# Shadow Implications: What does measuring the photon ring imply for gravity?

Avery E. Broderick Perimeter Institute for Theoretical Physics, 31 Caroline
Street North, Waterloo, ON, N2L 2Y5, Canada  Department of Physics and
Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, ON,
N2L 3G1, Canada  Waterloo Centre for Astrophysics, University of Waterloo,
Waterloo, ON N2L 3G1 Canada  Kiana Salehi  Perimeter Institute for Theoretical
Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada  Department
of Physics and Astronomy, University of Waterloo, 200 University Avenue West,
Waterloo, ON, N2L 3G1, Canada  Waterloo Centre for Astrophysics, University of
Waterloo, Waterloo, ON N2L 3G1 Canada  Boris Georgiev Perimeter Institute for
Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada
Department of Physics and Astronomy, University of Waterloo, 200 University
Avenue West, Waterloo, ON, N2L 3G1, Canada  Waterloo Centre for Astrophysics,
University of Waterloo, Waterloo, ON N2L 3G1 Canada

###### Abstract

With the imaging and characterization of the horizon-scale images of M87* and
Sgr A* by the Event Horizon Telescope (EHT), it has become possible to resolve
the near-horizon region of astrophysical black holes. As a result, there has
been considerable interest in the implications of the measurement of the
shadow size, i.e., the asymptotic photon ring. We explore the general
implications of such a measurement, identifying what is and, more importantly,
is not constrained by such measurements, with applications to EHT and future
instruments. We consider a general spherically symmetric metric, which
effectively applies for a polar observer (appropriate for M87*) in the slow
rotation limit. We propose a nonperturbative, nonparametric spacetime-domain
characterization of shadow size and related measurements that makes explicit
the nature and power (or lack thereof) of shadow-size-based constraints, and
facilitates comparisons among observations and targets.

Keywords

##  1 Introduction

With the first images of M87* and Sgr A*, the Event Horizon Telescope has
opened a window onto strong gravity in the vicinity of the event horizon of
known astrophysical black holes (Event Horizon Telescope Collaboration et al.,
2019a, b, c, d, e, f, 2022a, 2022b, 2022c, 2022d, 2022e, 2022f). Clearly
visible in the images of both sources is the black hole shadow, the locus of
rays that intersect the photon sphere (and therefore the event horizon). This
is a purely gravitational feature, and therefore has been the subject of
significant interest (Hilbert, 1917). Taken as a measurement of the black hole
mass, these observations represent the first time that photon dynamics (the
strong lensing of photons) was used to directly weigh a black hole, and
confines the mass of the EHT targets into the most compact regions in history.

For both sources, the size of the shadow matches that predicted by general
relativity on the basis of stellar dynamical estimates, providing an important
confirmation in the strong-gravity regime111Attempts to measure the shadow
shape are unlikely to be fruitful in the near future: M87* is viewed nearly
from along the putative spin axis (Broderick et al., 2022a), Sgr A* is
obscured by a scattering screen and the intervening putative accretion flow
(Event Horizon Telescope Collaboration et al., 2022f). Thus, we will focus on
the shadow size here. (Event Horizon Telescope Collaboration et al., 2022f,
2019f). However, attempts to quantify the implications for potential
deviations from general relativity have made use of either parameterized
deviations (Johannsen, 2013; Rezzolla & Zhidenko, 2014; Psaltis et al., 2020)
or explicit alternative metrics (Kocherlakota et al., 2021; Event Horizon
Telescope Collaboration et al., 2022f). Both of these approaches make strong
underlying assumptions that impose strong limits on the interpretation of any
results.

Parameterized metric expansions typically suffer from the inherently non-
linear nature of general relativity: near the event horizon all terms in the
typical expansions become similarly important, obscuring what is a limit and
what is an assumption regarding the class of alternative metrics (Psaltis et
al., 2020; Völkel et al., 2021). Wherein these assumptions are explicitly
avoided by construction (e.g., Rezzolla & Zhidenko, 2014), the resulting
parameter constraints are necessarily strongly correlated — a natural
consequence of a large-dimensional parameter space and a single measurement —
and therefore difficult to interpret practically. Armed with strong priors,
e.g., from gravitational wave experiments or theoretical arguments, this may
not be catastrophic. However, given the novel nature of the EHT images of M87*
and Sgr A*, and the extreme mass-scale disparity between EHT and current
gravitational wave targets, there is significant value in independent
gravitational tests.

Explicit alternative metrics (e.g., those in Kocherlakota et al., 2021)
provide a physically motivated set of strong priors on the metric deviations
by design, and in so doing avoid the arbitrariness associated with a
parameterized model. However, these constraints are only meaningful within the
context of the specific alternative metric under construction, for which the
Bayesian prior is unknown and usually assumed to be small. More importantly,
there is no guarantee that neighboring metrics, i.e., “small” deviations from
the alternative under consideration, are similarly constrained for the same
reasons that plague parametric approaches. Hence, the results from explicit
alternatives are typically only interpretable within a narrow context,
requiring the onerous reconstruction of images for every metric under
consideration.

Here we present an alternative scheme in which to characterize shadow size
measurements that restates these in terms of direct measurements of the
properties of the metric in an appropriate gauge. As such, these translate the
empirical image-domain measurements performed at infinity to a gravitational
domain. By casting the constraint as a measurement of the metric properties
directly, this scheme has two key features:

  1. 1.

It is nonparametric, thus avoiding the complications of strongly correlated
parameters while maintaining the general nature of the limit.

  2. 2.

It is nonperturbative, and therefore does not require any notion of
“smallness” and is more naturally applicable in the highly-nonlinear near-
horizon regime.

Importantly, by expressing the constraint from shadow size limits in terms of
an appropriately specified metric, these are more useful to the gravitational
community. It is no longer necessary to generate full images to compare with
the shadow size; rather the computation of the metric components within a
convenient gauge at a particular location is sufficient to bring an
alternative theory into contact with the EHT and future mm-VLBI constraints.

We will not address the observational problem of measuring the shadow size,
which is complicated by the fact that it is only the surrounding luminous
plasma that is visible. Event Horizon Telescope Collaboration et al. (2019f)
and Event Horizon Telescope Collaboration et al. (2022f) both calibrate their
shadow size measurements with simulated images generated within general
relativity or a narrow set of nearby alternatives, rendering interpretation of
the size constraints rather more complicated. More direct methods to extract
higher-order images, and therefore infer the diameter of the critical curve
that bounds the shadow have been proposed (Broderick et al., 2022a; Johnson et
al., 2020). However, the measurement particulars and their intrinsic
uncertainties are beyond the narrow scope adopted here: what would we learn
from such a measurement?

We will make the simplifying assumptions of spherical symmetry and
stationarity. Even within this restricted class of spacetimes, we are able to
elucidate which aspects of the spacetime remain unconstrained by a shadow
sizes measurement. For M87*, which is viewed from nearly the polar axis (see,
e.g., Event Horizon Telescope Collaboration et al., 2019e; Broderick et al.,
2022a, and references therein), spherical symmetry may be relaxed (see, e.g.,
Salehi & Broderick 2023, in preparation). Our conclusions regarding the
implications of shadow sizes also hold unchanged in the slow rotation limit
for such viewers.

In Section 2, we present a formalism for specifying the observed shadow size
in spherically symmetric spacetimes, describe our new scheme for
characterizing shadow size measurements, discuss the relationship to
parameterized and explicit alternative approaches, and discuss the
applicability of these results to slowly rotating spacetimes viewed from the
spin axis. We collect the implications of the EHT M87* and Sgr A* campaigns
and place these in Section 3. These are placed these into the broader context
of a variety of non-shadow constraints in Section 4. Conclusions are collected
in Section 5. Unless otherwise specified, we set $G=c=1$.

##  2 Shadows from General Spherically Symmetric Spacetimes

###  2.1 Definitions and Properties of a General Metric

We begin with laying out the formalism of generating shadow sizes outside of
the limitation of general relativity with a general spherically symmetric
metric. Expressed without loss of generality in areal coordinates, the metric
may be written as,

$ds^{2}=-N(r)^{2}dt^{2}+\frac{B(r)^{2}}{N(r)^{2}}dr^{2}+r^{2}d\Omega^{2}.$ (1)

This metric has two arbitrary real functions of radius, $N(r)$ and $B(r)$,
that set the $tt$ and $rr$ components of the metric. We assume asymptotic
flatness, i.e., $\lim_{r\rightarrow\infty}N(r)=1-{\mathcal{O}}(1/r)$ and
$\lim_{r\rightarrow\infty}B(r)=1$. By construction, we have enforced a metric
signature of $(-+++)$ throughout the region of the spacetime accessible to
external observers; without loss of generality, we will further assume
$N(r)>0$ and $B(r)>0$ everywhere in this region.

This spacetime admits two killing vector fields, and thus two constants of the
motion for null geodesics, which we choose to be the energy and angular
momentum,

$e=p_{t}=-N^{2}\frac{dt}{d\tau}~{}~{}\text{and}~{}~{}\ell=p_{\phi}=r^{2}\frac{d\phi}{d\tau},$
(2)

and due to the spacetime symmetry, all null geodesics are integrable, with the
standard properties (e.g., the redshift $1+z=N^{-1}(r)$, etc.).222Henceforth,
we will set $e=1$ and $b=\ell/e$, which due to the assumption of asymptotic
flatness is the impact parameter at infinity. All black hole spacetimes of
this form, by which we mean spacetimes with an event horizon, have
$N^{2}(r_{h})=0$ for some horizon radius $r_{h}$ by definition. Additionally,
all such spacetimes also contain an unstable circular photon orbit (hereafter,
abbreviated to simply “photon orbit”), i.e., a radius at which photons execute
a circular orbit about the black hole, located at,

$r_{\gamma}=\frac{N(r_{\gamma})}{N^{\prime}(r_{\gamma})},$ (3)

where $N^{\prime}(r)$ is the radial derivative of $N(r)$. That an
$r_{\gamma}\geq r_{h}$ exists that satisfies this condition follows from the
assumption of asymptotic flatness and the existence of an event horizon at
some finite $r_{h}$ (see Appendix C).

###  2.2 Shadows in Perturbed Spacetimes

For the innermost shadow, all null geodesics that are outward propagating at
$r_{\gamma}$ will have begun on the horizon (see Appendix C). Thus, the
boundary of the black hole shadow is associated with those null geodesics that
are tangent to the photon orbit at $r_{\gamma}$. This occurs when the photon
angular momentum is equal to a critical value,
$b_{\gamma}=r_{\gamma}/N(r_{\gamma})$. Identifying $b_{\gamma}$ with the
impact parameter at infinity and using the definition of $r_{\gamma}$, the
shadow radius is

$R=\frac{M}{N^{\prime}(r_{\gamma})}.$ (4)

As shown in Salehi & Broderick (2023, in preparation) and Appendix A, this
continues to hold unchanged for polar observers of slowly spinning black holes
(i.e., up to order $a$, where $a$ is the dimensionless black hole spin). As a
result, there is a simple, one-to-one relationship between the observed shadow
size and a property of the metric at a specific location. From this simple
result a number of profound conclusions immediately follow.

###  2.3 Characterizing Shadow Size Measurements

First, because $R$ depends solely on $N^{\prime}(r_{\gamma})$,
$N^{\prime}(r_{\gamma})$ is a convenient way in which to characterize the
constraints imposed by a shadow size measurement. While this may appear to be
a trivial redefinition given Equation 4, the interpretation of
$N^{\prime}(r_{\gamma})$ is fundamentally gravitational: it is a direct
measurement of spacetime geometry at a dynamically important location for all
massless fields, and thus for all electromagnetic and gravitational wave
observations.

The fact that shadow-size measurements constrain $N^{\prime}(r_{\gamma})$, and
not $N(r_{\gamma})$, a point upon which we expound below, suggests a natural
framework within which to begin describing near-horizon phenomena generally:

$N(r_{\gamma}),~{}N^{\prime}(r_{\gamma}),~{}N^{\prime\prime}(r_{\gamma}),~{}\dots$
(5)

from which the near-photon-orbit behavior of $N(r)$ can be constructed via
Taylor series. Note that because $R$ depends solely upon
$N^{\prime}(r_{\gamma})$, for shadow size measurements this framework is
nonperturbative and nonparametric – measurements of shadow size may be
translated into measurements of a (derivative of a) metric coefficient,
without making any assumptions about the size and form that any difference
from GR might take.

However, characterizing strong gravity probes in this way is conceptually
complicated by the unknown value of $r_{\gamma}$, i.e., the constrained
quantity is the value of $N^{\prime}(r)$ at the photon orbit, wherever that
may be for a particular spacetime. While at first this may appear an onerous
restriction, it is natural in that the location of the photon orbit is gauge
invariant (even if the value of $r_{\gamma}$ may not be). Explicit examples of
applying this constraint will be provided in Section 3.

###  2.4 $\psi$-$\psi^{\prime}$ Representation

An alternative, possibly more familiar representation is in terms of
deviations from Schwarzschild, i.e., setting

$N^{2}(r)=1-\frac{2M}{r}+2\psi(r)$ (6)

where in the linearized regime $\psi(r)$ would be the perturbation to the
gravitational potential. When $\psi(r_{\gamma})$ is small, $N(r_{\gamma})$ and
$N^{\prime}(r_{\gamma})$ are simple rearrangements of terms. However, absent
such a guarantee, we have

$\displaystyle N(r_{\gamma})$
$\displaystyle=\sqrt{1-\frac{2M}{r_{\gamma}}+2\psi(r_{\gamma})}$ (7)
$\displaystyle N^{\prime}(r_{\gamma})$
$\displaystyle=\left[\frac{M}{r_{\gamma}^{2}}+\psi^{\prime}(r_{\gamma})\right]\frac{1}{N(r_{\gamma})},$

with which constraints on $R$ can be converted into a joint constraint on
$\psi(r_{\gamma})$ and $\psi^{\prime}(r_{\gamma})$:

$\displaystyle\psi(r_{\gamma})$
$\displaystyle=\frac{M}{r_{\gamma}}+\frac{r_{\gamma}^{2}}{2R^{2}}-\frac{1}{2}$
(8) $\displaystyle\psi^{\prime}(r_{\gamma})$
$\displaystyle=\frac{r_{\gamma}}{R^{2}}-\frac{M}{r_{\gamma}^{2}}.$

Despite appearing more complicated on its face, expressing $N(r)$ in terms of
$\psi(r)$ does present one simplification that we will make use of in what
follows: if the values of two perturbing potentials and their derivatives
match at the $r_{\gamma}$ for one of them, then $r_{\gamma}$ is a photon orbit
for both and the corresponding shadow sizes are identical. That is, consider
two perturbing potentials, $\psi_{1}(r)$ and $\psi_{2}(r)$. For $\psi_{1}(r)$,
let $r_{\gamma}$ be the radius of a photon orbit. Then, if

$\displaystyle\psi_{2}(r_{\gamma})=\psi_{1}(r_{\gamma})~{}~{}\text{and}~{}~{}\psi^{\prime}_{2}(r_{\gamma})=\psi^{\prime}_{1}(r_{\gamma}),$
(9)

$r_{\gamma}$ will be a photon orbit in the spacetime defined by $\psi_{2}(r)$.
This follows immediately from the equality of the corresponding
$N(r_{\gamma})$ and $N^{\prime}(r_{\gamma})$ from Equation 7, and the use of
Equation 3. Moreover, because the shadow size is set by
$N^{\prime}(r_{\gamma})$, this is sufficient to guarantee that $R$ is the same
for both spacetimes (see Equation 4).

As an important specific case, if $\psi(3M)=0$ and $\psi^{\prime}(3M)=0$, then
the photon orbit radius and shadow size are identical to those for
Schwarzschild, almost333We do further assume that there is not an event
horizon or second photon orbit at $r>3M$. These are weak assumptions and do
not impact the point being made: a large class spacetimes will have shadow
sizes that exactly match those from Schwarzschild. An explicit example of such
a spacetime is provided in Equation 11. regardless of the form of $\psi(r)$
for $r\neq 3M$.

###  2.5 Caveats for Known Metric Expansions

As is immediately evident from Equation 4, there is no constraint on
$N(r_{\gamma})$ given a shadow size measurement. This simple fact has profound
consequences for shadow size interpretations: it does not follow that because
$R$ matches its general relativistic value that $\psi(r)$ must be small. This,
however, does not mean that shadow size measurements are not constraining; a
tight constraint on $R$ does indeed translate into a tight constraint on
$N^{\prime}(r_{\gamma})$. Moreover, the detection of any shadow is a
qualitative result that implies $N^{\prime}(r_{\gamma})>0$, eliminating all
metrics for which $N(r)$ is decreasing at $r_{\gamma}$. These subtleties are
often lost in the context of metric expansions, for which $N(r)$ and
$N^{\prime}(r)$ become correlated by construction. We review some examples
here.

####  2.5.1 Post-Newtonian Expansion

The post-Newtonian (PN) formalism employed in (Psaltis et al., 2020) and in a
restricted manner in Event Horizon Telescope Collaboration et al. (2022f)
presents an expansion of $\psi(r)$ of the form:

$\psi(r)=\frac{\kappa_{1}}{r^{2}}-\frac{\kappa_{2}}{r^{3}}+\frac{\kappa_{3}}{r^{4}}-\frac{\kappa_{4}}{r^{5}}\dots$
(10)

where the post-Newtonian coefficients, $\kappa_{i}$, can be related at large
$r$ to terms in the often used parameterized post-Newtonian formalism. In this
way, it is hoped that measurements of $R$ can be related to a broad range of
collected tests of general relativity on scales ranging from the laboratory to
the cosmos (Will, 2014; Baker et al., 2015)

Figure 1: Comparison of allowed range of $N(r_{\gamma})$,
$N^{\prime}(r_{\gamma})$ for a shadow size measurement with
$\sigma_{R}/R=10\%$ (gray band) and the curves by single-parameter
modifications to the metric. Explicitly, those associated with various PN
orders, JP metric orders, MGBK metric orders for $\gamma_{1}$ and
$\gamma_{4}$, and RZ metric with $a_{0}$ and $a_{1}$ varied, holding all other
metric expansion parameters fixed at zero (the latter are degenerate with
those the MGBK $\gamma_{1,2}$ and $\gamma_{1,3}$ curves, and are otherwise
hidden). Thin lines indicate values for which an event horizon does not exist
(affecting only the PN expansion examples); the small open and filled points
at which the curves terminate show when the photon orbit becomes infinite and
degenerate with the event horizon, respectively. The large black point
indicates the values corresponding to general relativity.

As illustrated in Figure 1, a perturbation defined by any single PN term
traces out a curve in the $N(r_{\gamma})$-$N^{\prime}(r_{\gamma})$ plane
traversed by the associated PN coefficient. Wherein this curve lies within the
band of allowed $N^{\prime}(r_{\gamma})$ given a measurement of $R$, the
values of the associated $\kappa_{i}$ are permissible, appearing to place a
constraint on the magnitude of the $\psi(r_{\gamma})$.

However, the addition of even a second PN term results in a band that covers
the entire physically-relevant quadrant of the
$N(r_{\gamma})$-$N^{\prime}(r_{\gamma})$ plane. Thus, it appears that even
with only two PN terms, no constraints are possible. This is, of course, not
true: the two PN coefficients are strongly correlated, and it is within the
context of that correlation, indicated rather more simply in Figure 1 by the
gray band, that the shadow size constraint is present.

Moreover, the measurement of a shadow size does not exclude large deviations
beyond $r_{\gamma}$. For example, consider the perturbation,

$\psi(r)=\kappa_{2}\frac{(3M-r)^{2}}{r^{5}},$ (11)

which is dominated by the 2PN term at $r\gg 3M$ with 2PN coefficient
$\kappa_{2}$. This explicitly satisfies the conditions that $\psi(3M)=0$ and
$\psi^{\prime}(3M)=0$, and thus has $r_{\gamma}=3M$ and $R=\sqrt{27}M$,
identical to those from Schwarzschild. This is true for any value of
$\kappa_{2}$. In this sense, there is no meaningful limit on $\kappa_{2}$ from
any measurement of the dynamics of massless fields near the photon orbit
without additional, typically strong, assumptions about the spacetime
geometry.

####  2.5.2 Other Metric Expansions

For completeness, in Figure 1, we also show the paths traced out by the
various other metric expansions considered in Section 5.1 of Event Horizon
Telescope Collaboration et al. (2022f). These expansions include the
spherically-symmetric restriction on the metrics proposed by Johannsen (2013,
hereafter JP), MGBK Vigeland et al. (2011, hereafter MGBK), and Rezzolla &
Zhidenko (2014, hereafter RZ), as described in Event Horizon Telescope
Collaboration et al. (2022f). Each exhibits a similar qualitative behavior to
the post-Newtonian expansions: the range of the inferred limits on
$N(r_{\gamma})$ are solely due to the priors imposed by the underlying
expansion themselves. Quantitative differences are present, further
highlighting the impact of these priors.

###  2.6 Comparison to Explicit Alternatives

Alternative metrics to Schwarzschild, e.g., Reissner-Nordström and those
associated with alternative gravity theories, present a similar story as those
associated with metric expansions. The details of the metric perturbation
appear to induce a limit on $N(r_{\gamma})$ through model-induced correlations
between $N(r_{\gamma})$ and $N^{\prime}(r_{\gamma})$. However, alternative
metrics differ in an important conceptual way: the correlations are a
consequence of the physical prior that the metric of interest applies and is
not an arbitrary truncation of an otherwise infinite series of terms. In this
sense, the constraints are meaningful within the narrow context of the
alternative metric.

Figure 2: Comparison of allowed range of $N(r_{\gamma})$,
$N^{\prime}(r_{\gamma})$ for a shadow size measurement with
$\sigma_{R}/R=10\%$ (gray band) and the curves associated with the various
spherically symmetric alternative spacetimes considered in Kocherlakota &
Rezzolla (2020). The small open points at which the curves terminate show when
the charge reaches an intrinsic limit, e.g., the solution no longer has an
event horizon. The large black point indicates the values corresponding to
general relativity.

In Figure 2 we show the regions of the
$N(r_{\gamma})$-$N^{\prime}(r_{\gamma})$ plane spanned by a sample of
alternative black hole metrics. Following Event Horizon Telescope
Collaboration et al. (2022f), we focus attention on a subset of representative
spherically-symmetric alternatives, though we expand this list to the twelve
listed in Table 1 of Kocherlakota & Rezzolla (2020). We do not make any
representation that these twelve are complete, but rather only that they are
illustrative.444Where a metric depends on more than one parameter, we evaluate
it at a fixed value of all but one. We refer the reader Event Horizon
Telescope Collaboration et al. (2022f) and Kocherlakota & Rezzolla (2020) for
details on the metrics themselves and their underlying assumptions.

Two things are immediately evident upon comparison with the parameterized
metric expansions. First, the imposition of physical constraints on the metric
itself typically limits the region in the
$N(r_{\gamma})$-$N^{\prime}(r_{\gamma})$ plane spanned by alternative metrics
significantly. In this sense, the parameterized metric expansions are more
agnostic, covering a wider variety of potential deviations from general
relativity. However, this is also a consequence in the difference in
interpretation: where the metric expansions need to be sensible only locally,
alternative spacetimes must be globally well-behaved.

Second, the general direction in the $N(r_{\gamma})$-$N^{\prime}(r_{\gamma})$
spanned by the alternative metrics differs from those for any single-parameter
exploration in the metric parameterizations in Figure 1. Of course, upon
permitting more than one parameter to vary in the metric expansions it is
possible to mimic the alternative spacetimes (e.g., see Section IV of
Kocherlakota & Rezzolla, 2020). However, this illustrates the difficulties
faced by single-parameter characterizations of the shadow-size constraints.

##  3 Implications of known shadow sizes

We now review the implications of measurements of the shadow sizes arising
from EHT observations of M87* and Sgr A*. While some uncertainty regarding the
methodology of such measurements may persist, we take these at face value
here, and assess the implications for deviations from gravity using the
$N(r_{\gamma})$-$N^{\prime}(r_{\gamma})$ formalism and more traditional metric
expansions and alternatives.

Figure 3: Representative constraints implied by the shadow size constraints
for M87* (left) and Sgr A* (right) published by the EHT. Illustrative curves
associated with the metric expansions in Figure 1 (blue) and alternative
spacetimes in Figure 2 (orange) are shown. See Figure 1 for a description of
line types. In both panels the relevant EHT measurements at $1\sigma$ and
$2\sigma$ are shown by the green bands. Table 1: Direct and Apparent
Implications of EHT Shadow Sizes for Various Metrics

|  Sole Varied Parameter | M87* 2017 |  | Sgr A* 2017
---|---|---|---|---
Basis/Metric | Paper VI | $\theta_{n=1}$ |  | Paper VI | Paper IV
This paper | $\sqrt{3}N(r_{\gamma})$ |  $\cdots$ |  $\cdots$ |  |  $\cdots$ |  $\cdots$
$\sqrt{27}N^{\prime}(r_{\gamma})$ | $0.95_{-0.12}^{+0.22}$ | $0.89_{-0.07}^{+0.16}$ |  | $1.04_{-0.10}^{+0.12}$ | $1.02_{-0.23}^{+0.17}$
PN | $\kappa_{2}$ | $[-2.7,0.7]$ | $[-3.2,0.7]$ |  | $[-0.5,0.7]$ | $[-4.2,0.7]$
$\kappa_{3}$ | $[-10.9,1.1]$ | $[-13.0,1.0]$ |  | $[-1.7,1.2]$ | $[-17.7,1.0]$
$\kappa_{4}$ | $[-44.1,1.8]$ | $[-54.1,1.8]$ |  | $[-5.5,2.2]$ | $[-77.6,1.8]$
JP | $\alpha_{12}$ | $[-1.2,2.0]$ | $[-0.4,2.2]$ |  | $[-1.1,0.5]$ | $[-1.3,2.8]$
$\alpha_{13}$ | $[-3.2,7.0]$ | $[-1.1,8.0]$ |  | $[-3.1,1.5]$ | $[-3.5,10.2]$
$\alpha_{14}$ | $[-8.2,26.0]$ | $[-3.2,30.5]$ |  | $[-7.8,4.6]$ | $[-8.7,40.5]$
MGBK | $\gamma_{1,2}$ | $[-3.2,3.2]$ | $[-0.8,3.5]$ |  | $[-2.9,0.9]$ | $[-3.6,4.2]$
$\gamma_{1,3}$ | $[-8.6,11.5]$ | $[-2.5,13.1]$ |  | $[-8.0,2.7]$ | $[-9.8,16.2]$
$\gamma_{4,2}$ | $[-5.2,4.0]$ | $[-1.3,4.4]$ |  | $[-4.8,1.3]$ | $[-5.9,5.0]$
$\gamma_{4,3}$ | $[-15.2,13.0]$ | $[-3.9,14.3]$ |  | $[-14.0,3.8]$ | $[-17.5,16.8]$
RZ | $a_{0}$ | $[-0.8,0.8]$ | $[-0.9,0.2]$ |  | $[-0.2,0.7]$ | $[-1.0,0.9]$
$a_{1}$ | $[-1.4,1.1]$ | $[-1.6,0.3]$ |  | $[-0.3,1.0]$ | $[-2.0,1.2]$
Reissner-Nordström | $0<\bar{q}\leq 1$ | $<0.86$ | $<0.51$ |  | $<0.84$ | $<0.90$
Bardeen | $\bar{q}_{m}\leq\sqrt{16/27}$ |  $\cdots$ | $<0.50$ |  |  $\cdots$ |  $\cdots$
Hayward | $\bar{l}\leq 1.06$ |  $\cdots$ | $<1.01$ |  |  $\cdots$ |  $\cdots$
Frolov ($\bar{l}=0.4$) | $\bar{q}\leq 0.79$ |  $\cdots$ | $<0.42$ |  | $<0.77$ |  $\cdots$
Kazakov-Solodhukin | $\bar{a}$ | $<1.65$ | $<1.75$ |  | $<0.79$ | $<1.95$
EMd-1 | $\bar{q}\leq\sqrt{2}$ | $<0.90$ | $<0.51$ |  | $<0.87$ | $<0.94$
E ae-1 | $0\leq\bar{c}_{13}<1$ | $<0.93$ | $<0.94$ |  | $<0.66$ | $<0.95$
E ae-2 ($\bar{c}_{13}=0.25$) | $0\leq\bar{c}_{14}\leq 2\bar{c}_{13}$ | $<0.33$ | $<0.16$ |  | $<0.31$ | $<0.36$
CFM A | $\beta<1$ |  $\cdots$ |  $\cdots$ |  |  $\cdots$ |  $\cdots$
CFM B | $1<\beta<5/4$ |  $\cdots$ |  $\cdots$ |  |  $\cdots$ |  $\cdots$
EEH ($\bar{\alpha}=0.25$) | $0<\bar{q}_{m}$ | $<0.86$ | $<0.51$ |  | $<0.84$ | $<0.90$
Bronnikov | $0\leq\bar{c}_{13}<1$ | $<0.86$ | $<0.51$ |  | $<0.84$ | $<0.90$

Note. — For all metrics, only the parameter listed is varied. Varying multiple
parameters typically results in no parameter constraint. Missing entries
correspond to no constraints. See Section 3.1 for how the various shadow size
estimates are produced.

###  3.1 Shadow Size Estimates

We consider four EHT shadow size measurements arising from the 2017 observing
campaign, two each for M87* and Sgr A*, differing in the particulars in how
they are produced. All involve two underlying measurements, that of the shadow
size with the EHT and a comparison mass measurement.

####  3.1.1 2017 M87* Paper VI

First, for M87*, we adopt the angular size of the gravitational radius, i.e.,
$\theta_{g}=GM/c^{2}D$ where $M$ and $D$ are the mass of and distance to M87*,
reported in Event Horizon Telescope Collaboration et al. (2019f) of $3.8\pm
0.4\,\mu{\rm as}$. Stellar dynamics measurements by Gebhardt et al. (2011)
produce a corresponding estimate of $\theta_{\rm
dyn}=3.62^{+0.60}_{-0.34}\,\mu{\rm as}$ (Event Horizon Telescope Collaboration
et al., 2019f). Taking the latter to define $M$, the former gives an estimated
Shadow radius of

$R/M=\sqrt{27}\frac{\theta_{g}}{\theta_{\rm
dyn}}=\sqrt{27}\left(1.05_{-0.20}^{+0.15}\right).$ (12)

This estimate presumes that the shadow size is indeed related to the EHT-
measured size by the canonical $\sqrt{27}$, which may differ due to spin
and/or the assumed astrophysics of the emitting region (Gralla et al., 2019;
Blandford & Globus, 2022). The latter concern is amplified by the fact that
the mass measurement presented in Event Horizon Telescope Collaboration et al.
(2019f) calibrates the relationship between the bright ring and black hole
mass using simulations that assume general relativity. Nevertheless, it forms
the basis for the general relativity tests reported in (Event Horizon
Telescope Collaboration et al., 2019f) and Psaltis et al. (2020), and thus we
include it here. We refer to this shadow size measurement as the M87* 2017
Paper VI estimate in Table 1.

####  3.1.2 2017 M87* $\theta_{n=1}$

Second, again for M87*, we use the size of the $n=1$ photon ring
$\theta_{n=1}=21.74\pm 0.10\,\mu{\rm as}$, generated via the secondary image
of the emission region, reported in Broderick et al. (2022b). This is
calibrated using numerical simulations to the shadow size, which finds a shift
between the two of $\Delta\theta=0.56\pm 0.32\,\mu{\rm as}$, resulting in a
spin-zero estimate of the shadow radius of $21.09\pm 0.33\,\mu{\rm as}$. Again
using the stellar dynamics estimate of $M$, this gives a shadow radius of

$R/M=\sqrt{27}\left(1.12_{-0.17}^{+0.10}\right).$ (13)

This estimate is less dependent on the underlying astrophysics, though does
rely upon the identification of the ring-like structure with the $n=1$ photon
ring. It is, however, again reliant upon the numerical simulations used to
relate the $n=1$ photon ring to the edge of the shadow. We refer to this
shadow size measurement as the M87* 2017 $\theta_{n=1}$ estimate in Table 1.

####  3.1.3 2017 Sgr A* Paper VI

Third, for Sgr A*, we make use of the estimate reported in Event Horizon
Telescope Collaboration et al. (2022f) via the $\delta$:

$R/M=\sqrt{27}\left(1+\delta\right)=\sqrt{27}\left(0.96\pm 0.10\right).$ (14)

This estimate depends on the specific calibration procedure employed in Event
Horizon Telescope Collaboration et al. (2022f), which differs from that used
for M87* in some important respects: the calibration is performed directly to
the shadow size, included in the calibration library are a handful of non-Kerr
spacetimes. However, it also makes astrophysical assumptions regarding the
emission region, and optimistic assumptions regarding the variability. As a
result, we consider this to be an optimistic constraint on the shadow size. We
refer to this shadow size measurement as the Sgr A* 2017 Paper VI estimate in
Table 1.

####  3.1.4 2017 Sgr A* Paper IV

Finally, for Sgr A*, we employ a similar procedure as that employed by M87*
based upon the mass estimate in Event Horizon Telescope Collaboration et al.
(2022d), $\theta_{g}=4.8_{-0.7}^{+1.4}\,\mu{\rm as}$, and the stellar dynamics
measurements by Do et al. (2019) and GRAVITY Collaboration et al. (2022) as
collated in (Event Horizon Telescope Collaboration et al., 2022f),
$\theta_{\rm dyn}=4.92\pm 0.3\,\mu{\rm as}$. The resulting estimate for the
shadow size is

$R/M=\sqrt{27}\frac{\theta_{g}}{\theta_{\rm
dyn}}=\sqrt{27}\left(0.98_{-0.14}^{+0.28}\right).$ (15)

This is significantly more conservative than the estimate from (Event Horizon
Telescope Collaboration et al., 2022f), with roughly twice the uncertainty. We
refer to this shadow size measurement as the Sgr A* 2017 Paper IV estimate in
Table 1.

###  3.2 Metric Expansions

The left panels of Figure 3 shows the implications for parameterized metric
perturbations of the 2017 M87* Paper VI and 2017 Sgr A* Paper IV shadow size
estimates. As in Figure 1, each metric parameterization imposes a strong prior
within the $N(r_{\gamma})$-$N^{\prime}(r_{\gamma})$ plane, inducing an
apparent constraint on $N(r_{\gamma})$ given the shadow measurement’s direct
constraint on $N^{\prime}(r_{\gamma})$.

The $1\sigma$ single-parameter limits, when all other perturbations are fixed
at zero, are collected in Table 1 for all four shadow size estimates. These
ranges are obtained simply by inspecting the range of parameter values for
which the curves in Figure 3 remain in the $1\sigma$ bands. Note in particular
that the limits from the 2017 Sgr A* Paper VI shadow size estimates
quantitatively match those in Event Horizon Telescope Collaboration et al.
(2022f).555A solitary exception is the constraint on $\gamma_{1,2}$ of the
MGBK expansion. However, the values in Table 1 quantitatively match those
inferred from Fig. 17 of Event Horizon Telescope Collaboration et al. (2022f),
and thus we attribute the mismatch to a typographical error in Table 3 of
Event Horizon Telescope Collaboration et al. (2022f). However, all of these
limits should be interpreted with significant care for the reasons described
in Section 2.5. In particular, the constraint on the magnitude of the
perturbation, at $r_{\gamma}$ or otherwise, is illusory.

###  3.3 Alternative Spacetimes

The implications for the alternative metrics in Kocherlakota & Rezzolla (2020)
are shown in the right panes of Figure 3 for the 2017 M87* Paper VI and 2017
Sgr A* Paper IV shadow size estimates. The $1\sigma$ limits on the alternative
metrics’ charges are collected in Table 1 for all four shadow size estimates.
Where appropriate, these agree with those reported in Event Horizon Telescope
Collaboration et al. (2022f) and Kocherlakota et al. (2021).

Generally, the global constraints on the alternative metrics, e.g., the
necessity of an event horizon, limits the range of $N^{\prime}(r_{\gamma})$
permitted substantially. While not universal, this does impose a typical scale
on the shadow size measurements that will be informative, roughly requiring
measurement precisions of a few percent.

As with the parameterized metric expansions, limits on $N(r_{\gamma})$ are
inferred from the form of the particular alternative metric under
consideration. Therefore, further empirical progress requires additional
observables that constrain quantities other than $N^{\prime}(r_{\gamma})$.

##  4 Beyond Shadow Sizes

A variety of astrophysical probes of the strong-gravity regime have either
already become possible, or will be possible in next decade. These are
frequently relevant for ostensibly stationary spacetimes, e.g., presumably
Kerr black holes, and we will consider that case. Here, we explore how they
may also be incorporated into measurements of $N(r_{\gamma})$,
$N^{\prime}(r_{\gamma})$, etc., and what additional underlying assumptions may
be necessary.

###  4.1 Light Echos

Echoes associated with time delays between the direct emission and higher
order images have been proposed as an alternate probe of the spacetime
(Moriyama et al., 2019; Hadar et al., 2021). These time delays are a result of
the additional path length of the photon trajectories associated with the
higher order images as the photons orbit about the black hole prior to
streaming toward the observer at infinity. Thus, the typical timescale is the
orbital time at the photon orbit, as measured by a distant observer.

Naively, one might imagine that this period is $2\pi r_{\gamma}$, and thus a
direct measure of the radius of the photon orbit. However, due to the
gravitational redshift, in practice the measurable orbital period is

$P=\frac{2\pi r_{\gamma}^{2}}{b_{\gamma}N^{2}(r_{\gamma})}=2\pi R.$ (16)

Hence, light echos and other orbital phenomena at the photon orbit provide
degenerate information to that contained within the measurement of the shadow
size.

###  4.2 Doppler Effects

The orbiting material responsible for the emission will be Doppler beamed and
shifted due to the bulk motions. This effect is responsible for the typically
asymmetric images appearing in simulated accretion flows and observed in M87*.
While the flux is dependent on the density and magnetization of the emitting
material, under the assumption of an approximately axisymmetric emission
region (at least on average), the ratio of fluxes on the approaching and
receding sides will depend solely on the dynamics of the orbiting plasma and
spectral index of the synchrotron emission.

Using the Lorentz invariance of $I_{\nu}/\nu^{3}$, where $I_{\nu}$ is the
observed intensity at a location on the sky at frequency $\nu$, the flux ratio
is,

${\mathcal{F}}=\left(\frac{\nu_{r}}{\nu_{a}}\right)^{\alpha+3},$ (17)

where we have assumed the emitted intensity is $\propto\nu^{-\alpha}$, and
$\nu_{a}$ and $\nu_{r}$ are the photon frequencies in the frame of the
emitting plasma that is approaching or receding, respectively. These
frequencies are related to the photon four-vector, $k^{\mu}$, and plasma four-
velocity, $u^{\mu}$, at the point of emission (which we will assume is located
at $r_{\gamma}$) for a given location on the observing screen ($R$), and are
therefore sensitive to astrophysical assumptions made about the accretion flow
and emission mechanism.

For concreteness, if we presume that the emission arises from a thin, modestly
sub-Keplerian accretion disk, i.e., the disk angular velocity as measured by a
distant observer is

$\Omega^{2}=\kappa^{2}\frac{N(r)N^{\prime}(r)}{r},$ (18)

which at the photon orbit reduces to $\kappa^{2}/R^{2}$, where the sub-
Keplerian factor is $\kappa\sim 0.9$ in GRMHD simulations (Narayan et al.,
2012; Porth et al., 2019). Near the edge of the shadow, for an observing
inclination of $i$ the observed frequencies for the approaching and receding
sides are

$\nu_{a,r}=\nu_{\rm obs}u^{t}\left(1\mp\kappa\sin i\right)$ (19)

which, up to the multiplicative factor $\nu_{\rm obs}u^{t}$, is independent of
$N(r_{\gamma})$ and $N^{\prime}(r_{\gamma})$. As a consequence, under the
above astrophysical assumptions, measurements of $\mathcal{F}$ does not
provide an additional constraint on the underlying spacetime.666Of course, it
does provide insight into $\kappa$ and $i$, both of which are of considerable
astrophysical interest!

A substantially different astrophysical picture of the near-horizon emission
region may result in different spacetime dependencies. However, in that case,
the gravitational measurements are fundamentally subject to the currently
large astrophysical uncertainties. While not insurmountable with the inclusion
of additional observations (e.g., multi-wavelength views, variability studies,
spectral energy distributions, etc.) a full discussion of how to perform joint
astrophysical/gravitational analyses is beyond the scope of this paper.

###  4.3 Redshift Measurements

The gravitational redshift from emission near the photon orbit is dependent
solely on $N(r_{\gamma})$:

$1+z=1/N(r_{\gamma}).$ (20)

As with the approach/receding flux ratio discussed above, this is critically
dependent upon the dynamical state of the emission region, and thus mixes the
astrophysical and gravitational uncertainties.

Relativistically broadened iron fluorescence lines (Fe K$\alpha$) have been
detected in a number of quasars (de La Calle Pérez et al., 2010; Tanaka et
al., 1995). These X-ray lines are the combined emission from a number of
radii, and are impacted by the X-ray corona, the rapid orbital motions, and
the gravitational redshift, which generates a characteristic broad “red wing”
that extends to lower X-ray energies. Typically, this emission is assumed to
arise outside of the innermost stable circular orbit (ISCO; Reynolds &
Begelman, 1997), which for Schwarzschild lies well beyond $r_{\gamma}$. We
discuss the uncertain relationship between the ISCO and $r_{\gamma}$ in
Section 4.5.2, and here note only that this makes direct comparisons with the
redshifts from these systems and $N(r_{\gamma})$ challenging.

For M87* and Sgr A*, the redshift has a further problem: the need for spectral
features from which to estimate a redshift. The observed spectral energy
distributions of M87* and Sgr A* exhibit no features beyond the broad-band
optically thick to thin transition near EHT wavelengths. The nearly virialized
accretion flows believed to be present in these sources is expected to reach
temperatures above $10^{10}\,{\rm K}$ near the horizon. At these temperatures,
were the accreting gas in local thermodynamic equilibrium (LTE), all atomic
species are photoionized, and most nuclei are photodissociated. However, the
low densities within the accretion flows, which are that are directly
responsible for their radiative inefficiency, also imply that the gas will
typically not be in LTE, opening the possibility that tightly bound atomic
species (e.g., Fe) may persist in the near-horizon region. In addition,
nuclear gamma-ray lines will be present. Finding and measuring either of these
would require next generation instruments.

###  4.4 Multiple Photon Rings

The shadow is bounded by an infinite sequence of higher-order images of the
accretion flow, often referred to as “photon rings”. Each higher-order photon
ring is the result of an additional half-orbit executed by the null geodesic
prior to streaming toward a distant observer. Necessarily, each ring is
distinctly located in the image (no position in the image can contribute to
more than one null geodesic, and, therefore, more than one image). These are
purely geometric features, depending only upon the strong lensing within the
spacetime, and therefore present a natural probe of general relativity
(Broderick et al., 2022a).

The ability to separate and extract individual ring-like structures within
images has recently been developed (e.g., Broderick et al., 2020, 2022b) and
future experiments may be able to resolve multiple such rings (Johnson et al.,
2020; Broderick et al., 2022a). Thus, measuring the size of multiple photon
rings presents a natural and practical extension to the notion of measuring
the shadow size, and is necessarily probing the region near the photon orbit.

In Appendix D we estimate the radii of high-order photon rings relative to
that of the shadow, finding,

$R_{n}-R\approx fe^{-\gamma n},$ (21)

where $f$ is a function of $r_{\gamma}$, $N(r_{\gamma})$,
$N^{\prime}(r_{\gamma})$, and $N^{\prime\prime}(r_{\gamma})$, and

$\gamma=\pi\frac{N^{3/2}(r_{\gamma})}{N^{\prime}(r_{\gamma})}\left[-\frac{N^{\prime\prime}(r_{\gamma})}{B^{2}(r_{\gamma})}\right]^{1/2}$
(22)

is the Lyapunov exponent that defines the self-similar ring structure. Note
that in addition to $N(r_{\gamma})$ and $N^{\prime}(r_{\gamma})$, $\gamma$
depends on $N^{\prime\prime}(r_{\gamma})$ and $B(r_{\gamma})$ through the
combination $N^{\prime\prime}(r_{\gamma})/B^{2}(r_{\gamma})$. This is a
consequence of the fact that the finite order photon ring radii are dictated
by the dynamics of photons very nearby, but outside of the photon orbit.
Because $\gamma$ also describes the rate at which the radii of these
trajectories grow, it naturally depends on the second derivative of the
effective potential, $N^{\prime\prime}(r_{\gamma})$, and the notion of radial
distance, $B(r_{\gamma})$, at $r_{\gamma}$.

The absolute normalization must ultimately be computed numerically and may
differ among spacetimes. However, the relative sizes are fully fixed by
$\gamma$, and thus it is possible to measure $\gamma$ directly with a shadow
size and two photon ring radii:

$\gamma=\ln\left(\frac{R_{n}-R}{R_{n+1}-R}\right).$ (23)

Alternatively, measuring three photon ring radii permits removing $R$
altogether,

$\gamma=\ln\left(\frac{R_{n+1}-R_{n}}{R_{n+2}-R_{n+1}}\right).$ (24)

As with the detection of a shadow, the detection of a single photon ring has
profound qualitative implications, requiring $N^{\prime\prime}(r_{\gamma})$ to
be positive.777More properly, positive in an appropriate neighborhood of
$r_{\gamma}$, the size of which depends on the particular order photon ring
under consideration. Otherwise, the photon orbit would be stable, and the
associated null geodesics not reach distant observers to generate a ringlike
structure in the images.

Figure 4: Constraints on $N^{\prime\prime}(r_{\gamma})/B^{2}(r_{\gamma})$
arising from a 10% measurement of $\gamma$. For reference, the curves
associated with parameterized metrics (left) and alternative theories (right)
are shown. Line types and points are the same as in Figure 1 (left) and Figure
2 (right).

Examples of the implications of a $10\%$ measurement of $\gamma$ are shown in
Figure 4, roughly the precision required to distinguish the $n=1$ and $n=2$
photon rings. Because measurements of $\gamma$ are necessarily coupled with a
high-precision measurement of $R$, $N^{\prime}(r_{\gamma})$ is effectively
fixed, resulting in an additional constraint in the
$N(r_{\gamma})$-$N^{\prime\prime}(r_{\gamma})/B^{2}(r_{\gamma})$ plane. The
qualitative differences between the interpretation of such a measurement for
parameterized metric expansions and explicit metric alternatives is similar to
that for the shadow size. Again, the measurement presents a fundamentally
degenerate constraint. Thus, even multiple photon ring measurements, it is not
possible to uniquely determine $N(r_{\gamma})$.

###  4.5 Gravitational Waves

With the detection of GW150914, gravitational waves have become an important
probe of the near-horizon region of merging stellar-mass black holes. Future
space-based interferometers (e.g., LISA Amaro-Seoane et al., 2017) and pulsar
timing monitoring experiments promise to expand these tests to supermassive
black holes that are directly comparable to EHT observations of Sgr A* and
M87*, respectively(Hobbs et al., 2010).

Comparison of these limits is complicated by the fact that gravitational wave
observations necessarily require the specification of the dynamical sector of
any putative gravity theory. Therefore, it is generally insufficient to
postulate alternative stationary spacetimes, as we have done in Equation 1, as
doing so does not inform dynamical phenomena of any alternative theory. With
this caveat, there are two situations in which may nevertheless be useful to
characterize results in the way proposed here: gravitational wave ringdowns
and extreme mass ratio inspirals (EMRIs), which we treat in turn.

####  4.5.1 Gravitational Wave Ringdown

The late-time evolution of high-angular momentum quasinormal modes (QNMs) is
related in general relativity to the shadow size (Jusufi, 2020; Stefanov et
al., 2010; Yang, 2021). This is a consequence of these modes being associated
with high-frequency, azimuthally propagating massless perturbation, which is
necessarily governed by the same dynamics as photons near the photon orbit. As
a consequence, generically the angular frequency,
$\omega=\omega_{R,l}+i\omega_{I,l}$ of the quasinormal mode with azimuthal
quantum number $l$,

$\lim_{l\rightarrow\infty}\omega_{I,l}=\frac{N^{\prime}(r_{\gamma})\gamma}{2\pi}~{}~{}\text{and}~{}~{}\lim_{l\rightarrow\infty}\frac{\omega_{R,l}}{l}=N^{\prime}(r_{\gamma}),$
(25)

which are equivalent to Equation I.1 of Yang (2021). Thus, observations of
high-frequency quasinormal modes result in spacetime constraints that are
directly comparable to those from shadow sizes and multiple photon ring
measurements. Of particular interest is that these can be combined to
separately measure $\gamma$:

$\gamma=2\pi\lim_{l\rightarrow\infty}l\frac{\omega_{I,l}}{\omega_{R,l}}.$ (26)

While the above expressions invoke the high-$l$ limit, in practice, for
Schwarzschild the approximations are good to better than 10% by $l=2$,
improving rapidly thereafter (see, e.g., Berti et al., 2009). Of course, this
does not confer any guarantees for alternative spacetimes. Nevertheless, for
illustrative purposes we consider the implications of the LIGO QNM
measurements.888The LIGO sources are decidedly not spin-zero black holes. The
$\omega_{R}$ differ by $\sim 30\%$ from their Schwarzschild counterparts for
the typical merger remnant spins. We defer a full discussion of the spinning
case until Salehi & Broderick (2023, in preparation).

Figure 5: Constraints on $N(r_{\gamma})$, $N^{\prime}(r_{\gamma})$, and
$N^{\prime\prime}(r_{\gamma})/B^{2}(r_{\gamma})$ arising from the LIGO QNM
constraints reported in Abbott et al. (2021). The magenta shaded region shows
the 68% confidence region. Illustrative curves associated with the metric
expansions in Figure 1 (blue) and alternative spacetimes in Figure 2 (orange)
are shown. See Figure 1 for a description of line types.

Abbott et al. (2021) reports constraints from LIGO on the low-order QNMs for
19 merger events. For each event, analysis of the ringdown portion of the
post-merger gravitational waveform yields an estimate of the QNM frequency
($f=\omega_{R,2}/2\pi$) and decay timescale ($\tau=1/\omega_{I,2}$). Comparing
to the expected values given the full inspiral-merger-ringdown waveform
analyses finds that in aggregate the fractional shifts in these quantities are
$\delta\hat{f}=0.03_{-0.21}^{+0.23}$ and $\delta\hat{\tau}=0.16\pm 0.60$,
respectively, where we have converted the 90% region to $1\sigma$ assuming
Gaussian statistics. The implied aggregate measurement on $R$ is

$R=\frac{\sqrt{27}}{1+\delta\hat{f}}=\sqrt{27}(0.97_{-0.18}^{+0.25}),$ (27)

and the implied $\gamma$ is, then,

$\gamma=\frac{\pi}{(1+\delta\hat{f})(1+\delta\hat{\tau})}=1.9_{-0.3}^{+3.3}.$
(28)

Note that the uncertainties on $R$ and $\gamma$ are correlated through
$\delta\hat{f}$. The corresponding limits on $N(r_{\gamma})$,
$N^{\prime}(r_{\gamma})$, and $N^{\prime\prime}(r_{\gamma})$ are shown in
Figure 5 in comparison to various parameterized metric expansions and
alternative metrics.

####  4.5.2 Gravitational Wave Inspiral

EMRIs have the significant virtue of the secondary being a perturbation on the
spacetime of the much more massive primary (see, e.g., Babak et al., 2017, and
references therein). Therefore, as with QMNs, the dynamics of the EMRI system
can be analyzed within the context of a stationary background spacetime,
subject to the caveat regarding the dynamical sector of the underlying gravity
theory. However, there is an additional complication: the inspiral typically
occurs at $r>r_{\gamma}$.

A short discussion of the circular orbits of massive paricles in arbitrary
spherically symmetric spacetimes may be found in Appendix E. An implicit
expression for the radius of the ISCO, which terminates the inspiral regime,
is given in Equation E8. While it is clear that the detection of a photon ring
ensures that stable circular orbits do not exist a $r_{\gamma}$ (see Appendix
E), general statements about the ordering and proxmity of ISCO and photon
orbit depend on the particular form of $N(r)$. As a result, some assumption
regarding the extrapolation of $N(r)$ away from $r_{\gamma}$ is necessary to
make any estimate of the implications of inspiral waveform measurements for
deviations from general relativity.

##  5 Conclusions

The EHT images of M87* and Sgr A* provide a new, direct window into the
properties of astrophysical black holes. Shadow size measurements, and more
generally, measurements of the size of photon rings, provide a means to
directly probe the spacetime geometry of black holes. However, interpreting
the gravitational implications of these measurements requires some care due to
the nonlinear nature of general relativity near black hole event horizons and
limited information contained in a single (or handful) of size measurements.
This is evident even in the limited case of spherically symmetric, stationary
spacetimes that we address.

For such spacetimes, significant qualitative conclusions may be reached
already based solely upon the detection of various image features. The
detection of a shadow immediately implies that $N^{\prime}(r_{\gamma})>0$. The
detection of any photon ring implies that $N^{\prime\prime}(r_{\gamma})<0$.
These hold independent of the shadow or photon ring size.

The size of the shadow is directly related to $N^{\prime}(r_{\gamma})$:
measuring the shadow size is synonymous with measuring the radial derivative
of the $tt$-component of the metric in areal coordinates. In this sense,
precise shadow size measurements generate precise metric constraints. However,
there is no constraint on $N(r_{\gamma})$ from the shadow size alone, and thus
shadow size measurements by themselves do not provide any limit on the
magnitude of a putative deviation in the $tt$-component of the metric, i.e.,
the perturbing potential $\psi(r)$. In this sense, precise shadow measurements
are uninformative.

When translated to spacetime parameters, either via parameterized metric
expansions or explicit alternative spacetimes, precise shadow size
measurements can appear to impose strong constraints on both $N(r_{\gamma})$
and $N^{\prime}(r_{\gamma})$. The constraint on $N(r_{\gamma})$ is a direct
consequence of the prior that has been adopted via the choice of the
underlying metric. In the case of parameterized metric expansions, for which
there is little significance to this prior, the strength of the attendant
limit on $N(r_{\gamma})$ is illusory. Indeed, it is straightforward to
generate examples of perturbed metrics that are otherwise consistent with all
existing constraints for Sgr A* and M87*, including the recent EHT shadow
sizes, that have nearly arbitrary magnitude metric perturbations at the photon
orbit. Claims in the literature that shadow size measurements limit a specific
metric expansion coefficient, or a linear combination of coefficients, must be
understood within a narrow context for a similar reason.

Nevertheless, because shadow size measurements do impose a strong measurement
on some facet of the underlying spacetime, we propose an alternative way to
characterize their gravitational implications: the values of $N(r)$,
$N^{\prime}(r)$, $N^{\prime\prime}(r)/B^{2}(r)$, etc., measured at
$r_{\gamma}$. This series has the virtue of being a nonparametric and
nonperturbative description of the shadow size measurement — in spherical
symmetry the identifications are exact. However, they are complicated by the a
priori unknown value of $r_{\gamma}$, i.e., properties of the metric are
constrained at an important dynamical location in the spacetime whose location
is otherwise unknown.

Despite the unknown $r_{\gamma}$, this proposal presents a particularly useful
basis for comparing near-horizon measurements, including redshifts, photon
echos, photon rings, shadow sizes, and black hole ringdowns, all of which are
dominated by massless particle dynamics near the event horizon, and therefore
the photon orbit. It is also convenient theoretically, providing a more
natural quantity for theoretical comparison and obviating the need for full
ray-tracing and radiative transfer simulations and/or mode spectrum
computation.

This basis remains poorly connected to gravity measurements that probe very
different spatial scales. This is a natural consequence of the afore-mentioned
nonlinearity expected near the photon orbit: the unavoidable price of adopting
a nonperturbative way to characterize near-horizon observations is difficulty
in making comparisons to perturbative characterizations of other measurements.
This complicates, e.g., quantitatively relating EHT shadow size measurements
to solar system tests or observations of the inspiral phase of black hole
mergers outside of a particular gravity theory.

We have focused on spherically symmetric spacetimes for simplicity. However,
to linear order in spin, all of the results obtained for Schwarzschild
continue to apply for polar observers (the relevant inclination for M87*). We
leave the expansion of the nonparametric, nonperturbative characterization to
integrable rotating spacetimes with arbitrary spins for future work.
Nevertheless, even the simple cases explored here, elucidate the power and
limitations of measurements of the shadow and photon ring sizes.

This work was supported in part by 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
Canada and by the Province of Ontario through the Ministry of Economic
Development, Job Creation and Trade. A.E.B. thanks the Delaney Family for
their generous financial support via the Delaney Family John A. Wheeler Chair
at Perimeter Institute. A.E.B. receives additional financial support from the
Natural Sciences and Engineering Research Council of Canada through a
Discovery Grant.

##  Appendix A Slowly Rotating Space times

When viewed from the polar axis, approximately appropriate for M87*, by
symmetry all axisymmetric spacetimes must produce circular shadows. While
limiting ourselves to this viewing angle reduces the impact of spacetime
properties, it does simplify the relationship between the shadow size and the
underlying spacetime geometry.

In Salehi & Broderick (2023, in preparation), the shadows of a broad class of
integrable, asymptotically flat, axisymmetric, spinning spacetimes is studied
in detail. In additional, these spacetimes reduce to spherical symmetry in the
limit of zero spin, i.e., $a=0$. Each may be written in the form

$\displaystyle ds^{2}=$
$\displaystyle-N(r)^{2}\left[1+a^{2}f_{t}(r,\cos\theta;a^{2})\right]dt^{2}$
(A1)
$\displaystyle+\frac{B(r)^{2}}{N(r)^{2}}\left[1+a^{2}f_{r}(r,\cos\theta;a^{2})\right]dr^{2}$
$\displaystyle+r^{2}\left[1+a^{2}f_{\theta}(r,\cos\theta;a^{2})\right]d\theta^{2}$
$\displaystyle+r^{2}\sin^{2}\theta\left[1+a^{2}f_{\phi}(r,\cos\theta;a^{2})\right]d\phi^{2}$
$\displaystyle+2ag(r,\cos\theta;a^{2})dtd\phi,$

for some set of functions $f_{t}(r,\cos\theta;a^{2})$,
$f_{r}(r,\cos\theta;a^{2})$, $f_{\theta}(r,\cos\theta;a^{2})$,
$f_{\phi}(r,\cos\theta;a^{2})$, and $g(r,\cos\theta;a^{2})$, where we have
explicitly enforced the reduction to spherical symmetry when $a=0$ and
reflection symmetry across the equatorial plane.

Because of this latter symmetry, shadow sizes observed by polar viewers must
be independent of the sign of the spin. Expanding the above confirms that the
lowest-order spin correction enters at $a^{2}$ Salehi & Broderick (2023, in
preparation). As a consequence, up to corrections of $a^{2}$, the shadow size
is that given in Equation 4.

##  Appendix B Explicit Metric Expansions

Here we collect explicit expansions for the various metric expansions. In all
cases, to produce the single-parameter curves in Figure 1 it is easier to
solve for the coefficient as a function of $r_{\gamma}$. Care must be taken
when more than one photon orbit exists (which occurs for the MGBK metric when
only $\gamma_{4,2}$ is varied). All of the expansions described here are
similar in character and reduce to expansions in $1/r$ for small parameter
values. We adopt the form of the metric expansions used in Event Horizon
Telescope Collaboration et al. (2022f).

###  B.1 Post-Newtonian

Combining Equation 6 and Equation 10 gives the expression for $N^{2}(r)$ for
the post-Newtonian metric expansion:

$N^{2}(r)=1-\frac{2M}{r}+\sum_{n=1}^{\infty}\frac{2\kappa_{n}(-M)^{n+1}}{r^{n+1}}.$
(B1)

###  B.2 JP Metric

From Johannsen (2013),

$N^{2}(r)=\left(1-\frac{2M}{r}\right)\left(1+\sum_{n=2}^{\infty}\frac{\alpha_{1n}M^{n}}{r^{n}}\right)^{-2}.$
(B2)

###  B.3 MGBK Metric

From Vigeland et al. (2011),

$N^{2}(r)=\left(1-\frac{2M}{r}\right)\Bigg{[}1-\sum_{n=2}^{\infty}\frac{\gamma_{1,n}M^{n}}{r^{n}}\\\
-2\left(1-\frac{2M}{r}\right)\sum_{n=2}^{\infty}\frac{\gamma_{4,n}M^{n}}{r^{n}}\Bigg{]}.$
(B3)

When only varying $\gamma_{4,2}$, this metric admits up to three solutions for
$r_{\gamma}$. Where appropriate, the largest $r_{\gamma}$ is adopted, which is
evident in Figure 1.

###  B.4 RZ Metric

From Rezzolla & Zhidenko (2014), up to $a_{1}$,

$N^{2}(r)=\left(1-\frac{r_{0}}{r}\right)\left[1-\epsilon\frac{r_{0}}{r}+(a_{0}-\epsilon)\frac{r_{0}^{2}}{r}^{2}+a_{1}\frac{r_{0}^{3}}{r^{3}}\right].$
(B4)

Imposing $r_{0}=2M$ and $\epsilon=-(1-2M/r_{0})=0$, as done in Event Horizon
Telescope Collaboration et al. (2022f), reduces this to

$N^{2}(r)=\left(1-\frac{2M}{r}\right)\left[1+4a_{0}\frac{M^{2}}{r^{2}}+8a_{1}\frac{M^{3}}{r^{3}}\right],$
(B5)

which is degenerate with the MGBK metric expansion with $\gamma_{1,2}=4a_{0}$
and $\gamma_{1,3}=8a_{1}$.

##  Appendix C Shadows and Photon Orbits

We review some facts about the shadow and their relationship to the photon
orbit in arbitrary spacetimes. We begin with the equation of motion for a null
geodesic in an arbitrary stationary and spherically symmetric spacetime, which
we obtain from the null condition and the fact that $E$ and $L=Eb$ are
conserved:

$\dot{r}^{2}=\frac{1}{B^{2}(r)}\left[1-\frac{b^{2}}{r^{2}}N^{2}(r)\right],$
(C1)

where $\dot{r}=dr/d\tau$. An effective potential, $V_{\rm
eff}(r)=N^{2}(r)/r^{2}$, may be specified in the normal way, yielding

$\dot{r}^{2}=\frac{1}{B^{2}(r)}\left[1-b^{2}V_{\rm eff}(r)\right].$ (C2)

Turning points occur when $V_{\rm eff}(r)=1/b^{2}$. If $V_{\rm eff}(r)$ has a
maximum, there will be a $b$ for which all null geodesics with smaller $b$
will have no turning points between the event horizon and the observer at
infinity. The collection of such trajectories comprise the shadow.

At a photon orbit, two conditions apply: $\dot{r}=0$ (i.e., the orbit is
circular) and $\ddot{r}=0$ (i.e., it stays circular), and thus

$b_{\gamma}=\frac{1}{\sqrt{V_{\rm
eff}(r_{\gamma})}}=\frac{r_{\gamma}}{N(r_{\gamma})}$ (C3)

and

$\left[\frac{1}{V_{\rm
eff}(r)}\right]^{\prime\prime}_{r_{\gamma}}=\left[\frac{r^{2}}{N^{2}(r)}\right]^{\prime}_{r_{\gamma}}=0,$
(C4)

which are equivalent to Equation 4 and Equation 3, respectively.

Note that these conditions may be satisfied at more than one location,
corresponding to multiple photon orbits. That there is at least one unstable
photon orbit for asymptotically flat, black hole spacetimes is guaranteed by
two facts: 1. asymptotic flatness gives $V_{\rm eff}(r)>0$ and
$V^{\prime}_{\rm eff}(r)<0$ at very large $r$, and 2. $V_{\rm eff}(r_{h})=0$
vanishes at some $r_{h}$ that defines the event horizon. Thus, $V_{\rm eff}$
is positive and decreasing at large $r$ and vanishes at small $r$, requiring
via the mean value theorem that at some intermediate point $V_{\rm eff}(r)$
has a maximum. Subsequently, photon orbits may appear in pairs, one stable
(maximum of $V_{\rm eff}(r)$) and one unstable (minima of $V_{\rm eff}(r)$).
Because the shadow is necessarily associated with the global maximum of
$V_{\rm eff}(r)$, by Equation C3, its size is set by the smallest
$b_{\gamma}$. Thus, henceforth, we will restrict ourselves to the photon orbit
with the smallest shadow, which we call $r_{\gamma}$ and $b_{\gamma}$,
respectively.

##  Appendix D Radii of Photon Rings

The higher order images of the near-horizon emission region form a sequence of
typically ring-like image features (Broderick & Loeb, 2006; Johnson et al.,
2020; Broderick et al., 2022a, see, e.g.). Each additional image is associated
with photon trajectories that execute an additional half orbit about the black
hole, and therefore the size of these rings are probes of spacetime about the
photon orbit. Here we derive the radius of the higher-order photon rings in
the limit of high photon-ring order.

For null geodesics near the photon orbit, i.e., for which

$r=r_{\gamma}+\delta r,$ (D1)

Equation C1 becomes after Taylor expanding about $r=r_{\gamma}$ to lowest
order,

$\dot{\delta
r}^{2}\approx-\frac{b_{\gamma}^{2}}{2B^{2}(r_{\gamma})}\left[\frac{N^{2}(r)}{r^{2}}\right]^{\prime\prime}_{r_{\gamma}}\delta
r^{2},$ (D2)

where the first two terms vanish due to Equation C3 and Equation C4 and
because the inner turning point (i.e., the minimum $\delta r$) is small, we’ve
replaced $b$ with $b_{\gamma}$. Combining this radial equation of motion with
Equation 2, gives

$\pi\frac{d\delta r}{d\phi}=\gamma\delta r$ (D3)

where

$\gamma=\pi\frac{N^{3/2}(r_{\gamma})}{N^{\prime}(r_{\gamma})}\left[-\frac{N^{\prime\prime}(r_{\gamma})}{B^{2}(r_{\gamma})}\right]^{1/2}$
(D4)

is the Lyapunov exponent identified in Johnson et al. (2020). Solving this, we
obtain,

$\delta r=\delta r_{0}e^{\gamma\phi/\pi},$ (D5)

i.e., upon every half orbit the geodesic grows by a factor of $e^{\gamma}$.
This matches Equation 7 of Johnson et al. (2020) in the limit of
Schwarzschild.

At some $\delta r_{\rm max}\sim M$, the perturbative expansion of the radial
equation of motion breaks down, and the photon trajectory streams to the
observer at infinity. Prior to this point, it will have executed

$n\approx 2\gamma^{-1}\ln\left(\delta r_{\rm max}/\delta r_{0}\right),$ (D6)

half orbits, where the preceding orbits from a distant source to the vicinity
of the photon orbit are now included, and thus contribute to the $n$th-order
photon ring. Describing the details of the transition is not necessary to
obtain the relative locations of the photons on a distant observing screen;
rather the photon ring sizes are completely controlled by $b$, which is, in
turn, set by the inner turning point, i.e., by $\delta r_{0}$. At $\delta
r_{0}$, $\dot{r}=0$, and therefore,

$b^{2}\approx
b^{2}_{\gamma}+\frac{1}{2}\left[\frac{r^{2}}{N^{2}(r)}\right]^{\prime\prime}_{r_{\gamma}}\delta
r_{0}^{2},$ (D7)

where again the linear term vanishes as a result of Equation C4. Identifying
the radius of the $n$th order photon ring, $R_{n}$, with the $b$ associated
with the $\delta r_{0}$ that corresponds to $n$ half orbits, we find

$R_{n}-R\approx\frac{N^{\prime}(r_{\gamma})}{4}\left[\frac{r^{2}}{N^{2}(r)}\right]^{\prime\prime}_{r_{\gamma}}\delta
r_{\rm max}^{2}e^{-\gamma n}.$ (D8)

The prefactor is common to all order photon rings. The shift relative to the
shadow size decreases as $e^{-\gamma n}$, from which we recover that the
shadow is bounded by the asymptotic photon ring corresponding to
$n\rightarrow\infty$. This matches Equation 12 of Johnson et al. (2020).

##  Appendix E Circular Orbits of Massive Particles

General expressions for circular orbits for massive particles may be
constructed for the family of spacetimes described by Equation 1 in a fashion
similar to that used to obtain $r_{\gamma}$. We again require $\dot{r}$ and
$\ddot{r}$ vanish. The first condition gives

$\dot{r}^{2}=\frac{1}{B^{2}(r)}\left[E^{2}-N^{2}-\frac{N^{2}}{r^{2}}L^{2}\right]=0,$
(E1)

where $E=-u_{t}$ and $L=u_{\phi}$ are the conserved specific energy and
specific angular momentum. From this, the second gives

$\ddot{r}=\frac{N(r)}{B^{2}(r)}\left[N^{\prime}(r)+L^{2}\left(\frac{N^{\prime}(r)}{r^{2}}-\frac{N(r)}{r^{3}}\right)\right]=0,$
(E2)

where we employed the previous condition. Together, these imply that for
circular orbits,

$L\equiv u_{\phi}=\pm\sqrt{\frac{r^{3}N^{\prime}(r)}{N(r)-rN^{\prime}(r)}},$
(E3)

with associated energy,

$E\equiv u_{t}=-\sqrt{\frac{N^{3}}{N(r)-rN^{\prime}(r)}}.$ (E4)

From these, the angular velocity as measured by a distant observer is,

$\Omega=\frac{u^{\phi}}{u^{t}}=-\frac{N^{2}(r)}{r^{2}}\frac{L}{E}=\pm\sqrt{\frac{N(r)N^{\prime}(r)}{r}}.$
(E5)

The stability of these circular orbits is determined by the response to
perturbations (again similar to the analysis of photon orbits in Appendix D),

$\dot{\delta r}^{2}=-\omega^{2}\delta r^{2}$ (E6)

where

$\omega^{2}=\frac{N(r)}{B^{2}(r)}\left[N^{\prime\prime}(r)\left(1+\frac{L^{2}}{r^{2}}\right)+\frac{3N^{3}(r)}{r^{4}}\frac{L^{2}}{E^{2}}\right]\delta
r^{2}.$ (E7)

When $\omega^{2}>0$, the perturbation is oscillatory and the orbits are
stable; when $\omega^{2}<0$ the perturbation grows exponentially. Note that at
$r_{\gamma}$, $L^{2}\rightarrow\infty$ and $\omega^{2}<0$ if
$N^{\prime\prime}(r_{\gamma})<0$, i.e., if any photon rings are observed then
timelike geodesics are also unstable at the photon orbit.

Transitions from stable to unstable circular orbits occur when $\omega^{2}=0$.
The radius of the innermost stable circular orbit is the minimum $r_{\rm I}$
for which

$r_{\rm I}=-\frac{3N(r_{\rm I})N^{\prime}(r_{\rm I})}{3N^{\prime 2}(r_{\rm
I})-N(r_{\rm I})N^{\prime\prime}(r_{\rm I})},$ (E8)

which is an analogous condition to that for $r_{\gamma}$ in Equation 3.
However, in the absence of knowledge about the particular form of $N(r)$ away
from $r_{\gamma}$, it is difficult to place any further general conditions on
$r_{\rm I}$ relative to $r_{\gamma}$.

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