Spins of supermassive black holes M87* and SgrA* revealed from the size of dark spots in Event Horizon Telescope Images

We reconstructed dark spots in the images of supermassive black holes SgrA* and M87* provided by the Event Horizon Telescope (EHT) collaboration by using the geometrically thin accretion disk model. In this model, the black hole is highlighted by the hot accretion matter up to the very vicinity of the black hole event horizon. The existence of hot accretion matter in the vicinity of black hole event horizons is predicted by the Blandford-Znajek mechanism, which is confirmed by recent general relativistic MHD simulations in supercomputers. A dark spot in the black hole image in the described model is a gravitationally lensed image of an event horizon globe. The lensed images of event horizons are always projected at the celestial sphere inside the awaited positions of the classical black hole shadows, which are invisible in both cases of M87* and SgrA*. We used the sizes of dark spots in the images of SgrA* and M87* for inferring their spins, 0.650.75, accordingly.


I. INTRODUCTION
The first images of Supermassive Black Holes (SMBHs) M87* [1][2][3][4][5][6] and SgrA* [7][8][9][10][11][12], obtained by the Event Horizon Telescope (EHT) collaboration, opened the road for the verification of modified gravity. It must be stressed that this is a unique possibility for an experimental verification of modified gravity theories. The strong field limit means the applicability of gravity theory at the limiting values of the Newtonian gravitational potential GM/r c 2 , where G is the Newtonian gravitational constant, M is the mass of the gravitating object, r is the characteristic radius of the gravitating object, and c is the velocity of light. The most intriguing features of EHT images are the sizes and forms of the dark spots and the surrounding luminous rings with bright spots.
It is natural to suppose that luminous rings are produced by the emission of hot accretion plasma. The radii of luminous rings in both EHT images very nearly coincide with the corresponding sizes of classical black hole shadows [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Meanwhile, the sizes of dark spots in both FIG. 1. Reconstruction of the Shwarzschild black hole event horizon silhouette using 3D trajectories of photons (multicolored curves), which start very near the black hole event horizon and are registered by a distant observer (by a distant telescope). The event horizon silhouette always projects at the celestial sphere within the classical black hole shadow with a radius 3 √ 3. Meanwhile, the corresponding radius of event horizon silhouette is r h 4.457 [50,51].

II. EVENT HORIZON SILHOUETTE ≡ LENSED IMAGE OF THE BLACK HOLE EVENT HORIZON
In numerical calculations of test particle trajectories in the Kerr metric, it is useful to use the Boyer-Lindquist coordinate system [54] with dimensional coordinates; for example, the dimensional radial length unit is GM/c 2 . The corresponding event horizon radius of the Kerr black hole with spin in the range 0 ≤ a ≤ 1. All photon trajectories in our Figures were calculated by using both differential (A1)-(A4) and integral (C1)-(C3) forms of equations of motion in the Kerr metric. The form of a dark event horizon silhouette is numerically calculated from photon trajectories, which start near the event horizon and are registered by a distant observer (see details in Appendix A). In all our figures, the dashed red ring is an event horizon radius in the absence of gravity, while the red arrow is the direction of black hole rotation axes. Figure 1 presents the reconstruction of a Shwarzschild black hole event horizon silhouette using 3D trajectories of photons (multicolored curves), which start very near the black hole event horizon and are registered by a distant observer. (by a distant telescope).  The dark spot in the described black hole image is a locus of points with an absence of photons, which start very near the black hole event horizon and are registered by a distant observer. A superposition of the modeled dark spot with the EHT image of SgrA* is shown in Figure 7.

V. CONCLUSIONS
Here, we evaluated the spins of SMBHs SgrA* and M87* by using the sizes of dark spots in their EHT images. We made numerical calculations of the corresponding dark spots by using trajectories of photons, which start very near the event horizon and are registered by a distant observer. We had in mind the Blandford-Znajek mechanism for heating the accretion plasma near the black hole event horizon.
The resulting dependence of the dark spot size, r h (a), on the black hole spin parameter a is shown in Figure 9, respectively, for SgrA* and M87*.

Appendix A: Geodesics in Kerr Metric
We used the classical Brandon Carter equations of motion for test particles in the Kerr metric with the famous integrals of motion (see [54,[66][67][68][69][70][71][72] for details): In these equations, τ is a proper time (for µ 0) or an affine parameter of particle (for µ = 0), the effective radial potential R(r) in (A1) is and the effective polar (latitudinal) potential Θ(θ) in (A1), which controls the motion in the polar (latitude) direction, is As usual, dimensionless orbital parameters were used, defining the motion of massive test particles, γ = E/µ, λ = L/E, and q 2 = Q/E 2 , respectively. The motion of massless particles (like photons) depends on only two parameters, λ = L/E and q 2 = Q/E 2 . These parameters are related with the horizontal and vertical impact parameters, α and β, viewed by the distant observer at the celestial sphere (see, e.g., [13,69] for more details),

Appendix B: Classical black hole shadow
The outer boundary of the classical black hole shadow in the Kerr metric may be determined by the joint resolution of equations for photon spheres, R(r) = 0 and [rR(r)] = 0 (see, e.g., [13,71] for more details). The corresponding mutual solution of these two equations may be written in parametric form (λ, q) = (λ(r), q(r)) if the black hole is viewed from its equatorial plane, Examples of the classical black hole shadows, projected at the celestial sphere, are shown in Figure 10 for different spin parameters a in the Kerr metric and possible inclinations of SMBHs M87* and SgrA* rotation axes at the celestial sphere.

Appendix C: Brandon Carter equations of motion in the Kerr metric in the integral form
In our calculations, we used the standard integral form of Brandon Carter equations of motion in the Kerr metric [66,71]: The integrals in (A2)-(C3) are the path integrals along the curved trajectory. A major specific feature of these integrals is the changing of the sign at each turning point, both in the radial and polar directions. As a result, these path integrals monotonically grow along the curved particle trajectory. In the absence of turning points, these path integrals become ordinary ones. For example, the path integral in (C1) can be written in the absence of turning points through the ordinary integral as In this equation, r s and θ s are the radial and polar starting points of the particle, respectively, and r 0 and θ 0 are the finishing points along the curved trajectory.
The second example is with one turning point in the polar direction, θ min (λ, q), (an extremum of latitudinal potential Θ(θ)). The corresponding integral Equation (C1) can be written in this case through the ordinary integrals as The third specific case is the particle trajectory with one turning point in the polar direction, θ min (λ, q), and one turning point in the radial direction, r min (λ, q). The path integral in (C1) in this case can be written through the ordinary integrals as