Constraints on black-hole charges with the 2017 EHT observations of M87*

Our understanding of strong gravity near supermassive compact objects has recently improved thanks to the measurements made by the Event Horizon Telescope (EHT). We use here the M87* shadow size to infer constraints on the physical charges of a large variety of nonrotating or rotating black holes. For example, we show that the quality of the measurements is already sufficient to rule out that M87* is a highly charged dilaton black hole. Similarly, when considering black holes with two physical and independent charges, we are able to exclude considerable regions of the space of parameters for the doubly-charged dilaton and the Sen black holes.

Our understanding of strong gravity near supermassive compact objects has recently improved thanks to the measurements made by the Event Horizon Telescope (EHT). We use here the M87* shadow size to infer constraints on the physical charges of a large variety of nonrotating or rotating black holes. For example, we show that the quality of the measurements is already sufficient to rule out that M87* is a highly charged dilaton black hole. Similarly, when considering black holes with two physical and independent charges, we are able to exclude considerable regions of the space of parameters for the doubly-charged dilaton and the Sen black holes.

I. INTRODUCTION
General relativity (GR) was formulated to consistently account for the interaction of dynamical gravitational fields with matter and energy, the central idea of which is that the former manifests itself through modifications of spacetime geometry and is fully characterized by a metric tensor. While the physical axioms that GR is founded on are contained in the equivalence principle [1,2], the Einstein-Hilbert action further postulates that the associated equations of motion involve no more than second-order derivatives of the metric tensor.
The strength of the gravitational field outside an object of mass M and characteristic size R, in geometrized units (G = c = 1), is related to its compactness C := M/R, which is ∼ 10 −6 for the Sun, and takes values ∼ 0.2 − 1 for compact objects such as neutron stars and black holes. Predictions from GR have been tested and validated by various solar-system experiments to very high precision [2,3], setting it on firm footing as the best-tested theory of classical gravity in the weak-field regime. It is important, however, to consider whether signatures of deviations from the Einstein-Hilbert action, e.g., due to higher derivative terms [4], could appear in measurements of phenomena occurring in strongfield regimes where C is large. Similarly, tests are needed to assess whether generic violations of the equivalence principle occur in strong-fields due, e.g., to the presence of additional dynamical fields, such as scalar [5] or vector fields [6], that may fall off asymptotically. Agreement with the predictions of GR coming from observations of binary pulsars [7], and of the gravitational redshift [8] and geodetic orbit-precession [9] of the star S2 near our galaxy's central supermassive compact object Sgr A by the GRAVITY collaboration, all indicate the success of GR in describing strong-field physics as well. In addition, with the gravitational-wave detections of coalescing binaries of compact objects by the LIGO/Virgo collaboration [10] and the first images of black holes produced by EHT, it is now possible to envision testing GR at the strongest field strengths possible.
While the inferred size of the shadow from the recently obtained horizon-scale images of the supermassive compact object in M87 galaxy by the EHT collaboration [11] was found to be consistent to within 17% for a 68% confidence interval of the size predicted from GR for a Schwarzschild black hole using the a priori known estimates for the mass and distance of M87* based on stellar dynamics [12], this measurement admits other possibilities, as do various weak-field tests [2,13]. Since the number of alternative theories to be tested using this measurement is large, a systematic study of the constraints set by a strong-field measurement is naturally more tractable within a theory-agnostic framework, and various such systems have recently been explored [14,15]. This approach allows for tests of a broad range of possibilities that may not be captured in the limited set of known solutions. This was exploited in Ref. [13], where constraints on two deformed metrics were obtained when determining how different M87* could be from a Kerr black hole while remaining consistent with the EHT measurements.
However, because such parametric tests cannot be connected directly to an underlying property of the alternative theory, here we use instead the EHT measurements to set bounds on the physical parameters, i.e., angular momentum, electric charge, scalar charge, etc., -and which we will generically refer to as "charges" (or hairs) -that various wellknown black-hole solutions depend upon. Such analyses can be very instructive [16][17][18] since they can shed light on which underlying theories are promising candidates and which must be discarded or modified. At the same time, they may provide insight into the types of additional dynamical fields that may be necessary for a complete theoretical description of physical phenomena, and whether associated violations of the equivalence principle occur.
More specifically, since the bending of light in the presence of curvature -either in static or in stationary spacetimesis assured in any metric theory of gravity, and the presence of large amounts of mass in very small volumes can allow for the existence of a region where null geodesics move on spherical orbits, an examination of the characteristics of such photon regions, when they exist, is a useful first step. The projected asymptotic collection of the photons trajectories that are captured by the black hole -namely, all of the photon trajectories falling within the value of the impact parameter at the unstable circular orbit in the case of nonrotating black holes -will appear as a dark area to a distant observer and thus represents the "shadow" of the capturing compact object. This shadow -which can obviously be associated with black holes [19][20][21][22][23][24], but also more exotic compact objects such as gravastars [25,26] or naked singularities [27,28] -is determined entirely by the underlying spacetime metric. Therefore, the properties of the shadow -and at lowest order its size -represent valuable observables common to all metric theories of gravity, and can be used to test them for their agreement with EHT measurements.
While the EHT measurement contains far more information related to the flow of magnetised plasma near M87*, we will consider only the measurement of the size of the bright ring. Here we consider various spherically symmetric black-hole solutions, from GR that are either singular (see, e.g., [29]) or non-singular [30][31][32], and string theory [33][34][35][36][37]. Additionally, we also consider the Reissner-Nordström (RN) and the Janis-Newman-Winicour (JNW) [38] naked singularity solutions, the latter being a solution of the Einstein-Klein-Gordon system. Many of these solutions have been recently summarised in Ref. [15], where they were cast in a generalised expansion of static and spherically symmetric metrics. Since angular momentum plays a key role in astrophysical scenarios, we also consider various rotating black-hole solutions [39][40][41][42] which can be expressed in the Newman-Janis form [43] to facilitate straightforward analytical computations. It is to be noted that this study is meant to be a proof of principle and that while the constraints we can set here are limited, the analytical procedure outlined below for this large class of metrics is general, so that as future observations become available, we expect the constraints that can be imposed following the approach proposed here to be much stronger.

II. SPHERICAL NULL GEODESICS AND SHADOWS
For all the static, spherically symmetric spacetimes we consider here, the definition of the shadow can be cast in rather general terms. In particular, for all the solutions considered, the line element expressed in areal-radial polar coordinates (t,r, θ, φ) has the form 1 and the photon region, which degenerates into a photon sphere, is located atr =:r ps , which can be obtained by solving [13] The boundary of this photon sphere when observed from the frame of an asymptotic observer, due to gravitational lensing, appears to be a circle of size [13] r sh =r ps f (r ps ) .
On the other hand, the class of Newman-Janis stationary, axisymmetric spacetimes we consider here [43], which are geodesically integrable (see, e.g., [22,44,45]), can be expressed in Boyer-Lindquist coordinates (t, r, θ, φ) as where f = f (r, θ) and Σ(r, θ) := r 2 + a 2 cos 2 θ and ∆(r) := Σ(r, θ)f (r, θ) + a 2 sin 2 θ. In particular, these are the stationary generalisations obtained by employing the Newman-Janis algorithm [43]) for "seed" metrics of the form (1) with g(r) = 1 2 . The Lagrangian L for geodesic motion in the spacetime (4) is given as 2L := g µνẋ µẋν , where an overdot represents a derivative with respect to the affine parameter, and 2L = −1 for timelike geodesics and 2L = 0 for null geodesics. The two Killing vectors ∂ t and ∂ φ yield two constants of motion in terms of which the geodesic equation for photons can be separated into where we have introduced first ξ := L/E, and then I := η + (a − ξ) 2 . Also, η is the Carter constant, and the existence of this fourth constant of motion is typically associated with the existence of an additional Killing-Yano tensor (see for example [23,47]).
In particular, we are interested here in spherical null geodesics (SNGs), which satisfyṙ = 0 andr = 0 and are not necessarily planar; equivalently, SNGs can exist at locations where R(r) = 0 and dR(r)/dr = 0. Since these are only two equations in three variables (r, ξ, η), it is convenient, for reasons that will become evident below, to obtain the associated conserved quantities along such SNGs in terms of their radii r as (see also [48]), The condition that Θ(θ) ≥ 0, which must necessarily hold as can be seen from Eq. (7), restricts the radial range for which SNGs exist, and it is evident that this range depends on θ. This region, which is filled by such SNGs, is called the photon region (see, e.g., Fig. 3.3 of [19]). The equality Θ(θ) = 0 determines the boundaries of the photon region, and the (disconnected) piece which lies in the exterior of the outermost horizon is of primary interest since its image, as seen by an asymptotic observer, is the shadow. We denote the inner and outer surfaces of this photon region by r p− (θ) and r p+ (θ) respectively, with the former (smaller) SNG corresponding to the location of a prograde photon orbit (i.e., ξ SNG (r p− ) > 0), and the latter to a retrograde orbit.
It can be shown that all of the SNGs that are admitted in the photon region, for both the spherically symmetric and axisymmetric solutions considered here, are unstable to radial perturbations. In particular, for the stationary solutions, the stability of SNGs at a radius r = r SNG with respect to radial perturbations is determined by the sign of ∂ 2 r R, and when ∂ 2 r R(r SNG ) > 0, SNGs at that radius are unstable. The expression for ∂ 2 r R reads To determine the appearance of the photon region and the associated shadow, as seen by asymptotic observers, we can introduce the usual notion of celestial coordinates (α, β), which for any photon with constants of the motion (ξ, η) can be obtained, for an asymptotic observer present at an inclination angle i with respect to the spin-axis of the compact object as in [49]. For photons on an SNG, we can set the conserved quantities (ξ, η) to the values given in Eq. 8 above to obtain [47,48] Recognizing that β = ± Θ(i), it becomes clear that only the SNGs with Θ(i) ≥ 0 determine the apparent shadow shape.
Since the photon region is not spherically symmetric in rotating spacetimes, the associated shadow is also not circular in general. It can be shown that the band of radii for which SNGs can exist narrows as we move away from the equatorial plane, and reduces to a single value at the pole, i.e. in the limit θ → π/2, we have r p+ = r p− (see e.g., Fig. 3.3 of [19]). As a result, the parametric curve of the shadow boundary as seen by an asymptotic observer lying along the pole is perfectly circular, α 2 sh + β 2 sh = η SNG (r p±,π/2 ) + ξ 2 SNG (r p±,π/2 ). We can now define the characteristic areal-radius of the shadow curve as [50] r sh,A :

III. SHADOW SIZE CONSTRAINTS FROM THE 2017 EHT OBSERVATIONS OF M87*
Measurements of stellar dynamics near M87* were previously used to produce a posterior distribution function of the angular gravitational radius θ g := M/D, where M is the mass of and D the distance to M87*. The 2017 EHT observations of M87* can be similarly used to determine such a posterior [11]. These observations were used to determine the angular diameterd of the bright emission ring that surrounds the shadow [11]. In Sec. 5.3 there, using synthetic images from general-relativistic magnetohydrodynamics (GRMHD) simulations of accreting Kerr black holes for a wide range of physical scenarios, the scaling factor α =d/θ g was calibrated. For emission from the outermost boundary of the photon region of a Kerr black hole, α should lie in the range 9.6 − 10.4.
The EHT measurement picks out a class of best-fit images ("top-set") from the image library, with a mean value for α of 11.55 (for the "xs-ring" model) and 11.50 (for the "xsringauss" model), when using two different geometric crescent models for the images, implying that the geometric models were accounting for emission in the top-set GRMHD images that preferentially fell outside of the photon ring. Using the distribution of α for these top-set images then enabled the determination of the posterior in the angular gravitational radius P obs (θ g ) for the EHT data. It is to be noted that this posterior was also determined using direct GRMHD fitting, and image domain feature extraction procedures, as described in Sec. 9.2 there, and a high level of consistency was found across all measurement methods. Finally, in Sec. 9.5 of [11], the fractional deviation in the angular gravitational radius δ was introduced as where θ g and θ dyn were used to denote the EHT and the stellardynamics inferences of the angular gravitational radius, respectively. The posterior on δ -as defined in Eq. (32) of [11] -was then obtained (see Fig. 21 there), and its width was found to be δ = −0.01 ± 0.17, for a 68% credible interval. This agreement of the 2017 EHT measurement of the angular gravitational radius for M87* with a previously existing estimate for the same, at much larger distances, constitutes a validation of the null hypothesis of the EHT, and in particular that M87* can be described by the Kerr black-hole solution.
Since the stellar dynamics measurements [12] are sensitive only to the monopole of the metric (i.e., the mass) due to negligible spin-dependent effects at the distances involved in that analysis, modeling M87* conservatively using the Schwarzschild solution is reasonable with their obtained posterior. Then, using the angular gravitational radius estimate from stellar dynamics yields a prediction for the angular shadow radius θ sh = r sh /D as being θ sh = 3 √ 3θ dyn . The 2017 EHT measurement, which includes spin-dependent effects as described above and which probes near-horizon scales, then determines the allowed spread in the angular shadow diameter as, θ sh ≈ 3 √ 3(1 ± 0.17) θ g , at 68% confidence levels [13]. Finally, since both angular estimates θ sh and θ g make use of the same distance estimate to M87*, it is possible to convert the 1-σ bounds on θ sh to bounds on the allowed shadow size for M87*.
That is, independently of whether the underlying solution be spherically symmetric (in which case we will considerr sh ) or axisymmetric (r sh,A ), the shadow size of M87* must lie in the range 3 √ 3(1 ± 0.17)M [13], i.e., (see gray-shaded region in Fig. 2 where we have introduced the maximum/minimum shadow radii r sh, EHT-max /r sh, EHT-min inferred by the EHT, at 68% confidence levels. Note that the bounds thus derived are consistent with compact objects that cast shadows that are both significantly smaller and larger than the minimum and maximum shadow sizes that a Kerr black hole could cast, which lie in the range, 4.83 M − 5.20 M (see, e.g., [13,51]).
An important caveat here is that the EHT posterior distribution on θ g was obtained after a comparison with a large library of synthetic images built from GRMHD simulations of accreting Kerr black holes [11]. Ideally, a rigorous comparison with non-Kerr solutions would require a similar analysis and posterior distributions built from equivalent libraries obtained from GRMHD simulations of such non-Kerr solutions. Besides being computationally unfeasible, this approach is arguably not necessary in practice. For example, the recent comparative analysis of Ref. [17] has shown that the image libraries produced in this way would be very similar and essentially indistinguishable, given the present quality of the observations. As a result, we adopt here the working assumption that the 1-σ uncertainty in the shadow angular size for non-Kerr solutions is very similar to that for Kerr black holes, and hence employ the constraints (14) for all of the solutions considered here.

IV. NOTABLE PROPERTIES OF VARIOUS SPACETIMES
As mentioned above, a rigorous comparison with non-Kerr black holes would require constructing a series of exhaustive libraries of synthetic images obtained from GRMHD simulations on such non-Kerr black holes. In turn, this would provide consistent posterior distributions of angular gravitational radii for the various black holes and hence determine how δ varies across different non-Kerr black holes, e.g., for Sen black holes. Because this is computationally unfeasiblethe construction of only the Kerr library has required the joint effort of several groups with the EHTC over a good fraction of a year -we briefly discuss below qualitative arguments to support our use of the bounds given in Eq. 14 above as an approximate, yet indicative, measure.
To this end, we summarize in Table I the relevant properties of the various solutions used here. First, we have considered here solutions from three types of theories, i.e., the underlying actions are either (a) Einstein-Hilbert-Maxwell-matter [29-33, 38, 39, 42], (b) Einstein-Hilbert-Maxwell-dilatonaxion [34,35,41], or (c) Einstein-Hilbert-Maxwell-Maxwelldilaton [37]. This careful choice implies that the gravitational piece of the action is always given by the Einstein-Hilbert term and that matter is minimally coupled to gravity. As a result, the dynamical evolution of the accreting plasma is expected to be very similar to that in GR, as indeed found in Ref. [17]. Second, since a microphysical description that allows one to describe the interaction of the exotic matter present in some of the regular black-hole spacetimes used here [30,31] -which typically do not satisfy some form of the energy conditions [42,53] -with the ordinary matter is thus far lacking, it is reasonable to assume that the interaction between these two types of fluids is gravitational only. This is indeed what is done in standard numerical simulations, either in dynamical spacetimes (see, e.g., [54]), or in fixed ones [16,55]. Third, since the mass-energy in the matter and electromagnetic fields for the non-vacuum spacetimes used here is of the order of the mass of the central compact object M , while the total mass of the accreting plasma in the GRMHD simulations is only a tiny fraction of the same, it is reasonable to treat the spacetime geometry and the stationary fields as unaffected by the plasma. Fourth, we have also been careful not to use solutions from theories with modified electrodynamics (such as nonlinear electrodynamics). As a result, the electromagnetic Lagrangian in all of the theories considered here is the Maxwell Lagrangian (see, e.g., the discussion in [15] and compare with [20]). This ensures that in these spacetimes light moves along the null geodesics of the metric tensor (see, e.g., Sec. 4.3 of [29] and compare against Sec. 2 of [56]). Therefore, we are also assured that ray-tracing the radiation emitted from the accreting matter in these spacetimes can be handled similarly as in the Kerr spacetime.
Finally, under the assumption that the dominant effects in determining the angular gravitational radii come from variations in the location of the photon region and in location of the inner edge of the accretion disk in these spacetimes, it is instructive to learn how these two physical quantities vary when changing physical charges, and, in particular, to demonstrate that they are quantitatively comparable to the corresponding values for the Kerr spacetime.
For this purpose, we study the single-charge solutions used here and report in Fig. 1 the variation in the location of the photon spheres (left panel) and innermost stable circular orbit (ISCO) radii (right panel) as a function of the relevant physical charge (cf. left panel of Fig. 1 in the main text). Note that both the photon-sphere and the ISCO radii depend exclusively on the g tt component of the metric when expressed using an areal radial coordinater (see, e.g., [13,15]). To gauge the effect of spin, we also show the variation in the locations of the equatorial prograde and retrograde circular photon orbits and the ISCOs in the Kerr black-hole spacetime, expressed in terms of the Cartesian Kerr-Schild radial coordinate r CKS , which, in the equatorial plane, is related to the Boyer-Lindquist radial coordinate used elsewhere in this work r simply via [57] It is apparent from Fig. 1 that the maximum deviation in the photon-sphere size from the Schwarzschild solution occurs for the EMd-1 black hole and is ≈ 75%, while the size of the prograde equatorial circular photon orbit for Kerr deviates by at most ≈ 50%. Similarly, the maximum variation in the ISCO size also occurs for the EMd-1 solution and is ≈ 73%, while the prograde equatorial ISCO for Kerr can differ by ≈ 66%.

V. CHARGE CONSTRAINTS FROM THE EHT M87* OBSERVATIONS
We first consider compact objects with a single "charge," and report in the left panel of Fig. 2 the variation in the shadow radius for various spherically symmetric black hole solutions, as well as for the RN and JNW naked singularities 3 . More specifically, we consider the black-hole solutions given by Reissner-Nordström (RN) [29], Bardeen [30,42], Hayward [31,58], Kazakov-Solodhukin (KS) [33], and also the asymptotically-flat Einstein-Maxwell-dilaton (EMd-1) with Note that the shadow radii tend to become smaller with increasing physical charge, but also that this is not universal behaviour, since the KS black holes have increasing shadow radii (the singularity is smeared out on a surface for this solution, which increases in size with increasingl).
Overall, it is apparent that the regular Bardeen, Hayward, and Frolov black-hole solutions are compatible with the present constraints. At the same time, the Reissner-Nordström and Einstein-Maxwell-dilaton 1 black-hole solutions, for certain values of the physical charge, produce shadow radii that lie outside the 1-σ region allowed by the 2017 EHT observations, and we find that these solutions are now constrained to take values in, 0 <q 0.90 and 0 <q 0.95 respectively. Furthermore, the Reissner-Nordström naked singularity is entirely eliminated as a viable model for M87* and the Janis-Newman-Winicour naked singularity parameter space is restricted further by this measurement to 0 <ν 0.47. Finally, we also find that the KS black hole is also restricted to have charges in the rangel < 1.53. In addition, note that the nonrotating Einstein-Maxwell-dilaton 2 (EMd-2) solution [37] -which depends on two independent charges -can also produce shadow radii that are incompatible with the EHT observations; we will discuss this further below. The two EMd black-hole solutions (1 and 2) correspond to fundamentally different field contents, as discussed in [37].
We report in the right panel of Fig. 2 the shadow areal radius r sh,A for a number of stationary black holes, such as Kerr [39], Kerr-Newman (KN) [40], Sen [41], and the rotating versions of the Bardeen and Hayward black holes [42]. The data refers to an observer inclination angle of i = π/2, and we find that the variation in the shadow size with spin at higher inclinations (of up to i = π/100) is at most about 7.1% (for i = π/2, this is 5%); of course, at zero-spin the shadow size does not change with inclination. The shadow areal radii are shown as a function of the dimensionless spin of the black hole a := J/M 2 , where J is its angular momentum, and for representative values of the additional parameters that characterize the solutions. Note that -similar to the angular momentum for a Kerr black hole -the role of an electric charge or the presence of a de Sitter core (as in the case of the Hayward black holes) is to reduce the apparent size of the shadow. Furthermore, on increasing the spin parameter, we recover the typical trend that the shadow becomes increasingly noncircular, as encoded, e.g., in the distortion parameter δ sh defined in [24,50] (see Appendix A). Also in this case, while the regular rotating Bardeen and Hayward solutions are compatible with the present constraints set by the 2017 EHT observations, the Kerr-Newman and Sen families of black holes can produce shadow areal radii that lie outside of the 1-σ region allowed by the observations.
To further explore the constraints on the excluded regions for the Einstein-Maxwell-dilaton 2 and the Sen black holes, we report in Fig. 3 the relevant ranges for these two solutions. The Einstein-Maxwell-dilaton 2 black holes are nonrotating but have two physical charges expressed by the coefficients 0 <q e < √ 2 and 0 <q m < √ 2, while the Sen black holes spin (a) and have an additional electromagnetic chargē q m . Also in this case, the gray/red shaded regions refer to the areas that are consistent/inconsistent with the 2017 EHT observations. The figure shows rather easily that for these two black-hole families there are large areas of the space of parameters that are excluded at the 1-σ level. Not surprisingly, these areas are those where the physical charges take their largest values and hence the corresponding black-hole solutions are furthest away from the corresponding Schwarzschild or Kerr solutions. The obvious prospect is of course that as the EHT increases the precision of it's measurements, increasingly larger portions of the space of parameters of these black holes will be excluded. Furthermore, other solutions that are presently still compatible with the observations may see their corresponding physical charges restricted.

VI. CONCLUSIONS
As our understanding of gravity under extreme regimes improves, and as physical measurements of these regimes are now becoming available -either through the imaging of supermassive black holes or the detection of gravitational waves from stellar-mass black holes -we are finally in the position of setting some constraints to the large landscape of non-Kerr black holes that have been proposed over the years. We have used here the recent 2017 EHT observations of M87* to set constraints, at the 1-σ-level, on the physical charges -either electric, scalar, or angular momentum -of a large variety of static (nonrotating) or stationary (rotating) black holes.
In this way, when considering nonrotating black holes with a single physical charge, we have been able to rule out, at 68% confidence levels, the possibility that M87* is a nearextremal Reissner-Nordström or Einstein-Maxwell-dilaton 1 black hole and that the corresponding physical charge must be in the range, RN: 0 <q 0.90 and EMd-1: 0 <q 0.95. We also find that it cannot be a Reissner-Nordström naked singularity or a JNW naked singularity with large scalar charge, i.e., only 0 <ν 0.47 is allowed. Similarly, when considering black holes with two physical charges (either nonrotating or rotating), we have been able to exclude, with 68% confidence, considerable regions of the space of parameters in the case of the Einstein-Maxwell-dilaton 2, Kerr-Newman and Sen black holes. Although the idea of setting such constraints is an old one (see, e.g., [14,15,18,21,22]), and while there have been recent important developments in the study of other possible observational signatures of such alternative solutions, such as in X-ray spectra of accreting black holes (see, e.g., [59]) and in gravitational waves [56,60], to the best of our knowledge, constraints of this type have not been set before for the spacetimes considered here.
As a final remark, we note that while we have chosen only a few solutions that can be seen as deviations from the Schwarzschild/Kerr solutions since they share the same basic Einstein-Hilbert-Maxwell action of GR, the work presented here is meant largely as a proof-of-concept investigation and a methodological example of how to exploit observations and measurements that impact the photon region. While a certain degeneracy in the shadow size induced by mass and spin remains and is inevitable, when in the future the relative difference in the posterior for the angular gravitational radius for M87* can be pushed to 5%, we should be able to constrain its spin, when modeling it as a Kerr black hole. Furthermore, since this posterior implies a spread in the estimated mass, one can expect small changes in the exact values of the maximum allowed charges reported here. Hence, as future observations -either in terms of black-hole imaging or of gravitationalwave detection -will become more precise and notwithstanding a poor measurement of the black-hole spin, the methodology presented here can be readily applied to set even tighter constraints on the physical charges of non-Einsteinian black holes.
and Compute Canada (http://www.computecanada. ca). We thank the staff at the participating observatories, correlation centers, and institutions for their enthusiastic support. This paper makes use of the following ALMA data: ADS/JAO.ALMA#2016.1.01154.V. ALMA is a partnership of the European Southern Observatory (ESO; Europe, representing its member states), NSF, and National Institutes of Natural Sciences of Japan, together with National Research Council (Canada), Ministry of Science and Technology (MOST; Taiwan), Academia Sinica Institute of Astronomy and Astro-physics(ASIAA; Taiwan), and Korea Astronomy and Space Science Institute (KASI; Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, Associated Universities, Inc. (AUI)/NRAO, and the National Astronomical Observatory of Japan (NAOJ). The NRAO is a facility of the NSF operated under cooperative agreement by AUI. This paper has made use of the following APEX data: Project ID T-091.F-0006-2013. APEX is a collaboration between the Max-Planck-Institut für Radioastronomie (Germany), ESO, and the Onsala Space Observatory (Sweden). The SMA is a joint project between the SAO and ASIAA and is funded by the Smithsonian Institution and the Academia Sinica. The JCMT is operated by the East Asian Observatory on behalf of the NAOJ, ASIAA, and KASI, as well as the Ministry of Finance of China, Chinese Academy of Sciences, and the National Key R&D Program (No. 2017YFA0402700) of China. Additional funding support for the JCMT is provided by the Science and Technologies Facility Council (UK) and participating universities in the UK and Canada. The LMT is a project operated by the Instituto Nacionalde Astrofísica,Óptica, y Electrónica (Mexico) and the University of Massachusetts at Amherst (USA), with financial support from the Consejo Nacional de Ciencia y Tecnología and the National Science Foundation. The IRAM 30m telescope on Pico Veleta, Spain is operated by IRAM and supported by CNRS (Centre National de la Recherche Scien-tifique, France), MPG (Max-Planck-Gesellschaft, Germany) and IGN (Instituto Geográfico Nacional, Spain). The SMT is operated by the Arizona Radio Observatory, a part of the Steward Observatory of the University of Arizona, with financial support of operations from the State of Arizona and financial support for instrumentation development from the NSF. The SPT is supported by the National Science Foundation through grant PLR-1248097. Partial support is also provided by the NSF Physics Frontier Center grant PHY-1125897 to the Kavli Institute of Cosmological Physics at the University of Chicago, the Kavli Foundation and the Gordon and Betty Moore Foundation grant GBMF 947. The SPT hydrogen maser was provided on loan from the GLT, courtesy of ASIAA. The EHTC has received generous donations of FPGA chips from Xilinx Inc., under the Xilinx University Program. The EHTC has benefited from technology shared under open-source license by the Collaboration for Astronomy Signal Processing and Electronics Research (CASPER). The EHT project is grateful to T4Science and Microsemi for their assistance with Hydrogen Masers. This research has made use of NASA's Astrophysics Data System. We gratefully acknowledge the support provided by the extended staff of the ALMA, both from the inception of the ALMA Phasing Project through the observational campaigns of 2017 and 2018. We would like to thank A. Deller and W. Brisken for EHT-specific support with the use of DiFX. We acknowledge the significance that Maunakea, where the SMA and JCMT EHT stations are located, has for the indigenous Hawaiian people.
Software: AIPS [62], ParselTongue [63], GNU Parallel [64], eht-imaging [65], Difmap [66], Numpy [67], Scipy [68], Pandas [69], Astropy [70,71], Jupyter [72], Matplotlib [73], THEMIS [74], DMC [75], polsolve [76], GPCAL [77]. models for M87*, its shadow is less distorted from a circle than that of an extremal Kerr black hole (see Fig. 4). In other words, the examples just made highlight the importance of using the appropriate bounds on a sufficiently robust quantity when using the EHT measurement to test theories of gravity. Failing to do so may lead to incorrect bounds on the blackhole properties. For instance, Ref. [21] is able to set bounds on the parameter space of the uncharged, rotating Hayward black hole by imposing bounds on the maximum distortion of the shape of its shadow boundaries, albeit using a different measure for the distortion from a circle [see Eq. (58) there], whereas we have shown that this is not possible, upon using the bounds 4.31 M − 6.08 M for the size of their shadows (cf. right panel of Fig. 2).