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

$R_{*}=(1+\mu)R_{H},$ (11)

where $R_{H}=2M$ is the radius of the event horizon. For $\mu\ll 1$, the
additional time delay is $\sim 2M\ln(1/\mu)$, which is $\sim 40\,\ln(1/\mu)$ 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, $(1+z)\sim\mu^{-1/2}$,
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 $1/(1+z)^{4}\sim\mu^{2}$ 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 $T_{\infty}$, then the radiation emitted by the surface will
have a temperature $T_{\rm local}=(1+z)T_{\infty}$ in the local rest frame.
The radiation emerging from the surface will have a flux equal to $\sigma_{\rm
SB}T_{\infty}^{4}(1+z)^{4}$, 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 Galaxy, i.e., several billion
years old. Over much of that time, it must have accreted gas at a rate equal
at least666 There is clear evidence that Sgr A∗ went through episodes of much
larger $\dot{M}$ a few hundred years ago (Ponti et al., 2010; Clavel et al.,
2013), and there are suggestive arguments for enhanced accretion over the last
millions of years (e.g., Mou et al., 2014). On the time scale of the age of
the Galaxy, if Sgr A∗ acquired much of its mass by accretion, it would have
had to accrete at an average rate of more than $10^{-4}M_{\odot}{\rm
yr^{-1}}$, i.e., orders of magnitude larger than the conservative rates we
have been assuming. Such large average accretion rates are routinely predicted
by cosmological models of galaxy and supermassive black hole evolution. to the
present $\dot{M}$. The time needed to achieve the steady state condition
implicit in equation (9), namely, $\eta\dot{M}c^{2}=L_{\infty}=(\sigma
T_{\infty}^{4})(4\pi R_{\rm app}^{2})$, or equivalently $T_{\rm
local}=(1+z)T_{\infty}$, is far shorter than the age of Sgr A∗ for almost any
model.

The one exception is if $\mu\ll 1$, 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 $\mu^{-1}$ and can become
arbitrarily long. The physical reason is that the region between $R=R_{*}$ and
the photon sphere, $R_{\rm ph}=3M$, 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
$\mu\gtrsim 10^{-14}$. If the surface is even closer to the horizon radius
than this limit, i.e., if $R_{*}\lesssim R_{H}+10^{-2}$ 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 $\mu$. The argument goes as follows. Because Sgr A∗ accretes mass
continuously, its horizon radius $2M$ increases with time. In order to
maintain $R_{*}=(1+\mu)R_{H}$, the surface also needs to expand. However, if
$\mu$ 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 $\dot{M}\sim 10^{-11}M_{\odot}{\rm yr^{-1}}$ (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 $\mu\gtrsim 10^{-23}$, i.e., if
$R_{*}\gtrsim R_{H}+10^{-11}$ 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 $\mu$ in the range $10^{-23}<\mu<10^{-14}$
for which Sgr A∗ is presently 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 which (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
$T_{\infty}$ 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 ($>10^{6}M_{\odot}$) object has not been
studied and merits further attention. Models of spherical accretion on neutron
stars ($M=1.4M_{\odot}$) 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∗ ($M=4\times
10^{6}M_{\odot}$) 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., $\mu\ll 1$, the
volume between $R_{*}$ and $R_{\rm ph}=3M$ acts like an enclosed cavity, with
radiation allowed to escape only over a small solid angle $\sim\mu$. 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
presently 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∗.

###  4.2 Reflecting Surface

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 wave-vector $k^{\mu}$ becomes an outward-moving ray with
$k^{r}$ reversed and the other components of $k^{\mu}$ unchanged. (ii) If the
intensity of the incoming ray is $I_{\nu}$, the outgoing ray has an intensity
$AI_{\nu}$, where $A\leq 1$ 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 magnetically
arrested disk (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 $(r,\theta)$ plane, time-averaged over the
simulation period $t=50,000-100,000M$. We set the electron temperature using
the prescription given in Event Horizon Telescope Collaboration et al. (2019b,
which is based on ) with parameter values, $R_{\rm high}=20$, $R_{\rm low}=1$.
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∗ (Wielgus et al., 2022). We
then compute a synthetic 230 GHz image for an observer at an inclination angle
of $i=60^{o}$.

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 $15\,\mu$as; this beam size corresponds to the
typical resolution that is achieved by the EHT using super-resolution image
reconstruction techniques.

The unblurred image in the top left panel in 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 $60^{o}$ inclination of the observer. These
two features are visible even in the $15\,\mu$as blurred image in the top
middle panel (the features merge if we blur with a $20\,\mu$as 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 the shadow
region of the original black hole image. When blurred, the resulting image,
shown in the second row middle panel, 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., $A=0.3,~{}0.1$, 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
re-radiated as part of the thermalized emission discussed in Section 4.1. For
any value of $(1-A)\gtrsim 0.1$, 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, $R_{\rm high}=20$, $R_{\rm low}=1$, observer
inclination $i=60^{o}$, and surface radius $R_{*}=2.5M$. The values of $R_{\rm
high}$ 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 gas dynamics 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. Also, we have focused here on a spherically symmetric spacetime
around a non-spinning 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.

Figure 15: Comparison of a synthetic image of a hot accretion flow around a
black hole with corresponding images when the central object has a reflecting
surface. Top-Left: Model image of Sgr A∗ at 230 GHz as seen by an observer at
inclination angle of $60^{o}$, computed by ray-tracing a GRMHD simulation of a
hot accretion flow around a non-spinning black hole. The color scale
corresponds to brightness temperature (K). The image shows a relatively
diffuse elliptical feature corresponding to equatorial emission from the
accretion flow, a sharp circular lensing ring, and a dark shadow region in the
middle. Top-Middle: Shows the image on the left blurred with a gaussian beam
with FWHM $=15\,\mu$as. Top-Right: Image reconstruction with SMILI top set
(see Paper III for details) using synthetic visibility data generated from the
model shown in Top-Left. Second Row Left: Model image when the black hole
horizon is replaced by a reflecting surface with radius $R_{*}=2.5M$ and
albedo $A=1$. Additional features appear in the image, especially a thick
bright secondary ring inside the shadow region. Second-Middle: Blurred (FWHM
$=15\,\mu$as) version of the image on the left. Second-Right: SMILI
reconstruction using synthetic visibilities generated from the model in
Second-Left. Comparison with the Top-Right image shows that the shadow region,
which is prominent in the upper image (black hole case), is largely filled in
by surface reflection. This change is potentially distinguishable with EHT
observations. Third Row: Similar to the Second Row, but for albedo $A=0.3$.
Compared to the $A=1$ model, in this case the difference from the black hole
image due to the presence of a surface is only marginally detectable. Bottom
Row: Model with surface albedo $A=0.1$. Here the image is indistinguishable
from the black hole image (Top row) at the resolution and sensitivity of the
EHT 2017 data.

####  4.2.2 Constraints from EHT Images

Figures 13, 14 and 17 in Paper III show a range of images of Sgr A∗ obtained
by applying different image reconstruction techniques to the EHT 2017 data.
The vast majority of images show a ring-like morphology with a pronounced dark
shadow region at the center. These images are visibly different from the
synthetic blurred image shown in the second row middle panel in our Figure 15.
We can thus exclude this particular model using EHT observations.

The right four panels in Figure 15 show image reconstructions of the four
synthetic models shown in the left panels using one of the image
reconstruction methods described in Paper III. For each model, synthetic
visibility data were generated with the same $(u,v)$-coverage as in the April
7 EHT observations of Sgr A∗, and the appropriate amount of noise was added to
match the noise present in the real data. These synthetic visibilities were
then analysed using the SMILI top set (see Paper III for details) and the
resulting images are shown in the panels on the right in Figure 15. The SMILI
reconstructions are fairly similar to the $15\,\mu$as-blurred images shown in
the corresponding central panels, though the SMILI images appear to be
slightly more blurred. More interestingly, the SMILI image of the $A=1$
reflecting surface model (second row right panel) is quite different from the
reconstruction of the black hole model (top row right panel). This confirms
our expectation that a surface which reflects in-falling radiation with 100%
efficiency can potentially be ruled out by EHT 2017 observations of Sgr A∗
(modulo the many caveats mentioned in Section 4.2.1). In the case of the
$A=0.3$ model, and especially the $A=0.1$ model, the SMILI reconstructions do
not differ much from the black hole image, hence it would be hard to
distinguish such models using the current EHT data.

Another way of comparing models is to measure the brightness depression in the
shadow region of the image. For instance, Table 13 in Paper III presents
estimates of a parameter $f_{c}$, which measures the ratio of the brightness
at the center of the ring image to the mean brightness around the ring. This
quantity is a measure of the brightness depression in the central shadow
region of the image. From Figure 22 in Paper III, the estimate based on SMILI
is $f_{c}\approx 0.2$, with not much probability that $f_{c}>0.3$. This
implies that the image intensity in the shadow region in Sgr A∗ is very likely
$<30\%$ of the mean intensity around the ring.777The upper limit on $f_{c}$ is
significantly lower when estimated via Themis (see Table 13 in Paper III) or
by the modeling methods described in Paper IV. To be consistent with Figure
15, here we focus on only the SMILI results.

Such a degree of flux depression is inconsistent with the blurred synthetic
image and SMILI reconstructed image in the second row of Figure 15. That
particular model could be potentially ruled out. However, the dynamic range of
images based on the current EHT 2017 data is too low, and its angular
resolution is too poor, to constrain a weakly reflecting surface in Sgr A∗,
such as the models in the third and fourth rows of Figure 15.

An array with more stations and with larger bandwidth (e.g., the proposed Next
Generation Event Horizon Telescope, Doeleman et al. 2019; Raymond et al. 2021)
would solve the sensitivity problem. However, to improve the angular
resolution substantially, it will be necessary to expand the telescope array
into space by sending one or more radio dishes into large-radius orbits around
the Earth (Palumbo et al., 2019; Roelofs et al., 2020b; Fish et al., 2020;
Gurvits et al., 2021; Kudriashov et al., 2021; Fromm et al., 2021). With such
an expanded array, we might be able to observe images with sufficient angular
resolution to see some of the details revealed in the various panels in the
left column of Figure 15 and to check for the presence of a surface in Sgr A∗.
Furthermore, if a surface is present, we might be able to determine the albedo
and the surface radius.

Chael et al. (2021) show that, with better sensitivity and angular resolution
than the present EHT is able to provide, the observed image of a hot accretion
flow around a supermassive black hole could be used to delineate the inner
edge of the accretion disk. Their interest is to use this technique to
identify the edge of the event horizon. However, their method could equally
well be used to measure the radius of a potential surface. This might provide
a direct estimate of $R_{*}$.

Additionally, it may be possible to observe time-delayed echoes from a
reflecting surface. In the case of time-variable emission from the accretion
flow, e.g., sub-millimeter or infrared flares, the observer would see both the
primary signal from the emitting gas element as well as a delayed reflected
copy of the same radiation. This is the electromagnetic analog of
gravitational wave echoes that have been searched for in LIGO/Virgo
observations of merging stellar-mass black holes (Abedi et al., 2017;
Westerweck et al., 2018). However, here it could be done with spatially
resolved images, with all the rich detail they can provide. We do not pursue
electromagnetic echoes further, but note that their presence could potentially
be explored already using existing image-integrated light curve data.

###  4.3 Surfaceless Horizonless Compact Objects

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 due
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; , and 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, can form as the
nonempty end-state of the gravitational collapse of a (non-thermal) 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 the naked singularity present at $r=0$ does not interact with
matter or radiation (classical gravity).

##  5 Metric tests from shadow size

In Section 3.5, we used the prior information on the mass-to-distance ratio of
the Sgr A∗ black hole to calculate the predicted size of its shadow and
compared the result to the size inferred from the EHT images and visibility-
domain model fitting. We based this prediction on the Kerr metric and found
that there is no evidence for any violations of the theory of General
Relativity. Our goal in this section is to use these bounds on plausible
deviations in the shadow size that are still consistent with the imaging data
in order to place constraints on deviations of the parameters of the
underlying black-hole metric.

We will follow two complementary approaches. First, in Section 5.1, we will
constrain the parameters of stationary metrics that are agnostic to the
underlying physical theory. These have been designed in a way that they reduce
to the Kerr metric, when the deviation parameters are set to zero, but remain
free of pathologies for a wide range of parameter values (Psaltis et al.,
2020a). Although these metrics do not arise from any particular modification
to gravity, they allow us to explore phenomenologically a very broad range of
possibilities, which can be mapped afterwards to parameters of a fundamental
theory. Second, in Section 5.2, we will constrain the parameters of stationary
metrics that are generated by various matter distributions and/or those that
arise from specific modifications to the theory of gravity, and which depend
on additional generalized charges (Kocherlakota et al., 2021). Although the
latter represent only particular types of deviations from General Relativity,
they allow us to translate directly the constraints from the EHT images to
bounds on physical parameters. Finally, in Section 5.3, we will compare the
constraints on the various metrics in terms of their asymptotic post-Newtonian
parameters in order to demonstrate that, fundamentally, the bounds imposed by
the EHT imaging observations of Sgr A∗ depend weakly on the particular metrics
used to describe deviations from Kerr.

Figure 16: Numerical constraints on various deviation parameters for the JP
and MGBK metrics as a function of the dimensionless black-hole spin placed by
the requirement that the predicted size of the black-hole shadow is consistent
with the size inferred for Sgr A∗. Here the shading shows the regions of the
parameter space that are ruled out by the bound derived from the calibration
based on the Keck mass measurement, the eht-imaging algorithm, and the GRMHD
simulation library, as an example. The left (right) panels show constraints
for the JP (MGBK) metrics, while the different symbols/colors in the top and
bottom panels show the effect of the secondary deviation parameter and the
inclination of the observer, respectively. In the top panels we set the
inclination angle to $i=42^{\circ}$ and set the secondary deviation parameter
to zero in the bottom panels. In all panels the black dashed lines correspond
to the analytic constraints for non-spinning black holes. The non-monotonic
behavior of the constraints are due to the fact that both the size and the
shape of the shadows are affected by the deviation parameters. The measurement
of the shadow size constrains primarily the parameters that quantify
deviations in the $tt-$components of the various metrics, as expressed in
areal coordinates, and the resulting bounds depend only weakly on black-hole
spin or observer inclination.

Throughout this section, for clarity of presentation, we will use the bounds
$\delta=-0.04^{+0.09}_{-0.10}$ (Keck) and $\delta=-0.08^{+0.09}_{-0.09}$
(VLTI) on the fractional deviation inferred from the predicted shadow size, as
calculated for the fiducial analyses that use the Keck and VLTI priors on the
mass-to-distance ratio, the eht-imaging imaging method, and the GRMHD library
for quantifying the theoretical uncertainties. Where it is not possible to
show both bounds, we will show the Keck bound as an example. This constraint
depends weakly on the choice of priors and techniques. In particular, the
fiducial bounds correspond to the following constraints on shadow size:

$\begin{split}3\sqrt{3}(1-0.14)M&\lesssim\tilde{r}_{\text{sh}}\lesssim
3\sqrt{3}(1+0.05)M,\\\ 4.5M&\lesssim\tilde{r}_{\text{sh}}\lesssim
5.5M,\end{split}$ (12)

for Keck and

$\begin{split}3\sqrt{3}(1-0.17)M&\lesssim\tilde{r}_{\text{sh}}\lesssim
3\sqrt{3}(1+0.01)M,\\\ 4.3M&\lesssim\tilde{r}_{\text{sh}}\lesssim
5.3M,\end{split}$ (13)

for VLTI, where we have set $G=c=1$, as we will do throughout this section.
For the different metrics employed in this section, we calculate the
fractional diameter deviation of the metric as the relative difference between
the analytic shadow diameter and the Schwarzschild diameter, namely:

$\delta_{\rm metric}=\frac{\tilde{d}_{\mathrm{metric}}}{6\sqrt{3}}-1\,,$ (14)

where $\tilde{d}_{\rm metric}$ is the median shadow diameter, i.e., the locus
of critical impact parameters.

Throughout this section we will not consider constraints on the circularity of
the shadow. This is due to the sparse interferometric coverage of 2017
observations, which may lead to significant uncertainties in circularity
measurements that we do not quantify here. In addition to measurement
uncertainty, we also do not quantify the theoretical uncertainty between the
circularity of the shadow and that of the observational feature. However, in
future EHT observations with additional telescopes the circularity of the
shadow may potentially be used for constraints on deviations from the Kerr
metric.

###  5.1 Constraints on Metrics with Parametric Deviations

According to the no-hair theorem in General Relativity, the only stationary,
asymptotically flat, Ricci-flat spacetime that is free of pathologies888By
pathologies we mean closed timelike loops, naked singularities, and non-
Lorentzian signatures outside of the event horizon. 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 (c.f. 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 2013b for a detailed study of the pathologies of several
parametrized metrics and Kocherlakota & Rezzolla 2022 for an analysis of
theoretically-allowed parameter spaces of the RZ metric). For this reason,
several parametrized metrics have been developed in the last decade that allow
for general deviations from the Kerr metric while minimizing pathologies
mostly by relaxing the assumption of Ricci flatness. These parametrized
metrics are completely agnostic to an underlying physical theory and,
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 2013a); 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 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 non-spinning 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., $\alpha_{12}=0$ and
$\gamma_{4,2}=-\frac{1}{2}\gamma_{1,2}$, 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.

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
$\alpha_{13}$ and $\gamma_{1,2}$ 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)

$r_{\mathrm{sh}}=\ \frac{r_{\mathrm{ph}}}{\sqrt{-g_{tt}(r_{\mathrm{ph}})}},$
(15)

where

$r_{\mathrm{ph}}\equiv
2g_{tt}(r_{\mathrm{ph}})\left(\left.\frac{dg_{tt}}{dr}\right|_{r_{\mathrm{ph}}}\right)^{-1}\;,$
(16)

is the radius of the photon orbit.

For non-spinning black holes, the $tt-$component of the JP metric in areal
coordinates is (see Johannsen 2013b)

$g_{tt}^{\mathrm{JP}}=-\left(1-\frac{2}{r_{A}}\right)\left(1+\sum^{\infty}_{i=2}\frac{\alpha_{1i}}{r_{A}^{i}}\right)^{-2}$
(17)

where $\alpha_{1i}$ are deviation parameters and the subscript $A$ denotes the
fact that we use areal coordinates.

Figure 17: Analytic constraints on deviation parameters for the JP, MGBK, and
RZ metrics. The regions excluded by the Sgr A∗ constraints using Keck (VLTI)
are shown in blue (magenta), see also Equations 12 and 13. For each curve we
allow only one parameter to vary while setting the others to zero. The
original $g_{tt}$ components of the metrics were used for this plot, not their
expansions.

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

$g_{tt}^{\mathrm{MGBK}}=-\left(1-\frac{2}{r_{A}}\right)\left[1-\gamma_{1}(r_{A})-2\gamma_{4}(r_{A})\left(1-\frac{2}{r_{A}}\right)\right]$
(18)

where $\gamma_{1}(r_{A})$ and $\gamma_{4}(r_{A})$ are defined by

$\gamma_{A}=\sum^{\infty}_{n=2}\frac{\gamma_{A,n}}{r_{A}^{n}},$ (19)

$A=1$ or 4, and $\gamma_{A,n}$ are the deviation parameters.

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

$g_{tt}^{\mathrm{RZ}}=-x\left[1-\epsilon(1-x)+(a_{0}-\epsilon)(1-x)^{2}+\tilde{A}(x)(1-x)^{3}\right]$
(20)

where

$\begin{split}x&\equiv 1-\frac{r_{0}}{r_{A}},\\\
\tilde{A}(x)&=\frac{a_{1}}{1+\frac{a_{2}x}{1+\frac{a_{3}x}{...}}},\end{split}$
(21)

and $r_{0}$ is the coordinate radius of the infinite redshift surface
(identified with the horizon if no pathologies exist). The parameters
$\epsilon,\,a_{1},\,a_{2},\,...$ 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

$\epsilon=-\left(1-\frac{2}{r_{0}}\right).$ (22)

For simplicity we assume $r_{0}=2$ 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 that 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.

Table 3: Constraints on Deviation Parameters of Various Metrics

Parametrized metric | Constraints
---|---
JP | $\begin{array}[]{lcl}-1.1\lesssim&\alpha_{12}\lesssim 0.5\\\ -3.1\lesssim&\alpha_{13}\lesssim 1.5\\\ -7.8\lesssim&\alpha_{14}\lesssim 4.6\\\ \end{array}$
MGBK | $\begin{array}[]{lcl}-4.0&\lesssim\gamma_{1,2}\lesssim 1.4\\\ -4.8&\lesssim\gamma_{4,2}\lesssim 1.3\\\ -8.0&\lesssim\gamma_{1,3}\lesssim 2.7\\\ -14.0&\lesssim\gamma_{4,3}\lesssim 3.8\end{array}$
RZ | $\begin{array}[]{lcl}-0.2\lesssim&a_{0}\lesssim 0.7\\\ -0.3\lesssim&a_{1}\lesssim 1.0\\\ \end{array}$

  * •

Notes: 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 parametrized metrics that deviate from Kerr (see
Johannsen & Psaltis 2011; Vigeland et al. 2011; Rezzolla & Zhidenko 2014 for
details on these metrics).

Figure 18: The dependence of the fractional shadow diameter deviation from the
Schwarzschild value (left) on the relevant physical “charges” of various
nonspinning black hole metrics and (right) on the black-hole spins, for fixed
values of the charges and for an observer inclination $i=\pi/2$. The white
regions correspond to shadow sizes that are consistent at the 68% level with
the 2017 EHT observations for Sgr A∗. As in the case of the Kerr and of the
parametric metrics, the spin of the black hole introduces only minor
corrections to the predicted shadow size and, hence, to the metric
constraints. Current EHT imaging observations of Sgr A∗ are inconsistent with
some metrics when their physical charges are comparable to their maximum
theoretically-allowed values. We use the median shadow diameter to
characterize the size of the noncircular shadows cast by the spinning black
holes (right), as done in Section 3.

###  5.2 Constraints on Specific Compact Object Spacetime Metrics

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 build-up
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;
Sen 1992; García et al. 1995; Kallosh et al. 1992; see also the discussion in,
e.g., Magueijo 2003 and 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., Sec. 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 semi-analytic 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).999These spacetimes have also been obtained as solutions in
other theories (see, e.g., Ayón-Beato & García 1998, 2000; Held et al. 2019).
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 19: We show here the constraints on two EMda solutions from different
EMda theories. In the top plot we show the constraints on the parameter space
of a nonspinning black hole from an EMda theory with two $U(1)$ gauge fields
(Kallosh et al., 1992) whereas in the bottom plot we show the constraints on
the parameter space of a spinning Sen black hole from an EMda theory with a
single $U(1)$ gauge field (Sen, 1992). Figure 20: Same as Figure 18 but for
metrics that describe various naked singularities (denoted by a star) and a
wormhole.

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).101010For the Kazakov-
Solodukhin (KS; Kazakov & Solodukhin 1994) black hole, the theoretically-
permitted range of the relevant parameter is noncompact, $\bar{\alpha}>0$, and
we show the range $0<\bar{\alpha}\leq\sqrt{2}$ in Figure 18. Similarly to the
case of the Kerr and of 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.

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 $\bar{q}>1$, and denote
it by RN∗. These spacetimes admit photon spheres only for
$1<\bar{q}\leq\sqrt{9/8}$, 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
$0<\hat{\bar{\nu}}<1$. However, the JNW naked singularities only cast shadows
when $0<\hat{\bar{\nu}}\leq 0.5$, 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 $M_{0}$ family of static
spacetimes containing a compact region $r_{A}<r_{A,\text{b}}\equiv 2M/M_{0}$
filled by an anisotropic fluid, where $M$ is the 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
$r_{A,\text{b}}<3M$ or equivalently when $M_{0}\geq 2/3$ contributed by the
exterior Schwarzschild spacetime.

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 that of a Schwarzschild black hole (see Fig. 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∗ (cf. Section 4).

Figure 20 shows the bounds imposed by the EHT images of Sgr A∗ on the 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, traversible
Morris-Thorne (MT) metrics (Morris & Thorne, 1988) in General Relativity, for
which the $tt$-components of the metric are determined by the “redshift
function” $\Phi$ as $g_{tt}=-\exp{(2\Phi)}$. For wormholes in General
Relativity to be traversible, 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
$-1+r_{A}~{}d\Phi/dr_{A}=0$. If we restrict to the simplest case of an MT
wormhole with a single unstable circular null geodesic,111111We note that
there exist wormhole spacetimes that could be particularly difficult to
distinguish from a black hole using the present considerations (see, e.g.,
Morris et al. 1988). which can then be identified as the location of the
photon sphere, e.g., by setting $\Phi=-r_{A,\text{t}}/r_{A}$ as in Bambi
(2013), we find that the (Keplerian/ADM) mass definition forces
$r_{A,\text{t}}=M$. This implies a shadow radius of
$r_{\text{sh}}=e~{}M\approx 2.72~{}M$ or equivalently $\delta\approx-0.48$,
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
modelled as a Kerr black hole, if we are able to achieve a precision of
$|\delta|<0.07$ (Johannsen & Psaltis 2010; see also Fig. 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 non-minimal 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.

Table 4: Post-Newtonian Coefficients of Parametrized Metrics

Parametrized metric | $\kappa_{1}$ | $\kappa_{2}$ | Constraints
---|---|---|---
JP | $-\alpha_{12}$ | $-2\alpha_{12}+\alpha_{13}$ | $\begin{array}[]{lcl}-0.3&\lesssim\kappa_{1}\lesssim 0.7\\\ -3.1&\lesssim\kappa_{2}\lesssim 1.5\\\ \end{array}$
MGBK | $-\frac{1}{2}\gamma_{1,2}-\gamma_{4,2}$ | $-\gamma_{1,2}+\frac{1}{2}\gamma_{1,3}-4\gamma_{4,2}+\gamma_{4,3}$ | $\begin{array}[]{lcl}-0.3&\lesssim\kappa_{1}\lesssim 0.9\\\ -4.0&\lesssim\kappa_{2}\lesssim 1.3\\\ \end{array}$
RZ | $2a_{0}$ | $4\left(a_{0}-\frac{a_{1}}{1+\frac{a_{2}}{1+\frac{a_{3}}{...}}}\right)$ | $\begin{array}[]{lcl}-0.3&\lesssim\kappa_{1}\lesssim 0.9\\\ -4.0&\lesssim\kappa_{2}\lesssim 1.3\\\ \end{array}$

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

Spacetime | Charge Range | Constraints | $\kappa_{1}$ | $\kappa_{2}$
---|---|---|---|---
RN | $0<\bar{q}\leq 1$ | $0<\bar{q}\leq 0.84$ | $\bar{q}^{2}/2$ | $0$
RN* | $1<\bar{q}$ | $\times$ | $\bar{q}^{2}/2$ | $0$
Schwarzschild | $-$ | $-$ | $0$ | $0$
Bardeen | $0<\bar{q}<\sqrt{16/27}$ | $-$ | $0$ | $-3\bar{q}^{2}/2$
Frolov | $0<\bar{l}<\sqrt{16/27},0<\bar{q}\leq 1$ | see, e.g., Figure 18 | $\bar{q}^{2}/2$ | 0
Hayward | $0<\bar{l}<\sqrt{16/27}$ | $-$ | $0$ | $0$
MT Wormhole | $-$ | $\times$ | $-1$ | $-2/3$
KS | $0<\bar{\alpha}$ | $0<\bar{\alpha}\leq 0.79$ | $-\bar{\alpha}^{2}/4$ | $0$
EMd-1† | $0<\bar{q}<\sqrt{2}$ | $0<\bar{q}<0.87$ | $\bar{q}^{2}/2$ | $\bar{q}^{4}/8$
EMd-2† | $0<\bar{q}_{\text{e}}\leq\sqrt{2}-\bar{q}_{\text{m}}<\sqrt{2}$ | Figure 19 (top) | $\left(\bar{q}_{\text{m}}^{2}+\bar{q}_{\text{e}}^{2}\right)/2$ | $\left(\bar{q}_{\text{m}}^{2}-\bar{q}_{\text{e}}^{2}\right)^{2}/8$

  * Notes: We denote by crosses and dashes spacetimes that are entirely ruled out and that are unaffected by the EHT measurements.

###  5.3 Comparisons Between Metric Constraints

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 back hole itself. For the parametrized
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.

Figure 21: Constraints on the post-Newtonian coefficients $\kappa_{1}$ (top
panel) and $\kappa_{2}$ (bottom panel) of the Sgr A∗ metric imposed by the EHT
images. We include three parametrized metrics as well as several metrics that
are known solutions to particular modified gravity theories. For each curve,
we only allow one of the first two post-Newtonian coefficient to vary and set
the others to zero. The bounds on the post-Newtonian coefficients depend
weakly on the specific properties of the metric used to obtain them.

Translating directly the bounds on the parameters of one metric to those of
another is non-trivial 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
towards radial infinity. In particular, we write the $tt-$component in areal
coordinates in terms of the parametrized post-Newtonian (PPN) expansion

$g_{tt}=-1+\frac{2}{r_{A}}-2\left(\frac{\kappa_{1}}{r_{A}^{2}}\right)+2\left(\frac{\kappa_{2}}{r_{A}^{3}}\right)-2\left(\frac{\kappa_{3}}{r_{A}^{4}}\right)+\mathcal{O}(r_{A}^{-5})$
(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 $\delta$
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.

As before, we show only two particular cross sections of the multi-dimensional
parameter space for which the metric parameters were chosen such that only one
of the first two post-Newtonian deviation coefficients has a non-zero value.
As an example, for the JP metric we set $\alpha_{13}=2\alpha_{12}$ for the
$\kappa_{1}$ plot in Figure 21 and all deviation parameters other than
$\alpha_{12}$ and $\alpha_{13}$ to zero. This forces the $\kappa_{2}$ 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 decrease quite rapidly (see also Equations 29-33 of Psaltis et al.
2021). For the $\kappa_{2}$ plot for the JP metric we set $\alpha_{12}=0$ to
force $\kappa_{1}=0$ and allow only $\alpha_{13}$ to be non-zero. For the MGBK
metric we set $\gamma_{4,2}=-\gamma_{1,2}/4$, and set all parameters other
than $\gamma_{1,2}$ and $\gamma_{4,2}$ to zero for the $\kappa_{1}$ plot. For
the $\kappa_{2}$ plot we set $\gamma_{4,2}=-\gamma_{1,2}/2$, and all other
parameters to zero. For the RZ metric we set $r_{0}=2$, which in turn sets
$\epsilon=0$ as discussed in Section 5.1 and set $a_{0}=a_{1}$ for the
$\kappa_{1}$ plot, and only $a_{1}$ to be non-zero for the $\kappa_{2}$ 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 non-zero.
The full $tt-$component 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 $\sim 1$ and $\sim 5$ 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).

##  6 Comparisons To other tests of gravity

Figure 22: An illustration of the different aspects of the theory of gravity
probed by several examples of current tests in solar-system, compact-object,
and cosmological settings. A group of tests explore primarily the dynamics and
propagating modes of the theory, while others probe the stationary spacetimes
of isolated objects. A different group of tests probe vacuum spacetimes, while
others are sensitive to the coupling of matter to the gravitational fields.
Any particular modification to General Relativity may alter one, several, or
all of these aspects of the theory. Tests with horizon-scale images probe the
characteristics of the stationary vacuum spacetimes of compact objects.

###  6.1 The Gravitational Field Probed by the Image of Sgr A∗

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. 121212Note that these
tests depend also on the properties of stationary spacetimes and can,
therefore, provide information about them as well, as we will see below. 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.

Figure 23: A parameter space of tests of gravity with astrophysical and
cosmological systems (after Baker et al. 2015). For the Solar System and
pulsar tests, the straight lines connect the range of gravitational fields
that could affect, in principle, the outcome of each test, from the location
of the outermost probe to the location of the central massive object; the
dashed region of the cyan line indicates that the connection to the largest
curvatures is theory-specific. The green lines connect the range of
gravitational fields probed by two gravitational-wave tests with black-hole
inspirals. Filled areas show the typical range of gravitational fields probed
by (orange) cosmological, (green) gravitational-wave, and (magenta) black-hole
imaging tests. Even though different tests explore, in principle, different
aspects of the gravitational theory, as Figure 22 illustrates, they also probe
vastly different scales. In particular, the horizon-scale images of Sgr A∗
that we report here probe a previously unexplored region of this parameter
space of gravitational physics tests.

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 $\epsilon=GM/rc^{2}$. 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).

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 the 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, 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.

###  6.2 Comparing Gravitational Tests Across Scales

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 Parametrized Post-Newtonian
(PPN) expansions (Will, 2014), pulsar tests use post-Keplerian
parametrizations 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; Johannsen, 2013a; Vigeland et al., 2011; 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), parametrized 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 are 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.

###  6.3 Tests with the S2 Orbit

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, as well as 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 $f_{\rm
SP}$, such that the precession per orbit at the first post-Newtonian order can
be written as

$\Delta\phi_{\mathrm{1}}=f_{\rm SP}\frac{6\pi GM}{a(1-e^{2})c^{2}}\;,$ (24)

where $a$ and $e$ are the orbital separation and eccentricity of the orbit,
respectively. The best-fit value for $f_{\rm SP}$ was found to be consistent
with the predictions of the Schwarzschild metric, i.e., $f_{\rm SP}=1.1\pm
0.19$.

In the PPN formalism, the phenomenological parameter $f_{\rm SP}$ is related
to two of the first-order post-Newtonian parameters of the metric via (Will,
2014)

$f_{\rm SP}=\frac{1}{3}\left(2+2\gamma_{\rm S0-2}-\beta_{\rm S0-2}\right)\;,$
(25)

such that

$2\gamma_{\rm S0-2}-\beta_{\rm S0-2}=1.3\pm 0.57\;.$ (26)

Here the subscripts explicitly denote the fact that these parameters are not
universal constants but are specific to the metric of Sgr A∗ as measured at
the location of the S0-2 orbit. In deriving this equation, we have also
assumed that the mass of the S0-2 star is negligible with respect to the
black-hole mass (see eq. [32] below).

We now assess the freedom these observations allow for possible deviations at
higher post-Newtonian orders and hence the leverage of the strong-field
imaging tests that we report here in constraining the metric of Sgr A∗. We
first write the precession per orbit at the second post-Newtonian orbit as
(Will, 2018)

$\Delta\phi_{2}=-6\pi\left[\frac{GM}{2a(1-e^{2})c^{2}}\right]^{2}(10-e^{2})\;.$
(27)

The ratio of the second to the first order post-Newtonian term for the S0-2
star is

$\frac{\Delta\phi_{2}}{\Delta\phi_{1}}=\frac{GM}{4a(1-e^{2})c^{2}}(10-e^{2})\simeq
7\times 10^{-4}\;.$ (28)

Because the first post-Newtonian term has been measured to an accuracy of
$\sim 20$% (see eq. [26]), the second post-Newtonian term would have to be
$\sim 0.2/7\times 10^{-4}\simeq 285$ times larger than the Schwarzschild
prediction in order for it to cause deviations detectable with current
instruments. We will consider this as a heuristic upper bound on possible
deviations at the second post-Newtonian order imposed by the precession of the
S0-2 orbit.

Had the metric of Sgr A∗ deviated from Schwarzschild by, e.g., a factor of 250
at the second post-Newtonian order, this would have still been undetectable by
the S0-2-precession test but would have led to a shadow size as large as $\sim
85GM/c^{2}\simeq 425~{}\mu$as (see eq. [33] of Psaltis et al. 2021). This
predicted size would have been at least a factor of 8 larger than the measured
size we report in Section 3.5 and, more importantly, would have been almost
two orders of magnitude larger than any potential uncertainty introduced by
systematics due to plasma physics (as captured by the $\alpha_{1}-1$ factor)
or due to our measurement methods (as captured by the $\alpha_{2}-1$ factor).
In other words, the horizon-scale images of Sgr A∗ provide substantial
constraints to potential deviations of the black-hole spacetime from the GR
predictions that could have evaded all prior bounds, beyond any astrophysical
uncertainties.

Figure 24: Comparison of the posterior distributions for the fractional
deviation $\delta$ from the Schwarzschild predictions, as inferred by the EHT
measurement of the size of the black-hole shadows in Sgr A∗ and M87. The
purple-shaded area shows the $\sim 8$% range predicted for the Kerr metric,
depending on the black-hole spin and observer inclination. The red-shaded area
shows the small range of posteriors for Sgr A∗, inferred with different
imaging and calibration algorithms (see Fig. 12). The solid and dashed lines
show the posteriors for the M87 black hole, when the stellar- and gas-dynamics
measurements of the mass-to-distance ratio have been used, respectively. The
negligible uncertainties in the mass measurement of Sgr A∗, which is the
result of the detection of relativistic effects in the orbit of the S0-2 star,
removes any ambiguity in the comparison with the Kerr predictions.

###  6.4 M87 Imaging Tests

The black hole at the center of the M87 galaxy has a mass that is
approximately 1500 times larger than the one in Sgr A∗. As a result,
observations of horizon-scale images from the M87 black hole probe similar
potentials but curvatures that are 6 orders of magnitude smaller than those of
Sgr A∗. In Event Horizon Telescope Collaboration et al. (2019c), we used the
2017 EHT images of the M87 black hole to derive constraints on possible
deviations of the inferred size of the black hole shadow from the
Schwarzschild prediction and, in Psaltis et al. (2020a) and Kocherlakota et
al. (2021), we used these measurements to place constraints on possible
deviations of metric parameters from Kerr.

Contrary to the case of Sgr A∗ that we report here, there were two independent
and distinct priors on the mass-to-distance ratio for the black hole in M87,
based on either stellar- (Gebhardt et al., 2011) or gas-dynamic measurements
(Walsh et al., 2013). Adopting the former resulted in an upper bound on
deviations from the Kerr predictions that was consistent with zero, within
$\sim 17$%. We opted to assign negligible prior likelihood to the latter
prior, as it would have led us to conclude that there is significant tension
between the Kerr predictions and the observations, and instead used the
measurements as a null hypothesis test, i.e., concluded that the EHT images
were not inconsistent with the Kerr predictions.

Figure 24 compares the posteriors on the deviation parameter $\delta$ obtained
here for Sgr A∗ to those reported earlier for the M87 black hole. In the case
of the image in Sgr A∗, we have a precise measurement of the mass-to-distance
ratio for the black hole based on the detection of relativistic effects in the
orbit of the S0-2 star, as discussed in Section 2. This removes any ambiguity
in our calculation of the Kerr predictions. Moreover, the uncertainties in the
mass-to-distance priors for Sgr A∗ are negligible compared to those in M87,
even if we only adopt the stellar-dynamic measurement for the latter. This
results in uncertainties on the bounds of the deviation parameter $\delta$
that are almost a factor of 2 smaller in Sgr A∗ compared to the M87 black
hole.

In both the Sgr A∗ and M87 cases, the inferred sizes of the black-hole shadows
are consistent with the Kerr predictions, even though the black holes span 3
orders of magnitude in mass and 6 orders in curvature scale. This serves as a
confirmation of the General Relativistic prediction that the spacetime
properties of black holes scale with their mass and can be further reinforced
by leveraging tests that involve stellar-mass black holes, as we discuss
bellow.

###  6.5 Gravitational Wave Tests

Observations of gravitational waves from coalescing black-hole binaries with
LIGO/Virgo provide strong-constraints on potential near-horizon modifications
of the predictions of the theory of gravity for black holes (Abbott et al.,
2016, 2019, 2021). Because of the frequency range of these ground-based
gravitational-wave detectors, the black-hole masses they are sensitive to are
in the $10-100~{}M_{\odot}$ range. As a result, compared to the tests with the
EHT black-hole images, existing gravitational wave observations probe similar
potentials but curvatures that are different by 8-16 orders of magnitude.

A second important difference arises from the fact that, fundamentally,
gravitational-wave observations measure the propagating gravitational modes of
the theory, whereas black-hole images measure electromagnetic modes
propagating on the black-hole spacetimes. It is possible that the number and
polarization of the propagating modes of the fundamental theory of gravity are
the same as those in General Relativity but the stationary metrics are not; in
fact, it is possible that the fundamental theory of gravity is General
Relativity but the stationary metrics of the supermassive compact objects in
the centers of galaxies are not described by the Kerr metric (see, e.g., Gair
et al. 2008). Alternatively, it is possible that the propagating modes of the
theory are very different than those in General Relativity but the stationary
spacetimes remain Kerr (Psaltis et al., 2008; Barausse & Sotiriou, 2008). In
this way, the gravitational-wave and imaging observations of black holes
provide complementary probes of potential modifications to General Relativity.

Figure 25: Comparison of the posterior distributions for the fractional
deviation $\delta$ from the Schwarzschild predictions, as inferred by the EHT
measurement of the size of the black-hole shadow in Sgr A∗ (red curve for the
fiducial priors) and M87 (green curve for the stellar-dynamics mass) as well
as by the LIGO/Virgo measurements of the inspiral phases of GW170608 (blue)
and GW190924 (orange). The posteriors corresponding to the latter two reflect
the prediction on the fractional deviation $\delta$ in the shadow size one
would have calculated based on the constraints imposed by the gravitational-
wave measurements, if the coalescing, stellar-mass sources had the same form
of metrics as those of the supermassive black holes observed with the EHT. The
gray-shaded area shows the $\sim 8$% range predicted for the Kerr metric,
depending on the black-hole spin and observer inclination. Even though tests
with gravitational waves and black-hole images span black-hole masses that are
different by 8 orders of magnitude, they are all consistent with the GR
predictions that all black holes are described by the same metric, independent
of their mass.

Because of this fundamental difference, however, in order to compare directly
gravitational-wave constraints to those of black-hole imaging, one needs to
make specific assumptions. Our main goal in this section is to leverage the
gravitational-wave tests in order to assess whether the black-hole metric
properties scale with mass, as predicted by General Relativity. For this
reason, we will focus on the inspiral phases of the observed gravitational
waves, as these are ones that are mostly sensitive to modifications in the
metrics of the coalescing black holes131313There is a multitude of other tests
of gravity that are possible with gravitational-wave observations, such as
those that place bounds on the mass of the graviton (Abbott et al., 2016;
Baker et al., 2017). Albeit extremely important, these tests are not directly
comparable to those we report here as imaging tests are sensitive only to the
stationary black-hole metrics and not to other aspects of the theory (see,
also, Völkel & Barausse 2020). Moreover, following Psaltis et al. (2021), we
will assume here that the propagating modes of the theory are
indistinguishable from those in General Relativity and assign any room for
potential deviations to changes in the underlying metrics of the black holes.
Unless the fundamental theory of gravity is finely tuned such that the
modifications in the radiative sector exactly cancel those in the metrics, for
the masses of the LIGO/Virgo black holes, our constraints will represent
broad-brush upper limits on potential metric deviations.

Under the assumptions outlined above, the LIGO/Virgo measurements of the
inspiral phases of coalescing black-hole binaries depend entirely on the
$tt-$components of the metrics, as expressed in areal coordinates (Carson &
Yagi, 2020; Cárdenas-Avendaño et al., 2020). This is a consequence of the fact
that the waveforms of the gravitational waves during the decay of quasi-
circular orbits are determined by the binding energies of the orbits and their
angular frequencies (see, e.g., eq. [9] of Carson & Yagi 2020), both of which
are determined by the $tt-$components of the metric (Ryan, 1995). This is the
same component of the metric that determines the size of the black-hole
shadows measured with the EHT (Psaltis et al., 2020a). Remarkably, because of
a coincidence related to the masses of the coalescing black holes, the
degeneracies between the constraints from inspiral measurements on the various
parameters of metrics that deviate from Kerr are nearly parallel to those of
the constraints imposed from the EHT imaging observations (Psaltis et al.,
2021). In other words, this coincidence allows us to use the LIGO/Virgo
constraints and make a prediction on the fractional deviation $\delta$ in the
shadow-size one would have calculated, if the gravitational-wave sources had
the same metrics as those of the supermassive black hole observed with the
EHT.

Figure 25 compares the results on the deviation parameter $\delta$ for the two
most constraining gravitational-wave events, GW170608 and GW190924 (Psaltis et
al., 2021), to those obtained in §3.5 for Sgr A∗, as well as those for the M87
black hole derived earlier (Event Horizon Telescope Collaboration et al.,
2019c). As discussed above, all observations are consistent with the
predictions of General Relativity, even though they utilize black holes with
masses that are different by 8 orders of magnitude. This lends support not
only to the Kerr nature of the black-hole spacetimes but also to the fact that
the fundamental theory of gravity does not have a scale between those probed
by stellar-mass and supermassive black holes.

###  6.6 Pulsar Timing Tests

The potentials and curvatures probed by pulsar timing tests may depend on the
underlying theory of gravity, as discussed above. In principle, in theories
without a characteristic scale (such as screening) between the orbital
separation of the binary and the size of the neutron star, pulsar tests probe
the coupling of matter to the gravitational field, which takes place in the
strong field regime of the neutron-star interior, i.e., at a potential of
order unity and curvature of order $10^{-10}$ cm-2. For such theories, the
horizon-scale images of Sgr A∗ probe a similar potential but a curvature that
is different by 15 orders of magnitude (see Fig. 23).

In contrast, theories with a characteristic scale between the orbital
separation of the binary and the size of the neutron star (or other similar
effects as in Yagi et al. 2016) are only probed at the potential of the
periapsis distance for tests involving the orbital period derivative and
orbital precession (for the double pulsar this is about $6\times
10^{4}GM/c^{2}$) or at the distance of minimum approach for the Shapiro delay
(for the double pulsar, this distance is about $1500GM/c^{2}$). In this case,
the horizon-scale images of Sgr A∗ probe similar curvatures but a potential
that is larger by 5 orders of magnitude.

Following these two approaches, we are going to discuss the constraints
imposed on the various deviation parameters in two complementary ways. First
is in a theory-agnostic way, in terms of the effective post-Newtonian
parameters of the metrics of the compact objects. Second is in terms of the
scalar-tensor gravity theory of Damour & Esposito-Farese (Damour & Esposito-
Farese, 1993), with second-order couplings between the scalar field and the
Einstein tensor defined by the parameters $\alpha_{a}\equiv\partial\ln
m_{a}/\partial\phi_{0}$ and
$\beta_{a}\equiv\partial\alpha_{a}/\partial\phi_{0}$, where the subscript $a$
corresponds to either the pulsar “p” or the companion “c”. In such a theory,
the effective 1PN parameters can be expressed in terms of the theory
parameters $\alpha_{a}$ and $\beta_{\alpha}$ as

$\hat{\gamma}_{ab}=1-\frac{2\alpha_{a}\alpha_{b}}{1+\alpha_{a}\alpha_{b}}$
(29)

and

$\hat{\beta}^{a}_{bc}=1+\frac{\beta_{a}\alpha_{b}\alpha_{c}}{2(1+\alpha_{a}\alpha_{b})(1+\alpha_{a}\alpha_{c})}\;,$
(30)

where the “hats” and subscripts emphasize the fact that the strong-field
equivalents to the PPN parameters are not universal constants.

In principle, there are at least five theory-independent post-Keplerian
parameters in a binary system that can be measured from pulsar timing (Damour
& Taylor, 1992). Together with the normal Keplerian parameters, they comprise
a set of inferred quantities that is larger than the free parameters in the
system (see Wex & Kramer 2020 for a recent review). The combination of any two
post-Keplerian parameters is used to determine the masses of the two objects
in the binary. Any additional measurement can then be used for testing GR and
a very broad class of alternative (boost-invariant) theories (Damour & Taylor,
1992; Will, 2014, 2018; De Laurentis et al., 2018). We will focus below on the
constraints imposed by the measurement of Shapiro delay, of the precession of
periapsis of the binary, and of the orbital period evolution caused by the
emission of gravitational waves.

Shapiro delay.— Some of the main constraints on deviations from General
Relativity come from the measurement of a Shapiro delay in binary pulsar
systems. Like the imaging of black holes, such experiments provide rare
opportunity to study the light-propagation near strongly self-gravitating
objects. Two parameters can be measured, the “shape” $s$ and the “range” $r$.
The Shapiro shape, $s$, can quite generally be identified with the sine of the
orbital inclination (i.e., $s=\sin i$, e.g. Kramer et al. 2021) 141414There is
an indirect dependence on the underlying metric via its relation to the masses
of the orbiting objects, as measured within the particular gravity theory. In
comparison, the Shapiro range $r$ relates to the companion mass, e.g., in
General Relativity one finds $r=Gm_{c}/c^{3}$.

For a theory agnostic constraint, one can relate $r$ to the 1PN parameter
$\gamma$ via

$r=\frac{\hat{G}_{0,c}m_{c}(1+\hat{\gamma}_{0c})}{2c^{3}}$ (31)

where $m_{c}$ is the mass of the companion star and the subscript on the PN
parameter makes explicit that this is the value for the coupling between a
test particle and the pulsar companion, i.e. they are not universal constants
but object-specific. The subscript in the gravitational constant $G$ also
denotes that this is the effective gravitational constant felt by a test
particle in the field of the companion.

Figure 26: Comparison of the current limits on various potential metric
deviation parameters at the first post-Newtonian order obtained from different
tests of gravity with astrophysical objects, as a function of the (top panel)
gravitational potential and (right panel) curvature probed by each test. For
visual clarity, each upper limit is calculated assuming that the deviations at
all other post-Newtonian orders are negligible; this plot, therefore,
represents only one cross section in the multi-dimensional parameter space of
plaussible deviations. Every tests probes a different combination of post-
Newtonian parameters, as shown on the upper panel. In most modifications to
General Relativity, these post-Newtonian parameters are not universal
constants but depend on the nature of the central object, its mass,
composition, scale, etc.

Precession of periapsis.— In GR, within the PPN framework, the rate of
precession of the periapsis $\dot{\omega}$ depends on the combination
$2+2\hat{\gamma}-\hat{\beta}$ (in the limit when the mass ratio is zero).
Incorporating the effects due to the presence of two orbiting objects for
which the strong-field coupling of matter to gravity cannot be neglected,
$\dot{\omega}$ becomes proportional to

$k=\left(2+2\hat{\gamma}_{pc}-\frac{m_{c}}{M}\hat{\beta}_{cc}^{p}-\frac{m_{p}}{M}\hat{\beta}_{pp}^{c}\right)\frac{\left(\hat{G}_{pc}Mn_{b}\right)^{2/3}}{c^{2}(1-e^{2})}\;.$
(32)

Here $M\equiv m_{c}+m_{p}$ is the sum of the companion and pulsar masses,
$m_{c}$ and $m_{p}$, respectively, and $e$ is the eccentricity of the orbit.

Orbital period derivative.— The orbital period of a binary system may change
due to a number of effects. In case that only gravity plays a role, a decay in
the orbital period results from the emission of gravitational waves. The
measurement of an orbital period derivative can then be used to confront a
given theory with its predictions. In General Relativity, to leading order,
the orbital period derivative is given by the quadrupole formula (Peters,
1964). In contrast, many alternative theories of gravity violate the strong
equivalence principle, resulting in the emission of gravitational dipolar
radiation. Observations of binary pulsars can provide strict limits on the
existence of dipolar radiation (Shao & Wex, 2016; Wex & Kramer, 2020). Indeed,
the Double Pulsar system provides currently the most precise test of the
General Relativistic quadrupolar description of gravitational waves,
validating the prediction at a level of $1.3\times 10^{-4}$ (95% conf. level,
Kramer et al. 2021).

Under the assumption that the radiative sector is negligibly different from
General Relativity, one can put, in a theory agnostic way, constraints on the
post-Newtonian parameters of the waveform, obtaining at 1PN order

$\hat{\beta}-\hat{\gamma}=\Delta\phi_{2}\frac{743+924\eta}{1344}\;,$ (33)

where $\Delta\phi_{2}$ is the 1PN order correction to the phase term of the
gravitational waveform.

Following this framework, and emphasizing the mentioned assumptions, we can
use the other precise measurements in the Double Pulsar system for $r$ and
$\dot{\omega}$ to derive constraints on the corresponding post-Newtonian
parameters at the first and second orders. At the 68% confidence limit,
particular combinations of these parameters that give rise to the the Shapiro
range, the periapsis advance and the orbital decay have been found to be
consistent with the General Relativistic predictions at a precision of
$3.4\times 10^{-3}$, $2.6\times 10^{-4}$, and $6.3\times 10^{-5}$,
respectively (Kramer et al., 2021).

Universality of free fall .— One can use pulsars also to test the strong
equivalence principle (SEP), as first shown by Damour & Schaefer (1991). The
discovery of a pulsar in a triple system (Ransom et al., 2014) allow a
variation of the Damour-Schäfer experiment in very constraining way (Freire et
al., 2012). In the triple system an inner pulsar - white dwarf system is
orbited by a second white dwarf in an outer orbit. By tracking the orbital
motion of the neutron star via pulsar timing, one can study how the inner two
objects with significantly different gravitational self-energy are falling in
the gravitational field of the third object. Archibald et al. (2018) presented
a limit on the strong-field equivalent of the Nordtvedt parameter of
$\eta<3\times 10^{-5}$. Following Damour & Schäfer, it relates to the strong-
field SEP violation parameter as

$\Delta=\eta\epsilon_{\rm grav}+\eta\textquoteright\epsilon_{\rm grav}^{2}$
(34)

where $\eta$ and $\eta^{\prime}$ are strong-field equivalents of the Nordtvedt
parameter and $\epsilon_{\rm grav}=E_{\rm grav}/mc^{2}$ is the normalized
Newtonian gravitational binding energy. At the first post-Newtonian order, the
Nordtvedt parameter is related to the post-Newtonian coefficients by
$|\eta|=|4\beta-\gamma-3|$. Ignoring higher terms, and using the results by
Archibald et al. (2018), leads to $|4\beta-\gamma-3|<1.0\times 10^{-5}$ (95%
C.L.). At the second post-Newtonian order, we make use of the Messenger limit
obtained in the Solar System (Genova et al., 2018) to constrain
$\eta^{\prime}<1.2\times 10^{-3}$ (95% C.L.).

###  6.7 Leveraging Gravitational Tests Across Different Scales

Figures 26 and 27 compare the constraints on the various metric deviation
parameters at the first and second post-Newtonian orders that are imposed by
the gravity tests discussed above. For visual clarity reasons alone, the
figures show only a particular cross section of the multi-dimensional
parameter space of plausible deviations: in constructing this figure, we have
assumed that all deviation parameters other than those plotted are negligible.
Barring any fine tuning of the fundamental theory that would lead to
fortuitous cancellations, the bounds plotted can be regarded as rough upper
limits. It is important to emphasize here that since each test probes a
different combination of the various post-Newtonian parameter, which are
indicated on the figure for those at the first post-Newtonian order, it is
mathematically impossible for any non-Kerr metric to evade simultaneously all
constraints purely by fortuitous cancellations.

The bounds on the post-Newtonian parameters that are imposed by the imaging
observations of Sgr A∗ and M87, the detection of periapsis precession in the
S2 orbit, the gravitational-wave observations, and the double pulsar have been
discussed in §5.3, Event Horizon Telescope Collaboration et al. (2019c), §6.3,
Psaltis et al. (2021), and §6.6, respectively. The figures also incorporate
the limits on the first- and second post-Newtonian parameters obtained by
Solar System tests, which we briefly discuss below.

The measurement of the solar Shapiro delay with the Cassini spacecraft
constrained the $\gamma_{\odot}-1$ parameter at the first post-Newtonian order
with an accuracy of $\sim 2\times 10^{-5}$ (Bertotti et al., 2003). At the
second post-Newtonian order, the best constraint comes from the deflection of
light measurements that are consistent with the Schwarzschild predictions to
an accuracy of $\sim 10^{-4}$. The second-order post-Newtonian correction for
light deflection at the solar surface is of order $\sim 3\times 10^{-6}$
(Bodenner & Will, 2003). Therefore, any second-order post-Newtonian
corrections to the solar metric cannot be larger than $\sim 10^{-4}/(3\times
10^{-6})\simeq 35$ times the Schwarzschild predictions.

The periastron precession of Mercury constraints the combination
$2\gamma_{\odot}-\beta_{\odot}-1$ with an accuracy of $\sim 2\times 10^{-5}$.
Given the strict limit on $\gamma_{\odot}$ for the Solar metric from Cassini,
it can be translated into an upper bound on $\beta$ alone. Effects due to the
second post-Newtonian order are subdominant compared to relativistic effects
involving other Solar-System bodies and are of order $\sim 7\times 10^{-8}$
smaller than the first-order effects (Will, 2018). As a result, we can
conclude that any second-order post-Newtonian corrections to the solar metric
cannot be larger than $\sim 2\times 10^{-5}/(7\times 10^{-8})\simeq 300$ times
the Schwarzschild predictions.

Figure 27: Same as Fig. 26 but for the constraints on the second post-
Newtonian order, assuming that the deviations at all other post-Newtonian
orders are negligible.

Examining the combined constraints shown in Figures 26 and 27 allows to draw
some general conclusions on modifications to the equilibrium metrics of
massive, isolated objects from the predictions of General Relativity that can
be accommodated within current observational bounds, barring any fortuitous
cancellations: (i) They may appear at the first post-Newtonian order and
attain up to order-unity magnitudes at the highest gravitational potentials of
compact objects but only in theories with coupling to matter that evades the
theory-specific bounds imposed by the double pulsar pulsar. (ii) They may
appear primarily at the second or higher post-Newtonian orders but with
magnitudes that are constrained only to within order-unity deviations from the
general relativistic predictions.

##  7 Summary

The Galactic Center black hole, Sgr A∗, is an ideal and natural laboratory for
testing the strong-field predictions of General Relativity. Monitoring of the
orbits of tens of stars within its radius of influence and the recent
detection of two relativistic effects in one of them has led to a precise
determination of the black-hole mass and distance from the Earth (Gravity
Collaboration et al., 2018a; Do et al., 2019; Gravity Collaboration et al.,
2020a). Compared to any other black hole on the sky observed in the
electromagnetic spectrum, these measurements lead to precise predictions for
the magnitudes of gravitational effect, which can then be tested against other
observations.

In this series of papers, we are reporting the first horizon-scale images of
Sgr A∗, obtained with the EHT at a wavelength of 1.3 mm (Paper II). The images
are characterized by a bright ring of emission surrounded by a deep brightness
depression (Paper III). This image structure is stable and remains present for
at least the $\sim 32$hr span of observations on 2017 April 5-6, which
corresponds to $\sim 60-500$ dynamical timescales at the radius of the
innermost stable circular orbit, depending on black-hole spin. Using a variety
of image-reconstruction and visibility-domain modeling tools, we measured the
diameter of the ring-like structure to be equal to $50\pm XX~{}\mu$as (make
consistent with Paper I).

The structure, size, and persistent nature of the black-hole image leads us to
identify the central brightness depression with the shadow that the black hole
is expected to cast on the emission from the accreting plasma. Using the depth
and size of this brightness depression as well as the observed broadband
spectrum of the source, we rule out the possibility that Sgr A∗ has either a
surface that fully absorbs and remits thermally the incoming energy flux or
that it has a reflecting surface at 1.3 mm with significant albedo.

Using an extensive suite of images and synthetic data based on time-dependent
and semi-analytic plasma models in a variety of spacetimes, we calibrated the
difference between the size of the observed emission ring and that of the
shadow as well as potential systematic effects introduced by the sparse
interferometric coverage of the EHT array and our analysis methods. We found
the magnitudes of these effects to be of order $\sim 10$%. This is subdominant
compared to the significantly larger effect caused by changing the spacetime
of Sgr A∗, which can be as large as an order of magnitude while remaining
consistent with the bounds imposed by the observation of the orbit of the S0-2
star in the spacetime of the same black hole.

Because in the Kerr metric the size of the shadow depends primarily on the
mass-to-distance ratio of the black hole, which is well determined by
monitoring of stellar orbits, we can compare directly the predicted Kerr size
to the observations, without any free parameters. We find that the Kerr
predictions are consistent with observations at the $\sim 10$% level. We then
use this strong-field inference to place bounds of order unity to the
parameters of metrics that deviate from Kerr.

The $\sim 4\times 10^{6}M_{\odot}$ mass of Sgr A∗ places these tests of the
Kerr metric in a region of the parameter space of gravitational objects that
has never been probed before in the strong field regime. This mass is
approximately 5 orders of magnitude larger than the masses of the objects
probed by LIGO/Virgo via the detection of gravitational waves and a factor of
1500 smaller than the mass of the black hole in M87. Leveraging the fact that
similar bounds on the strong-field predictions of General Relativity have been
placed by gravitational-wave and imaging observations across these 8 orders of
magnitude range in mass, we conclude that it is unlikely for the fundamental
theory of gravity to possess a scale in this range.

These conclusions are based predominantly on the identification of the central
brightness depression with the shadow of the black hole and, at the $\sim 10$%
level, on the calibration of the relative size of the bright ring of emission,
which we measure, to that of the shadow, which we infer. In the case of the
tests involving the image of the M87 black hole that we reported earlier
(Event Horizon Telescope Collaboration et al., 2019c; Psaltis et al., 2020a;
Kocherlakota et al., 2021), two issues related to that black hole left open
the possibility that the observed brightness depression might not have been
related to the black-hole shadow. First is the factor of $\sim 2$ difference
in the prior measurements of the black-hole mass based on stellar and gas
dynamics (Gebhardt et al., 2011; Walsh et al., 2013). It could have been
possible, in principle, for (i) the brightness depression to not be related to
the black-hole shadow but be generated, for example, at a larger distance from
the black hole and (ii) for the black-hole mass to be smaller than that
inferred from stellar dynamics in such a way that, by pure coincidence, the
size of the brightness depression is equal to the size inferred
observationally. This is the approach taken, e.g., by Gralla et al. (2019)
(see also Gralla et al. 2019, 2020). Second is the fact that the time spread
of the M87 observations was comparable to the dynamical timescale at its
innermost stable circular orbit, allowing for the possibility that the image
structure observed was transient and did not correspond to the persistent
image of the bright ring surrounding the black-hole shadow.

In the case of the Sgr A∗ images, neither of these considerations provide
reasonable alternatives to our interpretation. Indeed, the mass of the black
hole in Sgr A∗ is known to such a degree of precision that leaves very little
room for uncertainties in the predicted diameter of the black-hole shadow. At
the same time, the relatively small value of this mass, compared to the M87
black hole, allows us to observe the image over tens to hundreds of dynamical
timescales and conclude that this structure is persistent and not transient.
With these potential sources of uncertainty under control, we measured the
size of the black-hole shadow in Sgr A∗ and found it to be in agreement with
the Kerr prediction, as we did for the case of the M87 black hole. In order to
argue that the observed brightness depression is not related to the black-hole
shadow, we would have to assign this consistency not only to coincidence but
also require that the same coincidence works for both M87 and Sgr A∗. Given
that the two black holes have masses that are different by a factor of $\sim
1500$, are accreting at widely different rates with one showing a prominent
jet that is missing from the other, and are probably observed at different
inclinations, we consider this alternative to be highly unlikely.

The authors thank the anonymous referee for comments that improved the
manuscript. The Event Horizon Telescope Collaboration thanks the following
organizations and programs: National Science Foundation (awards OISE-1743747,
AST-1816420, AST-1716536, AST-1440254, AST-1935980); the Black Hole
Initiative, which is funded by grants from the John Templeton Foundation and
the Gordon and Betty Moore Foundation (although the opinions expressed in this
work are those of the author(s) and do not necessarily reflect the views of
these Foundations); NASA Hubble Fellowship grant HST-HF2-51431.001-A awarded
by the Space Telescope Science Institute, which is operated by the Association
of Universities for Research in Astronomy, Inc., for NASA, under contract
NAS5-26555; the Academy of Finland (projects 274477, 284495, 312496, 315721);
the Agencia Nacional de Investigación y Desarrollo (ANID), Chile via
NCN$19\\_058$ (TITANs) and Fondecyt 1221421, the Alexander von Humboldt
Stiftung; an Alfred P. Sloan Research Fellowship; Allegro, the European ALMA
Regional Centre node in the Netherlands, the NL astronomy research network
NOVA and the astronomy institutes of the University of Amsterdam, Leiden
University and Radboud University; the Institute for Advanced Study; the China
Scholarship Council; Consejo Nacional de Ciencia y Tecnología (CONACYT,
Mexico, projects U0004-246083, U0004-259839, F0003-272050, M0037-279006,
F0003-281692, 104497, 275201, 263356); the Delaney Family via the Delaney
Family John A. Wheeler Chair at Perimeter Institute; Dirección General de
Asuntos del Personal Académico-—Universidad Nacional Autónoma de México
(DGAPA-—UNAM, projects IN112417 and IN112820); the European Research Council
Synergy Grant “BlackHoleCam: Imaging the Event Horizon of Black Holes" (grant
610058); the Generalitat Valenciana postdoctoral grant APOSTD/2018/177 and
GenT Program (project CIDEGENT/2018/021); MICINN Research Project
PID2019-108995GB-C22; the European Research Council for advanced grant
‘JETSET: Launching, propagation and emission of relativistic jets from binary
mergers and across mass scales’ (Grant No. 884631); the Istituto Nazionale di
Fisica Nucleare (INFN) sezione di Napoli, iniziative specifiche TEONGRAV; the
two Dutch National Supercomputers, Cartesius and Snellius (NWO Grant
2021.013); the International Max Planck Research School for Astronomy and
Astrophysics at the Universities of Bonn and Cologne; DFG research grant “Jet
physics on horizon scales and beyond” (Grant No. FR 4069/2- 1); Joint
Princeton/Flatiron and Joint Columbia/Flatiron Postdoctoral Fellowships,
research at the Flatiron Institute is supported by the Simons Foundation; the
Japanese Government (Monbukagakusho: MEXT) Scholarship; the Japan Society for
the Promotion of Science (JSPS) Grant-in-Aid for JSPS Research Fellowship
(JP17J08829); the Key Research Program of Frontier Sciences, Chinese Academy
of Sciences (CAS, grants QYZDJ-SSW-SLH057, QYZDJSSW-SYS008, ZDBS-LY-SLH011);
the Leverhulme Trust Early Career Research Fellowship; the Max-Planck-
Gesellschaft (MPG); the Max Planck Partner Group of the MPG and the CAS; the
MEXT/JSPS KAKENHI (grants 18KK0090, JP21H01137, JP18H03721, JP18K13594,
18K03709, JP19K14761, 18H01245, 25120007); the Malaysian Fundamental Research
Grant Scheme (FRGS) FRGS/1/2019/STG02/UM/02/6; the MIT International Science
and Technology Initiatives (MISTI) Funds; the Ministry of Science and
Technology (MOST) of Taiwan (103-2119-M-001-010-MY2, 105-2112-M-001-025-MY3,
105-2119-M-001-042, 106-2112-M-001-011, 106-2119-M-001-013,
106-2119-M-001-027, 106-2923-M-001-005, 107-2119-M-001-017,
107-2119-M-001-020, 107-2119-M-001-041, 107-2119-M-110-005,
107-2923-M-001-009, 108-2112-M-001-048, 108-2112-M-001-051,
108-2923-M-001-002, 109-2112-M-001-025, 109-2124-M-001-005,
109-2923-M-001-001, 110-2112-M-003-007-MY2, 110-2112-M-001-033,
110-2124-M-001-007, and 110-2923-M-001-001); the Ministry of Education (MoE)
of Taiwan Yushan Young Scholar Program; the Physics Division, National Center
for Theoretical Sciences of Taiwan; the National Aeronautics and Space
Administration (NASA, Fermi Guest Investigator grant 80NSSC20K1567, NASA
Astrophysics Theory Program grant 80NSSC20K0527, NASA NuSTAR award
80NSSC20K0645); the National Institute of Natural Sciences (NINS) of Japan;
the National Key Research and Development Program of China (grant
2016YFA0400704, 2017YFA0402703, 2016YFA0400702); the National Science
Foundation (NSF, grants AST-0096454, AST-0352953, AST-0521233, AST-0705062,
AST-0905844, AST-0922984, AST-1126433, AST-1140030, DGE-1144085, AST-1207704,
AST-1207730, AST-1207752, MRI-1228509, OPP-1248097, AST-1310896, AST-1555365,
AST-1614868, AST-1615796, AST-1715061, AST-1716327, AST-2034306); the Natural
Science Foundation of China (grants 11650110427, 10625314, 11721303, 11725312,
11873028, 11933007, 11991052, 11991053, 12192220, 12192223); NWO grant number
OCENW.KLEIN.113; a fellowship of China Postdoctoral Science Foundation
(2020M671266); the Natural Sciences and Engineering Research Council of Canada
(NSERC, including a Discovery Grant and the NSERC Alexander Graham Bell Canada
Graduate Scholarships-Doctoral Program); the National Youth Thousand Talents
Program of China; the National Research Foundation of Korea (the Global PhD
Fellowship Grant: grants NRF-2015H1A2A1033752, the Korea Research Fellowship
Program: NRF-2015H1D3A1066561, Basic Research Support Grant 2019R1F1A1059721,
2022R1C1C1005255); the Dutch Organization for Scientific Research (NWO) VICI
award (grant 639.043.513) and Spinoza Prize SPI 78-409; the YCAA Prize
Postdoctoral Fellowship. LM gratefully acknowledges support from an NSF
Astronomy and Astrophysics Postdoctoral Fellowship under award no.
AST-1903847. TK is supported by MEXT as "Program for Promoting Researches on
the Supercomputer Fugaku" (Toward a unified view of the universe: from large
scale structures to planets, JPMXP1020200109) and JICFuS. RPD and IN
acknowledge funding by the South African Research Chairs Initiative, through
the South African Radio Astronomy Observatory (SARAO, grant ID 77948), which
is a facility of the National Research Foundation (NRF), an agency of the
Department of Science and Innovation (DSI) of South Africa. We thank the
Onsala Space Observatory (OSO) national infrastructure, for the provisioning
of its facilities/observational support (OSO receives funding through the
Swedish Research Council under grant 2017-00648); the Perimeter Institute for
Theoretical Physics (research at Perimeter Institute is supported by the
Government of Canada through the Department of Innovation, Science and
Economic Development and by the Province of Ontario through the Ministry of
Research, Innovation and Science); the Spanish Ministerio de Ciencia e
Innovación (grants PGC2018-098915-B-C21, AYA2016-80889-P,
PID2019-108995GB-C21, PID2020-117404GB-C21); the University of Pretoria for
financial aid in the provision of the new Cluster Server nodes and SuperMicro
(USA) for a SEEDING GRANT approved towards these nodes in 2020; the State
Agency for Research of the Spanish MCIU through the “Center of Excellence
Severo Ochoa” award for the Instituto de Astrofísica de Andalucía (SEV-2017-
0709); the Toray Science Foundation; the Consejería de Economía, Conocimiento,
Empresas y Universidad of the Junta de Andalucía (grant P18-FR-1769), the
Consejo Superior de Investigaciones Científicas (grant 2019AEP112); the
M2FINDERS project which has received funding by the European Research Council
(ERC) under the European Union’s Horizon 2020 Research and Innovation
Programme (grant agreement No 101018682); the US Department of Energy (USDOE)
through the Los Alamos National Laboratory (operated by Triad National
Security, LLC, for the National Nuclear Security Administration of the USDOE
(Contract 89233218CNA000001); the European Union’s Horizon 2020 research and
innovation programme under grant agreement No 730562 RadioNet; Shanghai Pilot
Program for Basic Research, Chinese Academy of Science, Shanghai Branch (JCYJ-
SHFY-2021-013); ALMA North America Development Fund; the Academia Sinica;
Chandra DD7-18089X and TM6- 17006X; the GenT Program (Generalitat Valenciana)
Project CIDEGENT/2018/021. This work used the Extreme Science and Engineering
Discovery Environment (XSEDE), supported by NSF grant ACI-1548562, and
CyVerse, supported by NSF grants DBI-0735191, DBI-1265383, and DBI-1743442.
XSEDE Stampede2 resource at TACC was allocated through TG-AST170024 and TG-
AST080026N. XSEDE JetStream resource at PTI and TACC was allocated through
AST170028. The simulations were performed in part on the SuperMUC cluster at
the LRZ in Garching, on the LOEWE cluster in CSC in Frankfurt, and on the
HazelHen cluster at the HLRS in Stuttgart. This research was enabled in part
by support provided by Compute Ontario (http://computeontario.ca), Calcul
Quebec (http://www.calculquebec.ca) and Compute Canada
(http://www.computecanada.ca). CC acknowledges support from the Swedish
Research Council (VR). We thank the staff at the participating observatories,
correlation centers, and institutions for their enthusiastic support. This
paper makes use of the following ALMA data: ADS/JAO.ALMA#2016.1.01154.V. ALMA
is a partnership of the European Southern Observatory (ESO; Europe,
representing its member states), NSF, and National Institutes of Natural
Sciences of Japan, together with National Research Council (Canada), Ministry
of Science and Technology (MOST; Taiwan), Academia Sinica Institute of
Astronomy and Astrophysics (ASIAA; Taiwan), and Korea Astronomy and Space
Science Institute (KASI; Republic of Korea), in cooperation with the Republic
of Chile. The Joint ALMA Observatory is operated by ESO, Associated
Universities, Inc. (AUI)/NRAO, and the National Astronomical Observatory of
Japan (NAOJ). The NRAO is a facility of the NSF operated under cooperative
agreement by AUI. This research used resources of the Oak Ridge Leadership
Computing Facility at the Oak Ridge National Laboratory, which is supported by
the Office of Science of the U.S. Department of Energy under Contract No. DE-
AC05-00OR22725. We also thank the Center for Computational Astrophysics,
National Astronomical Observatory of Japan. Support for this work was also
provided by the NASA Hubble Fellowship grant HST-HF2-51431.001-A awarded by
the Space Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., for NASA, under contract
NAS5-26555. HO and GM were supported by Virtual Institute of Accretion (VIA)
postdoctoral fellowships from the Netherlands Research School for Astronomy
(NOVA). APEX is a collaboration between the Max-Planck-Institut für
Radioastronomie (Germany), ESO, and the Onsala Space Observatory (Sweden). The
SMA is a joint project between the SAO and ASIAA and is funded by the
Smithsonian Institution and the Academia Sinica. The JCMT is operated by the
East Asian Observatory on behalf of the NAOJ, ASIAA, and KASI, as well as the
Ministry of Finance of China, Chinese Academy of Sciences, and the National
Key Research and Development Program (No. 2017YFA0402700) of China and Natural
Science Foundation of China grant 11873028. Additional funding support for the
JCMT is provided by the Science and Technologies Facility Council (UK) and
participating universities in the UK and Canada. Simulations were performed in
part on the SuperMUC cluster at the LRZ in Garching, on the LOEWE cluster in
CSC in Frankfurt, on the HazelHen cluster at the HLRS in Stuttgart, and on the
Pi2.0 and Siyuan Mark-I at Shanghai Jiao Tong University. The computer
resources of the Finnish IT Center for Science (CSC) and the Finnish Computing
Competence Infrastructure (FCCI) project are acknowledged. JO was supported by
Basic Science Research Program through the National Research Foundation of
Korea(NRF) funded by the Ministry of Education(NRF-2021R1A6A3A01086420;
2022R1C1C1005255). We thank Martin Shepherd for the addition of extra features
in the Difmap software that were used for the CLEAN imaging results presented
in this paper. The computing cluster of Shanghai VLBI correlator supported by
the Special Fund for Astronomy from the Ministry of Finance in China is
acknowledged. This work was supported by the Brain Pool Program through the
National Research Foundation of Korea (NRF) funded by the Ministry of Science
and ICT (019H1D3A1A01102564). This research is part of the Frontera computing
project at the Texas Advanced Computing Center through the Frontera Large-
Scale Community Partnerships allocation AST20023. Frontera is made possible by
National Science Foundation award OAC-1818253. This research was carried out
using resources provided by the Open Science Grid, which is supported by the
National Science Foundation and the U.S. Department of Energy Office of
Science. The LMT is a project operated by the Instituto Nacional de
Astrófisica, Óptica, y Electrónica (Mexico) and the University of
Massachusetts at Amherst (USA). The IRAM 30-m telescope on Pico Veleta, Spain
is operated by IRAM and supported by CNRS (Centre National de la Recherche
Scientifique, France), MPG (Max-Planck-Gesellschaft, Germany) and IGN
(Instituto Geográfico Nacional, Spain). The SMT is operated by the Arizona
Radio Observatory, a part of the Steward Observatory of the University of
Arizona, with financial support of operations from the State of Arizona and
financial support for instrumentation development from the NSF. Support for
SPT participation in the EHT is provided by the National Science Foundation
through award OPP-1852617 to the University of Chicago. Partial support is
also provided by the Kavli Institute of Cosmological Physics at the University
of Chicago. The SPT hydrogen maser was provided on loan from the GLT, courtesy
of ASIAA. Support for this work was provided by NASA through the NASA Hubble
Fellowship grant #HST–HF2–51494.001 awarded by the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Inc., for NASA, under contract NAS5–26555. Jongho Park
acknowledges financial support through the EACOA Fellowship awarded by the
East Asia Core Observatories Association, which consists of the Academia
Sinica Institute of Astronomy and Astrophysics, the National Astronomical
Observatory of Japan, Center for Astronomical Mega-Science, Chinese Academy of
Sciences, and the Korea Astronomy and Space Science Institute. The EHTC has
received generous donations of FPGA chips from Xilinx Inc., under the Xilinx
University Program. The EHTC has benefited from technology shared under open-
source license by the Collaboration for Astronomy Signal Processing and
Electronics Research (CASPER). The EHT project is grateful to T4Science and
Microsemi for their assistance with Hydrogen Masers. This research has made
use of NASA’s Astrophysics Data System. We gratefully acknowledge the support
provided by the extended staff of the ALMA, both from the inception of the
ALMA Phasing Project through the observational campaigns of 2017 and 2018. We
would like to thank A. Deller and W. Brisken for EHT-specific support with the
use of DiFX. We acknowledge the significance that Maunakea, where the SMA and
JCMT EHT stations are located, has for the indigenous Hawaiian people. EHT,
ALMA, APEX, IRAM:30m, JCMT, LMT, SMA, ARO:SMT, SPT.

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