# Cosmic Censorship in Sgr A* and M87*: Observationally Excluding Naked
Singularities

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

###### Abstract

The imaging of Sagittarius A* (Sgr A*) and the supermassive black hole at the
center of Messier 87 (M87*) by the Event Horizon Telescope constrains the
location and nature of emission from these objects. Coupled with flux limits
from the near-infrared through the ultraviolet, the attendant size constraints
provide strong evidence for the absence of an accretion-powered photosphere,
and therefore for the existence of an event horizon about an astrophysical
black hole. Here, we demonstrate that a broad class of naked singularities are
also generically excluded, regardless of the nature and unknown physical
impact of singularity itself, subject to a single weak assumption about
locality. While we restrict our attention to static, spherically symmetric
spacetimes, we are nevertheless able to exclude a large number of commonly
invoked naked singularity spacetimes in this way.

Gravitation, Naked singularities, Radio continuum emission, High energy
astrophysics, Galactic center, Active galaxies, Strong gravitational lensing

##  1 Introduction

Singularities are synonymous with black holes, the inexorable final
destination of accreting material, and their resolution is among the chief
motivations of the search for a quantum theory of gravity. In the absence of
such a theory, they remain a mystery, presenting an as yet unknown boundary
condition for the spacetime and the fields that populate it, and a potential
source of unknown physical effects. For these reasons, predicting the future
development of a singular spacetime and its contents presents a significant
challenge. Nevertheless, the possibility of creating naked singularities,
i.e., singularities not hidden behind an event horizon, even if only in
principle, by over-spinning or over-charging general relativistic black holes
remains unsettling. Moreover, it remains unclear if it is possible to generate
naked singularities from generic initial conditions in general relativity or
alternate theories of gravity (see, e.g., Janis et al., 1968; Joshi et al.,
2011, 2014).

The formal problems associated with naked singularities threaten to render
general relativity on its own a poorly posed initial value theory. This
unpalatable state of affairs may be ameliorated in the observable universe if
all singularities were to be hidden behind horizons (i.e., within black holes
or behind the cosmic horizon). This idea, codified as the “cosmic censorship
hypothesis” introduced by Penrose (1969), effectively ensures that no point
within the universe that we can currently observe is within the future Cauchy
development of any singularity, and therefore unaffected by its unknown
properties. However, cosmic censorship remains unproven theoretically and is,
therefore, currently only addressable experimentally.

Observational arguments for the existence of astrophysical event horizons have
existed for two decades now. These typically invoke advection dominated
accretion flows (ADAFs), and the more general class of radiatively inefficient
accretion flows (RIAFS; Narayan et al., 1995, 1998; Blandford & Begelman,
1999; Yuan & Narayan, 2014). Such flows are theoretically anticipated for
accretion rates well below the Eddington rate, i.e., $\dot{M}<0.01\dot{M}_{\rm
Edd}=2\times 10^{-11}(M/10^{9}~{}{\rm M_{\odot}})~{}{\rm M_{\odot}}\,{\rm
yr}^{-1}$, due to the weak Coulomb coupling between the electrons (which
efficiently radiate) and the much more massive ions (which liberate the vast
majority of the gravitational binding energy; Narayan et al., 1998; Narayan &
McClintock, 2008; Yuan & Narayan, 2014). RIAFs, therefore, necessarily advect
a large fraction of $\dot{M}c^{2}$ as kinetic energy toward the central
object. Wherein an event horizon (or at least apparent horizon) is present,
this kinetic energy is lost, deposited within the black hole and increasing
its mass. However, in the absence of an event horizon, e.g., in the presence
of a surface, this energy will typically be thermalized and radiated. While
thermalization happens naturally in baryonic atmospheres, for compact surfaces
it is guaranteed by strong lensing for sufficiently high-redshift surfaces
(Broderick et al., 2009b).111See Broderick & Narayan (2007) for an example
where the surface does not thermalize rapidly, and the potential for
constraints on the exitence of such systems even in that case. The result is a
thermal bump in the spectral energy distribution (SED), the temperature and
luminosity of which is set by energy balance with $\dot{M}$ (Narayan et al.,
1997; Narayan & Heyl, 2002; Narayan & McClintock, 2008; Broderick et al.,
2009b, 2015).

These thermal bumps are explicitly seen in the X-ray spectra of accreting
neutron star X-ray binaries, there associated with emission from the boundary
flow onto the stellar surface, and conspicuously absent in black hole X-ray
binaries (Narayan et al., 1997; Narayan & Heyl, 2002; McClintock et al., 2004;
Narayan & McClintock, 2008). Supplemented with a long history of panchromatic
observations, a detailed effort to understand and model accretion and jet
launching in active galactic nuclei (AGN), the strongest constraints on
visible surface/photosphere emission have been produced using the low-
luminosity AGN Sagittarius A* (Sgr A*) and Messier 87* (M87*) (Broderick et
al., 2009b, 2015). Within these, the dominant systematic uncertainty was the
size of the emitting surface; a large surface can be cooler and therefore
escape detection by hiding underneath the bright emission at mm wavelengths
from the much hotter accretion flow.

The Event Horizon Telescope (EHT) has effectively retired this uncertainty for
Sgr A* (Event Horizon Telescope Collaboration et al., 2022a, b, c, d, e, f,
2024a, 2024b, hereafter Sgr A* Paper I-Sgr A* Paper VIII) and M87* (Event
Horizon Telescope Collaboration et al., 2019a, b, c, d, e, f, 2021a, 2021b;
Event Horizon Telescope Collaboration et al., 2023, hereafter M87* Paper
I-M87* Paper IX) by directly imaging those sources on angular scales that
resolve the putative event horizons. These images confirm (1) that unstable
circular photon orbits exist through the existence of the central brightness
depression, and (2) that the emission region on the scale of the circular
photon orbit is fully consistent with that anticipated by accretion onto
general relativistic black holes (for implications for black hole spacetimes
beyond general relativity see Kocherlakota et al. 2021; Sgr A* Paper VI;
Broderick et al. 2023; Salehi et al. 2024). Thus it is, EHT observations
combined with contemporaneous near-infrared (NIR), optical and ultraviolet
(UV) flux limits that provide the strongest empirical evidence currently for
the existence of astrophysical event horizons.

The above empirical case in favor of event horizons is predicated on some
degree of regularity near where the horizon would be, and may fail in the
presence of a naked singularity. For example, the unknown physical impact of
the singularity itself on the accreting material may prevent the creation of a
thermal photosphere, or accreting baryonic matter may simply disappear upon
impacting the singularity. Even should the accreted material remain for some
time, the gravitational redshift may never be sufficiently high to support the
argument that the emitting surface must approximately be in thermal
equilibrium due to strong lensing.

Despite the inherent uncertainties, a number of authors have embarked on
making theoretical predictions of the images from naked singularity spacetimes
(Joshi et al., 2011; Virbhadra & Ellis, 2002; Joshi et al., 2020; Eichhorn et
al., 2023; Nguyen et al., 2023; Saurabh et al., 2024; Deliyski et al., 2024;
Mishra et al., 2024; Chen et al., 2024a). Typically, these images include a
new family of lensed images, generated by in-going photon trajectories that
would otherwise be captured by the black hole. On the basis of the excess
interior flux such images would generate, EHT images of Sgr A* have already
been brought to bear on the existence of some naked singularity spacetimes
(Sgr A* Paper VI). Such limits presume that the intervening space between the
singularity and the emission region is transparent, a potentially dubious
assumption given the astrophysical properties of the EHT targets. However,
that generic, initially in-going null geodesics escape the near-singularity
region immediately raises the possibility of similar general behavior for
timelike geodesics, and thus may impact the fate of the accreting baryonic
material.

Given a single modest assumption about the physics of the singularity — the
direct unknown physical consequences of the singularity appear only within a
very small distance of the singularity itself (e.g., the Planck length) — we
show that for a very broad class of naked singularity spacetimes that the
creation of a dense baryonic (and thus thermal) atmosphere is inevitable. We
demonstrate this existence explicitly for static, spherically symmetric,
asymptotically flat spacetimes. For Sgr A* and M87* we further show that the
existence of such an atmosphere is conclusively excluded, and therefore
neither of these objects can harbor naked singularities.

We begin in Section 2 with a general discussion of static, spherically
symmetric, asymptotically flat spacetimes exhibiting naked singularities,
presenting a classification scheme that identifies where and why the spacetime
is singular. In Section 3 we address the general behavior of null and timelike
geodesics, and therefore the fate of accreting baryonic gas. The spectral
signatures and observational constraints for Sgr A* and M87* appear in Section
4 and Section 5, respectively. Finally, we collect concluding remarks in
Section 6. Unless otherwise stated we assume a metric signature of $-+++$ and
set $G=c=1$.

##  2 General Stationary Spherically Symmetric Naked Singularities

The metric of any spherically symmetric, asymptotically spacetime can be
written as,

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

where $r$ is the areal radius, and $N(r)$ and $B(r)$ are arbitrary functions
of radius that asymptote to unity as $r\rightarrow\infty$. Well-behaved event
horizons exist wherever $N(r)=0$; where $B(r)=\infty$ the coordinates becomes
singular. We turn now to the conditions under which Equation 1 describes a
spacetime with a naked singularity.

The Kretschmann scalar, $K\equiv R_{abcd}R^{abcd}$, for the spherically
symmetric spacetime described by Equation 1 is given by

$K=\left\\{\frac{1}{B}\left[\frac{\left(N^{2}\right)^{\prime}}{B}\right]^{\prime}\right\\}^{2}+2\left\\{\frac{\left(N^{2}\right)^{\prime}}{rB^{2}}\right\\}^{2}\\\
+2\left\\{\frac{1}{r}\left(\frac{N^{2}}{B^{2}}\right)^{\prime}\right\\}^{2}+\left\\{\frac{1}{r^{2}}\left(1-\frac{N^{2}}{B^{2}}\right)\right\\}^{2},$
(2)

(see, e.g., eqs. 14-18 of Gkigkitzis, 2014). Singular behavior in $K$
necessarily betrays spacetime singularities. Because in spherical symmetry,
$K$ is composed of the elements of the Reimann squared (the metric in Equation
1 is manifestly diagonal), the converse statement is true: all spacetime
singularities will produce singular $K$. When $B(r)=1$ everywhere (which is
often true), Equation 2 reduces to the more simple form,

$K=(f^{\prime\prime})^{2}+\frac{4(f^{\prime})^{2}}{r^{2}}+\frac{4f^{2}}{r^{4}},$
(3)

where $f\equiv 1-N^{2}$. For Schwarzschild, $f=2M/r$, and the above reduces to
the well-known result, $K=48M^{2}/r^{6}$, which is singular at $r=0$.

Therefore, the conditions upon the free functions in Equation 1 for the
spacetime to contain a naked singularity are:

  1. 1.

At some radius (or collection of radii), $r_{*}$, $K$ becomes singular,
implying that a spacetime singularity exists.

  2. 2.

For the outermost singularity, at all $r>r_{*}$ $N(r)$ and $B(r)$ are finite
positive-definite functions of radius.

###  2.1 Classification of Spherical Singularities

The condition that the spacetime is singular requires one of the four terms in
Equation 2 diverge at some radius $r_{*}$. We classify such behaviors by their
location and the lowest order of the derivative of the term that diverges.
First, by location, there are two possibilities,

P

If $r_{*}=0$, the singularity is “point-like”. This the only point-like
singularity in spherical symmetry.

S

If $r_{*}>0$, the singularity is “shell-like”. All non-point-like
singularities are shell-like in spherical symmetry.

Second, by the lowest order of the first term that diverges222It is possible
for a naked singularity to be type P0 but the first term that diverges is not
the last term in Equation 2 if $\lim_{r\rightarrow 0}N^{2}/B^{2}=1$.
Nevertheless, supplemented with the condition that a singularity does, in
fact, exist, this does not present a significant confusion.,

0

$N^{2}/r^{2}B^{2}\rightarrow\infty$ at $r_{*}$.

1

$N^{2}/r^{2}B^{2}$ is finite but $(N^{2}/B^{2})^{\prime}/r\rightarrow\infty$
or $(N^{2})^{\prime}/rB^{2}\rightarrow\infty$ at $r_{*}$.

2

$N^{2}/r^{2}B^{2}$, $(N^{2}/B^{2})^{\prime}/r$, and $(N^{2})^{\prime}/rB^{2}$
are finite but $[(N^{2})^{\prime}/B]^{\prime}/B\rightarrow\infty$ at $r_{*}$.

Thus a spacetime for which $N^{2}/r^{2}B^{2}\rightarrow\infty$ at $r_{*}=0$ is
classified as P0. Note that if any higher-order derivative term diverges at
the origin, then the zeroth must as well (see Appendix A), and thus P0 is the
only kind of “point-like” singularity in spherically symmetric, static
spacetimes.

We further subdivide the type-0 spacetimes based on what is responsible for
the divergence of $N^{2}/r^{2}B^{2}$.

z

If $N\rightarrow\infty$ at $r_{*}$ faster than $r^{-1}$, and $B^{-1}$ remains
finite, we define this to be a “redshift”-type singularity because the
redshfit ($1+z=N^{-1}$) goes to zero at $r_{*}$.

j

If $B\rightarrow 0$ at $r_{*}$ while $r$ and $N$ remain finite, we define this
to be a “Jacobian”-type singularity because $\sqrt{-g}=Br^{2}\sin\theta$ goes
to zero at $r_{*}$.

r

If $r\rightarrow 0$ at $r_{*}$ while $B$ and $N$ remain finite, we define this
to be a “radial”-type singularity, which necessarily only occurs for P0-type
singularities.

Table 1: Example Spacetimes with Naked Singularities  Name | Type | $N^{2}(r)$ | $B^{2}(r)$ | Null/Weak | Param. range |  Reference
---|---|---|---|---|---|---
RN | P0z | $\displaystyle 1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}$ | $1$ | Y/Y | $|Q|>M$ | Reissner (1916)
SHRN | P0z | $\displaystyle 1-\frac{2M}{r}+\frac{Q^{2}+s}{r^{2}}$ | $1$ | Y/Y | $Q^{2}+s>M^{2}$ | Astorino (2013)
EMD-1 | P0z | $\displaystyle 1-\frac{\sqrt{4M^{2}r^{2}+q^{4}}-q^{2}}{r^{2}}$ | $\displaystyle\frac{4r^{2}}{4r^{2}+q^{4}}$ | Y/Y | $q>\sqrt{2}$ | Kocherlakota & Rezzolla (2020)
E ae-1 | P0z | $\displaystyle 1-\frac{2M}{r}-\frac{27M^{4}c_{13}}{16(1-c_{13})r^{4}}$ | $1$ | Y/Y |  $c_{13}<0$ or $c_{13}>1$ | Kocherlakota & Rezzolla (2020)
E ae-2 | P0z | $\displaystyle 1-\frac{(2-c_{14})M}{r}-\frac{(2c_{13}-c_{14})(2-c_{14})^{2}M^{2}}{8(1-c_{13})r^{2}}$ | $1$ | Y/Y | $\displaystyle\frac{2c_{13}-c_{14}}{1-c_{13}}<-8$ | Kocherlakota & Rezzolla (2020)
EEH | P0z | $\displaystyle 1-\frac{2M}{r}+\frac{q_{m}^{2}}{r^{2}}-\alpha\frac{2q_{m}^{4}}{5r^{6}}$ | $1$ | Y/Y |  $\alpha\leq 0$, $q_{m}^{2}>M^{2}$ | Yajima & Tamaki (2001)
JNWaaThe function $s$ is defined implicitly by $r=s\\{1-2M/[(1-\nu)s]\\}^{\nu/2}$, where $s\in[s_{*},\infty)$ maps smoothly onto $r\in[0,\infty)$. Note that within the literature, the Janis-Newman-Winicour metric is also called the Fisher metric (Fisher, 1999), the Wyman solution (Wyman, 1981; Virbhadra, 1997), and the Buchdahl spacetime (Buchdahl, 1959; Bhadra & Nandi, 2001). | P0z/r | $\displaystyle\left(1-\frac{2M}{(1-\nu)s}\right)^{1-\nu}$ | $\displaystyle\left(1-\frac{2M}{(1-\nu)s}\right)^{2-\nu}$ | Y/Y |  $\nu>1$ or $0<\nu<1$ | Janis et al. (1968)
JMN-1 | P0r | $\displaystyle-\left(1-\frac{2M}{R_{b}}\right)\left(\frac{r}{R_{b}}\right)^{2M/(R_{b}-2M)}$ | $\displaystyle\left(\frac{r}{R_{b}}\right)^{2M/(R_{b}-2M)}$ | Y/Y | $0\leq M\leq 2R_{b}/5$ | Joshi et al. (2011)
JMN-2 | P0r | $\displaystyle\left[\frac{(1+\lambda)^{2}}{4\lambda\sqrt{2-\lambda^{2}}}\left(\frac{r}{R_{b}}\right)^{1-\lambda}-\frac{(1-\lambda)^{2}}{4\lambda\sqrt{2-\lambda^{2}}}\left(\frac{r}{R_{b}}\right)^{1+\lambda}\right]^{2}$ | $\displaystyle(2-\lambda^{2})N^{2}(r)$ | Y/Y |  $0\leq\lambda<1$ and $R_{b}>0$ | Joshi et al. (2014)
BSTbbNote that for the Bertrand singular spacetime, when $\beta=1$ it is the second term in Equation 2 that is the lowest order term that diverges. | P0r | $\displaystyle\frac{2\beta^{2}}{1+R_{b}/r}$ | $\displaystyle N^{2}(r)/\beta^{2}$ |  $\displaystyle\beta^{2}\leq\frac{12}{13}$/$\displaystyle\beta^{2}\leq\frac{2}{3}$ | $\beta,R_{b}\neq 0$ | Banik et al. (2017)
BINSccBorn-Infeld naked singularity. For compactness, we define $\bar{F}(z)\equiv\mbox{}_{2}F_{1}(1/4,1/2,5/4;-z)$, where $\mbox{}_{2}F_{1}(\dots)$ is the Gaussian hypergeometric function, and $M_{e}(a,\bar{Q})\equiv[4\bar{Q}^{2}-a+8\bar{Q}^{2}\bar{F}(16a\bar{Q}^{2}/(4\bar{Q}^{2}-a)^{2}]/6\sqrt{4\bar{Q}^{2}-a}$. | P0z | $\displaystyle 1-\frac{2M}{r}-\frac{2\bar{Q}^{2}}{3r^{2}+3\sqrt{r^{4}+a\bar{Q}^{2}}}+\frac{4\bar{Q}^{2}}{3r^{2}}\bar{F}\left(\frac{a\bar{Q}^{2}}{r^{4}}\right)$ | 1 | Y/Y |  $a<4\bar{Q}^{2}$ and $\displaystyle M<M_{e}(a,\bar{Q})$ | Chen et al. (2024b)
CNS | P0z | $\displaystyle\left(1+\frac{M}{r}\right)^{-2}+\frac{Q^{2}}{r^{2}}$ | 1 | Y/Y | $Q\neq 0$ | Viththani et al. (2024)
4D EGB | P0r | $\displaystyle 1+\frac{r^{2}}{2\gamma M^{2}}\left(1-\sqrt{1+\frac{8\gamma M^{3}}{r^{3}}}\right)$ | 1 | Y/Y | $\gamma>0$ | Deliyski et al. (2024)
4D EGBQ | S1 | $\displaystyle 1+\frac{r^{2}}{2\gamma M^{2}}\left(1-\sqrt{1+\frac{8\gamma M^{3}}{r^{3}}-\frac{4\gamma M^{2}Q^{2}}{r^{4}}}\right)$ | 1 | Y/Y |  $|Q|>0$ and $\gamma>1-Q^{2}/M^{2}$ | Fernandes (2020)

In the following, we will demonstrate many key ideas with the Reissner-
Nordström spacetime, which has the virtue of having or not having an event
horizon, depending on the electric charge, $Q$, for which

$N^{2}(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}},$ (4)

and $B(r)=1$. The associated $K$ is

$K=\frac{48M^{2}}{r^{6}}-\frac{96MQ^{2}}{r^{7}}+\frac{56Q^{4}}{r^{8}},$ (5)

which is singular only at $r_{*}=0$, and therefore is type P0. Because $B=1$
and $N\approx Q^{2}/r^{2}$ near $r=0$, this is further classified as P0z. When
$Q>M$ there is no event horizon and this metric provides a useful example. A
number of other naked singularity spacetimes are collected in Table 1, where
their type, $N^{2}$, $B^{2}$, relevant parameter ranges may be found.

###  2.2 Energy Conditions and Singular Spacetimes

For alternative gravity theories, the energy content of the spacetime may not
be relevant. Nevertheless, within the context of general relativity (and
theories that admit an equation of the form of the Einstein equation), the
null and weak energy conditions can provide insight into which classes of
singular spacetimes are relevant and which may be “sick”.

The consideration of energy conditions is motivated by Hawking (1992), which
proves that closed timelike curves (CTCs) cannot be produced by classical
fields in compact regions of non-singular spacetimes without violating the
weak energy condition and conjectures that a similar condition holds for
localized violations for quantum fields (the “chronology protection
conjecture”). While Hawking (1992) does not apply in spacetime regions
containing singularities, if we imagine that the naked singularity spacetime
is the late-time evolution from non-singular initial conditions, then the weak
energy condition is sufficient to ensure that CTCs do not appear prior to the
singularity’s formation. Therefore, insofar as exotic matter (negative rest
energy) and CTCs are unphysical, the null and weak energy conditions provide
some guidance as to which spacetimes may be relevant (apart, of course, having
a naked singularlity!).

The Ricci tensor for the metric in Equation 1 is diagonal and given by

$\displaystyle R^{t}_{t}=R^{r}_{r}$
$\displaystyle=-\frac{1}{2r^{2}B}\left[\frac{r^{2}(N^{2})^{\prime}}{B}\right]^{\prime}$
(6) $\displaystyle R^{\theta}_{\theta}=R^{\phi}_{\phi}$
$\displaystyle=\frac{1}{r^{2}}-\frac{1}{r^{2}B}\left(\frac{rN^{2}}{B}\right)^{\prime},$

from which we have the Ricci scalar is

$R=-\frac{1}{r^{2}B}\left[\frac{(r^{2}N^{2})^{\prime}}{B}\right]^{\prime}+\frac{2}{r^{2}}.$
(7)

The associated Einstein tensor is diagonal and given by

$\displaystyle G^{t}_{t}=G^{r}_{r}$
$\displaystyle=\frac{1}{r^{2}B}\left(\frac{rN^{2}}{B}\right)^{\prime}-\frac{1}{r^{2}}$
(8) $\displaystyle G^{\theta}_{\theta}=G^{\phi}_{\phi}$
$\displaystyle=\frac{1}{2r^{2}B}\left[\frac{r^{2}(N^{2})^{\prime}}{B}\right]^{\prime},$

from which we may immediately obtain the stress-energy tensor via
$T^{\mu}_{\nu}=G^{\mu}_{\nu}/8\pi$.

The weak and null energy conditions must be met by all observers within the
relevant class. Because the spacetimes under consideration are spherical
symmetric, without loss of generality, we restrict our attention to the
equatorial plane, within which, we may write the momentum of an arbitrary
observer as $p_{\mu}=(-e,X_{r},0,\ell)$, which must satisfy

$p^{\mu}p_{\mu}=-\frac{e^{2}}{N^{2}}+\frac{B^{2}}{N^{2}}X_{r}^{2}+\frac{\ell^{2}}{r^{2}}=-\mu^{2},$
(9)

where $\mu>0$ for timelike observers and $\mu=0$ for null observers. The
statement of the null and weak energy conditions,
$p_{\mu}T^{\mu}_{\nu}p^{\nu}\geq 0$ for all $p^{\mu}$, becomes,

$-\frac{\mu^{2}}{r^{2}}\left[\frac{1}{B}\left(\frac{rN^{2}}{B}\right)^{\prime}-1\right]\\\
+\frac{\ell^{2}}{r^{4}}\left\\{\frac{1}{2B}\left[\frac{r^{4}}{B}\left(\frac{N^{2}}{r^{2}}\right)^{\prime}\right]^{\prime}+1\right\\}\geq
0.$ (10)

That is, for the weak energy condition we require the coefficients of both
$\mu^{2}$ and $\ell^{2}$ to be non-negative for all $r>r_{*}$, while for the
null energy condition we require only the latter.

Again, it is instructive to consider the case when $B(r)=1$, for which we find

$8\pi
p_{\mu}T^{\mu}_{\nu}p^{\nu}=\mu^{2}\left(\frac{f}{r^{2}}+\frac{f^{\prime}}{r}\right)+\frac{\ell^{2}}{r^{2}}\left(\frac{f}{r^{2}}-\frac{f^{\prime\prime}}{2}\right),$
(11)

where $f$ is defined as in Equation 3. The weak energy condition requires, in
this case, that $f/r^{2}+f^{\prime}/r\geq 0$ and
$f/r^{2}-f^{\prime\prime}/2\geq 0$ at all $r>r_{*}$. The first implies that
for spacetimes exhibiting S1-type naked singularities and that obey the weak
energy condition, $f^{\prime}$ must diverge positively; were $f^{\prime}$ to
diverge negatively, $f/r$ must diverge as quickly, and the singularity would
be of type S0. Similarly, the second implies that for spacetimes exhibiting
S2-type naked singularities and that obey the weak and/or null energy
condition, $f^{\prime\prime}$ must diverge negatively (or be S0).

For our example spacetime, Reissner-Nordström, these conditions reduce to

$\frac{f}{r^{2}}+\frac{f^{\prime}}{r}=\frac{Q^{2}}{r^{4}}\geq
0~{}~{}\text{and}~{}~{}\frac{f}{r^{2}}-\frac{f^{\prime\prime}}{2}=\frac{2Q^{2}}{r^{4}}\geq
0,$ (12)

both of which are satisfied unconditionally. Therefore, the Reissner-Nordström
spacetime satisfies the weak energy condition (and thus the null energy
condition). In Table 1, we note which naked singularity spacetimes pass which
energy condition.

##  3 Null and Timelike Geodesics about Spherically Symmetric Naked
Singularities

By definition, for a spacetime to contain a naked singularity, timelike
geodesics that connect the singularity to infinity, $i_{+}$, must exist. This
does not mean that all timelike geodesics must extend from negative timelike
infinity, $i_{-}$, to $i_{+}$. Nor does it require that timelike geodesics not
become generally trapped by the singularity itself. In practice, however, for
all stationary, spherically symmetric spacetimes with naked singularities of
type P0z, P0r, or S0z, neither of these problems appear. Apart from a set of
finely tuned cases, comprising a set of measure zero in $(e,\ell)$ (see
below), generically null and timelike geodesics escape the near-singularity
region in finite time. We now turn to demonstrating and quantifying these
facts.

###  3.1 Effective Potentials and Inner Turning Points

All of the spacetimes described by Equation 1 admit two constants of motion
associated with the timelike and azimuthal symmetries. As done in Section 2.2,
these are typically expressed in terms of an energy, $p_{t}=e$, and angular
momentum, $p_{\phi}=\ell$. The spherical symmetry implies that all motion is
planar, and without loss of generality we may restrict our attention to motion
confined to the equatorial plane. Therefore, the final constant of motion is
the effective mass of the underlying particle.

For photons, which are massless, the null condition is gives,

$-\frac{e^{2}}{N^{2}}+\frac{B^{2}}{N^{2}}{p^{r}}^{2}+\frac{\ell^{2}}{r^{2}}=0,$
(13)

from which we have

${p^{r}}^{2}=\frac{1}{B^{2}}\left[e^{2}-\frac{N^{2}\ell^{2}}{r^{2}}\right]=\frac{1}{B^{2}}\left[e^{2}-V^{n}_{\rm
eff}(r,\ell)\right],$ (14)

where $V^{n}_{\rm eff}(r,\ell)=N^{2}\ell^{2}/r^{2}$ is the effective potential
for null particles. Whether or not photons will encounter an inner turning
point, and therefore extend from $\mathscr{I}_{-}$ to $\mathscr{I}_{+}$,
depends on which class of singularity is present and the photon’s angular
momentum.

For all type-0z and type-0r singularities (i.e., P0r, P0z, and S0z),
$V^{n}_{\rm eff}(r,\ell)$ diverges at $r_{*}$ by definition, and therefore for
any $e$ and $\ell\neq 0$, by continuity and asymptotic flatness, there must
exist some $r>r_{*}$ at which ${p^{r}}^{2}=0$, and hence photons generally
encounter an inner turning point. For other types of singularities, the
equation of motion of photons at the singularity may remain regular. We will
focus our attention on the type P0r, P0z, and S0z singularities henceforth
unless otherwise stated.

For massive particles, the equation of motion departs from that of photons by
the introduction of the particle mass, $m$,

${p^{r}}^{2}=\frac{1}{B^{2}}\left[e^{2}-V^{t}_{\rm eff}(r,\ell)\right],$ (15)

where

$V^{t}_{\rm eff}(r,\ell)=V^{n}_{\rm eff}(r,\ell)+N^{2}m^{2}$ (16)

is the effective potential for massive particles. Because $N^{2}$ is positive
definite for all $r>r_{*}$ (because the singularity is naked!) $V^{t}_{\rm
eff}(r,\ell)>V^{n}_{\rm eff}(r,\ell)$ generally. This has the significant
consequence that anywhere photons encounter an inner turning point, massive
particles with equal energy and angular momenta will do so as well at larger
radii. Massive particles with lesser energy will encounter an inner turning
point even further away from the singularity (see, e.g., the examples in
Figure 1). This fact has important consequences for the astronomical
phenomenology of accreting systems.

###  3.2 Time Delays at Inner Turning Points

Having established that for the nakedly singular spacetimes of interest (P0r,
P0z, and S0z) that all timelike geodesics with $\ell\neq 0$ will encounter a
radial turning point prior to the singularity, we now turn to the practical
question of how long this process takes as measured by distant observers.
Generally, the equations of motion for an outwardly moving massive particle
are,

$p^{t}=\frac{e}{N^{2}}~{}~{}\text{and}~{}~{}p^{r}=\sqrt{\frac{e^{2}-V^{t}_{\rm
eff}(r,\ell)}{B^{2}}}.$ (17)

from which we have the time to propagate from $r$ inward to $r_{\rm tp}$ and
back is, by symmetry,

$\Delta t=2\int_{r_{\rm tp}}^{r}dr\frac{p^{t}}{p^{r}}=2\int_{r_{\rm
tp}}^{r}dr\frac{eB}{N^{2}\sqrt{e^{2}-V^{t}_{\rm eff}(r,\ell)}}.$ (18)

For a black hole, the smallest value of $N^{2}(r)$ sets the timescale by
virtue of the gravitational redshift. Even in that case, the propagation time
diverges only logarithmically with the maximum redshift (see Appendix B). For
naked singularities, $N^{2}(r)$ never vanishes outside of $r_{*}$, and thus it
is the radical in the denominator, which vanishes at the turning point, that
sets the propagation timescale for massive particles.

The integral in Equation 18 may be evaluated approximately by Taylor expanding
the radical term about the turning point. That is, near $r_{\rm tp}$,

$e^{2}-V^{t}_{\rm eff}(r,\ell)={V^{t}_{\rm eff}}^{\prime}(r-r_{\rm
tp})+\dots,$ (19)

where ${V^{t}_{\rm eff}}^{\prime}$ is the first derivative with respect to $r$
of $V^{t}_{\rm eff}(r,\ell)$ evaluated at $r_{\rm tp}$ and we have suppressed
the dependence on $\ell$ for clarity and ignored the special case in which
${V^{t}_{\rm eff}}^{\prime}$ vanishes333For a given a particle energy, $e$,
${V^{t}_{\rm eff}}^{\prime}$ will vanish at the turning point for orbits with
angular momentum set by
$\ell=2N\left(e^{2}-N^{2}m^{2}\right)^{3/2}/\left[(N^{2})^{\prime}e^{2}\right]$,
where all quantities are evaluated at $r_{\rm tp}$. For infinitesimally larger
or smaller $e$ and/or $\ell$, ${V^{t}_{\rm eff}}^{\prime}\neq 0$. Because we
will be interested in the behavior of distributions of massive particles
(i.e., accretion flows), we will ignore higher order derivatives of
$V^{t}_{\rm eff}$ in the estimation of propagation timescales as such terms
are relevant for at most a set of measure zero in $(e,\ell)$.. Therefore, the
total time delay,

$\Delta t\gtrsim\frac{eB(r_{\rm tp})}{N^{2}(r_{\rm tp})}\sqrt{\frac{r-r_{\rm
tp}}{{V^{t}_{\rm eff}}^{\prime}}},$ (20)

is explicitly bounded with respect to $r_{\rm tp}$ (in stark contrast to the
behavior near horizons, which typically diverges logarithmically with the
distance from the horizon, see Appendix B). Thus, $\Delta t$ is generally
finite, typically of order the light crossing time of the system.

##  4 Spectral Signatures of Naked Singularities

Summarizing the previous section: P0r, P0z, and S0z singular spacetimes,
infalling massive particles will generally encounter an inner turning point
prior to reaching the singularity on a finite timescale as seen by a distant
observer. Importantly, this goes beyond proving the existence of a family of
outward-going timelike geodesics (which exist by definition for naked
singularities), but rather proves that generic infalling timelike geodesics
will necessarily exit the near-singularity region, avoiding the singularity
altogether. This result has significant consequences for the spectral
signatures of these spacetimes, set entirely by baryonic physics in the non-
singular portion of the spacetime.

We fashion an argument similar to Broderick et al. (2009b) and Broderick et
al. (2015), based upon the observed low-efficiency of the accretion flows in
Sgr A* and M87*, and constraints at shorter wavelengths on the thermalization
of the unradiated kinetic energy. This is predicated on four underlying
assumptions and observations.

Figure 1: Null (dotted) and timelike (solid) effective potentials for
Schwarzschild (dark red) and Reissner-Nordström (blue) spacetimes for
$Q=1.01M$ and $\ell=0.5\sqrt{27}$. The trajectory for an infalling particle
with $e=m$ at infinity is shown schematically in green, including the
reflection at the inner turning point (open point), associated accretion shock
when the outgoing and infalling flows collide, and the ultimate settling flow
(dashed) depositing material at the potential minimum (solid point).

1\. The accretion flows on Sgr A* and M87* are radiatively inefficient. That
is, only a small fraction of the liberated gravitational binding energy is
radiated during infall. For both sources this is both theoretically
anticipated and observed. For objects with accretion rates below 0.1%
Eddington, Coulomb scattering is insufficient to redistribute energy from the
ions, which by virtue of their mass accrue the majority of the gravitational
binding energy, and the electrons, responsible for the overwhelming majority
of the emission (Narayan et al., 1995, 1998; Narayan & McClintock, 2008; Yuan
& Narayan, 2014). As a result, the vast majority of the kinetic energy of the
accreting material is advected towards the central object. In practice, the
bolometric luminosity of Sgr A* ($\simeq 10^{36}~{}{\rm erg}\,{\rm s}^{-1}$)
and M87* ($\sim 10^{42}~{}{\rm erg}\,{\rm s}^{-1}$) are well below estimates
of $\dot{M}c^{2}$ ($\sim 3\times 10^{38}~{}{\rm erg}\,{\rm s}^{-1}$ and
$10^{44}~{}{\rm erg}\,{\rm s}^{-1}$, respectively), arrived at by Faraday
rotation limits and EHT observations.

2\. Interaction between inflowing and outflowing accretion streams efficiently
thermalizes bulk flows. After encountering the near-singularity turning point,
accreting gas will generate a local, if bound, outflow. For approximately
axisymmetric accretion flows, which encompasses all models under consideration
for Sgr A* and M87*, these outflows will collide with subsequently accreting
gas, though potentially at a distinct azimuthal location. The resulting,
relativistic counter-streaming gas flows are subject to multiple mechanisms
for thermalizing their bulk energy, chief among which are shocks that will
thermalize the energy on the shock-crossing time, which is less than or
similar to the typical orbital timescale. Importantly, these dissipation
mechanisms are dependent solely on the baryonic physics of the accreting
material, and independent of the details of the singular spacetime. Thus, it
is generally anticipated that the kinetic energy of the bulk flow fully
thermalizes on very short timescales.

3\. The thermalized region remains compact. Following thermalization, we will
assume that the hot, thermalized region remains compact, i.e., smaller than
the spacetime’s photon orbit (should one exist). This condition is a natural
consequence of the very short thermalization timescale at the shocks,
following which the gas will form into a settling flow toward the minimum of
$V^{t}_{\rm eff}(r,0)$, $r_{\rm tp,min}$ which is generally outside $r_{*}$.
Moreover, in the cases of Sgr A* and M87*, the shadows observed by EHT
indicate that the accretion flow remains well ordered down to the photon
orbit, implying that any accretion shock must appear inside this region in
those sources.

4\. The non-gravitational impact of the singularity is localized to its
immediate neighborhood. While the physics of the singularity is unknown, we
presume that its non-gravitational influence is felt only by material within
some small characteristic length scale, $l_{*}$ (e.g., the Planck length).
While it may be possible to engineer spacetimes with naked singularities with
$r_{\rm tp,min}-r_{*}<l_{*}$, such situations represent an extreme fine tuning
and we do not consider them further.

Subject to the above, the general result is an optically thick, compact
settling flow that efficiently radiates at a rate set by that at which energy
is advected inward. That is, the luminosity and temperature as seen from a
distant observer is

$L_{\infty}=4\pi R^{2}\sigma
T_{\infty}^{4}~{}~{}\text{and}~{}~{}T_{\infty}=\left(\frac{\eta_{\rm
ad}\dot{M}c^{2}}{4\pi R^{2}\sigma}\right)^{1/4},$ (21)

where $\eta_{\rm ad}\gtrsim 0.1$ is the fraction of the accreting rest mass
energy that has liberated during infall that is advected into the settling
flow, $R$ is the radius of the black hole shadow, and $\sigma$ is the Stefan-
Boltzmann constant.

##  5 Observational Constraints

The general development of an optically thick, baryonic photosphere around a
broad class of naked singularities (P0z, P0r, S0z) creates significant
challenges for attempts to probe these spacetimes by looking for lensing
signatures within the shadow (e.g., Sgr A* Paper VI, ). Photons traveling on
initially ingoing geodesics will be absorbed by the photosphere, which then
prevents the additional lensing components from reaching distant observers.
Nevertheless, strong empirical constraints on singular spacetime can be
obtained by looking for the thermal emission from these photospheres directly.

Because $T_{\infty}$ is set by $\dot{M}$, the associated thermal emission
spectrum need not peak near 230 GHz, the frequency at which EHT observed Sgr
A* and M87* in 2017 and 2018. Indeed, for the inferred accretion rates in
those sources, it peaks at much higher frequency, with the consequence that
EHT observations alone place only a weak constraint on the existence of an
additional shock-heated thermal component. However, when the size constraints
from EHT are supplemented with spectral flux measurements in the NIR, optical
and UV, such a component can be excluded altogether. We now consider each
target in turn, for which we summarize the state of the EHT observations and
associated implied ranges for $T_{\infty}$, and compare to high-frequency flux
measurements.

###  5.1 Sgr A*

Figure 2: SED of Sgr A* from a radially structured, non-thermal synchrotron
emitting region (black dotted) and a thermal bump due to an optically thick
settling flow (red) for $\dot{M}$ ranging from $10^{-9}~{}{\rm
M_{\odot}}\,{\rm yr}^{-1}$ to $10^{-7}~{}{\rm M_{\odot}}\,{\rm yr}^{-1}$. For
comparison, the 2017 EHT 230 GHz (1.3 mm) compact flux estimate (open blue
square, Wielgus et al., 2022), contemporaneous broadband SED (open black
points, Sgr A* Paper II, ), and historical NIR upper limits from Broderick et
al. (2009b) on the quiescent flux (gray closed points) are shown.

Sgr A* is the bright radio source associated with the putative supermassive
black hole at the center of the Milky Way. It’s spectral energy distribution
(SED), shown in Figure 2, is well described by a radiatively inefficient
accretion flow that emits via synchrotron. Below a THz, Sgr A* exhibits an
inverted SED, with a spectral index ($F_{\nu}\propto\nu^{-\alpha}$) of
$\alpha=-0.4$, characteristic of a self-absorbed, radially structured
synchrotron source (Blandford & Königl 1979; Broderick & Loeb 2006; Broderick
et al. 2009a; Yuan et al. 2004; Sgr A* Paper II). From near-infrared (NIR)
through the X-ray wavelengths, Sgr A*’s SED is well described by a power law
with $\alpha=1.25$, typical of optically thin nonthermal synchrotron sources
(Sgr A* Paper II).

The mass of and distance to Sgr A* are the best known of any black hole
candidate, established most accurately via the orbits of individual stars that
pass within $\sim 10^{2-3}~{}{\rm AU}$ of it, finding $M=(4.297+0.013)\times
10^{6}~{}{\rm M_{\odot}}$ and $D=8.277\pm 34~{}{\rm kpc}$ (GRAVITY
Collaboration et al., 2022; Do et al., 2019). However, that Sgr A* is
supermassive has been established in multiple ways. It lies at the center of a
stellar cusp, the structure and dynamics of which require a supermassive
central object (Genzel et al., 2010). It sits still very nearly at the minimum
of the Galactic potential, with a velocity toward the North Galactic pole of
$-0.85\pm 0.75~{}{\rm km}\,{\rm s}^{-1}$; if this is the result of dynamical
friction it requires $M>>1\times 10^{6}~{}{\rm M_{\odot}}$ (Reid & Brunthaler,
2020). Similarly, the distance to Sgr A* has been further verified by direct
parallax measurements of masers in the Galactic center (Reid et al., 2019).
This mass and distance is fully consistent with that inferred by EHT through
directly imaging the 1.3 mm emission on Schwarzschild scales, which found
$4.0^{+1.1}_{-0.6}\times 10^{6}~{}{\rm M_{\odot}}$ (; ).

The EHT images of Sgr A* provide at least three lines of strong evidence for
the canonical near-virial accretion flow model. First, the presence of a dark
shadow surrounded by a bright ring of emission matches quantitatively that
anticipated by gravitational lensing about a $4\times 10^{6}~{}{\rm
M_{\odot}}$ black hole, and in particular the existence of a circular photon
orbit (Sgr A* Paper VI; Kocherlakota et al. 2021; Broderick et al. 2023).
Second, the shape and amplitude of the power spectrum of brightness
fluctuations matches those associated with the turbulence responsible for
angular momentum transfer through the accretion flow in by general
relativistic magnetohydrodynamic (GRMHD) simulations (Sgr A* Paper V; Georgiev
et al. 2022). Third, the linear polarization map is consistent with a
predominantly toroidal magnetic field, again consistent with that anticipated
by GRMHD simulations (; ; Broderick & Loeb 2006). Therefore, while the EHT
images of Sgr A* do not preclude any interior emission444The EHT is sensitive
only to emission with brightness temperatures exceeding $\sim 10^{8}~{}{\rm
K}$, and thus would not be able to detect, e.g., the thermal emission from a
settling flow surrounding a naked singularity (Broderick & Narayan, 2006).,
they do confirm the absence of any obstruction beyond the photon orbit.

At $L=10^{36}~{}{\rm erg}\,{\rm s}^{-1}$, the total bolometric luminosity of
Sgr A* is well below its Eddington limit of $6\times 10^{44}~{}{\rm erg}\,{\rm
s}^{-1}$. Despite this, the high observed Faraday rotation measure implies
cold plasma densities at distances of $10M$-$100M$ of $10^{6}~{}{\rm
cm}^{-3}$, and corresponding to accretion rates of $\dot{M}\sim 10^{-8}~{}{\rm
M_{\odot}}\,{\rm yr}^{-1}\approx 10^{-7}\dot{M}_{\rm Edd}$, well into the RIAF
regime (Agol, 2000; Quataert & Gruzinov, 2000; Marrone et al., 2007; Yuan &
Narayan, 2014). Similar conclusions follow from the direct modeling of the EHT
images, which recover bolometric luminosities of $7-9\times 10^{35}~{}{\rm
erg}\,{\rm s}^{-1}$ with associated accretion rates of $0.5-1\times
10^{-8}~{}{\rm M_{\odot}}\,{\rm yr}^{-1}$ (Sgr A* Paper V). Thus, the
empirically derived radiative efficiency is at most $\eta\equiv
L/\dot{M}c^{2}\sim 10^{-3}$, implying that the vast majority of the liberated
gravitational potential energy is advected inward with the accretion flow.

From Equation 21, the temperature of an optically thick settling flow in Sgr
A* would be,

$T_{\infty}\sim 10^{4}\left(\frac{\eta_{\rm
ad}}{0.1}\right)^{1/4}\left(\frac{\dot{M}}{10^{-8}~{}{\rm M_{\odot}}\,{\rm
yr}^{-1}}\right)^{1/4}~{}{\rm K},$ (22)

which produces a thermal spectrum that peaks at optical wavelengths, shown for
$\eta_{\rm ad}=0.1$ and various $\dot{M}$ in Figure 2.555Variations in the
central mass (and thus $R$) or the distance have negligible impact in
comparison to the direct dependence on $\dot{M}$ for both Sgr A* and M87*. For
comparison, we also show the SED contemporaneous with the 2017 EHT observation
campaign with a broken power-law fit approximating the self-absorbed
synchrotron spectrum. We supplement these with the historical limits on the
quiescent NIR flux listed in Table 2 of Broderick et al. (2009b). Optical flux
estimates are not available due to the large extinction in the direction of
the Galactic center. Nevertheless, the NIR flux limits conclusively exclude
the existence of such an additional thermal component.

###  5.2 M87*

Figure 3: Spectral energy density of M87* from a structured, non-thermal
synchrotron emitting region (black dotted) and a thermal bump due to an
optically thick settling flow (red) for $\dot{M}$ ranging from $10^{-3}~{}{\rm
M_{\odot}}\,{\rm yr}^{-1}$ to $10^{-1}~{}{\rm M_{\odot}}\,{\rm yr}^{-1}$. For
comparison, the 2017 EHT 230 GHz (1.3 mm) compact flux estimate (open blue
square, M87* Paper VI, ), contemporaneous broadband SED (open black points,
EHT MWL Science Working Group et al., 2021), and historical NIR/Optical/UV
fluxes from Broderick et al. (2015) (gray closed points) are shown.

The environment and behavior of M87* differs from Sgr A* in many relevant
aspects. It is located at the center of the giant elliptical galaxy, Messier
87, which is itself among the brightest galaxies in the Virgo cluster (Zhu et
al., 2014). A bright, broadband radio source, M87* sits at the bottom of a
prominent jet, a narrowly collimated relativistic outflow which extends to
$\sim 30~{}{\rm kpc}$ (de Gasperin et al., 2012). The origin of this jet is a
topic of intense study, with both of the two main competing theoretical models
powered by rotation and invoking twisted mangetic fields, either embedded
within an orbiting accretion flow (Blandford & Payne, 1982) or in a black hole
horizon (Blandford & Znajek, 1977)666The Blandford-Znajek model invokes black
hole spin, and thus violates the underlying assumption of spherical symmetry.
Nevertheless, we neglect spin further in our treatment here.. Both models
require a surrounding accretion flow to source and confine the magnetic field,
and thus $\dot{M}$ is naturally related to the jet power.

Numerous independent estimates find a total jet power range of $\sim
10^{44}~{}{\rm erg}\,{\rm s}^{-1}$-$10^{45}~{}{\rm erg}\,{\rm s}^{-1}$ (see
§2.2 of Broderick et al., 2015). These estimates range across physical scale,
and therefore temporal delay, from the $30~{}{\rm kpc}$ radio structure
powered by the jet to HST-1, presumed to be powered by a jet shock (de
Gasperin et al., 2012; Stawarz et al., 2006). This may be compared to
$\dot{M}$ inferred from jet launching models, which under conservative
assumptions are related by

$\dot{M}\gtrsim\frac{L_{\rm jet}}{2c^{2}}\approx 10^{-3}~{}{\rm
M_{\odot}}\,{\rm yr}^{-1},$ (23)

(see discussion in §2.1 of Broderick et al., 2015). This in close agreement
with the $\dot{M}$ inferred from the Faraday rotation within the inner $40M$,
assuming that the Faraday screen lies within an RIAF (Kuo et al., 2014).

The mass of M87* has been measured primarily in two ways. First, via the
dynamics of the orbiting stellar population as inferred from their cumulative
spectra. This measurement is sensitive to assumptions regarding the dark
matter cusp, and most recently found to be $6.14^{+1.07}_{-0.62}\times
10^{9}~{}{\rm M_{\odot}}$ (Gebhardt et al., 2011). Second, through the
dynamics of orbiting gas, which has typically found much smaller masses (Walsh
et al., 2013), though see Jeter et al. (2019) and Jeter & Broderick (2021).
The larger, stellar dynamics mass is consistent with that inferred from the
size of the bright ring observed by the EHT, $M=(6.5\pm 0.7)\times
10^{9}~{}{\rm M_{\odot}}$, and the one that we make use of here (; ). The
distance is estimated either by using surface brightness fluctuations (Bird et
al., 2010) or from the tip of the red giant branch (Blakeslee et al., 2009;
Cantiello et al., 2018). We follow M87* Paper VI and combine these to arrive
at the estimate $16.8^{+0.8}_{-0.7}~{}{\rm Mpc}$.

As with Sgr A*, the EHT observations of M87* are broadly consistent with the
underlying astrophysical picture of the source. The brightness asymmetry is in
good agreement with the dynamics of the orbiting material and the orientation
of the jet on larger scales. The timescale over which EHT images of M87*
evolve is consistent with those found in GRMHD simulations, i.e., images of
M87* appear similar on neighboring days and differs on a week or longer (; ;
Event Horizon Telescope Collaboration et al. 2024c). The magnetic field
geometry of the jet footprint in GRMHD simulations produces “twisty” linear
polarization patterns that are quantitatively similar to that observed in EHT
images (; ). The size of the inner shadow is indicative of the strong
gravitational lensing at the circular photon orbit, and like Sgr A*, provides
strong evidence that there is no interfering settling flow outside of this
radius.

The SED of M87* is well described by a self-absorbed synchrotron jet source
that transitions from optically thick to thin near a THz, shown in Figure 3
(Blandford & Königl, 1979). Below the transition frequency, the SED is
essentially flat ($\alpha=0$); above the transition the SED is well fit by a
power law with $\alpha=1.25$. The isotropic bolometric luminosity of M87* is
approximately $10^{42}~{}{\rm erg}\,{\rm s}^{-1}$, with the bulk of that
appearing in the sub-mm (EHT MWL Science Working Group et al., 2021). Were
this completely due to accretion, for $\dot{M}=10^{-3}~{}{\rm M_{\odot}}\,{\rm
yr}^{-1}$, the radiative efficiency would be $\eta\sim 10^{-2}$, reducing
further if the jet is powered in part by some other source (e.g., spin of the
naked singularity). These estimates are in good agreement with the GRMHD
modeling of M87* based on the EHT images, which find $\dot{M}$ ranging from
$3\times 10^{-2}~{}{\rm M_{\odot}}\,{\rm yr}^{-1}$ to $10^{-4}~{}{\rm
M_{\odot}}\,{\rm yr}^{-1}$ and an corresponding radiative efficiency range of
$10^{-4}$ to $10^{-1}$ (M87* Paper V). Thus, like in Sgr A*, the majority of
the liberated gravitational potential energy in M87* is advected toward the
black hole.

For M87*, Equation 21 gives the equilibrium temperature for a putative
optically thick settling flow to be

$T_{\infty}\sim 4\times 10^{3}\left(\frac{\eta_{\rm
ad}}{0.1}\right)^{1/4}\left(\frac{\dot{M}}{10^{-3}~{}{\rm M_{\odot}}\,{\rm
yr}^{-1}}\right)^{1/4}~{}{\rm K}.$ (24)

At this temperature, the associated thermal spectrum peaks in the NIR, though
extends well into optical wavelengths. This is shown for various $\dot{M}$ in
Figure 3. Measurements of the SED of M87* made contemporaneously with the 2017
EHT campaign are also shown, along with a broken power-law fit approximating a
self-absorbed synchrotron source. We include the historical measurements from
Broderick et al. (2015) to provide some sense of the variation with time.

Because M87* is not obscured at optical and UV wavelengths, the measured SEDs
extend across nearly the entire range of frequencies relevant for the thermal
peak. Note that with the exception of the EHT compact flux measurement, all of
the flux measurements are contaminated with emission from beyond $10M$ (e.g.,
HST-1). Nevertheless, even with potential sizable contributions from outside
the immediate vicinity of M87*, the additional thermal component is
conclusively excluded.

##  6 Conclusions

Naked singularities differ from other black hole foils in the potential for
the introduction of unknown and uncontrolled physics, the effects of which
must somehow be circumscribed. Nevertheless, generally, a wide class of naked
singularity spacetimes (P0z, P0r, and S0z) will contain inner turning points
for timelike geodesics. As a result, accreting baryonic gas will reverse
direction in a short time as seen far from the object, rapidly encounter
subsequently accreting gas, shock, and ultimately produce a settling flow
outside of the singularity. This process occurs entirely within the non-
singular part of the spacetime on scales that have already been observed; that
it is independent of the unknown physics at the singularity corresponds to the
very modest assumption that the singularity’s impact is confined to matter
that comes within some small distance of it, e.g., comparable to the Planck
length.

That accreting material impinging upon the settling flow will thermalize the
remainder of its kinetic energy, i.e., that gained during infall, is
guaranteed by known baryonic physics. Therefore, for this class of naked
singularity spacetimes, an enveloping, optically thick, thermally emitting,
photosphere, is an inevitable consequence. This photosphere may block
initially ingoing null geodesics, rendering tests based on directly observing
lensed emission inside the shadow challenging. However, for sources that are
sufficiently radiatively inefficient and compact, this photosphere can be
bright, and therefore its existence observationally probed.

Sgr A* and M87* satisfy both of these conditions. Both sources are radiatively
inefficient, i.e., only a small fraction of the liberated gravitational
potential energy is radiated. This is anticipated by theoretical models of the
accretion process that have received significant empirical support from EHT
observations and directly demonstrated via comparison of the source broadband
luminosity to estimates of the accretion rate. EHT observations of both
sources also constrains the distance at which departures can occur from a
black hole accretion flow to within the circular photon orbit. That is, in
both sources any putative baryonic photosphere would need to be compact.

The inferred $\dot{M}$ in both sources would generate an emitting photosphere
with a temperature, as measured by a distant observer, $\sim 10^{4}~{}{\rm
K}$, and therefore peaking in the NIR, optical, or UV. The presence of such a
thermal component can be conclusive excluded by flux measurements made
contemporaneously with the EHT campaign in 2017. That is, neither Sgr A* nor
M87* can be a type P0z, P0r, or S0z naked singularity spacetime. Practically,
this set of classes encompass all but one of the asymptotically flat,
spherically symmetric, naked singularity spacetimes that we could find in the
literature (listed in Table 1).

There are at least three natural ways for naked singularities to evade the
observational constraints we present. First, the singularity may be of a
different type, i.e., S1, S2, or P0j. We note, however, that simply being a
different type is not sufficient. While timelike geodesics in P0r, P0z, and
S0z singular spacetimes have generic behaviors that result in now-excluded
observational consequences, it is clearly possible for other types of naked
singularity spacetimes to do so. In Appendix C, we explicitly demonstrate that
a type S1 naked singularity spacetime (and the only example we could find in
the literature) may be excluded by the argument described here. Of course, we
have also neglected any observable impact from the singularity itself.

Second, we have not considered the impact of spin. Neither the class of naked
singularity spacetimes we exclude nor the specific examples listed in Table 1
are spinning. Nevertheless, the formal possibility remains that the
introduction of spin may make qualitative differences in the strength of
constraints. Incorporating spin complicates the question of the ultimate fate
the accreting gas, substantially increasing of the size of the class of
geodesics that must be surveyed. Therefore, We leave an investigation into
general classes of spinning spacetimes for future work (e.g., that presented
by Johannsen et al. 2016 and considered in Salehi et al. 2024).

Third, the unknown physics introduced by the singularity may include non-local
interactions that extend past the inner turning point. Were this to be the
case, the chief advantage of the constraints we present, that they derive
wholly form non-singular regions within the spacetime, would be eliminated.
However, we consider this situation to be a rather different problem.

Finally, while we have argued that neither Sgr A* nor M87* can harbor a naked
singularity of the proscribed types, this says nothing about whether or not
naked singularities can exist in other sources. Natural extensions of our
argument can be made to other AGN and X-ray binaries in low-luminosity states
(e.g., XTE J1118+480). The main limitation will be, in the absence of images
that resolve the gravitational radius, the need for an additional assumption
regarding the compactness of the putative additional thermal component.
Nevertheless, future ground-based (Blackburn et al., 2019; Doeleman et al.,
2023) and space-based arrays promise to extend the reach of direct imaging to
a much larger population of sources (Pesce et al., 2021, 2022), and with it
strong evidence excluding naked singularities in many additional sources.

We thank Aaron Held for helpful discussions on the potential of indirect
detection of naked singularities. 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. receives
additional financial support from the Natural Sciences and Engineering
Research Council of Canada through a Discovery Grant.

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##  Appendix A Point-like Singularity Classes

For P-type singularities, necessarily the divergence of higher-order
derivative terms imply that $N^{2}/r^{2}B^{2}$ diverges, and thus all P-type
singularities are P0.

We begin the proof of this by first looking at the case when $B=1$, for which
the order class is set by the lowest-derivative term of $f/r^{2}$,
$f^{\prime}/r$, or $f^{\prime\prime}$ that diverges. However, for a
singularity to be anything other than P0-type, $f$ must vanish at $r=0$
sufficiently rapidly. But in that case, by L’Hopital’s rule,

$\lim_{r\rightarrow 0}\frac{f^{\prime}}{r}=2\lim_{r\rightarrow
0}\frac{f}{r^{2}},$ (A1)

and thus it cannot be type P1. Similarly, if it is not P1, $f^{\prime}$ must
vanish at $r=0$ but then,

$\lim_{r\rightarrow 0}f^{\prime\prime}=\lim_{r\rightarrow
0}\frac{f^{\prime}}{r},$ (A2)

and it cannot be P2. Thus, if a spacetime has a P-type singularity, has $B=1$,
and is not type P0, then it cannot be P1 or P2, yielding a contradiction.
Because P-type singularities with $B=1$ exist, this immediately implies that
all such singularities must be type P0.

We proceed with $B\neq 1$ with similar line of argument. A P-type singularity
that is not type P0 requires that $N^{2}/B^{2}$ vanish at $r=0$ faster than
$r^{2}$. But in that case,

$\lim_{r\rightarrow
0}\frac{(N^{2}/B^{2})^{\prime}}{r}=\frac{1}{2}\lim_{r\rightarrow
0}\frac{N^{2}/B^{2}}{r^{2}},$ (A3)

and the first 1st-derivative term must also be finite. If $B$ is nonzero at
$r=0$, then, $N^{2}$ must vanish at $r=0$ and by L’Hopital’s rule,

$\lim_{r\rightarrow 0}\frac{(N^{2})^{\prime}/r}{B^{2}}=\lim_{r\rightarrow
0}\frac{N^{2}/r^{2}}{B^{2}}.$ (A4)

If $B$ vanishes at $r=0$ then $N^{2}/r^{2}$ must also vanish at $r=0$ (which
is a stronger statement than $N^{2}$ vanishing), and the above still applies.
Therefore, the second 1st-derivative term must also be finite. That is, if a
spacetime is not type P0, it cannot be type P1.

We repeat this argument to show that a spacetime that is neither P0 nor P1, it
cannot be P2. For the spacetime to not be type P1, $(N^{2})^{\prime}$ must
vanish at $r=0$ at least as fast as $r$, and therefore, by L’Hopital’s rule,

$\lim_{r\rightarrow
0}\frac{[(N^{2})^{\prime}/B]^{\prime}}{B}=\lim_{r\rightarrow
0}\frac{[(N^{2})^{\prime}/B]/r}{B},$ (A5)

which immediately implies that it cannot be type P2. Finally, as before,
because P-type singularities exist, this immediately implies that all such
singularities must be type P0.

##  Appendix B Horizon Approach Timescale

For comparison with the timescale estimates in Section 3.2, here we make a
similar estimate for general spherically symmetric, stationary black hole
spacetimes. Such spacetimes are also described by Equation 1, though defined
by the vanishing of $N^{2}$ at some radius $r_{h}$. Therefore, near this
radius,

$N^{2}(r)=(N^{2})^{\prime}(r-r_{h})+\dots,$ (B1)

where $(N^{2})^{\prime}$ is evaluated at $r_{h}$ (suppressed for clarity).
Thus, the timescale for reaching a radius $r=r_{h}+s$ for small $s$ is,

$\displaystyle\Delta t_{h}$
$\displaystyle\approx\int_{r_{h}+s}^{r}dr\frac{eB}{(N^{2})^{\prime}(r-r_{h})\sqrt{e^{2}-V^{t}_{\rm
eff}}}$ (B2)
$\displaystyle\approx\frac{eB}{(N^{2})^{\prime}\sqrt{e^{2}-V^{t}_{\rm
eff}}}\ln\left(\frac{r-r_{h}}{s}\right).$

As noted in Section 3.2, this diverges logarithmically with $s$.

##  Appendix C Excluding 4D EGBQ

Despite not being type-P0, the 4D charged Einstein-Gauss-Bonnet (4D EGBQ)
naked singularity spacetime is still excluded for all parameter values
($\gamma$, $M$, and $Q$), and provides a natural example of the application of
the general argument made here to beyond the P0 class.

Figure 4: Effective potentials for three representative cases of the 4D EGBQ
naked singularity with comparisons to the astrophysically relevant range of
energies for accreting particles. Shown are $V^{t}_{\rm eff}$ for orbits with
$\ell=\sqrt{12}M$ in a black hole spacetime with ($\gamma=0.8$, $Q=0.2$, dark
red) with the radius of the horizon marked by the black point and the
potential shown by a dotted line inside, orbits with $\ell$ given by that at
the outermost ISCO (light blue dot) in a naked singularity spacetime
($\gamma=1$, $Q=0.2$, light blue), and orbits with $\ell=0$ in a naked
singularity spacetime ($\gamma=1$, $Q=1$, dark blue). For comparison, in green
the values of $e^{2}$ at the ISCO of the first singular spacetime (dashed) and
$e^{2}=1$ (solid) are indicated. While all examples have singularities at
$r_{*}>0$, only in the last case is the singularity visible in the plotted
range (large dark blue point).

For the 4D EGBQ spacetime,

$N^{2}(r)=1+\frac{r^{2}}{2\gamma M^{2}}\left(1-\sqrt{D}\right),$ (C1)

where

$D\equiv 1+\frac{8\gamma M^{3}}{r^{3}}-\frac{4\gamma M^{2}Q^{2}}{r^{4}}.$ (C2)

An event horizon exists when $\gamma\leq 1-Q^{2}/M^{2}$, and possesses a naked
singularity otherwise. The location of the singularity is set by the condition
that $D=0$, and is the largest positive root of

$r_{*}^{4}+8\gamma M^{2}r_{*}-4\gamma M^{2}Q^{2}=0.$ (C3)

Because $\lim_{r\rightarrow\infty}D=1$, by definition $D$ must be positive
definite for all $r>r_{*}$, with two important consequences.

First, at $r=r_{*}$, $N^{2}(r_{*})=1+r_{*}^{2}/(2\gamma M^{2})>1$. A particle
falling from infinity that is initially marginally bound has $e^{2}=m^{2}$,
and thus even with $\ell=0$, $e^{2}-V^{t}_{\rm eff}(r_{*},0)<0$ and there is
an inner turning point for some $r>r_{*}$. For accreting particles on orbits
with $\ell\neq 0$, $V^{t}_{\rm eff}(r_{*},\ell)>V^{t}_{\rm eff}(r_{*},0)$, and
thus we concluded that an inner turning point always exists for physical
accretion flows. The effective potential for an example radial orbit is shown
by the dark blue line in Figure 4, along with the $e=1$ line, illustrating
this behavior.

Second, $(N^{2})^{\prime}(r_{*})<0$, and hence $N^{2}(r_{*})$ is a local
maximum (in fact, it can be shown that $\lim_{r\rightarrow
r_{*}}(N^{2})^{\prime}(r)=-\infty$. Therefore, for accretion flows that
effectively transport energy and angular momentum so that matter plunges from
the outer innermost stable circular orbit (ISCO), $e^{2}=V^{t}_{\rm
eff}(r_{\rm ISCO},\ell_{\rm ISCO})<m^{2}$ and $\ell_{\rm ISCO}\neq 0$. Hence,
again, for such flows we generally have an inner turning point. The effective
potential for such a case is shown by the light blue line in Figure 4, where
$\ell=3.32M$ is the specific angular momentum at the ISCO, marked by the light
blue dot.

Two practical effects may increase the specific energy of infalling orbits
above unity. First, the accreting matter may begin far from the black hole
with non-zero kinetic energy, e.g., the Bondi radius is not at infinity.
However, such an increase is small – neither Sgr A* nor M87* are accreting
from a distant, relativistically hot plasma, and in both sources the
brightness temperatures measured by the EHT, $\sim 10^{10}~{}{\rm K}$, are
less than the anticipated virial temperatures. Moreover, even were this the
case, and some accreting particles were sufficiently to surmount $V^{t}_{\rm
eff}(r_{*},0)$, a significant fraction of particles would necessarily be
launched on orbits with $e<1$, and would therefore begin to form the inner
settling flow, that when created would interact with and capture the energy
from subsequently accreting particles of all masses.

Figure 5: Excluded region (blue) in the $\gamma$-$Q$ parameter space of the 4D
EGBQ naked singularity spacetime. The black region in the lower-left shows
parameter values for which the 4D EGBQ metric describes a black hole. The
dotted red line shows the approximation in Equation C5. Note that the axes are
logarithmic.

Second, the act of heating the settling flow via accretion necessarily
increases $e$ locally. Again, this effect is small in Sgr A* and M87*, neither
of which are expected to have relativistically hot settling flows (i.e., the
$kT_{\infty}\ll m_{p}c^{2}$). It remains formally possible for the thermal
energy to be sufficient to raise the specific energy of the settling flow from
the minimum to above $V^{t}_{\rm eff}(r_{*},0)$. For small charges
($(Q/M)^{6}\ll 64\gamma$), this condition is dominated by the minimum of the
effective potential, which for large $\gamma$ occurs at
$r\approx\gamma^{1/3}M$ and is $V^{t}_{\rm eff}(r,0)\approx 1-\gamma^{-1/3}$.
For large charges ($(Q/M)^{6}\gg 64\gamma$), it is dominated by the effective
potential at the singularity, which occurs at $r_{*}\approx(4\gamma
M^{2}Q^{2})^{1/4}$ and $V^{t}_{\rm eff}(r_{*},0)\approx\gamma^{-1/2}|Q/M|$.
Therefore, the energy gap to be overcome by the thermal energy is,

$V^{t}_{\rm eff}(r_{*},0)-V^{t}_{\rm
eff}(r,0)=\begin{cases}\gamma^{-1/3}&(Q/M)^{6}\ll 64\gamma\\\
\gamma^{-1/2}|Q/M|&(Q/M)^{6}\gg 64\gamma,\end{cases}$ (C4)

which implies that for the accretion-heating of the settling flow to drive
matter into the singularity, the coupling constant $\gamma$ must satisfy,

$\gamma\gg
10^{26}\begin{cases}\displaystyle\left(\frac{T_{\infty}}{10^{4}~{}{\rm
K}}\right)^{-3}&\displaystyle\frac{Q}{M}<2\times
10^{4}\left(\frac{T_{\infty}}{10^{4}~{}{\rm K}}\right)^{-1/2}\\\
\displaystyle\left(\frac{T_{\infty}}{10^{4}~{}{\rm
K}}\right)^{-2}\left(\frac{Q}{M}\right)^{2}&\displaystyle\text{otherwise.}\end{cases}$
(C5)

This approximate constraint on $\gamma$ matches that obtained by direct
numerical computation, shown in Figure 5, and effectively excludes the
entirety of the physically credible parameter space.