches that include accretion disks, whether made of plasma or dark matter.    
 
As an alternative to singular black holes, regular black holes are solutions of the gravitational field equations without a
 singularity inside the event horizon. Once again, Bardeen was pionner when constructed the very first regular metric that describes a regular black hole <cit.>. The Bardeen black hole is spherical, i.e., a nonrotating black hole, and is described by a mass function that depends on the radial coordinate. Inside the event horizon, the Bardeen metric hides a de
 Sitter core instead of a Schwarzschild singularity. A de Sitter core inside the event horizon avoids either the point singularity of spherical black holes <cit.> or the ring singularity in geometries with axial symmetry <cit.>. As is well-known, the de
 Sitter core provides energy conditions violations in order to avoid consequences of the singularities theorems.[See Wald's book <cit.> for detailed studies on the theorems.] Then, the Bardeen regular black hole violates the strong energy condition, and, according to our work <cit.>, those with rotation ignore the weak energy condition.   
 
Besides a mass function that depends on the radial coordinate, regular black holes possess a mass function that depends on the parameter related directly to the regular geometry or the absence of a singularity. In Ayn-Beato and Garcia's work
 <cit.>, such a parameter is a charge, and the Bardeen regular black hole is conceived of as a solution of general
  relativity coupled to a nonlinear electrodynamics for those authors. However, recently, our work <cit.>
   suggested another interpretation to the Bardeen metric. By using a generalized uncertainty principle (GUP), we
    computed quantum corrections to the Hawking temperature for the Schwarzschild black hole. Thus, we pointed out
     that, at
     second order, Bardeen's regular black hole may be view as a quantum-corrected  Sch-warzschild black hole. This new
      interpretation led to a constraint between a metric parameter, namely, the parameter in the mass function that produces
       regularity, and the GUP parameter. With the aid of such a relation, the GUP parameter will be estimated using the
        black hole shadow.  The regular black holes used in this shadow calculation come from a class of rotating regular
         black holes with a cosmological constant published in our article <cit.>.
  
GUPs appear, for example, in candidates for quantum gravity theories and in the deformed quantum mechanics,
 where the canonical commutation relations are modified in order to provide a minimal length scale and,
 consequently, its contribution to empirical results, like in the modified hy-drogen-atom spectrum
  <cit.>, Lamb shift, Landau Levels and in the
       scanning tunneling microscope  <cit.>.[See Ref. <cit.>
for a review on GUPs.] The dimensionless  GUP  parameter (also called quantum gravity parameter)  deforms
 the Heisenberg relation, and with  β_0=0, where  β_0 is the mentioned parameter, Heisenberg's 
 uncertainty relation is recovered. There is a  debate on the value of  the GUP parameter <cit.>. Assuming that the GUP
   parameter is   β_0 ∼ 1 implies that effects of β_0 are hard or too small to be detected. But without such
    an a priori assumption, it is possible  to obtain upper bounds on the GUP parameter by using recent
     experiments. Like Ref. <cit.>, where the  upper bounds were built with the aid of the light deflection
      and perihelion precession,  and Ref. <cit.>, where gravitational waves  were adopted, in this work an option in
       the strong gravitational field regime (the shadow of M87*) is used in order to obtain an upper bound on β_0.
        In general, as we will see,
      gravitational systems offer worse upper bounds than quantum options, like the Lamb shift, Landau Levels or the
       scanning tunneling microscope used in Ref. <cit.>.  
  
 As I said, the deviation from circularity, reported by the collaboration, will be used in order to estimate the  GUP
  parameter. It is worth emphasizing that the class of rotating regular black holes used here and presented in Ref.
   <cit.> generalizes earlier Kerr-like regular solutions because it possesses a general mass function m(r) and
    a cosmological constant. The shadow of that class has its silhouette  presented here for the first time in the literature.
     With the M87* parameters, by assuming that the angle between the black hole rotation axis and the observer is 
     θ_obs=17^∘ (in agreement with observed jets supposedly aligned with the rotation axis <cit.>), it is indicated the β_0<10^90  as upper bound on the GUP parameter. 

The structure of this paper is as follows: in Section 2, the geodesic equations for the class of  rotating regular black holes   and equations that provide the shadow's silhouette were derived. In Section 3, two shadow's observables are indicated, oblateness and root-mean-square distance from the average radius of the shadow, which gives the deviation from circularity, such that the latter was computed in order to provide an upper bound on the GUP parameter in Section 4. In Section 5, the final remarks are made.[In this work, I adopted geometric units such that G=c=1, where G is the gravitational constant, and c is the speed of light in vacuum. For the evaluation of the GUP parameter, the M87* data were used, and, consequently, G and c were restored.]




§ THE SHADOW OF ROTATING REGULAR BLACK HOLES

Let us obtain the shadow of rotating regular black holes with a cosmological constant in this section. Firstly, the spacetime metric and its geodesic equations are shown, then the silhouette equations are computed.  



 §.§ Spacetime metric and geodesic equations

In  this article, shadows are obtained from a class of rotating regular black holes developed
 in Ref. <cit.>. The spacetime metric of that class—using the  Boyer-Lindquist coordinates—is given by

    ds^2    =     -1/Σ(Δ_r - Δ_θ a^2sin^2θ) dt^2
           - 2a/ΞΣ[(r^2 + a^2)Δ_θ - Δ_r] sin^2θ dt dϕ
           + Σ/Δ_rdr^2 + Σ/Δ_θdθ^2
            + sin^2θ/Ξ^2Σ[(r^2+a^2)^2Δ_θ-Δ_ra^2sin^2θ] dϕ^2,

where


    Δ_θ = 1 + Λ/3 a^2cos^2θ, Σ = r^2 + a^2cos^2θ, Ξ = 1 + Λ/3 a^2,

and

    Δ_r = (r^2 + a^2)( 1 - Λ/3r^2) - 2m(r)r.

The constant Λ is the cosmological constant, a is the rotation parameter and, for the class of regular black holes studied here, the black hole mass depends on the radial coordinate r, namely,

    m(r)=M(1+(r_0/r)^q)^-3/q.

The mass function (<ref>) provides black holes without a singularity.[Aspects of regular black holes
 with the mass function (<ref>) were studied: thermodynamics in Ref <cit.>, accretion of
  perfect fluids in Ref. <cit.>, and cosmic strings in Ref. <cit.>.] For different values of the integer q, we have well-known regular black holes. For example, for the spherical case, q=2 provides the Bardeen black hole
   <cit.>, and q=3 produces the Hayward regular metric <cit.>, but the mass function also gives
    axisymmetric regular black holes or Kerr-like black holes. The parameter M is the mass parameter (for large values of r, m(r)∼ M), and r_0 is a length parameter that provides regular metrics, parameter conceived of as a microscopical constant related to both the GUP parameter and the Planck length according to our work
 Ref. <cit.> (in Section 4 such a parameter will be briefly discussed).

Due to cosmological observations <cit.>, I will focus on the positive cosmological constant in this article. In this case, regarding r_0≪ M, the function Δ_r provides three roots: the inner horizon r_-, the event horizon r_+, and the cosmological horizon r_++.  Then, the spacetime structure reads

    r_-<r_+<r_++.
 

In order to construct the black hole shadows of the class of rotating regular black holes given by
 Eq. (<ref>), the geodesic equations are needed. Geodesics for the Kerr metric were obtained by Carter
  <cit.>, who showed the separability of the geodesic equations. Carter argued that a test particle in the Kerr
   spacetime  possesses four constants of motion along geodesics. Accordingly, one has the two Killing  vector fields ξ_t and ξ_ϕ with their respective constants, the mass of the test particle and the fourth constant, which is called Carter
    constant (indicated by K). Both the Kerr-anti-de Sitter and Kerr-de Sitter spacetimes present those same four constants  for test particles along geodesics, whether in the general relativity realm <cit.> or in the brane world, as we  indicated in Ref. <cit.>. The class of rotating regular black holes given by (<ref>) also presents these constants.     

The geodesic equations for the spacetime (<ref>) are obtained from the Hamilton-Jacobi equation:

    ∂ S/∂σ=1/2g^μν∂ S/∂ x^μ∂ S/∂ x^ν,

where σ is a parameter related to the affine parameter τ (i.e., τ = δσ, with δ playing the role of mass of the test particle along the geodesic). The function S is the Jacobi action, which is related to the generalized momentum through 

    p_μ≡∂ S/∂ x^μ.

From the two Killing vector fields ξ_t and ξ_ϕ given by the geometry with axial symmetry (<ref>), we have the particle's constants of motion E and L, namely, energy and angular momentum, respectively,

    p_t=-E        p_ϕ = L.

Following Carter, it is assumed that S may be written as

    S=±1/2δ^2 σ-Et+L ϕ+S_θ (θ)+S_r(r),
 
where plus and minus mean the de Sitter and anti-de Sitter cases, respectively,[From the condition g_μνẋ^μẋ^ν=±δ^2, the signal plus and minus are due to the norm of timelike vectors in de Sitter and
 anti-de Sitter spacetimes, respectively.] and expressions for S_θ and S_r will be omitted here. As is known in
  general relativity, from the Lagrangian ℒ, the generalized momentum is defined as p_μ≡∂ℒ/∂ẋ^μ= g_μνẋ^ν, where dot means ordinary derivative with
   respect to the parameter σ. Therefore, from that definition and Eq.(<ref>), substituted into Eq.
    (<ref>), one has the geodesic equations for the metric (<ref>), which in the coordinate basis are 

    Σṫ    =   (r^2+a^2)P/Δ_r-a/Δ_θ[aE sin^2θ-(1+Λ/3a^2)L], 
    
    Σṙ   =   √(ℛ), 
    
    Σθ̇    =   √(Θ),  
    
    Σϕ̇   =    aP/Δ_r-1/Δ_θ[aE-(1+Λ/3a^2)cosec^2θ L],

where Σ is given by Eq. (<ref>). The functions P, ℛ and Θ are written as

    P     =    (r^2+a^2)E - (1+Λ/3a^2)a L,
     
    
    ℛ    =     P^2-Δ_r(±δ^2r^2+K) ,
     
    
    Θ    =     Q - cos^2θ[ a^2(±Δ_θδ^2-E^2) 
    
    
           + (1+Λ/3a^2)^2cosec^2θ L^2] .

As I said, the parameter δ represents the mass of the particle along the geodesic, so δ=0 in the case studied here. The constant Q  is related to Carter's constant K, that is to say,

    Q = Δ_θK-[(1+Λ/3a^2)L-aE]^2,

which vanishes for equatorial orbits. 



 §.§ The shadow's silhouette

In particular, for the shadow phenomenon, only photons orbits, or null geodesics, will be adopted (δ=0). And as the cosmological context is considered in this article, the cosmological constant will be assumed positive, i.e., Λ >0. Following <cit.>, two new parameters are defined as

    ξ=L/E        η=K/E^2,

which are constants in the shadow's silhouette. Such a silhouette is given by unstable photon orbits with r=r_p constant outside the event horizon, i.e., r_p>r_+. In such orbits, photons may either fall into the black hole or escape to the observer position. Thus, according to geodesic equation ṙ, we have ℛ(r_p)=ℛ'(r_p)=0 in order to provide the unstable orbits (the symbol ' means derivative with respect to r). Typically, for a black hole with rotation  r_p=r_p- and r_p=r_p+ (with r_p-≤ r_p+), i.e., there are both a minimum and a maximum value for r_p, and the edge of the black hole shadow should be built for those values of r_p. That is, as we will see, the left and the right sides of the shadow can be different for rotating black holes due to the spacetime dragging. On the other hand, for the Schwarzschild black hole r_p-=r_p+=3M (in the Schwarzschild case, M is the black hole mass or the Arnowitt-Deser-Misner mass), and the shadow is perfectly symmetrical.  The equations that involve ℛ(r_p) and its derivative lead to

    η(r_p)=16r_p^2 Δ_r(r_p)/Δ'_r (r_p)^2,

and

    ξ (r_p)=(r_p^2+a^2)Δ_r'(r_p)-4r_pΔ_r(r_p)/Ξ a Δ'_r(r_p).

As we will see, these conserved quantities are part of the equations that draw the shadow.


  

It is worth emphasizing that the metric (<ref>) is not necessarily asymptotically flat, it is either
 asymptotically de Sitter or anti-de Sitter for Λ≠ 0. Thus, a given observer is not at infinity describing the
  black hole shadow, indeed he/she is in the domain of outer communication, which is the region defined in
   between the event horizon r_+ and the cosmological horizon r_++ in the de Sitter case. Therefore, the observer
    will be located at the point with coordinates (r_obs,θ_o), according to Fig. <ref>. In terms of the
     observer position, because of the axial symmetry, the black hole shadow depends only on the radial and
      polar coordinates. The coordinate θ_o stands for the observer angle, which is the position in which the shadow
       is observed in relation to the rotation axis, and the parameter r_obs is the distance from the black hole. Following
       <cit.>, it is adopted the orthonormal tetrad e_a^μ=(e_0^μ,e_1^μ,e_2^μ,e_3^μ) in order
        to describe the shadow silhouette. That is, the null congruences coming from r_p-≤ r ≤ r_p+ reaches the
         observer and are projected onto the tetrad, then the shadow phenomenon is described by using e_a^μ. Such a tetrad is written with the aid of the coordinate basis vectors (∂_t,∂_r, ∂_θ, ∂_ϕ), therefore we have

    e_0   =   (r^2+a^2)∂_t + aΞ∂_ϕ/√(Δ_rΣ)|_(r_obs,θ_o), 
    
    e_1   =   √(Δ_θ/Σ)∂_θ|_(r_obs,θ_o),
    
    e_2   =   -Ξ∂_ϕ+asin^2θ∂_t/√(Δ_θΣ)sinθ|_(r_obs,θ_o), 
    
    e_3   =   -√(Δ_r/Σ)∂_r|_(r_obs,θ_o).

According to Grenzebach et al. <cit.>, the direction of e_3 points toward the black hole (see  Fig.
 <ref>), and e_0 is the observer's four-velocity, then in this case the observer is not necessarily at rest. The
  tetrad is chosen such that e_0± e_3 is tangential to the principal null congruence direction for a metric like
   (<ref>).




The null congruence, that is to say, the light rays that come from the region defined by r_p, are conceived of as curves γ such that their tangent vectors are given by 

    γ̇=ṫ∂_t+ṙ∂_r+θ̇∂_θ+ϕ̇∂_ϕ

 in the coordinate basis. According to Ref. <cit.>, at the observer position, γ̇ is described by the tetrad e_a^μ, i.e.,

    γ̇=ζ(-e_0+sinαcosβ e_1+sinαsinβ e_2+cosα e_3  ),

where the new angles α and β are celestial coordinates as indicated in Fig. <ref>. The description of the black hole shadow is made by using these coordinates. The factor ζ is obtained from Eqs. (<ref>)-(<ref>) and the tetrad equations, such that

    ζ=-(r^2+a^2)E-aΞ L/√(Δ_rΣ)|_(r_obs,θ_o).
  
From the factor ζ, by comparing terms in Eqs. (<ref>) and (<ref>) with the aid of the equations for the tetrad, the celestial coordinates are straightforwardly written as

    cosα = Σṙ/(r^2+a^2)E-aΞ L|_(r_obs,θ_o),


    sinβ = √(Δ_θ)sinθ/√(Δ_r)Ξsinα(Δ_rΣϕ̇/(r^2+a^2)E-aΞ L-aΞ) |_(r_obs,θ_o).

Then, using the geodesic equations for ṙ and ϕ̇, substituting them into Eqs. (<ref>)-(<ref>), we obtain  simple expressions for α and β, namely,

    sinα   =   √(Δ_r η(r_p))/(r^2+a^2)-aΞξ(r_p)|_(r_obs,θ_o), 
    
    sinβ   =   ( Ξξ(r_p)^2 θ-a )sinθ/√(Δ_θη(r_p))|_(r_obs,θ_o).

As we can see, the equations that describe the celestial coordinates depend on the observer position (r_obs,θ_o) and parameters of the photon orbits, also known as photon sphere. 



In order to describe the shadow's silhouette it is appropriate to define the Cartesian coordinates

    x(r_p)   =   -2tan(α(r_p)/2)sin(β(r_p) ), 
    
    
    y(r_p)   =   -2tan(α(r_p)/2)cos(β(r_p) ).

Therefore, the parametric equations (<ref>) and (<ref>) draw the shadow's silhouette. According to Fig. <ref>, the shadow is symmetrical to the x-axis. For a Kerr black hole with mass M,
 the silhouette depends on the rotation parameter a and the observer polar angle θ_o, considering an observer at
  infinity. On the other hand, for the de Sitter case, the observer is in the domain of outer communication (r=r_obs),
   but still far away from the black hole, and the silhouette depends on the cosmological constant as well. In particular,
    according to Fig. <ref>, the shadow will depend on the  parameters of the mass function (<ref>)
     for regular black holes as well. As we will see, the parameter r_0, which generates regular metrics, can increase the distortion or the deviation from
      circularity and decrease the size of the shadow.   

As is known from the Kerr metric, the black hole shadow (its form or silhouette) is strongly sensible to the parameters a
 and θ_o. However, for the class of regular black holes given by (<ref>), we have two new
 parameters that modify the shadow: q and r_0. Here it is shown just cases in which the rotation parameter is smaller than the black
  hole mass, i.e., a^2<M^2, and, in order to produce shadows compatible with M87*, it is adopted r_0≪ M,
   which, alongside   a^2<M^2, provides the spacetime structure with three horizons indicated by (<ref>).   
    As  we can see from Fig. <ref> and Fig. <ref>, for θ_o ≠ 0, the more rotation, the more 
     moved in the positive x-direction the shadow is. Such a motion points toward the rotation direction. In particular,  
      the silhouette is  deformed on the left (assuming that the rotation is from left to right) for large values of a, 
      θ_o and for r_0.  This  difference between the left  and the right sides is due to the photon orbits and the
       spacetime dragging. On the left,  photons travel in the same direction of the black hole rotation, on the right they travel
        in the opposite direction. Points on the left and on the right in the shadow's silhouette are given by r_p- and
         r_p+, respectively. In Fig.  <ref>, we see the parameter r_0 can increase the shadow deformation for
          large values of θ_o. On the other hand, small values of θ_o can produce highly symmetrical shadows
           even for large values of a. Assuming that the M87* is observed at 17^∘, we note that the deviation from
            circularity is small, and, for large values of θ_o, interestingly large values of r_0 decrease the shadow
             size, according to Fig. <ref> (see the second and the third shadows).  





§ OBLATENESS AND DEVIATION FROM CIRCULARITY

In this section, two observables are built, oblateness and deviation from circularity. But following the Event Horizon Telescope collaboration, I will use the latter in order to compute an upper bound on the GUP parameter. 







 §.§ Oblateness

Following Refs. <cit.>, let us construct two observables for the black hole shadow. The first one indicates the difference between the x and y axes. Such a difference D is the oblateness. The second one, which was used for the Event Horizon Telescope collaboration, provides the shadow's deviation from circularity using the root-mean-square of the shadow's radius.


In order to evaluate D, one adopts an approximation for the Cartesian celestial coordinates x and y due to the large distance of the observer, r_obs=(16.8 ± 0.8)Mpc, which turns the value of α into a small quantity. Thus, Eqs. (<ref>) and (<ref>) are rewritten as     

    x(r_p)   ≃   -sin(α(r_p))sin(β(r_p) ),
    
    
    y(r_p)   ≃   -sin(α(r_p))cos(β(r_p) ).



The oblateness is simply defined as

    D=Δ x/Δ y=x_R-x_L/2y_m,

in which

    x_L=-sinα(r_p-)         x_R=sinα(r_p+),
 
where r_p- is solution of sinβ=1, and r_p+ is given by sinβ=-1. According to Fig. (<ref>), the values x_L and x_R (left and right) indicate maximum and minimum values assumed by the shadow silhouette on the x-axis. On the other hand, y_m is obtained from (<ref>) and its maximum, i.e., dy(r_p)/dr_p=0 leads to

    sinαd/dr_pcosβ+cosβd/dr_psinα  = 0.
   
And the roots of Eq. (<ref>) provide the value of r at which

    y_m=y(r_p').

The parameter r=r_p' (r_p-<r_p'<r_p+) indicates the silhouette's maximum value on the y-axis. And, as we can see in all shadows shown here, ± y_m are maximum and minimum values of y due to axial symmetry. Moreover, as we will see, the parameters x_L, x_R, and y_m will be useful in order to construct the deviation from circularity using the root-mean-square of the shadow's radius. 

By using Eq. (<ref>) for members of the class of rotating regular black holes studied here, Fig. <ref> indicates a strong dependence between oblateness and the rotation parameter a. The more rotation, the more deformed the shadow is. Moreover, the parameter r_0, which is related to the GUP parameter, modifies D, decreasing the oblateness. 




 §.§ Deviation from circularity

According to the Event Horizon Telescope collaboration <cit.>, a measure for the shadow deformation is provided by the deviation from circularity, which is conceived of as deviation from the root-mean-square of the radius l(r_p). Following Bambi et al. <cit.> (but here another parameterization is adopted[Bambi et al. <cit.> use the angle defined by l and the x-axis in order to parameterize the shadow radius.]), the shadow radius l(r_p) is defined as

    l(r_p)=√((x-x_c)^2+(y-y_c)^2),

with x_c=x(r_p') and y_c=0 (see Fig. <ref>). The average radius (root-mean-square) is given by

    l_RMS=√(1/(r_p+-r_p-)∫_r_p-^r_p+l(r_p)^2 dr_p).

According to the cited authors, who follow the report from the collaboration, the deviation from circularity is conceived of as the root-mean-square distance from the average radius l_RMS, that is to say, 

    Δ C = √(1/(x_R-x_L)∫_r_p-^r_p+ (l(r_p) - l_RMS)^2 dr_p).

However, as I said, another parametrization is adopted and suggested here, demanding that (x_R-x_L) is in the
 denominator of (<ref>). Like the oblateness, Δ C increases with the black hole rotation and the observation
  angle (see Fig. <ref>). The Event Horizon Telescope reported Δ C <0.1 assuming a Kerr metric as the geometry
   of M87*. Here Kerr-like objects are adopted, and using the M87* parameters the deviation from circularity given by Eq.
    (<ref>) will provide  an upper bound on the GUP parameter.
    
    
    



§ ESTIMATING THE QUANTUM GRAVITY PARAMETER

In order to constrain the GUP parameter, it is necessary to relate it to the spacetime metric. In Ref. <cit.>, it is applied a GUP in order to calculate quantum corrections to the black hole temperature. From the GUP

    x p≥ℏ(1+β_0 l_p^2/ℏ^2 p^2),

where l_p=√(ℏ G/c^3)≈10^-35 m is the Planck length, and β_0 is the so-called GUP parameter or the dimensionless quantum gravity parameter (as we can see, it is straightforward that for β_0 → 0 we have the standard uncertainty relation), we derived the quantum-corrected temperature for the Schwarzschild black hole and, interestingly, such a temperature is the Hawking temperature for the Bardeen regular black hole (up to second order approximation in l_p/r_+). Thus, the Bardeen metric was interpreted as a quantum-corrected Schwarzschild black hole under the assumption that the GUP parameter could be related to a metric parameter (r_0, the parameter that generates regular black holes introduced in Section 2), namely,

    r_0 = β_0^1/2 l_p/3.

Assuming that the relation (<ref>) is valid for the entire class of regular black holes, for any q in Eq. (<ref>), whether spherical or axisymmetric black holes, then the shadow of M87* can provide an upper bound on r_0 and, consequently, on β_0. That is to say, the deviation from circularity (<ref>) depends on the spacetime metric and its parameters like r_0, thus, by using the relation (<ref>), Δ C will depend on β_0 as well. Consequently, the upper bound on the GUP parameter can be obtained by using Δ C (β_0) and the constraint
provided by the Event Horizon Telescope Collaboration for the deviation from circularity.   

    


Therefore, by using Δ C (β_0), given by Eq. (<ref>), and assuming the M87* parameters, i.e., M=(6.5 ± 0.7 )× 10^9M_⊙ and r_obs= (16.8 ± 0.8)Mpc, the deviation from circularity reported by the Collaboration, Δ C (β_0)< 0.1, imposes

    β_0 <10^90,

for a_* ≲ 0.99 (with a_*=a/M) and θ_o=17^∘, for q=1,2 or q=3 in the mass function
 (<ref>). As we can see in Fig. <ref>, the constraint on the deviation from circularity gives an upper
  bound on the GUP parameter, i.e., curves that satisfy  Δ C (β_0)< 0.1 with a_* ≲ 0.99 are possible
   only for values given by Eq. (<ref>). On the other hand, for a_* > 0.99, the  deviation from circularity is
    always Δ C (β_0) >0.1. Thus, for θ_o=17^∘, the parameter a_* > 0.99  is ruled out according
     to M87* shadow. As I pointed out in Introduction, the value θ_o=17^∘ is conceived of as the angle
      between the jet and the observer, assuming  that the jet direction is orthogonal to the M87* equatorial plane. The range
       for the rotation parameter presented here is in agreement with studies like <cit.>, where the Kerr metric is
        adopted in order to describe the M87* shadow. 
  
      

As we can see in Table <ref>, β_0<10^90 is the worst value compared to other upper bounds
 provided by different approaches, like the Lamb shift, Landau levels and scanning tunneling microscope in Ref.
  <cit.>, or using light deflection, pulsar PRS B 1913+16 and perihelion precession in Ref. <cit.>, or from the
   gravitational waves in Ref. <cit.>. According to Das and Vagenas <cit.>, the scanning tunneling
    microscope delivers the best one, β_0<10^21. On the other hand, like the present work, the authors of Refs.
     <cit.> and  <cit.> present upper bounds based on gravitational phenomena, which
      rendered worse values than those provided by quantum systems. Quantum approaches have provided more stringent
       upper bounds on the GUP parameter.
       
 



§ FINAL REMARKS

The Event Horizon Telescope announced the first image of a black hole. According to the collaboration, the black hole
 shadow is well-described by the Kerr metric. However, it is argued that other options are still possible within the M87*
  parameters. Bambi et al. <cit.>, for example, did not rule out a superspinar as a candidate for the
  geometry that produces the M87* shadow. In this article, a class of rotating regular black holes (thought of as a slight
   deviation from the Kerr metric) is adopted in order to describe the reported shadow.

The class of rotating regular black holes studied here presents a parameter according to which regular geometries or black
 holes without a singularity are generated. In our work <cit.>, such a parameter from the spacetime metric
  was linked to the GUP parameter. GUPs appear in theories of quantum gravity and their parameters, which deform or
   generalize the Heisenberg principle, have been estimated. From a reported deviation from circularity of the M87*
    shadow (<10%), an upper bound on the GUP parameter was computed. Alongside gravitational approaches adopted
     in order to obtain  an upper bound on the GUP parameter, the value obtained here, β_0 < 10^90, indicates that
      quantum options  to estimate such a parameter are better alternatives than the gravitational options like gravitational
       waves, light  deflection, pulsar PRS B 1913+16, perihelion precession, and  the black hole shadow studied here.
        In general, quantum approaches provide more stringent upper bounds on the GUP parameter to date. 




§ ACKNOWLEDGMENTS

This study was financed in part by the Coordenao de Aperfeioamento de Pessoal de Nvel Superior-Brasil (CAPES)-Finance Code 001. I thank Vilson Zanchin's research group for comments and suggestions during a seminar presentation at the UFABC. I also thank an anonymous referee for important comments and suggestions.



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