First Sagittarius A* Event Horizon Telescope Results. VI. Testing the Black Hole Metric

The mass and distance of Sgr A* have been extensively studied by analyzing the dynamics of the central stellar cluster in the innermost 10'' of the Galactic center (Genzel et al. 1997, 2000, 2003a, 2010; Ghez et al. 1998, 2000; Eckart et al. 1999, 2002; Gezari et al. 2002; Schödel et al. 2007; Martins et al. 2008; Morris et al. 2012; Do et al. 2013; Jia et al. 2019). Near-infrared (NIR) observation with 8–10 m class telescopes supported by adaptive optics (AO) revealed the orbits of individual stars in the innermost arcsec (the so-called S stars), in particular the star S0-2. ¹⁴⁹ For this star, the combined fit for orbital elements and black hole parameters (mass, distance, projected position in the sky, proper motions, and radial velocity) has provided the most precise estimates for Sgr A*'s mass and distance so far (Schödel et al. 2003; Eisenhauer et al. 2005; Ghez et al. 2005b, 2008; Gillessen et al. 2009a, 2009b, 2017; Meyer et al. 2012; Boehle et al. 2016; Chu et al. 2018; Hees et al. 2019; O'Neil et al. 2019; Gravity Collaboration et al. 2021a).

S0-2 is a star with an apparent 2.2 μm (NIR K-band) magnitude of m_K = 14, an orbital period of P ≈ 16 yr, and a semimajor axis of a = 125 mas (or ∼10³ au at an 8 kpc distance); thus, it is the brightest star with a comparatively close orbit and short period at the Galactic center. The study of S0-2's orbit has predominantly been conducted with two sets of instruments, the two 10 m telescopes of the Keck Observatory, and the Very Large Telescope (VLT) of the European Southern Observatory (ESO), using its individual telescopes as well as GRAVITY, an interferometer combining all four 8.2 m telescopes of the VLT (VLTI). The orbit of S0-2 provides some of the best evidence for the existence of a black hole. S0-2 has concluded an entire revolution between 2002 and 2018 covered by observations and has allowed the Keck and VLTI teams to test relativistic effects like the gravitational redshift or the Schwarzschild precession (Gravity Collaboration et al. 2018a, 2019, 2020a; Amorim et al. 2019; Do et al. 2019) and to constrain alternative theories of gravity (Hees et al. 2017; Della Monica et al. 2022; De Martino et al. 2021) and variations of the fine-structure constant (Hees et al. 2020).

In order to measure S0-2's projected position in the sky, an astrometric reference frame has to be established. Menten et al. (1997) proposed the idea to use a group of SiO maser stars at the Galactic center—visible both in the radio and in the NIR—with positions and proper motions determined by interferometric astrometry at radio wavelengths. These masers allow it to establish a reference frame in the NIR. For both imaging instruments (Keck II/NIRC2, VLT/NACO) the field of view (10'' and 14'', respectively) is not large enough to capture the S stars and the seven masers in the same pointing. Instead, a dither pattern of pointings overlapping with one another is observed. Astrometric measurements in the central field are then executed via secondary astrometric standards, either in the form of matching coordinate lists or by generating mosaic images. In this process, systematic astrometric errors occur owing to the geometric distortions of the camera optics and field dependence of the point-spread function caused by anisoplanatism of the AO correction and higher-order aberrations of the optics (Yelda et al. 2010; Plewa et al. 2015; Sakai et al. 2019).

The VLTI team included interferometric data in their analysis starting in 2016. VLTI/GRAVITY provided high-precision distance measurements to Sgr A* during S0-2's closest approach with ∼1 mas resolution and ∼40 μas astrometric precision.

This subset of the interferometric data is not affected by the systematic uncertainties of the reference frame because the projected position of S0-2 is directly referenced to the projected (center of light) position of Sgr A*. However, also VLTI/GRAVITY data have systematic uncertainties, mainly due to aberrations of the optical trains of the individual telescopes (Gravity Collaboration et al. 2021b).

Information on the third dimension of the stellar orbits is obtained in the form of radial velocities, which in the case of S0-2 can be determined by observing the 2.167 μm H i (Brγ) and the 2.11 μm He i lines with integral field spectrographs like VLT/SINFONI or Keck/OSIRIS.

In their latest publication on the measurement of the gravitational redshift (Do et al. 2019, Table 1) the Keck team found for the distance a value of R₀ = 7959 ± 59 ± 32 pc (for the fit that leaves the redshift parameter free). They also published a posterior version with the assumption that general relativity is true (redshift parameter set to unity), R₀ = 7935 ± 50 pc, which is practically equivalent within the uncertainties. Their estimates for the black hole mass are M = (3.975 ± 0.058 ± 0.026) × 10⁶ M_⊙ and M = (3.951 ± 0.047) × 10⁶ M_⊙, respectively.

Table 1. The Inferred Shadow Diameter of Sgr A* in μas

	GRMHD	Analytic	Analytic
		Kerr	Non-Kerr
eht-imaging	${48.9}_{-5.1}^{+5.2}$	${50.0}_{-5.6}^{+5.6}$	${49.9}_{-4.9}^{+5.2}$
SMILI	${48.1}_{-5.2}^{+6.3}$	${49.1}_{-5.7}^{+6.5}$	${49.2}_{-5.4}^{+6.2}$
DIFMAP	${46.9}_{-5.2}^{+4.9}$	${47.8}_{-5.7}^{+5.1}$	${47.8}_{-5.2}^{+4.9}$

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In the publication on the detection of the Schwarzschild precession (Gravity Collaboration et al. 2020a), the VLTI team found R₀ = 8246.7 ± 9.3 pc and M = (4.261 ± 0.012) × 10⁶ M_⊙. Their latest paper on the mass distribution in the Galactic center (Gravity Collaboration et al. 2022) changes these values slightly. Their Table B.1 is an overview of recently published VLTI values for the black hole mass and distance; they also give an estimate of their systematics due to aberrations in GRAVITY's optics: R₀ = 8277 ± 9 ± 33 pc (Gravity Collaboration et al. 2021b). For the mass they find M = (4.297 ± 0.012 ± 0.040) × 10⁶ M_⊙. Additionally, the team provided a file with the posterior chains of their Bayesian analysis (S. Gillessen 2022, private communication), assuming general relativity to be true, which has median values of R₀ = 8278 ± 10 pc and M = (4.298 ± 0.013) × 10⁶ M_⊙.

It is interesting to point out that a third, independent estimate for the distance to Sgr A* has been provided by the Bessel project, a study of the Milky Way structure with VLBI astrometry: R₀ = 8.15 ± 0.15 kpc (Reid et al. 2009, 2014, 2019). This value for the distance is in marginally better agreement with the VLTI results.

Here, the two values considered for the distance are R₀ = 7935 ± 50 ± 32 pc and R₀ = 8277 ± 9 ± 33 pc. Mass and distance set a characteristic scale of the orbit in its projection on the sky and are highly correlated. The values for θ_g ≡ GM/Dc² as derived from the posterior distributions are θ_g = 5.125 ± 0.009 ± 0.020 μas (VLTI) and θ_g = 4.92 ± 0.03 ± 0.01 μas (Keck). For the VLTI value, the systematics were derived by error propagation according to $M\propto {R}_{0}^{2}$ (Gravity Collaboration et al. 2022). For the Keck value, a dedicated jackknife analysis was conducted to quantify the systematics stemming from the reference system (T. Do 2022, private communication). We show the posteriors in Figure 1. The discrepancy between the values of the two studies is about 4%.

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The spectral energy distribution (SED) of Sgr A* is shown in Figure 2. It has been compiled from the large body of literature, starting as early as 1992. Points show SED values taken from Zylka et al. (1992), Telesco et al. (1996), Falcke et al. (1998), Cotera et al. (1999), An et al. (2005), Dodds-Eden et al. (2009, 2011), Schödel et al. (2011), Nowak et al. (2012), Bower et al. (2015), Liu et al. (2016), Stone et al. (2016), Zhang et al. (2017), von Fellenberg et al. (2018), Witzel et al. (2018), Bower et al. (2019), Haggard et al. (2019), and Gravity Collaboration et al. (2020b). We present the radio part of the SED (Falcke et al. 1998; An et al. 2005; Bower et al. 2015, 2019; Liu et al. 2016) in a binned version (for a more detailed version showing all historic literature values in the radio to submillimeter regime, including some epochs of heightened variability, see Paper II). The steepening of the SED slope at centimeter wavelengths (Falcke et al. 1998) is clearly visible between 10 and 20 GHz and the submillimeter. From THz frequencies to the mid-IR Sgr A* has not be detected, and we have included lower and upper limits.

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The SED shows variability in all observable parts. Especially in the NIR and X-ray regime Sgr A* is strongly variable with regular flux density changes of factors of tens and hundreds, respectively, within 10–20 minutes (e.g., Baganoff et al. 2001; Genzel et al. 2003b; Eckart et al. 2004; Ghez et al. 2005a; Do et al. 2009; Dodds-Eden et al. 2011; Witzel et al. 2012; Neilsen et al. 2013, 2015; Ponti et al. 2017; Fazio et al. 2018). However, at radio frequencies and in the millimeter/submillimeter regime the variability is comparatively minor, with typical excursions of about 50% or less of the mean flux density (Falcke 1999; Herrnstein et al. 2004; Marrone et al. 2008; Dexter et al. 2014; Brinkerink et al. 2015; Subroweit et al. 2017; Fazio et al. 2018; Goddi et al. 2021; Murchikova & Witzel 2021).

Here we are focusing on the NIR properties of Sgr A*, in particular on limits for a steady component that is not varying on timescales of minutes and hours. Figure 2 shows the percentiles of the observed flux density distributions at 2.2 μm (VLT/NACO and KECK/NIRC2) and 4.5 μm (SPITZER/IRAC), as well as the corresponding spectral indices that change with flux density ¹⁵⁰ (Witzel et al. 2018). Additionally, we present the same percentiles for the flux density distribution measured with VLTI/GRAVITY at 2.2 μm (Gravity Collaboration et al. 2020b). While the VLT and KECK data are confusion limited and noise dominated at the low end of flux density distribution, resulting in nondetections of the source against the background, Gravity Collaboration et al. (2020b) report a clear detection of Sgr A* at all times. Because this detected source is variable at all times, their 5th percentile of the variable flux density distribution represents a conservative upper limit for any steady source component that may lie underneath.

We focus here on the 2017 April 7 data set because it satisfies three important criteria: the ALMA array, which leads to the highest signal-to-noise ratio data, participated in the observation; there is no evidence for an X-ray flare or large excursion in the 1.3 mm flux; and the interferometric coverage samples the visibility amplitude minima in the u–v plane, which are critical for establishing an accurate image size measurement. We also note that the analysis presented in Paper III for the 2017 April 6 data provides consistent results. The measurement uncertainties are obtained from modeling these data with imaging and model-fitting tools, as discussed in Event Horizon Telescope Collaboration et al. 2022b and 2022c, hereafter Papers III and IV, respectively. Here we quantify these results using characterization tools, as we describe below.

We use the CHaracterization Algorithm for Radius Measurements (CHARM), which is based on the feature extraction algorithm that was employed in Event Horizon Telescope Collaboration et al. (2019c) and improved further in Özel et al. (2021). Briefly, the algorithm (i) chooses a trial center for a potential ring-like feature; (ii) uses a rectangular bivariate spline interpolation to obtain radial cross sections of the filtered image brightness at 128 equidistant azimuthal orientations starting from the trial center; (iii) measures, in each radial cross section, the distance of the location of peak brightness from the trial center and identifies the ring diameter as two times the median value of this distance; (iv) iterates the location of the trial center and steps (i)−(iii) such that the variance in the diameter along different cross sections is minimized; and (v) measures a median FWHM of the ring by fitting an equivalent asymmetric Gaussian to each radial cross section such that the corresponding integrated brightness of the cross section of the filtered image is equal to that of the Gaussian. We then define the fractional width as the FWHM of the ring in units of the ring diameter.

We show in Figure 3 the fractional width and diameter measurements obtained for eht-imaging, SMILI, and DIFMAP top-set images for the April 7 Sgr A* data (see Paper IV for the details of these three imaging algorithms). Even though we apply CHARM to all of the top-set images, without employing clustering filters (e.g., to select only ring-like images), we find that the 68th and 95th percentile contours for the ring parameters form compact regions for each algorithm. This indicates that there is a discernible brightness depression in each image that is surrounded by a bright region that has a robust characteristic size.

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The gray bands in Figure 3 mark the effective limit of the fractional width that can be measured with imaging methods because of the finite resolution of the EHT array. The pink bands show the expected anticorrelation between the ring FWHM and the measured diameter ${\hat{d}}_{{\rm{m}}}$ that arises from the Gaussian broadening of an infinitesimally thin ring of diameter d. To first order in the fractional ring width, this anticorrelation follows (see Appendix G of Event Horizon Telescope Collaboration et al. 2019a)

$$\begin{eqnarray}&&{\hat{d}}_{{\rm{m}}}=d-\displaystyle \frac{1}{4\mathrm{ln}2}\displaystyle \frac{{\mathrm{FWHM}}^{2}}{d}.\end{eqnarray} \tag{ 2 }$$

Because some of the inferred fractional widths are relatively large, in calculating the actual shaded areas in Figure 3 we do not make this first-order approximation but rather employ a numerical evaluation of the complete expression.

In Figure 4, we compare the fractional widths and mean diameters inferred for Sgr A* with the three imaging algorithms. Even though there appear to be small differences in the mean diameter, all of the contours lie along the expected anticorrelation. This suggests that the differences are simply caused by the various algorithmic choices and do not reflect inconsistencies between them.

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We also use the image diameter and fractional width obtained from fitting analytic models to the visibility data (see Paper IV). In particular, we focus on the mG-ring model described in Paper IV, which comprises a Gaussian broadened ring with flux enhancements on the ring with m-fold azimuthal symmetry and an additional central Gaussian floor component. We use the posteriors obtained from the fitting algorithm Comrade (P. Tiede 2022, in preparation). In Figure 5, we show the posterior over the diameter and the fractional width obtained from fitting the mG-ring model to the April 7 data. The narrow posterior in diameter for this model reflects primarily the insufficient degree of model complexity in the model, as can be seen in the synthetic data analysis below (see also discussion in Psaltis et al. 2020b). Nevertheless, the inferred diameter is consistent with those of the imaging methods, given the expected anticorrelations.

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In this section, we use simulated black hole images to quantify the correction factor α₁, which is the ratio between the diameter of the peak brightness of the image and the diameter of the black hole shadow. We employ three different types of models to explore a range of effects related to the plasma properties, spacetime characteristics, and different numerical realizations of the turbulent flow.

The first category of images comprises ∼180,000 snapshots of GRMHD accretion flow simulations discussed in Paper V. The simulations cover a range of black hole spins (a = −0.94, −0.5, 0.0, 0.5, 0.94), observer inclinations (i = 10°, 30°, 50°, 70°, and 90°), magnetically arrested disk (MAD) and standard and normal evolution (SANE) magnetic field configurations, and thermal electron distributions with temperature prescriptions characterized by R_high = 10, 40, and 160. For each combination in these sets of parameters, we also considered snapshots calculated with two different GRMHD simulation algorithms, KHARMA (Prather et al. 2021) and BHAC (Porth et al. 2017), and corresponding images calculated using two different covariant radiation transport schemes, ipole (Mościbrodzka & Gammie 2018) and BHOSS (Younsi et al. 2012, 2016).

The second set comprises ∼4000 images from covariant plasma models in the Kerr metric that go beyond some assumptions of GRMHD. These employ analytic calculations that are agnostic to the particular microprocesses responsible for angular momentum transport and particle heating. The particular parameters of these models are discussed in detail in Özel et al. (2021).

The third category includes ∼200,000 images from analytic models that explore a range of black hole metrics that either are parametrically different from the Kerr metric or represent other known solutions to the field equations (Younsi et al. 2021). For the former, we employ the Johannsen–Psaltis (JP) metric (Johannsen & Psaltis 2011; Johannsen 2013b), which enables parametric deviations from Kerr and recovers the Kerr spacetime when its deviation parameters vanish, while still guaranteeing many of the basic properties of the Kerr metric (i.e., it is Petrov Type-D, free of pathologies, etc.). For the latter, we utilize the EMDA (Kerr–Sen) metric (García et al. 1995), which is a solution to the field equations of a modified gravity theory with additional scalar degrees of freedom. The plasma model is the same covariant analytic model of Özel et al. (2021), and the model library spans different black hole spins, observer inclinations, magnetic field configurations, plasma parameters, and, where appropriate, metric parameters, as discussed in Younsi et al. (2021). We refer to these models as analytic non-Kerr.

Using the covariant radiation transport code BHOSS (Younsi et al. 2012, 2016), Figure 6 presents a selection of illustrative simulated 1.3 mm Sgr A* images from five different non-Kerr spacetimes, together with an image from a GRMHD simulation of a Kerr black hole. The field of view in all panels is 150 μas in both directions, with the brightest pixel value in each panel normalized to unity. We show in the top row mean images from covariant MHD simulations averaged over a time window of 5000 GM/c³, with snapshots every 10 GM/c³ (∼3.5 minutes for Sgr A*). The Kerr GRMHD simulation parameters are as follows: MAD magnetic field configuration, a = 0.9375, i = 30°, R_low = 1, and R_high = 10 (see Paper V for further details of the modeling). The top middle panel shows an image of accretion onto a nonrotating dilaton black hole (Mizuno et al. 2018). The top right panel presents the image from a simulation of accretion onto a boson star (Olivares et al. 2020; Fromm et al. 2021). The boson star image represents one example of a compact object without an event horizon or an unstable photon orbit, thereby lacking a central brightness depression or a photon ring in its image. We do not consider such configurations in the calibration procedure discussed here but explore them in detail in Section 4.

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We present in the bottom row of Figure 6 images from non-Kerr spacetimes with the background semianalytic accretion flow model as specified in Özel et al. (2021) and Younsi et al. (2021). These spacetimes are the JP and the Kerr–Sen (EMDA) metrics, as well as a spinning traversable wormhole spacetime (Teo 1998; Harko et al. 2009). The JP metric for this example is nonspinning, with deformation parameters chosen to push the unstable photon orbit very close to the event horizon (hence the smaller central brightness depression). The Kerr–Sen spacetime parameters (axion and dilaton field couplings) have been chosen to produce an image with a photon ring larger than is possible with a Kerr black hole. Finally, the rotating wormhole spacetime is chosen to have a throat radius equal to the event horizon radius of a Kerr black hole with the same spin (a = 0.9375). In all of the examples with a central brightness depression, the size of the ring-like image scales with that of the black hole shadow.

We convolve all of the images in the three categories with an n = 2, 15 Gλ Butterworth filter to mimic the resolution of the EHT array. We then apply the characterization algorithm CHARM to all of these images to measure the median diameter ${D}_{\mathrm{im}}$ of the bright ring of emission, with respect to the analytically calculated center of the black hole shadow. We also calculate the shadow diameter in each spacetime; for Kerr, we use the analytic approximation derived in Chan et al. (2013). We then define the calibration factor α₁ as the ratio of the median diameter to the diameter of the shadow. We will refer to the difference α₁ − 1 as the fractional diameter difference. If the peak emission in the bright ring coincides with the shadow boundary, then the fractional diameter difference would be equal to zero.

We show in Figure 7 the distribution over the fractional diameter difference for the three types of images. As discussed in Özel et al. (2021) and Younsi et al. (2021), the distribution peaks at small positive values of α₁ − 1, indicating that the peak of the bright ring is slightly larger than the boundary of the black hole shadow.

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We turn to quantifying the correction factor α₂ and its uncertainty that arises from applying imaging and model-fitting tools to infer the size of a ring-like image. To this end, we first characterized all simulations discussed in Section 3.2 based on image morphology and size, degree of variability, spacetime metric, and plasma model. We then randomly selected segments and snapshots from each category. We assigned a random position angle in the sky to each image and generated synthetic EHT data from them using the VLBI synthetic data generation pipeline SYMBA (Janssen et al. 2019; Roelofs et al. 2020b; Natarajan et al. 2022). SYMBA accounts for the effects of interstellar scattering through the Galactic disk, as well as several realistic atmospheric, instrumental, and calibration effects. In addition, we designated a last category in which synthetic data were generated from a small number of snapshots but with several different realizations of all the measurement uncertainties. This yielded a total of 145 synthetic data sets.

We carried out blind image reconstructions and mG-ring fits to all the synthetic data using the same EHT imaging pipelines as those applied to Sgr A* data, separating into teams that did not have any prior knowledge of the synthetic data characteristics. As for the case of the real data, imaging teams generated a top set of reconstructions for each synthetic data set, using the exact same set of algorithmic parameters as those used for the real Sgr A* data. We applied CHARM to the entire top-set image reconstructions (for a total of 145 data sets × 2000 top-set parameters × 3 algorithms) and to the ground-truth images to measure the calibration factor α₂. Modeling teams applied the snapshot fitting procedure with an mG-ring model and returned their posteriors for the model diameter, which we used to calculate the α₂ calibration factor.

In the majority of cases, the set of reconstructions that correspond to the full range of top-set parameters or posteriors yielded a narrow range of diameters and widths for the ring features, indicating a robust inference of the prevalent features with little sensitivity to the choice of regularizers. However, in <30% of the data sets, the features of the images varied significantly within the top-set parameters, leading to an uncertainty in the ring diameter that is ∼3–8 times larger than what is measured in the Sgr A* data (see Figure 3). This primarily happens when the image size, position angle, and asymmetry of the ground-truth image that led to the particular synthetic data set conspire in a way to remove any prominent salient features in the visibility domain and the image reconstruction is dominated by the priors rather than any unique features in the data that an imaging or model-fitting algorithm can pick up on. More quantitatively, we define the spread in the diameter for all the reconstructions of a given synthetic data set by using the metric

$$\begin{eqnarray}&&\mathrm{diameter}\ \mathrm{spread}=\displaystyle \frac{{d}_{85}-{d}_{15}}{{d}_{50}},\end{eqnarray} \tag{ 3 }$$

where d₈₅, d₅₀, and d₁₅ refer to the 85th, 50th, and 15th percentiles of diameters in a given distribution, respectively. The spread in the top-set reconstructions of the actual Sgr A* data using this metric is 0.06–0.1 (see Figure 3). We place a conservative limit of diameter spread of less than 0.2 for the synthetic data reconstructions and include only the data sets that fulfill this criterion in our derivation of the α₂ calibration parameters.

Figures 8 and 9 show the distributions of the fractional diameter difference α₂ − 1 for the imaging reconstructions and mG-ring model fits, respectively, of the synthetic data sets discussed above. The trend in Figure 7, i.e., the slight offset between the peaks of the distributions calculated for the different imaging methods, follows the one we see in the reconstruction of the actual EHT Sgr A* data very closely (see Figure 3). This result reinforces our conclusion that the small differences in the inferred diameters between different algorithms are primarily caused by the different methodologies, prior, and regularizer choices in those methods (see Paper III). The same is true for the trend in the mG-ring results, albeit corresponding to more marked differences.

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We use the combination of the measurements and calibrations discussed in the previous sections to infer the diameter of the boundary of the black hole shadow, ${d}_{\mathrm{sh}}={\hat{d}}_{m}/({\alpha }_{1}\,{\alpha }_{2})$. The posterior over the shadow diameter is given by

$$\begin{eqnarray}\begin{array}{rcl}P({d}_{\mathrm{sh}}| \hat{d}) & = & C\displaystyle \int d{\alpha }_{1}\displaystyle \int d{\alpha }_{2}\,{ \mathcal L }[\hat{d}| {d}_{\mathrm{sh}},{\alpha }_{1},{\alpha }_{2}]\\ & & \times \,P({d}_{\mathrm{sh}})P({\alpha }_{1})P({\alpha }_{2}),\end{array}\end{eqnarray} \tag{ 4 }$$

where ${ \mathcal L }[\hat{d}| {d}_{\mathrm{sh}},{\alpha }_{1},{\alpha }_{2}]$ is the likelihood of measuring a ring diameter $\hat{d}$ given the model parameters, P(α₁) and P(α₂) are the distributions of the calibration parameters, and C is an appropriate normalization constant. P(d_sh) is the prior over the shadow diameter, which we assume to be flat over a range that is much broader than that of the posteriors.

We show in Figure 10 the posteriors over the shadow diameter as inferred from the three image-domain algorithms and for the different theoretical calibrations discussed in Section 3.2. In Table 2, we report the most likely values of the black hole shadow diameter for Sgr A*, as well as the 68th percentile credible levels. Finally, in Figure 11 we overlay the inferred shadow boundaries on the average EHT image of Sgr A* obtained from the 2017 April 7 data (Event Horizon Telescope Collaboration et al. 2022b). In this plot, the solid lines show the range 46.9–50.0 μas of the most likely values, and the dashed lines show the envelope of the 68th percentile credible intervals across the different methods, spanning 41.7–55.6 μas.

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Table 2. Schwarzschild Deviation Parameter δ for Sgr A*

	θg Prior	GRMHD	Analytic	Analytic
			Kerr	Non-Kerr
eht-	VLT(I)	$-{0.08}_{-0.09}^{+0.09}$	$-{0.05}_{-0.11}^{+0.09}$	$-{0.07}_{-0.09}^{+0.10}$
imaging	Keck	$-{0.04}_{-0.10}^{+0.09}$	$-{0.02}_{-0.11}^{+0.11}$	$-{0.02}_{-0.09}^{+0.10}$
SMILI	VLT(I)	$-{0.10}_{-0.10}^{+0.12}$	$-{0.08}_{-0.11}^{+0.13}$	$-{0.08}_{-0.10}^{+0.12}$
	Keck	$-{0.06}_{-0.10}^{+0.13}$	$-{0.04}_{-0.11}^{+0.13}$	$-{0.04}_{-0.10}^{+0.13}$
DIFMAP	VLT(I)	$-{0.12}_{-0.08}^{+0.10}$	$-{0.10}_{-0.10}^{+0.09}$	$-{0.10}_{-0.09}^{+0.09}$
	Keck	$-{0.08}_{-0.09}^{+0.09}$	$-{0.07}_{-0.11}^{+0.10}$	$-{0.07}_{-0.09}^{+0.09}$
mG-ring	VLT(I)	$-{0.17}_{-0.10}^{+0.11}$	$-{0.14}_{-0.13}^{+0.10}$	$-{0.17}_{-0.09}^{+0.13}$
	Keck	$-{0.13}_{-0.11}^{+0.11}$	$-{0.12}_{-0.11}^{+0.11}$	$-{0.12}_{-0.11}^{+0.10}$

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Using the uncertainties discussed above, we obtain the posterior over the deviation parameter δ by

$$\begin{eqnarray}\begin{array}{rcl}P(\delta | \hat{d}) & = & C\displaystyle \int d{\alpha }_{1}\displaystyle \int d{\alpha }_{2}\displaystyle \int d{\theta }_{{\rm{g}}}\,{ \mathcal L }[\hat{d}| {\theta }_{{\rm{g}}},{\alpha }_{1},{\alpha }_{2},\delta ]\\ & & \times \,P(\delta )P({\theta }_{{\rm{g}}})P({\alpha }_{1})P({\alpha }_{2}).\end{array}\end{eqnarray} \tag{ 5 }$$

Here C is an appropriate normalization constant, ${ \mathcal L }[\hat{d}| {\theta }_{{\rm{g}}},{\alpha }_{1},{\alpha }_{2},\delta ]$ is the likelihood of measuring a ring diameter $\hat{d}$ given the model parameters, which we identify with the distributions of measurements from the imaging and visibility-domain methods, P(θ _g ) denotes the prior in θ _g given by stellar-dynamics measurements, and P(α₁) and P(α₂) are obtained from the calibration procedures outlined in Sections 3.2 and 3.3.

As discussed earlier, we consider two separate priors for θ _g denoted by Keck and VLTI; three different measurements of the ring diameter from imaging methods (together with their corresponding α₂ calibrations) denoted by eht-imaging, SMILI, and DIFMAP; and three different sets of snapshot images for the α₁ calibration denoted by GRMHD, Analytic Kerr, and Analytic JP. We assume a flat prior in the fractional deviation δ, with limits that cover a range that is sufficiently broad not to affect the posteriors. We perform the two integrals in Equation (5) numerically and show the resulting posteriors in the deviation parameter δ in Figure 12.

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We repeat the same procedure for the measurements obtained from mG-ring fits to the Sgr A* data. We show the corresponding result for the deviation parameter δ in Figure 13.

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We present in Table 2 the means and 68th percentile credible levels for the posteriors we obtain for the deviation parameter δ using different combinations of black hole mass priors, theoretical models used for calibration, and the imaging and model-fitting methods used on the Sgr A* data. All of the posteriors are consistent with each other and with no deviation from the general relativistic predictions. We choose the eht-imaging +Keck+GRMHD and eht-imaging +VLTI+GRMHD combinations as the two fiducial cases to calculate constraints on the individual metric parameters in the remainder of this paper.

Accretion in Sgr A* is believed to occur via a hot accretion flow ¹⁵¹ (Yuan & Narayan 2014). Now that the EHT image of Sgr A* (Paper III) has revealed a brightness temperature well in excess of 10⁹ K, the evidence for the presence of very hot gas is particularly compelling.

The radiative luminosities of hot accretion flows are generally far less than $\dot{M}{c}^{2}$ (Narayan et al. 1995; Yuan & Narayan 2014), where $\dot{M}$ is the mass accretion rate. Therefore, the accreting gas in these systems reaches the compact object at the center with a considerable amount of thermal and kinetic energy. If the compact object is a black hole, this energy simply disappears through the event horizon. On the other hand, if the object has a surface, the energy will be thermalized and reradiated (once the system reaches steady state), giving a large surface luminosity that should be visible to a distant observer. Observations can thus tell the difference between an event horizon and a thermalizing surface.

In the previous paragraph, and also in the rest of Section 4, we assume that (i) matter in the compact object at the center of Sgr A* satisfies energy conservation; (ii) it obeys the laws of thermodynamics, in particular, it approaches statistical equilibrium in steady state; and (iii) it couples to and radiates in all electromagnetic modes. These assumptions can be considered "natural" minimal principles, but they can be violated in extreme models. For example, the shell-like black hole mimicker described in Danielsson et al. (2021) can be designed either not to produce any electromagnetic radiation or to radiate only in a handful of modes, thereby violating assumption (iii). It is not possible to constrain such models using astronomical observations in electromagnetic bands, though in certain cases it may be possible to distinguish them via gravitational waves (Abbott et al. 2021; see, e.g., Chirenti & Rezzolla 2007, for the case of gravastars). Note that even very exotic objects would satisfy our assumptions, including (iii), if only a small fraction of the accreted baryonic gas survives on their surface as normal matter. To be optically thick in the electromagnetic bands of interest to us, the skin of normal matter should have a surface density as little as 1 g cm⁻², which corresponds to just 10⁻¹⁴ of the total mass of Sgr A*. An exotic object would need to convert all accreted gas on its surface to electromagnetically inactive material if it is to escape detection by electromagnetic observations.

For a spherically symmetric spacetime, matter that starts from rest at infinity and then accretes via a radiatively inefficient mode to come to rest on a surface at radius R_* will release thermal energy as measured at infinity equal to a fraction η of the rest-mass energy of the gas, where (the following expression is obtained for the Schwarzschild metric; Broderick & Narayan 2006)

$$\begin{eqnarray}&&\eta =1-{\left(1-\displaystyle \frac{2M}{{R}_{* }}\right)}^{1/2}\gtrsim \displaystyle \frac{M}{{R}_{* }},\end{eqnarray} \tag{ 6 }$$

and we use geometrized units: G = c = 1. If the released thermal energy is radiated back to infinity—we emphasize that this is unavoidable once the object reaches steady state—the extra luminosity from the thermalizing surface will be typically much larger than the luminosity radiated by the hot accretion flow itself. This feature can be exploited to distinguish black holes, which by definition have an event horizon, from other kinds of compact objects that have a surface. In the context of stellar-mass black holes, this argument provides a convenient way of distinguishing black holes from neutron stars (Narayan & Yi 1995; Narayan et al. 1997; Garcia et al. 2001; Done & Gierliński 2003; McClintock et al. 2004; see Narayan & McClintock 2008 for a review).

In the case of Sgr A*, the argument proceeds differently. In essence, the observed submillimeter radiation provides a lower limit on the mass accretion rate, ${\dot{M}}_{\min }$, regardless of whether the radiation is produced by inflowing hot gas or an outflowing jet. Therefore, given an assumed radius R_* of the surface, we can estimate the minimum surface luminosity that should be observed at infinity,

$$\begin{eqnarray}&&{L}_{\infty }\gt \eta {\dot{M}}_{\min }{c}^{2}.\end{eqnarray} \tag{ 7 }$$

As we show below, the surface radiation should appear in the infrared, where observations provide strong upper limits on the luminosity of Sgr A*. These limits lie far below the predicted minimum surface luminosity, implying that Sgr A* does not have a radiating surface. Versions of this argument have been made in previous papers in the context of Sgr A* (see Narayan & McClintock 2008, for a review). A similar argument also applies to the supermassive black hole in M87 (Broderick et al. 2015; Event Horizon Telescope Collaboration et al. 2019c). In related work, Lu et al. (2017) argued that the absence of flashes of radiation from stars crashing on supermassive black hole candidates in galactic nuclei requires these candidates to be true black holes with event horizons.

4.1.1. EHT Limit on the Radius of the Surface

In the case of Sgr A*, a somewhat weak link in the argument outlined above was the hitherto lack of a strong upper limit on the radius R_* of a putative surface in Sgr A*. Since the surface luminosity for a given $\dot{M}$ scales as ${L}_{\infty }\propto \dot{M}(M/{R}_{* })$ (Equations (6) and (7)), one could make the predicted luminosity small by arbitrarily increasing R_*/M, thereby evading observational limits. This loophole has now been closed by EHT observations.

Using a maximally conservative analysis of EHT 2017 visibility data, and without any model assumptions, Paper II estimates the FWHM of the image of Sgr A* to lie in the range 39–87 μas. With a conversion factor, GM/c² D ≈ 5 μas (see Figure 1), this corresponds to an image diameter <18M.

The observed 230 GHz radiation in Sgr A* is from the hot accretion flow, not from the surface (which should radiate in the infrared). Any surface must lie interior to the 230 GHz emitting hot accretion flow and should have an apparent diameter smaller than 18M. Thus, from the analysis in Paper II, we set the following upper limit on the apparent radius of the surface as viewed by a distant observer: R_app < 9M.

Paper III presents image reconstructions of Sgr A* based on the EHT 2017 data. Table 7 in that paper summarizes the results of fitting a ring model to image reconstructions based on several methods. Using the imaging results from DIFMAP, eht-imaging, and SMILI and combining the ring analyses with REx and VIDA (see Paper III for details), the average ring diameter estimate is d = 51.3 ± 2.0 μas, and the ring width estimate is w = 29.6 ± 3.6 μas (these results correspond to descattered images from April 7 data). We take (d + w) = 80.9 ± 4.1 μas as a reasonable proxy for the apparent outer diameter of the source. Using the 95% confidence upper limit, (d + w) < 88 μas, we obtain R_app < 8.8M (95% CL). Paper III obtains a tighter constraint using the Bayesian imaging method Themis, while Paper IV similarly reports tighter constraints by fitting mG-ring models (based on Johnson et al. 2020) directly to visibility data. To be conservative, we do not use these limits.

The analyses described in the previous paragraph treat d and w as uncorrelated quantities. However, as the careful analysis in Section 3.1 of the present paper shows, there is a strong anticorrelation between the estimated values of d and w, such that their sum (d + w) is quite tightly constrained. The dotted and dashed curves in Figure 4 correspond to (d + w) = 90 and 80 μas, or equivalently R_app = 9M and 8M, respectively. Clearly, from this analysis, R_app < 8M is a safe upper limit (at about 95% confidence).

To be very safe, we choose as a conservative upper limit on the apparent radius of a surface in Sgr A* R_app < 9M. For a Schwarzschild spacetime, gravitational deflection of rays causes the apparent radius of a spherical surface as viewed by an observer at infinity to be larger than the true areal radius R_*. The relation between the two is

$$\begin{eqnarray}\begin{array}{rcl}{R}_{\mathrm{app}} & = & 3\sqrt{3}M,\qquad \qquad \qquad {R}_{* }\leqslant 3M,\\ & = & {R}_{* }{\left(1-\displaystyle \frac{2M}{{R}_{* }}\right)}^{-1/2},\,{R}_{* }\gt 3M.\end{array}\end{eqnarray} \tag{ 8 }$$

Our upper limit, R_app < 9M, then corresponds to R_* < 8M. In the discussion below, we consider the full range of allowed R_* values, from the event horizon radius R_H = 2M to the upper limit, namely, 2M < R_* ≤ 8M.

4.1.2. Predicted Spectrum of Surface Radiation

Paper V discusses hot accretion flow models of Sgr A* based on extensive GRMHD simulations. The models indicate that the mass accretion rate in Sgr A* is typically $\dot{M} \tilde {10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ (similar to estimates reported in, e.g., Falcke et al. 1993; Yuan et al. 2003; Chael et al. 2018; Ressler et al. 2020), but with a broad distribution that extends from ${\dot{M}}_{\min } \tilde {10}^{-9}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ to ${\dot{M}}_{\max } \tilde {10}^{-6}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$. The models at the lower end of this range are actually ruled out by various constraints (see Paper V); nevertheless, we stick to ${\dot{M}}_{\min }={10}^{-9}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ as a safe and conservative lower limit on the mass accretion rate. Equations (6) and (7), combined with our upper limit on R_*, then show that the surface luminosity measured at infinity must be ≳10³⁷ erg s⁻¹.

Meanwhile, we know that the hot accretion flow in Sgr A* produces synchrotron radiation at submillimeter wavelengths with a luminosity ∼5 × 10³⁵ erg s⁻¹, shown by the green curves in Figure 14. Even in the absence of any independent estimate of $\dot{M}$, just the fact that accretion results in this much radiation implies a certain minimum energy flow onto the surface. Since the accreting gas generally moves radially inward, relativistic beaming causes more radiation to impinge on the central object compared to what escapes to infinity. Thermalization of this infalling radiation would then give a surface luminosity greater than ¹⁵² 5 × 10³⁵ erg s⁻¹. Any additional energy released by the mechanical and thermal energy of the infalling gas (this is expected to dominate in most scenarios) would further increase the surface luminosity. We therefore treat 5 × 10³⁵ erg s⁻¹ as an even more conservative lower bound on the surface luminosity of Sgr A* than that discussed in the previous paragraph.

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A key feature of radiation emitted from a central surface in a hot accretion flow is that it will appear in a different region of the electromagnetic spectrum than the emission from the hot accreting gas and jet. The latter dominates in the radio and submillimeter bands (Figure 14, green curves). Meanwhile, the radiating gas at the surface, being optically thick, will radiate like a blackbody to a very good approximation (McClintock et al. 2004; Broderick & Narayan 2006, 2007).

The temperature of this radiation, measured at infinity, is given by

$$\begin{eqnarray}\begin{array}{rcl}{T}_{\infty } & = & {\left(\displaystyle \frac{{L}_{\infty }}{4\pi {R}_{\mathrm{app}}^{2}{\sigma }_{\mathrm{SB}}}\right)}^{1/4}\\ & \approx & 2100\,{\rm{K}}{\left(\displaystyle \frac{{L}_{\infty }}{{10}^{36}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right)}^{1/4}{\left(\displaystyle \frac{9M}{{R}_{\mathrm{app}}}\right)}^{1/2},\end{array}\end{eqnarray} \tag{ 9 }$$

where σ_SB is the Stefan–Boltzmann constant. For the estimates of L_∞ and R_app derived earlier, the predicted radiation should be in the NIR and optical bands. If we define the characteristic frequency ν_* of the blackbody radiation by h ν_* = kT_∞, the SED at infinity takes the form

$$\begin{eqnarray}&&\nu {L}_{\nu }=\displaystyle \frac{15}{{\pi }^{4}}\,{L}_{\infty }\displaystyle \frac{{(\nu /{\nu }_{* })}^{4}}{\exp (\nu /{\nu }_{* })-1},\quad {\nu }_{* }\equiv \displaystyle \frac{{{kT}}_{\infty }}{h}.\end{eqnarray} \tag{ 10 }$$

The left panel in Figure 14 shows predicted SEDs of surface radiation from Sgr A*, if the object has a surface with a radius R_* = 2.5M; we choose this radius as a fiducial model for illustration. The three solid black curves correspond to mass accretion rates $\dot{M}=[{10}^{-7},\,{10}^{-8},\,{10}^{-9}]\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$, respectively, the last of which is the conservative lower limit from Paper V mentioned earlier. The dashed blue curve corresponds to the absolute lower limit on the surface luminosity, L_∞ = 5 × 10³⁵ erg s⁻¹, discussed above.

The right panel of Figure 14 shows another sequence of models in which we vary the surface radius R_*. Taking the previously mentioned conservative mass accretion rate estimate of 10⁻⁹ M_⊙ yr⁻¹, we consider surface radii R_* = [2M, 3M, 4M, 6M, 8M], respectively.

In all the models shown in the two panels of Figure 14, the predicted surface emission (black and blue curves) is spectrally well separated from the synchrotron emission of the hot accretion flow (green curve).

Therefore, this predicted signature of surface emission is easy to identify via observations, making it possible to develop a robust test for the presence of a thermalizing surface.

4.1.3. Observational Limit on Surface Luminosity

4.1.4. Discussion and Caveats

Compared to previous discussions of this topic, what has improved is that, thanks to the EHT image of Sgr A*, we are now able to limit ourselves safely to surface radii R_* < 8M, whereas in earlier works much larger radii were considered (as large as 1000M in Narayan 2002 and 100M in Broderick et al. 2009). Moreover, the infrared constraints are also now very much stronger (Figure 14). Correspondingly, the argument for the absence of a thermalizing surface is substantially strengthened.

The discrepancy between the maximum steady infrared luminosity that Sgr A* can possibly have (the 5th percentile thick red line and large green circle in Figure 14) and the minimum possible luminosity it could theoretically have and still possess a thermalizing surface (the dashed blue line) is too large to be circumvented with small fixes to model details. This statement is true even if we use the 50th percentile of the infrared observations (the middle red line and middle green circle in Figure 14), which would be equivalent to counting all the observed infrared radiation, including the flares, as surface emission. If we wish to consider models of Sgr A* with a surface, we have to find a weakness in one of the links in the underlying logic of the argument. An easy way out is to give up one of the basic physics assumptions listed in the third paragraph of Section 4.1. Here we consider other less drastic possibilities.

Could the predicted infrared radiation from a hypothetical surface in Sgr A* be obscured by foreground matter such as dust? This is highly unlikely since the radiation from the infrared flares is clearly visible, and that radiation comes from hot external gas (not from the surface) at radii within 10M (Gravity Collaboration et al. 2018b). It is hard to imagine an obscuring medium that allows flare emission to make it through but blocks radiation from the surface.

Another minor worry may be quickly dealt with. Because of spacetime curvature, radiation from a surface at areal radius R_* in a Schwarzschild spacetime takes longer to reach a distant observer compared to a ray that travels in flat spacetime. Could this delay be so large that surface radiation has not yet reached us? Let us write

$$\begin{eqnarray}&&{R}_{* }=(1+\mu ){R}_{H},\end{eqnarray} \tag{ 11 }$$

where R_H = 2M is the radius of the event horizon. For μ ≪ 1, the additional time delay is $\sim 2M\mathrm{ln}(1/\mu )$, which is $\sim 40\,\mathrm{ln}(1/\mu )$ s in the case of Sgr A*. Even if the logarithm is as large as 100 (corresponding to R_* being located a Planck length above the event horizon), the extra time delay is only about an hour.

A related worry is that the gravitational redshift, (1 + z) ∼ μ^−1/2, between the surface and infinity might dilute the observed luminosity sufficiently to make the surface radiation invisible. Abramowicz et al. (2002) noted that this effect causes the radiation luminosity that reaches the observer to be reduced by a factor of 1/(1 + z)⁴ ∼ μ² compared to what is emitted at the surface. They claimed that, if (1 + z) were large enough, no detectable radiation would reach the observer and it would be impossible to distinguish an event horizon from a surface.

However, gravitational redshift is not an issue for the line of argument we have presented in this paper because we expressed everything in terms of energy and luminosity as measured at infinity; in such a framework, all redshift factors drop out. For instance, if the radiation observed at infinity has a temperature T_∞, then the radiation emitted by the surface will have a temperature T_local = (1 + z)T_∞ in the local rest frame. The radiation emerging from the surface will have a flux equal to ${\sigma }_{\mathrm{SB}}{T}_{\infty }^{4}{(1+z)}^{4}$, and the corresponding luminosity is larger than what reaches infinity by precisely the factor of (1 + z)⁴ noted by Abramowicz et al. (2002). The only question is whether the system has enough time to heat up to such a high local temperature. We discuss this important issue next.

Sgr A* is presumably as old as the Milky Way, i.e., several billion years old. Over much of that time, it must have accreted gas at a rate equal at least ¹⁵⁴ to the present $\dot{M}$. The time needed to achieve the steady-state condition implicit in Equation (9), namely, $\eta \dot{M}{c}^{2}={L}_{\infty }=(\sigma {T}_{\infty }^{4})(4\pi {R}_{\mathrm{app}}^{2})$, or equivalently T_local = (1 + z)T_∞, is far shorter than the age of Sgr A* for almost any model.

The one exception is if μ ≪ 1, i.e., if the surface R_* is extremely close to the event horizon. In this limit, as Lu et al. (2017) argued, the time required to achieve steady state scales as μ⁻¹ and can become arbitrarily long. The physical reason is that the region between R = R_* and the photon sphere, R_ph = 3M, traps radiation. This volume has a large thermal capacity and therefore takes a long time to reach steady state. Applying this logic to Sgr A*, Lu et al. (2017) concluded that the absence of infrared radiation in Sgr A* rules out a thermalizing surface only if μ ≳ 10⁻¹⁴. If the surface is even closer to the horizon radius than this limit, i.e., if R_* ≲ R_H + 10⁻² cm, then the steady-state condition will be invalid. Their argument thus provides an upper limit on R_*.

Using completely different reasoning, Carballo-Rubio et al. (2018) set a lower limit on μ. The argument goes as follows. Because Sgr A* accretes mass continuously, its horizon radius 2M increases with time. In order to maintain R_* = (1 + μ)R_H , the surface also needs to expand. However, if μ is too small, the required expansion speed is greater than the speed of light in the local frame, which is unphysical. Using a conservative estimate of $\dot{M} \tilde {10}^{-11}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ (which is two orders of magnitude smaller than the lower limit given in Paper V and more than seven orders of magnitude less than the likely average accretion rate over the life of Sgr A*), Carballo-Rubio et al. (2018) conclude that Sgr A* can avoid the faster-than-light conflict only if μ ≳ 10⁻²³, i.e., if R_* ≳ R_H + 10⁻¹¹ cm. Note that this rules out models in which the surface lies a Planck length (10⁻³³ cm) above the event horizon, as gravastar models (Mazur & Mottola 2001; Chapline 2003) often implicitly assume.

Combining the arguments in the previous two paragraphs, we are left with an interesting class of models with μ in the range 10⁻²³ < μ < 10⁻¹⁴ for which Sgr A* is currently allowed to have a thermalizing surface and yet not be ruled out by infrared constraints. This gap in model space merits further investigation.

Another issue worth serious discussion is the assumption that the surface will radiate like a blackbody. Since we are considering an object that (i) is in steady state and therefore in thermal equilibrium (by our assumptions), (ii) is likely nearly isothermal in the sense that the redshifted temperature T_∞ is independent of radius inside the object, and (iii) has an enormous optical depth, it seems unavoidable that the emission must be close to a blackbody. (For instance, stars radiate roughly like blackbodies because of their large optical depths and would be perfect blackbodies if they were isothermal.) Any deviations from a perfect blackbody in the putative surface radiation in Sgr A* might thus be expected to be minor. However, the specific case of radiation produced by energy release from matter falling on the surface of a compact supermassive (>10⁶ M_⊙) object has not been studied and merits further attention. Models of spherical accretion on neutron stars (M = 1.4 M_⊙) studied by Shapiro & Salpeter (1975) suggest that modest deviations from a perfect blackbody are expected in that case; their models show some hardening of the thermal spectrum plus the appearance of a power-law spectral component extending to higher frequencies. If the corresponding effects in the case of a surface in Sgr A* (M = 4 × 10⁶ M_⊙) are similarly modest, then our blackbody assumption is quite safe. Note that Shapiro & Salpeter (1975) did not include ray deflections and strong lensing in their model.

The argument for a blackbody spectrum is very strong in one particular limit. When the surface has a radius R_* close to R_H , i.e., μ ≪ 1, the volume between R_* and R_ph = 3M acts like an enclosed cavity, with radiation allowed to escape only over a small solid angle ∼ μ. The cavity then behaves like a textbook isothermal "furnace" with a tiny pinhole for escaping radiation. In this limit, the radiation that reaches a distant observer will be indistinguishable from a perfect blackbody (Broderick & Narayan 2006).

If the quiescent infrared radiation in Sgr A* corresponds to blackbody emission from a surface, it should be completely unpolarized. On the other hand, if the radiation is produced by synchrotron emission in optically thin (weak) flares, we might expect a certain degree of linear polarization. Bright infrared flares in Sgr A* show clear evidence for strong linear polarization (Eckart et al. 2006; Gravity Collaboration et al. 2020c), but there is currently no information on the degree of polarization of the weak emission below the 5th percentile. Sensitive polarimetry could be used in the future to explore this regime and might help to reduce even further the maximum level of blackbody emission allowed in Sgr A*.

In this subsection, we focus again on the possibility that Sgr A* may have a surface, but now we explore models in which the surface reflects incident radiation. We assume that, in the rest frame of the surface at some fixed areal radius R_*, the following properties hold: (i) any inward-moving ray that is incident with wavevector k^μ becomes an outward-moving ray with k^r reversed and the other components of k^μ unchanged; and (ii) if the intensity of the incoming ray is I_ν , the outgoing ray has an intensity AI_ν , where A ≤ 1 is the albedo of the surface. The motivation for considering such a model is that it makes interesting predictions that an interferometer like the EHT might be able to observe.

4.2.1. Synthetic Images Based on GRMHD Simulations

As an illustration of the effects we expect from a reflecting surface, we use a long-duration simulation of a hot accretion flow in the MAD state around a black hole of spin a_* = 0 (Narayan et al. 2021). We take the profiles of density, pressure, four-velocity, and magnetic field in the poloidal (r, θ)-plane, time-averaged over the simulation period t = 50,000M–100,000M. We set the electron temperature using the prescription given in Event Horizon Telescope Collaboration et al. (2019b, which is based on Mościbrodzka et al. 2016) with parameter values R_high = 20 and R_low = 1. We scale the density, and proportionately the gas pressure and magnetic energy density, such that the observed 230 GHz flux density is equal to 2.4 Jy, as measured during the 2017 EHT observations of Sgr A* (M. Wielgus et al. 2022, in preparation). We then compute a synthetic 230 GHz image for an observer at an inclination angle of i = 60°.

The top left panel in Figure 15 shows the 230 GHz image of the above model, assuming that the object at the center is a Schwarzschild black hole. The image is computed using the ray-tracing code HEROIC (Zhu et al. 2015; Narayan et al. 2016). The top middle panel shows the same image blurred with a Gaussian beam of FWHM equal to 15 μas; this beam size corresponds to the typical resolution that is achieved by the EHT using super-resolution image reconstruction techniques.

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The unblurred image in the top left panel of Figure 15 shows the usual features. The sharp circular ring is the photon ring produced by strong gravitational lensing by the black hole. The diffuse elliptical feature is the image of equatorial emission from the accretion flow, flattened in the vertical direction because of the 60° inclination of the observer. These two features are visible even in the 15 μas blurred image in the top middle panel (the features merge if we blur with a 20 μas beam, the nominal resolution of the EHT). Most importantly, a dark shadow region is clearly seen in the middle of even the blurred image.

The second row in Figure 15 shows the effect of including a reflecting surface with albedo A = 1 (100% reflection) at a radius R_* = 2.5M (selected as an example). In addition to the diffuse disk emission and sharp photon ring already described in the top left image, we find additional components that are caused by reflection. The thick bright ring at the center of the image corresponds to radiation from the equatorial accretion flow that is reflected from the side of the surface facing the observer. The thin ring (close to the original photon ring) is from rays that reflect off the far side of the surface and are then lensed around the compact object. Interestingly, the new features from reflection, especially the first one, appear in the shadow region of the original black hole image. When blurred, the resulting image, shown in the middle panel of the second row, has much of the shadow region filled in. This fairly dramatic effect is potentially distinguishable by the EHT.

The third and fourth rows in Figure 15 correspond to models with albedos A = 0.3 and 0.1, respectively. The image of the A = 0.3 model, when blurred, is only marginally different from the black hole image (top middle panel), while the blurred A = 0.1 model is indistinguishable from the black hole image.

The implication of these test images is that models in which Sgr A* has a reflecting surface with perfect albedo, A = 1, could potentially be distinguished by the EHT 2017 observations, but models with only partial albedo, e.g., A = 0.3 and 0.1, are harder to distinguish from the case of a black hole. Interestingly, in the latter models, a fraction (1 − A) of the radiation that falls on the surface must be absorbed and will presumably be reradiated as part of the thermalized emission discussed in Section 4.1. For any value of (1 − A) ≳ 0.1, this thermally reprocessed emission will lie well above the infrared limits discussed in Section 4.1.3 and shown in Figure 14. These models could thus be ruled out by that argument.

Note that several arbitrary choices were made in the above models: spin a_* = 0, temperature ratios, R_high = 20, R_low = 1, observer inclination i = 60°, and surface radius R_* = 2.5M. The values of R_high and i were chosen to lie near the center of the corresponding ranges considered in Paper V. As it happens, for spin a_* = 0, GRMHD-based models with these parameter values are fairly consistent with observations (see Paper V). Varying the parameters will certainly affect the predictions for the effect of surface reflection. The results may not change excessively since we pin the 230 GHz flux to 2.4 Jy. Nevertheless, we caution that the results presented in Section 4.2.2 below are for a preliminary toy model and are in the nature of a proof of concept. More detailed investigations are needed before we can draw firm conclusions.

An additional caveat is that, in this toy model, we have taken the flow solution to be the same as in a simulation that was run with a black hole event horizon at the center (Narayan et al. 2021). We simply truncated that solution at R = R_*. As mentioned in Section 4.1.4, the problem of self-consistently solving the gasdynamics and radiation field for a supermassive object with a surface has not yet been studied.

Another caveat is that we have considered only the case of specular reflection. Diffuse reflection, where radiation incident on the surface is reflected isotropically (or with a more complicated angular distribution), is also worth exploring. In that case, the surface reflected intensity will not be restricted to a few narrow features in the image, but will be spread more uniformly over the entire shadow region. This would eliminate any truly dark regions in the center of the image, conceivably making it easier to constrain such models.

Additionally, we considered a time-averaged steady image, whereas in reality we expect the image to fluctuate, which can lead to interesting time correlations of features. In addition, we have focused here on a spherically symmetric spacetime around a nonspinning object. Once we allow the central object to rotate, we will need to solve for the corresponding spacetime, which in general will not belong to the Kerr family of solutions.

4.2.2. Constraints from EHT Images

As examples of horizonless compact objects without surfaces, we consider black hole mimickers such as (mini-)boson stars, for which synthetic images from covariant MHD simulations of radiatively inefficient accretion flows have recently been obtained in Olivares et al. (2020). It was shown there that a region with a central brightness depression could appear in the final observed image of an unstable boson star (model A there), despite the absence of an unstable photon orbit in that spacetime. These features are the result of an effective low-density region that appears in the center of the spacetime owing to a centrifugal barrier. The unstable boson star has a significantly smaller intrinsic source size than corresponding constraints of Sgr A*, thereby ruling it out as a candidate alternative. Similarly, for the stable boson star configuration considered in Olivares et al. (2020), there is a complete absence of a central brightness depression with the inner image being extremely bright, akin to a radiating surface (see, e.g., Fromm et al. 2021). Given the EHT constraints, a mini-boson star becomes an unlikely candidate as a black hole mimicker to describe Sgr A*, since their image morphologies are generally too compact and lack both a characteristic ring-like feature and a central brightness depression. However, a more extensive study of a boson star spin, compactness, and astrophysical setup should be considered to make this argument conclusive (e.g., Vincent et al. 2021).

While the size of the bright emission ring/central brightness depression will be necessary to rule out or constrain the black hole and non-black-hole models considered in Section 5 below, the very presence of a central brightness depression in the 2017 image of Sgr A* (see Paper III; Paper IV; Paper V) is sufficient to rule out various models for compact objects that do not possess photon spheres. For example, these observations rule out the possibility that Sgr A* is a nonspinning Joshi–Malafarina–Narayan-2 (Joshi et al. 2014; JMN-2) naked singularity since these exotic compact objects do not cast shadows (Shaikh et al. 2019). We note that the JMN-2 spacetime is an exact solution of the Tolman–Oppenheimer–Volkoff equations of general relativity, it can form as the nonempty end state of the gravitational collapse of a (nonthermal) perfect fluid from regular Cauchy data, and photons in the spacetime move on null geodesics of the metric tensor (see also the associated discussion in Section 5.2). We assume that the naked singularity present at r = 0 does not interact with matter or radiation (classical gravity).

According to the no-hair theorem in general relativity, the only stationary, asymptotically flat, Ricci-flat spacetime that is free of pathologies ¹⁵⁶ is the one described by the Kerr metric. We do not consider here the astrophysically irrelevant case of black holes with a net electric charge (see Section 5.2). As a result, introducing simple phenomenological deviations to the Kerr metric leads to pathologies that severely constrain our ability to make predictions for the size of the black hole shadow, especially in the case of spinning black holes (see Johannsen 2013a for a detailed study of the pathologies of several parameterized metrics and Kocherlakota & Rezzolla 2022 for an analysis of theoretically allowed parameter spaces of the RZ metric). For this reason, several parameterized metrics have been developed in the past decade that allow for general deviations from the Kerr metric while minimizing pathologies mostly by relaxing the assumption of Ricci flatness. These parameterized metrics are completely agnostic to an underlying physical theory; therefore, significant assumptions must be made for stability tests (see, e.g., Suvorov & Völkel 2021 for a quasi-normal-mode analysis of the RZ metric). Among these metrics, we choose three representative ones: the so-called JP metric (Johannsen & Psaltis 2011, which was further developed to ensure the presence of a Carter-like integral of motion in Johannsen 2013b), the Modified Gravity Bumpy Kerr (MGBK) metric (Vigeland et al. 2011), and the so-called RZ metric (Rezzolla & Zhidenko 2014, which was further developed to include the effects of spin in Konoplya et al. 2016). We will derive analytic constraints for all three metrics and will use numerical calculations to derive spin-dependent constraints for both the JP and MGBK metrics.

Earlier studies have demonstrated that, because of a near cancellation between the effects of frame dragging and those of the quadrupole moment of the spacetime, the spin of the black hole affects the shadow size only marginally (see, e.g., Johannsen & Psaltis 2010; Psaltis et al. 2020a). Therefore, the bounds on the deviation parameters imposed by the measurement of the black hole shadow in Sgr A* are also expected to depend weakly on black hole spin. We demonstrate this in Figure 16, which shows the limits on different deviation parameters of the JP and MGBK metrics as a function of black hole spin, for various observer inclinations, and for different values of the secondary deviation parameters. The horizontal dashed curves show the bounds for nonspinning black holes when the secondary deviation parameters are zero. For this figure, we have set the parameters that affect the g_tt -component of the metric at the r⁻² order to zero (i.e., α₁₂ = 0 and ${\gamma }_{\mathrm{4,2}}=-\tfrac{1}{2}{\gamma }_{\mathrm{1,2}}$; see also Section 5.2) so we can focus on higher-order effects. The resulting constraints are of order unity and weakly dependent on spin, inclination angle, and the values of secondary deviation parameters.

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The simulations used for this figure are described in Medeiros et al. (2020). In these simulations we assume that the geodesic equation holds for all metrics and solve for the trajectories of photons ignoring matter effects. We define the boundary of the black hole shadow as the critical impact parameter between photon trajectories that fall into the event horizon and those that escape to infinity. As was done in Section 3, we define the size of a shadow as its median radius and compare this measurement to the bounds on shadow size from the eht-imaging algorithm, the GRMHD simulation library, and the prior mass and distance measurements from Keck.

As found in Psaltis et al. (2020a), the measurement of the size of the black hole shadow places constraints of order unity primarily on parameters such as α₁₃ and γ_1,2 that depend weakly on the magnitude of the black hole spin. What is common between these parameters is that they describe deviations in the tt-component of the black hole metric, as expressed in areal coordinates.

Since spin has a relatively small effect on the predicted shadow size and, hence, on the metric constraints, we now focus on a more detailed exploration of the constraints on these three metrics when we set the spin to zero (i.e., in the limit of spherical symmetry). The radius of the shadow in this limit is given by (Psaltis et al. 2020a)

$$\begin{eqnarray}&&{r}_{\mathrm{sh}}=\displaystyle \frac{{r}_{\mathrm{ph}}}{\sqrt{-{g}_{{tt}}({r}_{\mathrm{ph}})}},\end{eqnarray} \tag{ 15 }$$

where

$$\begin{eqnarray}&&{r}_{\mathrm{ph}}\equiv 2{g}_{{tt}}({r}_{\mathrm{ph}}){\left({\left.\displaystyle \frac{{{dg}}_{{tt}}}{{dr}}\right|}_{{r}_{\mathrm{ph}}}\right)}^{-1}\end{eqnarray} \tag{ 16 }$$

is the radius of the photon orbit.

For nonspinning black holes, the tt-component of the JP metric in areal coordinates is (see Johannsen 2013a)

$$\begin{eqnarray}\begin{array}{rcl}{g}_{{tt}}^{\mathrm{JP}} & = & -\left(1-\displaystyle \frac{2}{{r}_{A}}\right)\\ & & \times \,{\left(1+\sum _{i=2}^{\infty }\displaystyle \frac{{\alpha }_{1i}}{{r}_{A}^{i}}\right)}^{-2},\end{array}\end{eqnarray} \tag{ 17 }$$

where α_1i are deviation parameters and the subscript A denotes the fact that we use areal coordinates.

The tt-component of the nonspinning MGBK metric in areal coordinates is (see Gair & Yunes 2011; Vigeland et al. 2011)

$$\begin{eqnarray}&&{g}_{{tt}}^{\mathrm{MGBK}}=-\left(1-\displaystyle \frac{2}{{r}_{A}}\right)\left[1-{\gamma }_{1}({r}_{A})-2{\gamma }_{4}({r}_{A})\left(1-\displaystyle \frac{2}{{r}_{A}}\right)\right],\end{eqnarray} \tag{ 18 }$$

where γ₁(r_A ) and γ₄(r_A ) are defined by

$$\begin{eqnarray}&&{\gamma }_{A}=\sum _{n=2}^{\infty }\displaystyle \frac{{\gamma }_{A,n}}{{r}_{A}^{n}},\end{eqnarray} \tag{ 19 }$$

A = 1 or 4, and γ_A,n are the deviation parameters.

Finally, the tt-component of the spherically symmetric RZ metric in areal coordinates is (see Rezzolla & Zhidenko 2014)

$$\begin{eqnarray}\begin{array}{rcl}{g}_{{tt}}^{\mathrm{RZ}} & = & -x\left[1-\epsilon (1-x)+({a}_{0}-\epsilon ){\left(1-x\right)}^{2}\right.\\ & & +\left.\,\tilde{A}(x){\left(1-x\right)}^{3}\right],\end{array}\end{eqnarray} \tag{ 20 }$$

where

$$\begin{eqnarray}\begin{array}{rcl}x & \equiv & 1-\displaystyle \frac{{r}_{0}}{{r}_{A}},\\ \tilde{A}(x) & = & \displaystyle \frac{{a}_{1}}{1+\tfrac{{a}_{2}x}{1+\tfrac{{a}_{3}x}{...}}},\end{array}\end{eqnarray} \tag{ 21 }$$

and r₀ is the coordinate radius of the infinite redshift surface (identified with the horizon if no pathologies exist). The parameters , a₁, a₂, ... are the deviation parameters. As done in Psaltis et al. (2021), we write all radii in terms of the mass of the black hole at infinity, which fixes one of the parameters

$$\begin{eqnarray}&&\epsilon =-\left(1-\displaystyle \frac{2}{{r}_{0}}\right).\end{eqnarray} \tag{ 22 }$$

For simplicity we assume r₀ = 2 throughout the rest of this section.

Using these analytic expressions, we calculate the size of the black hole shadow as a function of the various deviation parameters using Equation (15). We then apply the bounds on the shadow size imposed by the Sgr A* images and obtain constraints on the deviation parameters of the various metrics. The dashed lines in Figure 16 compare the analytic bounds with those obtained numerically for the spinning spacetimes.

Because we use only one measured quantity from the EHT image of Sgr A*, i.e., the size of the black hole shadow, but the tt-components of each metric depend on a series of deviation parameters, it follows that the EHT observations place, in fact, correlated constraints on these parameters. These can be thought of as subspaces in the multidimensional parameter space of deviations and are very difficult to visualize in full generality. In Figure 16, we showed one particular cross section of this parameter space in which the various parameters were combined such that deviations from Kerr appear only at the second or higher post-Newtonian order.

Figure 17 shows a different cross section of these constraints, where we have varied only one parameter at a time, setting all others equal to zero. The constraints derived from the Keck bounds are also summarized in Table 3. The deviation parameters that correspond to higher-order corrections (as denoted by the second integer in their subscripts) affect the size of the shadow less strongly than the lower-order parameters and are, therefore, less constrained. We will return to the magnitudes of these constraints in Section 5.3, after we discuss the bounds on metrics that correspond to solutions of particular modified gravity theories.

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Table 3. Constraints on Deviation Parameters of Various Metrics

Parameterized Metric	Constraints
JP	$\begin{array}{l}-1.1\lesssim {\alpha }_{12}\lesssim 0.5\\ -3.1\lesssim {\alpha }_{13}\lesssim 1.5\\ -7.8\lesssim {\alpha }_{14}\lesssim 4.6\end{array}$
MGBK	$\begin{array}{l}-4.0\lesssim {\gamma }_{\mathrm{1,2}}\lesssim 1.4\\ -4.8\lesssim {\gamma }_{\mathrm{4,2}}\lesssim 1.3\\ -8.0\lesssim {\gamma }_{\mathrm{1,3}}\lesssim 2.7\\ -14.0\lesssim {\gamma }_{\mathrm{4,3}}\lesssim 3.8\end{array}$
RZ	$\begin{array}{l}-0.2\lesssim {a}_{0}\lesssim 0.7\\ -0.3\lesssim {a}_{1}\lesssim 1.0\end{array}$

Note. Here and in Table 4 we use the bound derived from the calibration based on the Keck mass measurement, the eht-imaging algorithm, and the GRMHD simulation library, as an example. JP, MGBK, and RZ are parameterized metrics that deviate from Kerr (see Johannsen & Psaltis 2011; Vigeland et al. 2011; Rezzolla & Zhidenko 2014 for details on these metrics).

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Detecting possible deviations from the Kerr metric using the agnostic approach discussed above can be used to infer constraints on multiple asymptotic expansion coefficients of the spacetime, test the no-hair theorem, assess the Ricci flatness of black hole metrics, etc. In this section, we follow a complementary approach to determining whether specific fundamental principles of the theory of gravity are violated by considering explicitly theories that incorporate such violations by design, finding (stationary) solutions to the associated field equations that describe supermassive compact objects such as Sgr A*, and determining whether their images, when undergoing similar accretion processes, are compatible with those observed with the EHT.

Adopting this approach helps us assess the necessity of including additional fundamental fields (such as dilatons or axions) in the description of the classical theory and yield quantitative constraints on the amount of buildup of various fields in the vicinity of supermassive compact objects (see, e.g., Kocherlakota et al. 2021). This could be instructive of the astrophysical processes that may have produced them. Studying the images of available solutions allows us also to address questions related to the type of object that Sgr A* is, e.g., whether it is a naked singularity, a boson star, or a black hole. Finally, working with a specific theory enables a comparison of its predictions for a variety of other physical scenarios with already-existing or future observations, for its overall compatibility, as discussed in Section 6.

Alternative Black Holes.—As an example, the equivalence principle is a fundamental building block of the theory of gravity and has thus far been tested in various regimes by complementary experiments. It comprises three aspects (Dicke 2019; Will 2014): the weak equivalence principle (WEP), local Lorentz invariance (LLI), and local positional invariance (LPI). To demonstrate the scope of testing theories that violate the WEP and LPI with the EHT, we will consider here black hole solutions from two Einstein–Maxwell-dilaton-axion (EMda) theories (Gibbons & Maeda 1988; Garfinkle et al. 1991; Kallosh et al. 1992; Sen 1992; García et al. 1995; see also the discussion in, e.g., Magueijo 2003; Kocherlakota & Rezzolla 2020), which emerge as the low-energy effective descriptions of the heterotic string. This conservative choice allows us to be certain that (a) the form of the equations that describe the dynamics of accreting plasma flow around EMda black holes is identical to those in general relativity due to the minimal coupling of matter to Einstein–Hilbert gravity via the metric tensor, and (b) photons move on null geodesics of the metric tensor since electromagnetism is described by the (linear) Maxwell Lagrangian (see, e.g., Section 4.3 of Wald 1984).

Synthetic images of radiative inefficient accretion flows onto Gibbons–Maeda–Garfinkle–Horowitz–Strominger black holes (Gibbons & Maeda 1988; Garfinkle et al. 1991), which describe charged, static black holes in one of the EMda theories (henceforth the EMd-1 for brevity), have been constructed in Mizuno et al. (2018), using MHD simulations (see also Figure 6). It was demonstrated there that the final images of these EMd-1 black holes are comparable to those of the Schwarzschild/Kerr black holes. More recently, properties of images of (Kerr-)Sen black holes (Sen 1992), which are the spinning generalizations of the EMd-1 black holes, have been calculated and characterized in Younsi et al. (2021), when undergoing accretion that is described by the semianalytic model of Özel et al. (2021; see also Figure 6). To compare the features of Sen black holes against their general relativistic counterparts, we consider the Reissner–Nordstrom (RN; Reissner 1916; Nordström 1918) and the Kerr–Newman (KN; Newman et al. 1965) solutions, which describe charged black holes with and without spin, respectively.

We also consider solutions arising from various attempts to regularize the central singularities of classical black holes within general relativity. In particular, we consider solutions by Bardeen (1968), Hayward (2006), and Frolov (2016). ¹⁵⁷ Such solutions are typically nonempty and the matter present, in a stationary configuration, typically violates one or more energy conditions (Hawking & Ellis 1973; Curiel 2017). Studying the images of available solutions that can be used to model compact objects allows us to test for possible violations of components of the equivalence principle or of energy conditions. We also include here the static Kazakov & Solodukhin (1994; KS) solution, which attempts to smear out the central singularity onto a surface. Additionally, we also consider the spinning counterparts of the Bardeen and Hayward metrics (Abdujabbarov et al. 2016). To conduct the analysis below, we have implicitly assumed that the "ordinary" matter in the accretion flow does not interact with the background matter in the nonempty spacetimes we consider here.

Figure 18 shows the dependence of the deviation of the shadow size from the Schwarzschild prediction on the parameters (the "generalized charges" or simply charges henceforth) of the various metrics discussed above (see also Section IV of Kocherlakota & Rezzolla 2020 for further details). In this figure, each relevant physical parameter has been normalized to its maximum theoretically allowed value (see also Table 5). ¹⁵⁸ Similarly to the case of the Kerr and the parametric metrics, we find that the black hole spin introduces minor corrections to the size of the shadow, which allows us to focus on nonspinning spacetimes. Moreover, the current bounds imposed by the EHT images of Sgr A* place constraints of order unity to the charges of several of the spacetimes, which are comparable to their maximum values, by construction. Figure 19 focuses on the constraints that we can set on the relevant parameter spaces of all the black holes from the two EMda theories considered here: the top panel shows the charged, nonspinning "EMd-2" solution from one theory (Kallosh et al. 1992). From the other theory, we show in the bottom panel the parameter space for the spinning Sen black hole (Sen 1992; the EMd-1 black hole corresponds to the a = 0 line). As can be seen from these figures, we find no evidence of violations of the equivalence principle or of the presence of energy-conditions violating matter within the present context.

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Thus far we have considered in detail the possibility that Sgr A* is a supermassive black hole (described by different metrics), as well as the alternative that it possesses a material surface (Section 4). We have also considered the possibility that Sgr A* is a surfaceless horizonless compact object without a photon sphere, with focus on mini-boson stars and naked singularities, in Section 4.3. Here, using specific solutions, we will address whether the spacetime in its vicinity can be well modeled by that of a naked singularity with a photon sphere and, later on, by a wormhole. Since all of the naked singularity solutions we consider here arise from metric theories of gravity, with the electromagnetic sector being governed by the linear Maxwell Lagrangian, photons move on null geodesics of the metric tensor, as discussed above. Since we consider exact solutions to the classical theory, we assume that the background spacetimes are static and that the naked singularities at r = 0 do not interact with matter or radiation in any way.

Naked Singularities.—For an example of a naked singularity spacetime, we will consider the Reissner–Nordström metric (Reissner 1916; Nordström 1918), characterized by specific electromagnetic charges of $\bar{q}\gt 1$, and denote it by RN*. These spacetimes admit photon spheres only for $1\lt \bar{q}\leqslant \sqrt{9/8}$, and Figure 20 shows only this range, normalized to the maximum. The Janis–Newman–Winicour (JNW; Janis et al. 1968) naked singularity spacetime is a solution of the Einstein–Maxwell-scalar theory with a theoretically allowed scalar charge parameter range of $0\lt \hat{\bar{\nu }}\lt 1$. However, the JNW naked singularities only cast shadows when $0\lt \hat{\bar{\nu }}\leqslant 0.5$, as indicated in Figure 20. We will also consider a new class of naked singularities within general relativity, namely, the Joshi–Malafarina–Narayan-1 (JMN-1; Joshi et al. 2011) naked singularities. This class of solutions describes a one-parameter M₀ family of static spacetimes containing a compact region r_A < r_A,b ≡ 2M/M₀ filled by an anisotropic fluid, where M is the Arnowitt-Deser-Misner (ADM) mass of the spacetime. This spacetime can be attained at asymptotically late times as a result of gravitational collapse from regular initial data (Joshi et al. 2011; see also Section IV of Dey et al. 2019) and contains a photon sphere when r_A,b < 3M or equivalently when M₀ ≥ 2/3 contributed by the exterior Schwarzschild spacetime.

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Spherical Bondi–Michel accretion onto JMN-1 naked singularities has been studied in Shaikh et al. (2019), where it was found that the final images of JMN-1 naked singularities with photon spheres are indistinguishable from those of a Schwarzschild black hole. More strikingly, for the same approximate luminosity as Sgr A* at 200 GHz, accretion flows onto these singularities have spectra nearly identical to those of a Schwarzschild black hole (see Figure 6 therein), indicating that a JMN-1 naked singularity with a photon sphere may be one of the best possible black hole mimickers for Sgr A* (see Section 4).

Figure 20 shows the bounds imposed by the EHT images of Sgr A* on the physical charges of these metrics that describe naked singularities. With the exception of the Reissner–Nordström metric, which predicts shadow sizes that are significantly smaller than what is observed, the possibility that Sgr A* is a naked singularity cannot be ruled out based on the metric tests we describe in this section.

Wormholes.—As an example of a wormhole, we consider nonspinning, traversable Morris–Thorne (MT) metrics (Morris & Thorne 1988) in general relativity, for which the tt-components of the metric are determined by the "redshift function" Φ as ${g}_{{tt}}=-\exp (2{\rm{\Phi }})$. For wormholes in general relativity to be traversable, the spacetime must necessarily contain energy-condition-violating matter and lack event horizons. The location(s) of the circular null geodesic(s) in this spacetime can be obtained by solving − 1 + r_A dΦ/dr_A = 0. If we restrict to the simplest case of an MT wormhole with a single unstable circular null geodesic, ¹⁵⁹ which can then be identified as the location of the photon sphere, e.g., by setting Φ = − r_A,t/r_A as in Bambi (2013), we find that the (Keplerian/ADM) mass definition forces r_A,t = M. This implies a shadow radius of r_sh = e M ≈ 2.72 M, or equivalently δ ≈ − 0.48, which is immediately ruled out by the present considerations (see Figure 20).

Finally, as noted above in Section 4, improved angular resolution with space-VLBI would greatly help constrain possible metric deviations from the Kerr geometry. Notably, it would be possible to infer the spin of Sgr A*, when modeled as a Kerr black hole, if we are able to achieve a precision of ∣δ∣ < 0.07 (Johannsen & Psaltis 2010; see also Figure 18). Shadows of spinning MT wormholes (Teo 1998) have recently been considered in Shaikh (2018), where it was shown that their shape can vary considerably from that of a Kerr black hole and possibly be detected with future EHT or ngEHT measurements. Spacetimes admitting multiple circular null geodesics were considered by Wielgus et al. (2020), an example of which is given by black holes in a nonminimal Einstein–Maxwell-scalar theory (Gan et al. 2021). Presence of a persistent multi-ring structure in an EHT image would constitute a robust topological discriminant of this family of spacetimes, particularly with future observations at higher resolution and flux sensitivity.

In the discussion above, we used the inferred size of the black hole shadow in Sgr A* in order to place constraints on parameters of metrics that deviate from Kerr. For the metrics that are solutions to particular modifications to general relativity, these parameters (or "charges") correspond to particular properties of the theory or of the black hole itself. For the parameterized metrics, these parameters are phenomenological coefficients that are agnostic to any particular aspect of the underlying theory. Even though it might appear that these bounds are specific to the particular metric used, we will show here that they describe deviations from Kerr that are mathematically very similar to each other and nearly independent of the characteristics of the metric used.

First, as Figures 16 and 18 show, the constraints imposed by the measurement of the size of a black hole shadow depend weakly on the spin of the black hole, for all the metrics explored. As discussed in Section 5.1, for metrics with zero spin, it can be shown analytically that the measurements lead to constraints only on the parameters that enter the tt-component of the metric in areal coordinates. The consequence of these two statements is that, for all spins, the primary constraints imposed by the measurement of a shadow size will be only on one of the metric components, largely independent of the other metric details.

Translating directly the bounds on the parameters of one metric to those of another is nontrivial because of the usual coordinate and gauge ambiguities that are inherent to relativistic spacetimes. However, one avenue of making this comparison is by exploring the asymptotic behavior of these metrics toward radial infinity. In particular, we write the tt-component in areal coordinates in terms of the parameterized post-Newtonian (PPN) expansion

$$\begin{eqnarray}&&{g}_{{tt}}=-1+\displaystyle \frac{2}{{r}_{A}}-2\left(\displaystyle \frac{{\kappa }_{1}}{{r}_{A}^{2}}\right)+2\left(\displaystyle \frac{{\kappa }_{2}}{{r}_{A}^{3}}\right)-2\left(\displaystyle \frac{{\kappa }_{3}}{{r}_{A}^{4}}\right)+{ \mathcal O }({r}_{A}^{-5})\end{eqnarray} \tag{ 23 }$$

and connect without ambiguity the post-Newtonian coefficients of each metric to each particular parameter. We then translate the bounds of the parameters to constraints on these post-Newtonian coefficients and compare the results obtained with different metrics.

We emphasize that we do not use these post-Newtonian expansions in order to calculate black hole shadows, which would have been inappropriate given that the size of the shadow is comparable to the horizon radius. Instead, we calculate shadow sizes and place constraints on the particular parameters of each of the metrics that has been developed specifically for use in the strong-field regime. We only use the post-Newtonian coefficients as a mechanism to compare the asymptotic behavior of these metrics. Because each post-Newtonian coefficient at a given order N is proportional to the derivative of the metric coefficient with respect to 1/r_A at order N + 1, comparing post-Newtonian coefficients is equivalent to comparing the detailed functional forms of the metrics.

Tables 4 and 5 summarize the post-Newtonian coefficients at the various orders for the different metrics used in the previous sections (see, e.g., Psaltis et al. 2021). Using this correspondence between metric parameters and post-Newtonian coefficients, we show in Figure 21 the fractional deviation δ of the shadow size from the Schwarzschild prediction but plotted against the equivalent post-Newtonian coefficients for each metric, at the first and second order.

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Table 4. Post-Newtonian Coefficients of Parameterized Metrics

Parameterized Metric	κ1	κ2	Constraints
JP	− α12	− 2α12 + α13	$\begin{array}{l}-0.3\lesssim {\kappa }_{1}\lesssim 0.7\\ -3.1\lesssim {\kappa }_{2}\lesssim 1.5\end{array}$
MGBK	$-\tfrac{1}{2}{\gamma }_{\mathrm{1,2}}-{\gamma }_{\mathrm{4,2}}$	$-{\gamma }_{\mathrm{1,2}}+\tfrac{1}{2}{\gamma }_{\mathrm{1,3}}-4{\gamma }_{\mathrm{4,2}}+{\gamma }_{\mathrm{4,3}}$	$\begin{array}{l}-0.3\lesssim {\kappa }_{1}\lesssim 0.9\\ -4.0\lesssim {\kappa }_{2}\lesssim 1.3\end{array}$
RZ	2a0	$4\left({a}_{0}-\tfrac{{a}_{1}}{1+\tfrac{{a}_{2}}{1+\tfrac{{a}_{3}}{...}}}\right)$	$\begin{array}{l}-0.3\lesssim {\kappa }_{1}\lesssim 0.9\\ -4.0\lesssim {\kappa }_{2}\lesssim 1.3\end{array}$

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Table 5. Post-Newtonian Coefficients of Specific Compact Object Spacetime Metrics

Spacetime	Charge Range	Constraints	κ1	κ2
RN	$0\lt \bar{q}\leqslant 1$	$0\lt \bar{q}\leqslant 0.84$	${\bar{q}}^{2}/2$	0
RN*	$1\lt \bar{q}$	×	${\bar{q}}^{2}/2$	0
Schwarzschild	⋯	⋯	0	0
Bardeen	$0\lt \bar{q}\lt \sqrt{16/27}$	⋯	0	$-3{\bar{q}}^{2}/2$
Frolov	$0\lt \bar{l}\lt \sqrt{16/27},0\lt \bar{q}\leqslant 1$	see, e.g., Figure 18	${\bar{q}}^{2}/2$	0
Hayward	$0\lt \bar{l}\lt \sqrt{16/27}$	⋯	0	0
MT wormhole	⋯	×	−1	−2/3
KS	$0\lt \bar{\alpha }$	$0\lt \bar{\alpha }\leqslant 0.79$	$-{\bar{\alpha }}^{2}/4$	0
EMd-1†	$0\lt \bar{q}\lt \sqrt{2}$	$0\lt \bar{q}\lt 0.87$	${\bar{q}}^{2}/2$	${\bar{q}}^{4}/8$
EMd-2†	$0\lt {\bar{q}}_{{\rm{e}}}\leqslant \sqrt{2}-{\bar{q}}_{{\rm{m}}}\lt \sqrt{2}$	Figure 19 (top)	$\left({\bar{q}}_{{\rm{m}}}^{2}+{\bar{q}}_{{\rm{e}}}^{2}\right)/2$	${\left({\bar{q}}_{{\rm{m}}}^{2}-{\bar{q}}_{{\rm{e}}}^{2}\right)}^{2}/8$

Note. We denote by crosses and ellipses spacetimes that are entirely ruled out and that are unaffected by the EHT measurements, respectively.

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As before, we show only two particular cross sections of the multidimensional parameter space for which the metric parameters were chosen such that only one of the first two post-Newtonian deviation coefficients has a nonzero value. As an example, for the JP metric we set α₁₃ = 2α₁₂ for the κ₁ plot in Figure 21 and all deviation parameters other than α₁₂ and α₁₃ to zero. This forces the κ₂ term to be zero for this metric but does not set higher-order terms to zero. Nevertheless, the influence on the shadow size of the higher-order terms for this metric decreases quite rapidly (see also Equations (29)–(33) of Psaltis et al. 2021). For the κ₂ plot for the JP metric we set α₁₂ = 0 to force κ₁ = 0 and allow only α₁₃ to be nonzero. For the MGBK metric we set γ_4,2 = − γ_1,2/4 and set all parameters other than γ_1,2 and γ_4,2 to zero for the κ₁ plot. For the κ₂ plot we set γ_4,2 = − γ_1,2/2 and all other parameters to zero. For the RZ metric we set r₀ = 2, which in turn sets = 0 as discussed in Section 5.1, and set a₀ = a₁ for the κ₁ plot and only a₁ to be nonzero for the κ₂ plot. For the metrics discussed in Section 5.2 we include only metrics for which it is possible to allow only one of the first two PPN parameters to be nonzero. The full tt-components of the metrics are used for these calculations, not their expansions.

These figures demonstrate that the inferred size of the Sgr A* black hole shadow places bounds of order ∼1 and ∼5 on the first and second post-Newtonian coefficients of the underlying metric, with the specific values showing a weak dependence on the particular metric used to obtain these constraints. Constraints on higher-order PN components would be factors of a few less stringent at each increasing order (see Psaltis et al. 2021 for details).

There exist a number of key qualitative differences between the aspects of the theory of gravity probed by various tests of general relativity. For example, as Figure 22 illustrates, some of the tests are sensitive primarily to the dynamics and propagating modes of the gravitational fields, as is the case with pulsar timing, gravitational waves, and cosmology. ¹⁶⁰ Other tests involve primarily measurements of photons in stationary spacetimes, as is the case of black hole images and stellar orbits. Some tests involve orbits of massive particles, while others involve the propagation of photons in relativistic spacetimes. Moreover, tests performed with neutron stars and in cosmological settings also probe the coupling of matter to the gravitational field, whereas black hole and most solar system tests are only sensitive to the properties of vacuum spacetimes.

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Even within these qualitative distinctions, different tests probe vastly different regimes of gravitational potential and curvature, because of the large range of masses and length scales involved. This is illustrated in Figure 23, following Baker et al. (2015). The horizontal axis in this figure shows the gravitational potential probed by each test; in the case of a test at distance r from a Newtonian object of mass M, this dimensionless potential is equal to = GM/rc². The vertical axis shows the spacetime curvature probed by each test, defined as the square root of the Kretschmann scalar; for a test in the Schwarzschild spacetime of an object, this is equal to $\xi =\sqrt{48}{GM}/{r}^{3}{c}^{2}$ (see Baker et al. 2015).

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In order to highlight explicitly the fact that any test may probe a range of potentials and curvatures, we use straight lines to connect the smallest and largest potential and curvature that, in principle, may affect the outcome of each test. For example, in the case of the solar system test with the Cassini spacecraft (Bertotti et al. 2003), the Shapiro delay of radio signals was measured between Earth and the spacecraft, when the latter was between Jupiter and Saturn and as these signals grazed the surface of the Sun; this test, therefore, probes the entire range of gravitational potentials and curvatures from the solar surface to the location of the Cassini spacecraft. We also use dashed lines to connect regions of the parameter space that may affect the outcome of a test, but in a theory-specific manner. For example, in double-pulsar tests, the evolution of the orbital period caused by the emission of gravitational waves probes directly the potential and curvature at the orbital separation. However, in numerous modifications of the theory of gravity, an enhanced rate of emission of gravitational waves becomes possible because of the coupling of neutron star matter to the gravitational field at the highest potential and curvature (see, e.g., Damour & Esposito-Farese 1993). In other words, depending on the particular modification of gravity that is being tested, the test involving the evolution of the binary period may probe the entire range of field strengths covered by the solid and dashed cyan lines in Figure 23.

As we will discuss below, these qualitative and quantitative differences between tests of gravity complicate our ability to cross-compare and combine their results. However, these same differences also allow us to leverage the broad range of conditions that various tests probe in order to draw conclusions about the theory of gravity that could not have been reached by any test individually. For example, one of the key predictions of general relativity is that the spacetime properties of black hole scale with their mass. This is a prediction that we can test by comparing the results of gravitational-wave tests that probe stellar-mass black holes to those of the imaging tests that probe supermassive black holes. At the same time, general relativity predicts that, according to Birkhoff's theorem, the external spacetime of a slowly spinning object is independent of its internal structure and composition. We will test this prediction by comparing the results of black hole tests to those that involve pulsars or the Sun.

Because of their qualitative differences, every type of test of general relativity is performed with a unique theoretical framework that is optimal for the system under study. Solar system tests use PPN expansions (Will 2014), pulsar tests use post-Keplerian parameterizations and also a strong-field equivalent of the PPN-formulation (see, e.g., Wex & Kramer 2020), shadow tests use parametric post-Kerr metrics (Johannsen & Psaltis 2011; Vigeland et al. 2011; Johannsen 2013b; Rezzolla & Zhidenko 2014; Konoplya et al. 2016), gravitational-wave tests with inspirals use post-Newtonian (Khan et al. 2016), effective-one-body (Buonanno & Damour 1999, 2000), parameterized post-Einstein (ppE) frameworks (Yunes & Pretorius 2009), etc. Unfortunately, this plurality of methods restricts our ability to combine and leverage the results of different tests since, in many cases, the parameters of each framework are not directly related to each other.

In principle, there are two ways we can combine tests across different scales and systems. In one approach, we can use a particular class of theories (e.g., scalar-tensor gravity) and compare the constraint of the parameters of that class from different tests. This is the most direct approach that requires typically no additional assumptions to be made. However, it is limited to the particular alternative to general relativity that is described by the theory under study. In a second approach, as a practical solution, we can make simplifying assumptions (e.g., that the dynamics of the theory are the same as in general relativity but the stationary spacetimes are not) and constrain phenomenological parameters of the metrics of the objects involved.

In order to make the latter approach independent of coordinate systems or gauges, and focusing here on tests of stationary metrics, we often convert the bounds on the parameters of a particular framework to constraints on the effective post-Newtonian parameters of the metrics of the objects involved. Within the particular assumptions inherent to each test, this approach is formally correct, even if one uses tests in the strong-field regime, as long as the framework used to obtain these constraints is itself applicable in that regime. Moreover, in doing so, there is no implicit assumption that the derived post-Newtonian parameters are universal constants. Indeed, in most modifications to general relativity, the values of these parameters are specific to the situation under consideration, as they may depend on the strength of the gravitational field (curvature or potential) probed, the nature of the compact object (binary or not, with matter or pure vacuum, etc.), and the boundary conditions (the coupling of matter to the field at the center of the system, the asymptotic cosmological boundary conditions, the cosmic time of the test, etc.). This is the reason why it is important to measure potential deviations of such parameters in different astrophysical and cosmological settings that span a wide range of masses and physical conditions, as shown in Figure 23.

Sgr A* is unique in enabling us to probe the metric of the same black hole both at horizon scales, with the EHT images reported here, and at larger distances, with the orbits of S stars. In performing the imaging test in Section 3.5, we have already used the measurement of the mass-to-distance ratio for the black hole that was obtained through monitoring the S-star orbits. In spacetime terms, this Keplerian mass is simply the coefficient of the asymptotic, Newtonian expansion of the metric. However, recent measurements of relativistic effects in the stellar orbits resulted in constraints on the metric properties beyond the Newtonian regime, which we explore here.

The motions of several S stars have been monitored for almost three decades with adaptive optics instruments on VLT and Keck, and their orbits have been well determined (see, e.g., Ghez et al. 2008; Gillessen et al. 2009b). The detection of gravitational redshift (Gravity Collaboration et al. 2018a; Do et al. 2019) and of the precession of the periapsis (Gravity Collaboration et al. 2020a) in the S0-2 orbit has led to tests of the equivalence principle and of the Schwarzschild metric (Section 2; see also Hees et al. 2017; Amorim et al. 2019). Because in the gravitational test we report here with the Sgr A* images we explicitly assume the validity of the equivalence principle and only test the metric, we will focus on the connection of the imaging to the post-Newtonian tests of the metric using the precession of the S0-2 orbit.

In Gravity Collaboration et al. (2020a), the measured rate of precession of the S0-2 orbit was quantified through a phenomenological parameter f_SP, such that the precession per orbit at the first post-Newtonian order can be written as

$$\begin{eqnarray}&&{\rm{\Delta }}{\phi }_{1}={f}_{\mathrm{SP}}\displaystyle \frac{6\pi {GM}}{a(1-{e}^{2}){c}^{2}},\end{eqnarray} \tag{ 24 }$$

where a and e are the orbital separation and eccentricity of the orbit, respectively. The best-fit value for f_SP was found to be consistent with the predictions of the Schwarzschild metric, i.e., f_SP = 1.1 ± 0.19.

In the PPN formalism, the phenomenological parameter f_SP is related to two of the first-order post-Newtonian parameters of the metric via (Will 2014)

$$\begin{eqnarray}&&{f}_{\mathrm{SP}}=\displaystyle \frac{1}{3}\left(2+2{\gamma }_{{\rm{S}}0-2}-{\beta }_{{\rm{S}}0-2}\right),\end{eqnarray} \tag{ 25 }$$

such that