# Why Do You Think It is a Black Hole?

Galina Weinstein This work is supported by ERC advanced grant number 834735.
The Department of Philosophy, University of Haifa, Haifa, the
Interdisciplinary Center (IDC), Herzliya, Israel.

(March 2, 2021)

###### Abstract

This paper analyzes the experiment presented in 2019 by the Event Horizon
Telescope (EHT) Collaboration that unveiled the first image of the
supermassive black hole at the center of galaxy M87. The very first question
asked by the EHT Collaboration was: What is the compact object at the center
of galaxy M87? Does it have a horizon? Is it a Kerr black hole? In order to
answer these questions, the EHT Collaboration first endorsed the working
hypothesis that the central object is a black hole described by the Kerr
metric, i.e. a spinning Kerr black hole as predicted by classical general
relativity. They chose this hypothesis based on previous research and
observations of the galaxy M87. After having adopted the Kerr black hole
hypothesis, the EHT Collaboration proceeded to test it. They confronted this
hypothesis with the data collected in the 2017 EHT experiment. They then
compared the Kerr rotating black hole hypothesis with alternative explanations
and finally found that their hypothesis was consistent with the data. In this
paper, I describe the complex methods used to test the spinning Kerr black
hole hypothesis. I conclude this paper with a discussion of the implications
of the findings presented here with respect to Hawking radiation.

##  1 Introduction

In April 2019 an international collaboration of scientists (hereafter EHT
Collaboration) unveiled an image that shows an apparently blurred asymmetric
ring around a dark shadow. After comparison with computer models, the picture
was interpreted as showing a supermassive black hole at the center of the
galaxy Messier 87 (hereafter M87). This interpretation fitted well with theory
and simulation, according to which the M87 black hole is one of the two
largest supermassive black holes on the sky along with the Sagittarius A
(hereafter Sgr A) black hole (at the center of the Milky Way, near the border
of the constellations Sagittarius and Scorpius).

Shepherd Doeleman, the founding leader of the Event Horizon Telescope
(hereafter EHT) project, has recently explained that the EHT is “a global
Project that came together with a specific focused scientific objective in
mind, which was to make the first image of a black hole”. Doeleman further
remarked: “you can see here the horizontal bands showing the locations of
those stars that they [Eddington and his team] used to determine this […] and
now here we are just over hundred years later, thinking of doing the same kind
of experiment but around the black hole” [6].

The Arthur Eddington solar eclipse expeditions seem to have inspired the EHT
Collaboration to do their experiment. The 1919 eclipse expeditions confirmed
Einstein’s 1915 prediction of the deflection of light near the Sun. An
astrographic telescope was used for the photography of the field of stars
surrounding the Sun in the 1919 total eclipse. Eddington’s famous total solar
eclipse photograph shows the eclipsed Sun (a round large black “shadow”)
surrounded by a few displaced stars. The locations of the displaced stars are
indicated by a few white horizontal bands. Although the supermassive black
hole does not behave like an eclipsed Sun, light is nevertheless deflected by
both objects.

The very first question asked by the EHT Collaboration was: What is the
compact object at the center of galaxy M87? Does it have a horizon? Is it a
Kerr black hole? In order to answer these questions, the EHT Collaboration
first reconstructed images from data obtained in the experiment, as discussed
in Section 2. They then endorsed the working hypothesis that the central
object is a black hole described by the Kerr metric, i.e. a spinning Kerr
black hole as predicted by classical general relativity, as discussed in
Section 3. The EHT Collaboration chose this hypothesis based on previous
research and observations of the galaxy M87. After having adopted the Kerr
metric, they proceeded to test it. They confronted this hypothesis with the
data collected in the 2017 EHT experiment. In Sections 3-6, I describe the
complex methods used to test the Kerr black hole hypothesis. The EHT
Collaboration then compared the Kerr rotating black hole hypothesis with
alternative explanations and finally found that their hypothesis was
consistent with the data, as shown in Section 6.

I analyze the black hole image and the EHT experiment on the basis of the six
letters published by the EHT Collaboration in _The Astrophysical Journal
Letters_ in April 2019, which contain a complete exposition of the experiment
[10]-[15].

In section 7, I discuss a paper (letter) submitted at about the same time as
the six EHT [10]-[15] letters. The paper discusses the GW150914 gravitational
wave event. The authors also adopt the Kerr black hole hypothesis and test it.
In both cases – the EHT experiment and the gravitational wave experiment – the
Kerr black hole hypothesis is selected as the best explanation. I discuss the
philosophical implications of this selection in Section 8. I end this paper
with Section 9, discussing the philosophical significance of the findings
presented here with respect to Hawking radiation.

##  2 Evidence and interpretation of the evidence

In 2000, Heino Falcke et al. performed simulations of an optically thin
emitting accretion disk surrounding a maximally spinning supermassive Kerr
black hole with spin $a_{*}=0.998$ and mass $M_{BH}=2.6\times 10^{6}M\odot$
[solar masses ($M\odot$), $a_{*}$ is defined by equation (1) in Section 3].
They presented ray traced images and images seen for a wavelength of 1.3 mm
and wrote that “a marked deficit of the observed intensity inside the apparent
boundary” is produced “which we refer to as the ‘shadow’ of the black hole”.
They further noted that to simulate the images we need a Very Long Baseline
Interferometry (hereafter VLBI) array of radio telescopes located $\approx
8,000km$ apart [8], p. L14.

Event-horizon observations of the core of the galaxy M87 were conducted on the
two consecutive nights of April 5/6, 2017 and again four nights later on the
two consecutive nights of April 10/11, 2017. The observations were made at a
wavelength of 1.3 mm and from a VLBI array of radio telescopes located
$\approx 10,000km$ apart. Scan durations of M87 varied between three and seven
minutes and the quasar 3C 279 was also scanned as a calibrator source. The
compact object at the center of M87 was observed with eight telescopes at six
geographic sites and it was the first time that the small array called Atacama
Large Millimeter/submillimeter Array (hereafter ALMA) has been included in the
VLBI array. This whole array is referred to as the EHT.

After Correlation, the visibility $V_{ij}$ data were calibrated by three
calibration pipelines EHT-HOPS, rPICARD and AIPS (Python-based scripts using
modules from the EHT analysis toolkit, the eat library).111The radio signals
collected by each telescope are digitized and labeled with timestamps created
by atomic clocks at each site and then recorded to hard drives. The data are
read from each hard drive to a correlator and then processed by the
correlator’s software, which matches up and compares the data streams from
every possible pairing of the EHT’s eight telescopes. After correlation, the
complex visibility data $V_{ij}$ are obtained. $V_{ij}$ is the output of the
correlator and the fundamental data product which gives information of both
the amplitude and the phase of the fringes of the signal on a baseline
(distance between two telescopes/stations) $i$ and $j$. Phase errors (fringe-
fitting) and atmospheric (turbulence and variations) errors were corrected and
fixed – and the amount of data was consequently reduced so that noisy data was
removed. All three pipelines produced data – visibility amplitudes as a
function of baseline – with a slight anisotropy. One can see peaks in the
observed visibility amplitudes that appear across all four observed days.
These peaks have two minima, nulls, on either side and at the location of the
minima the visibility amplitudes of the data are very low. The first of the
nulls occurs at $\sim 3.4G\lambda$ (giga-lambda) and the second is observed at
$\sim 8.3G\lambda$. The high peak between these two nulls is at $\sim
6G\lambda$. A single pipeline output, the EHT-HOPS, was designated as the
primary data set of the engineering data release [11], p. 14; [12], p. 2, 5-8,
pp. 23-24.

Residual station-based amplitude and phase calibration errors nevertheless
persist after performing calibration steps and overwhelmingly dominate the
remaining thermal random Gaussian noise $\epsilon$. It is therefore possible
to construct data products that are insensitive to these systematic errors.
Such quantities are called closure quantities.222At high frequencies,
visibility amplitude and phase calibration are made extremely difficult.
Amplitude gain ($g_{i}$ and $g_{j}$) and phase ($\Phi_{i}$ and $\Phi_{j}$)
errors corrupt the measured visibility:
$V_{ij}=g_{i}g_{j}{\mathcal{V}}_{ij}e^{i(\Phi_{i}-\Phi_{j})}+\epsilon$, where
${\mathcal{V}}_{ij}$ stands for the unknown model visibility (the ideal
measured visibility with no atmospheric corruption). Radio waves from M87
reach station $i$ before reaching station $j$. Closure phase is the product of
three visibilities (from three stations). It is formed from baseline
visibilities on a closed triangle $ijk$. Consider three telescopes $ijk$. The
visibility we measure on baseline $ij$ is: ($u$,$v$)$ij$ plus phase error at
the station $(i-j)$. The measured visibility on baseline $jk$ is:
($u$,$v$)$jk$ plus phase error $(j-k)$ and the one measured on baseline $ki$
is: ($u$,$v$)$ki$ plus phase error $(k-i)$. Now the phase errors ($i-j$) $+$
($j-k$) $+$ $(k-i)$ are all canceled when the phase visibilities are added up,
leaving a quantity: $(u,v){ij}$ $+$ $(u,v){jk}$ $+$ $(u,v){ki}$ such that the
baselines “close”. The logarithmic closure amplitude is constructed from
combinations of visibilities measured on four stations and is insensitive to
variations in the amplitude gains [13], pp. 2-4.

In June 2018 the EHT Collaboration split up into four teams in different
regions of the world and each team chose a different imaging method and worked
in isolation for seven weeks on the data. The 2017 EHT April 11 data was
selected.333The reason for selecting the 2017 EHT April 11 data set was that,
on that observing day, the EHT array of radio antennas covered the largest
area for M87 and the amplitude calibration among stations was the most stable.
The teams neither talked nor crossed photos and that enabled them to avoid
getting into collective bias influencing their final images. Each one of the
four teams relied upon the judgment of its members to select a different
imaging method to convert the data into images: teams 1 and 2 used the
Regularized Maximum Likelihood (hereafter RML) method, while Teams 3 and 4
used a version of the CLEAN algorithm [13], p. 9.444The two imaging techniques
used in generating the 2017 EHT black hole images are based on the traditional
deconvolution CLEAN algorithm and the RML method. Imaging algorithms are
broadly categorized into two methodologies: inverse modeling (CLEAN) and
forward modeling (RML). 1) CLEAN: The sky is only sparsely sampled by the VLBI
array. The original CLEAN algorithm begins with a Fourier transform of the
sampled visibilities (the dirty image). The algorithm then proceeds to find
the points of highest intensity in the dirty image. The process is iterative
and iterations are continued until a desired noise level (a stopping
criterion) is achieved in the dirty image. Finally, CLEAN takes the
accumulated point source model and convolves the image with a restoring beam,
a clean beam, Gaussian beam of full width at half maximum (FWHM), instead of
the dirty beam. The algorithm adds the residuals of the dirty image to this
image to form the CLEAN image. 2) RML is based on maximum likelihood
estimation (MLE): finding the best fit image that minimizes a function that is
a sum of a chi-squared $\chi^{2}$ function corresponding to data and another
function called regularizer. Regularization may include smoothness (requiring
that the image be smooth) or sparsity) [10], p. 4; [13], p. 4.

In July, 2018, all four teams came together and revealed for the first time
the four images they had produced. Although not precisely identical, when
comparing images, all four teams obtained an image of a crescent with a
central dark area surrounded by an asymmetric ring. The initial blind imaging
stage indicated that the image was dominated by an angular ring diameter (an
angular crescent diameter) of $\theta_{d}\sim 40\mu as$ [13], p. 9.

The next stage of the experiment required the objective evaluation of the
fidelity of the images reconstructed in the first stage. That is, the second
stage involved selecting imaging parameters that are independent of expert
judgment.

Three imaging Python pipelines, DIFMAP (Difference Mapping), SMILI (Sparse
Modeling Imaging Library) and eht-imaging, have been designed. The SMILI and
eht-imaging pipelines are based on RML and DIFMAP is a CLEAN Python
script.555DIFMAP is a CLEAN Python script implemented in DIFMAP, a software
package originally written by Martin Shepherd in the 1990s. After loops of
cleaning and self-calibration, the DIFMAP script generates final cleaned
images. The SMILI and eht-imaging pipelines have been developed by the EHT
Collaboration. The SMILI Python-interfaced library is based on sparse
sampling, reconstruction of images from sparse data. NumPy and SciPy power the
main tasks of eht-imaging and the main imaging methods are powered by the
routine scipy.optimize.minimize. Other libraries used in creating the 2017 EHT
image are: Astropy, Pandas, and Matplotlib by which the black hole image is
visualized and processed. The pipelines employ an iterative imaging loop,
alternately imaging and self-calibrating the data (visibility amplitudes and
closure quantities)[13], p. 12.666The eht-imaging pipeline handles self-
calibration by minimizing: $\chi^{2}=\sum_{i<j}\frac{\left(\mid
V\mid_{ij}-g_{i}g_{j}\mid{\mathcal{V}}\mid_{ij}\right)^{2}}{\sigma_{ij}^{2}}$
for each baseline in the EHT array. $\mid V\mid_{ij}$ is the measured
visibility amplitude, $\mid{\mathcal{V}}\mid_{ij}$ stands for the model
visibility amplitude and $\sigma_{ij}^{2}$ is the uncertainty on $\mid
V\mid_{ij}$. While in progress, the imaging + self-calibrating pipelines
gradually remove $g_{i}$ and $g_{j}$ by going through a loop of multiple
iterations.

Four training geometric models of sources with different compact structure (a
ring, a crescent, a disk and two blobs) were considered. The models were
chosen such that their visibility amplitude data on EHT baselines matched
salient features of the observations, in particular the first visibility null
at at $\sim 3.4G\lambda$ and the second at $\sim 8.3G\lambda$. You might
rightly think that a disk can definitely not fit the 2017 EHT data. But all
four models produce images of objects having an apparent diameter of $\approx
50\mu as$. Further, all of the geometric compact objects have a total flux
density of $0.6$ Jy and also a $0.6$ Jy jet that is resolved by EHT baselines.
Synthetic data were generated from the model images of the four different
geometric models using two independent Python imaging pipelines, MeqSilhouette
and eht-imaging. While MeqSilhouette was used for comparison, eht-imaging’s
synthetic data were used for image-reconstruction.

Each one of the three pipelines, DIFMAP, eht-imaging and SMILI, has fixed
settings (such as the pixel size) and also parameters taken as
arguments.777DIFMAP has five parameters taken as arguments: the total flux
density; the stop condition (imaging stops when the desired flux density is
reached or when the decrease in the residual image rms noise dips below a
given threshold); a relative correction factor for ALMA in the self-
calibration process, the diameter of the cleaning region ($40$ to $100\mu as$)
and a parameter controlling the visibility errors. eht-imaging and SMILI both
have seven parameters as arguments: the total flux density; the fractional
systematic error on the measured visibilities, the FWHM of the circular
Gaussian initial image, and four regularization terms. Those imaging parameter
combinations for the three different imaging pipelines were derived by
conducting parameter space surveys on synthetic data. To determine a single
combination of the best performing hyper parameters for each pipeline, called
fiducial parameters, many similar-looking images were reconstructed from the
synthetic data (produced by eht-imaging).

The images were reconstructed using $50,000$ possible parameter combinations,
which also included values that were expected to produce poor reconstructions.
The images were then compared with images that were generated from the 2017
EHT data. Cross-correlation between synthetic images and 2017 EHT images was
computed and values were given to perfect correlation between images, no
correlation, and perfect anticorrelation. The fidelity of the synthetic images
was quantified by the FWHM Gaussian beam (blurring) between the synthetic
images and the 2017 EHT images. A value of the blurring, the FWHM Gaussian
beam, for each combination of imaging parameters was then computed, and the
combination of parameters was ranked according to this value. The top-ranked
combination of parameters was then denoted as the “fiducial parameters” of
that method. The Top Set of imaging parameter combinations for each pipeline
was selected.

The hyper-parameter selection procedure was subsequently tested. The purpose
of the test was to see whether or not the chosen parameters perform well
across geometric models, including geometric shapes that do not resemble the
crescent observed in the 2017 EHT images. The disk model was excluded from the
training set. All three imaging pipelines nonetheless produced from the
synthetic data an image with a disk shape that did not resemble any of the
other models. The fiducial parameters that best produced images of the disk
shape were those that described the crescent, the ring, and the double objects
models. By repeating the same process with each of the four geometric models,
the set of fiducial imaging hyper-parameters was cross-validated. The chosen
fiducial parameters were able to distinguish between the four geometric shapes
and perform well on average in reconstructing images of the four shapes, of
what was there.

The three pipelines were now fed with the April 11, 2017 EHT data set and
produced the same fundamental structure: an asymmetric ring around a dark
shadow. Fiducial images of a yellow orange asymmetric bright ring around a
central dark shadow (a crescent) were reconstructed from the April 11, 2017
EHT data set.888A Python module ehtplot with a color submodule has been
created, a perceptually uniform colourmap that is used in the reconstruction
and simulations of images. The colors of the images represent the brightness
temperature which does not necessarily correspond to any physical temperature
of the radio emitting plasma in the jets and accretion disk [13], pp. 8-9.
Reconstructed images of data sets from observations made on April 5, 6, and 10
were subsequently incorporated, while preventing the poorest reconstructions
of 2017 EHT April 10 from dominating the outcome. The best fiducial 2017 EHT
images, for each of the four observed days, from each of the three imaging
pipelines, were subsequently selected, the averages of which were taken and
restored to an equivalent resolution of the EHT array. The images produced
from the three pipelines are broadly consistent across all four observing days
and show a central shadow and a prominent asymmetric ring having an enhanced
brightness toward the south. The EHT Collaboration formed a single image,
which they called the _consensus image_ , from the average of each day. This
image is the famous black hole image of M87 [10], p. 5; [13], pp. 7-21, p. 22,
pp. 32-33, pp. 37-41; [15], p. 46; [33], p. pp. 5-6.

Having reconstructed images by DIFMAP, SMILI and eht-imaging, validation tests
were performed to assess their reliability. Recall that in April 2017,
observations of the calibrator 3C 279 were performed with the EHT. Images of
3C 279 were reconstructed and self-calibrated by DIFMAP, SMILI and eht-imaging
and then compared with the 2017 EHT M87 images. It was shown that station
gains in both cases were broadly consistent. It was then concluded that
consistency between the two sources M87 and 3C279 and among the three
different pipelines provides confidence that the gain corrections are not
imaging artifacts or missing structures in the 2017 EHT images [13], pp.
22-24.

Finally, reconstructed images were generated using SMILI, eht-imaging and
DIFMAP, the purpose of which was to extract $\theta_{d}$. $\theta_{d}$ was
extracted in the image domain of the fiducial images, for each of the four
observed days, from each imaging pipeline (SMILI, eht-imaging and DIFMAP). The
images were generated with the fiducial hyper-parameters. Unlike the SMILI and
eht-imaging images, the ones obtained by DIFMAP were restored with a FWHM
Gaussian $20\mu as$ beam. The measured image domain ring diameters
$\theta_{d}$ were nonetheless consistent among all imaging pipelines. Across
all days, the DIFMAP images recovered an average value of $\theta_{d}\approx
40\pm 3\mu as$ when restored with a $20\mu as$. The SMILI and eht-imaging
images recovered an average value of $\theta_{d}\approx 41\pm 2\mu as$.

It is written at the end of the fourth letter: “These first images from the
EHT achieve the highest angular resolution in the history of ground-based
VLBI” [13], pp. 29-31. This is certainly true. The 2017 EHT data were
subsequently interpreted.

##  3 Comparing the merits of two rival hypotheses

In the fifth letter, the EHT Collaboration is endorsing the hypothesis that at
the center of M87 there is a supermassive Kerr rotating black hole [14], p. 1.
The Kerr solution of the Einstein field equations describes the spacetime
geometry around astrophysical rotating black holes. An alternative hypothesis
for what could be the compact object at the center of M87 had been checked and
discarded before the 2017 EHT experiment was performed. This alternative is
the hypothesis that the compact object at the center of M87 has no horizon but
has a surface (such compact object could be very dense neutron stars and
supermassive stars).

In 2015 a team of scientists belonging to the EHT Collaboration made an
experiment the goal of which was to compare the merits of the two rival
hypotheses: a compact object with a surface and a black hole with a horizon;
and then choose which hypothesis can best explain the evidence. At the time in
which the experiment was conducted, the EHT consisted of only three stations
(radio telescopes). Were a compact object with a surface present at the center
of M87 instead of a black hole with a horizon, said the team, its photosphere
would be heated by the constant deposition of kinetic energy from the
accretion gas. In the absence of an event horizon, the photosphere would
therefore radiate, resulting in an additional component in the measured
spectrum. The surface luminosity is proportional to the average mass accretion
rate on the surface. The implication is that if the object at the center of
M87 has a horizon, the kinetic energy of the accreting gas must advect past an
event horizon, beyond which it is no longer visible to distant observers. In
simple terms, a compact object with a surface would appear brighter than a
black hole with a horizon. It was found that the existence of an observable
photosphere in the core of M87 was ruled out. “The implication is that the
kinetic energy of the gas is advected past an event horizon, beyond which it
is no longer visible to distant observers. In other words, M87* must have an
event horizon” [5], pp. 7-8, and the black hole hypothesis was selected.

In the fifth letter, it is written: “In this Letter we adopt the working
hypothesis that the central object is a black hole described by the Kerr
metric, with mass $M$ and dimensionless spin $a_{*}$”. After having adopted
the Kerr metric and the no-hair theorem, the EHT Collaboration proceeds to
test the Kerr metric by fitting GRMHD models to 2017 EHT data.999The Kerr
metric was proved to be the unique stationary, asymptotically flat, vacuum
solution with an event horizon. It was shown that the Kerr-Newman solution
involves just three free parameters: mass $M_{BH}$, angular momentum $J$ (spin
$a_{*}$) and charge $Q$. This is the no-hair theorem which states that non-
charged black holes are uniquely characterized by their mass $M_{BH}$ and
angular momentum $J$ and are described by the Kerr metric. The black hole is
governed by general relativity (GR). Magnetohydrodynamics (MHD) is the
framework that governs the dynamics of the accretion flow and jets around the
black hole. The plasma is treated as a fluid and numerical methods to
integrate the GRMHD equations are searched for.

A GRMHD Simulation Library has been generated from several different
codes.1010101) The Black Hole Accretion Code (BHAC) performs
magnetohydrodynamical simulations of an accretion flow onto a black hole. The
ions (protons) and the electrons in the accretion disk plasma travel quite a
long distance along magnetic field lines before being scattered. This and
other factors complicate the calculations and require more computational
power. 2) A system called H-AMR (Hierarchical Adaptive Mesh Refinement) is
used, which accelerates GRMHD calculations by implementing the AMR strategy: a
method that reduces the number of computations. 3) Two additional codes, a 3D
version of High Accuracy Relativistic Magnetohydrodynamics (iharm3D) and 4)
KORAL [Kod radiacyjny L (in Polish)], solve the GRMHD conservation laws by a
shock-capturing method [27], pp. 6–8, p. 31. Forty-three high-resolution,
three-dimensional Standard and Normal Evolution (SANE) and Magnetically
Arrested Disk (MAD) simulations, covering well the physical properties of
magnetized accretion flows onto Kerr black holes, were performed by varying
two dimensionless parameters:

1) The spin $a_{*}$:

$a_{*}\equiv\frac{Jc}{GM_{BH}^{2}},-1<a_{*}<1.$ (1)

where $G$ is the universal gravitational constant, $c$ is the velocity of
light.

If $G=c=1$, then: $a_{*}\approx\frac{J}{M_{BH}^{2}}$.

2) The magnetic flux $\phi$:111111The SANE and MAD models are an approximation
of the complex non-linear general relativistic dynamical system of a
magnetized accretion disk flow orbiting a rotating supermassive black hole.
Models with $\phi\approx 1$ in equation (2) in the text are conventionally
referred to as SANE where accretion is largely unaffected by the black hole
magnetic field. If the magnetic flux reaches $\phi\approx 15$, the large-scale
magnetic field accumulated by the accretion disk stops the accretion flow.
Models with $\phi\approx 15$ are conventionally referred to as MAD. At
$\phi>\approx 15$, numerical simulations show that the accumulated magnetic
flux erupts, pushes aside the accretion flow, and escapes [14], p. 4.

$\phi=\frac{\Phi_{BH}}{\sqrt{\dot{M}{R_{g}}^{2}}c},1\leq\phi<15.$ (2)

where $\Phi_{BH}$ is the magnetic flux, $\dot{M}$ is the mass accretion rate
(the rate at which mass is accreted onto a black hole) and $R_{g}$ is the
gravitational radius defined by equation (4) below.

The Simulation Library contains SANE models with $a_{*}=-0.94$, $-0.5$, $0$,
$0.5$, $0.75$, $0.88$, $0.94$, $0.97$, and $0.98$, and MAD models with
$a_{*}=-0.94$, $-0.5$, $0$, $0.5$, $0.75$, and $0.94$.

The location of the surface of the event horizon of the Kerr black hole is:

$r_{+}=R_{g}\left(1+\sqrt{1-a_{*}^{2}}\right).$ (3)

where $R_{g}$ is given by:

$R_{g}=\frac{G{M_{BH}}}{c^{2}}.$ (4)

In order to produce images from the simulations, the following parameters were
fixed in the GRMHD simulations:

1) PA. The position angle of the forward radio jet measured east of north.
Based on prior knowledge from observations, the chosen value was: $PA\approx
288^{0}$.

2) i. The observer inclination, the orientation of the observer through $PA$.
Based on prior knowledge from observations, images were generated at
$i=12^{0}$, $17^{0}$, $22^{0}$, $158^{0}$, $163^{0}$, and $168^{0}$ and a few
at $i=148^{0}$. Basically, images were generated at two major inclinations
$17^{0}$ and $163^{0}$.

3) $\mathbf{\theta_{g}}$. The image scale:

$\theta_{g}=\frac{G{M_{BH}}}{dc^{2}}.$ (5)

If $G=c=1$, then $\theta_{g}\approx\frac{M_{BH}}{d}$, where $d$ is the
distance to M87: $d=16.8\pm 0.8$ Mpc (Megaparsec) and $\theta_{g}$ represents
the angular gravitational radius.

For $M_{BH}$ “we use the most likely value from the stellar absorption-line
work, $M_{BH}\approx 6.2\times 10^{9}M\odot$ ([16]), see explanation in
Section 5].

Inserting the above value of $M_{BH}$ into equation (9) from Section 5:

$\theta_{d}=\frac{9.79{M_{BH}}}{d}\approx 35.5\mu as$ is calculated.

4) $\mathbf{F_{\nu}}$. The total compact flux density measured in Jy (Jansky).
The average flux density of $1.3mm$ ($230$GHz) emission is $0.6$ Jy;

5) $\mathbf{R_{high}}$. The temperature ratio of electrons to
protons.121212The plasma in the accretion disk is an accretion flow which is
composed of ions and electrons. Both species have the same temperature in the
funnel (the strongly magnetized regions of the accretion flow), but have a
substantially different temperature in the middle of the accretion disk (the
weakly magnetized regions). Assuming the gas is composed of non relativistic
ions with temperature $T_{i}$ and relativistic electrons with temperature
$T_{e}$, the ratio of the temperatures of the two species can be imposed in
terms of a single parameter $R_{high}$. Relativistic electrons emit radio
photons at $1.3mm$ wavelength observed by the EHT known as synchrotron
radiation. If the synchrotron radiation is emitted from weakly magnetized
regions: $T_{i}\approx T_{e}$. If the emission comes from the funnel:
$\frac{T_{i}}{T_{e}}\approx R_{high}$ [14], pp. 4-5, p. 12. Images were
generated at: ${R_{high}}=1$, $10$, $20$, $40$, $80$, and $160$.

After fixing the values of these parameters ($\frac{M_{BH}}{d}$, $i$, $PA$,
$F_{\nu}$, $R_{high}$) snapshots were drawn from the time evolution of the
simulation at a cadence of every: $10-50\frac{R_{g}}{c}$.

From the SANE and MAD simulations, more than 60,000 synthetic $1.3mm$ snapshot
images were produced by three general relativistic ray-tracing (GRRT) codes,
ipole, RAPTOR and BHOSS. The snapshot images created by GRRT codes show the
black hole with a variety of accretion flows and jets and depict a yellow-
orange asymmetric bright ring around a central dark shadow (a crescent).

Each snapshot image generates a single snapshot model (SSM) defined by three
parameters: $F_{\nu},\theta_{g},PA$. The snapshot images were compared with
the 2017 EHT April 6 data set.131313Model-fitting requires a large number of
scans. The number of scans obtained of M87’s compact object each night ranged
from seven on April 10 to twenty-five on April 6.

##  4 Models of accretion around a Kerr black hole

Comparison of models to data was performed by computing the distance
$\chi_{\nu}^{2}$ between the data and the snapshot image.141414The reduced
chi-squared statistic $\chi_{\nu}^{2}$.
$\chi_{\nu}^{2}\equiv\frac{\chi}{\nu}\equiv\frac{\chi}{N-M}$, where $\nu$
represents the degrees of freedom, $N$ is the number of data values and $M$ is
the number of free parameters. In the course of computing $\chi_{\nu}^{2}$,
the three parameters $F_{\nu},\theta_{g},PA$ and the gains $g_{i}$ at each
VLBI station are varied “in order to give each image every opportunity to fit
the data” [14], p. 8.

Variations in the three parameters $F_{\nu},\theta_{g},PA$ approximately
correspond to variations in the accretion rate, black hole mass, and
orientation of the black hole spin, respectively. In fitting GRMHD snapshot
images to data, the image is stretched by adjusting
$\theta_{g}\left(\frac{M_{BH}}{D}\right)$, re-scaled by varying $F_{\nu}$ and
rotated by changing $PA$. Varying the parameters $a_{*}$, $\phi$ and
$R_{high}$ can change the width and asymmetry of the photon ring and introduce
additional structures exterior and interior to the photon ring [14], pp. 7-8.

Fitting models to data was performed by two different methods: GENA (a genetic
algorithm)151515GENA fits models to data by finding the model parameters which
minimize a $\chi_{\nu}^{2}$ statistic. It implements the differential
evolution (DE) algorithm, an optimization algorithm inspired by natural
evolution theory. The major steps of DE are the following: A random population
of model parameters (individuals, parents) is created. Fitness of parameters
with the data (visibility amplitudes and closure phases) is computed. Fitness
is inversely proportional to $\chi_{\nu}^{2}$. The fitness of
$F_{\nu},\theta_{g},PA$ with the data is computed such that maximizing fitness
of parameters to data minimizes the value of $\chi_{\nu}^{2}$. In genetic
algorithms, based on their fitness, best-fit parameters (parents) are improved
(selected) by iterations in which mutation is used. GENA, however, utilizes a
Nondominated Sorting GA (NSGA), the NSGA-II, to explore the parameter space.
NSGA implements an elitist selection process for multi-objective optimization.
A population of parents is generated randomly. Offsprings are compared with
parents. If the offsprings do not dominate their parents, they are sorted,
moved to the next generation and the parents are replaced. If the offsprings
dominate their parents, they are not elite offsprings and are rejected. The
process of selecting the non-dominated offsprings continues until the initial
population of parents is replaced. GENA further uses the procedure of network
amplitude calibration of the residual station gains $g_{i}$, $g_{j}$ performed
by the eht-imaging pipeline.. And THEMIS, a Bayesian parameter estimation and
model comparison framework, which has been specifically developed for the EHT
experiment.161616THEMIS uses the (differential evolution) DE-MCMC (Markov
chain Monte Carlo) sampler in which many chains are run in parallel. An MCMC
sampler is a random walk method through parameter space for performing
Bayesian inference. It randomly samples the posterior probability distribution
of parameters and generates a sequence of random samples of parameters that
fits the data. The parameter is dependent upon the previous one in the chain.
DE-MCMC is especially efficient in sampling from models with highly correlated
parameters. THEMIS implements a parallel-tempering algorithm for DE-MCMC.
Parallel tempering is a method based on an analogy with statistical physics,
called thermodynamic integration, to calculate the Bayesian evidence. Many
MCMC chains are run in parallel on tempered versions of the original
likelihood function at different temperatures. Within THEMIS at each MCMC
step, the station gains $g_{i}$, $g_{j}$ are addressed by marginalizing over
the nuisance parameters. Gain amplitudes $g_{i}$ represent approximately
between $40$ and $143$ additional nuisance parameters per data set. The gain
parameters are subsumed into the likelihood and incorporated as model
parameters. Assuming Gaussian priors, the log-likelihood:
$\mathcal{L}=-\sum_{i<j}\frac{\left(\mid
V\mid_{ij}-g_{i}g_{j}\mid{\mathcal{V}}\mid_{ij}\right)^{2}}{2\sigma_{ij}^{2}}$
is maximized and a marginalization over all $g_{i}$ is performed [4], pp.
14-15, p. 35; [15], pp. 5-6, p. 25. THEMIS uses a Differential Evolution
Markov Chain Monte Carlo (DE-MCMC) sampler to produce Bayesian posterior
estimates for the three parameters $F_{\nu},\theta_{g},PA$ whilst GENA uses a
DE algorithm for producing best fit estimates for the three parameters [4], p.
4; pp. 15-16; [10], pp. 3-6, p. 10, p. 36; [13], pp. 8-9, p. 14, p. 28; [14],
p. 3, pp. 7-8; [15], p. 6, p. 15.

Computing the distance $\chi_{\nu}^{2}$ between the data and the snapshot
images is a difficult challenge. That is because of stochastic fluctuations,
associated with turbulence in the underlying accretion flow in the GRMHD
simulations that produce large variations in image structure. On the other
hand, data products are calibrated and self-calibrated. There is, therefore,
dominance of the stochastic image features over the observational noise. This
implies that individual snapshots are highly unlikely to provide an acceptable
fit with $\chi_{\nu}^{2}\approx 1$. This problem is solved as follows. An
average snapshot image from an ensemble of snapshot images is generated and
fitted to data by THEMIS Average Image Scoring (AIS). That is, models are
rejected if none of the snapshots are as similar to the average image as the
data. In the sixth letter it is noted that “the ‘true’ model is necessarily
accepted by the THEMIS-AIS procedure” [15], p. 34. That is, the model that
would fit the evidence would be the “true” model.

It is shown in the fifth letter that the distribution of $\frac{M_{Bh}}{d}$
from fitting snapshot images to data using THEMIS and GENA gives qualitatively
similar results. As better fits are required, the distribution of
$\frac{M_{Bh}}{d}$ from fitting snapshot images to 2017 EHT data narrows and
peaks close to $\frac{M_{Bh}}{d}\approx 3.6\mu as$. So, most models favor
$\frac{M_{Bh}}{d}\approx 3.6\mu as$. The exception is the $a_{*}=-0.94$ and
$R_{high}=1$ SANE model which favors a small $\frac{M_{Bh}}{d}\approx 2\mu as$
[14], p. 10.

The best fit $PA$ were then searched for. Recall that $PA$ approximately
corresponds to the orientation of the black hole spin $a_{*}$. The SANE and
MAD snapshot images were divided into two groups: the spin-away models171717In
which the black hole’s spin $a_{*}$ points away from Earth ($i>90^{0}$, and
$a_{*}>0$, or $i<90^{0}$ and $a_{*}<0$). and the spin-toward models181818In
which the black hole’s spin $a_{*}$ points toward Earth ($i>90^{0}$, and
$a_{*}<0$, or $i<90^{0}$ and $a_{*}>0$).. In each group, the accretion flow
either moves with the black hole’s spin or against it (prograde and
retrograde, respectively).191919If the accretion flow’s angular momentum and
that of the black hole $J$ are aligned, the accretion disk is prograde
($a_{*}\geq 0$) with respect to the black hole spin axis. But if the black
hole’s angular momentum $J$ is opposite that of the accretion flow, the
accretion disk is retrograde ($a_{*}<0$).

The values of the large-scale jet $PA$ were found by the fitting procedure
(THEMIS and GENA) to be $150^{0}$-$200^{0}$ east to north, consistent with the
spin-away models and inconsistent with the spin toward models. The two chosen
cases – prograde ($i=163^{0}$ and $a_{*}>0$) and retrograde ($i=17^{0}$ and
$a_{*}<0$) – were therefore the spin-away models, the ones in which the spin
of the black hole $a_{*}$ must always be moving clockwise (as seen from Earth)
and the bright section is at the bottom part of the ring. This means there is
a persistent asymmetry with the brightest region to the South [14], p. 8, p.
10.

It was found that the large-scale jet $PA$ lies on the shoulder of the spin-
away models but lies off the shoulder of the spin-toward models. The
conclusion was: the alternative GRMHD model images having a bright section of
the ring at the top – the spin-toward models, $a_{*}$ is moving anticlockwise
(as seen from Earth) – do not represent observations. The snapshot images with
the bright section of the ring at the bottom capture the qualitative features
found in the 2017 EHT 2017 April 6 image.

It is explained in the fifth letter that the ring is brighter at the bottom
because the plasma is moving toward us. Due to Doppler beaming, at the top,
the ring is less bright because the material is moving away from us. While the
approaching side of the plasma of the forward jet is Doppler boosted, the
receding side is Doppler dimmed, producing a surface brightness contrast [14],
p. 3.

The above finding implies that the sense of rotation of both the jet and the
funnel wall (the strongly magnetized region of the accretion flow) are
controlled by the black hole spin $a_{*}$ [14], p. 8. This was a hint that the
Blandford-Znajek process may be confirmed here [2]. More on this below.

After performing the AIS test, only very few models were rejected. In the
fifth letter it is said that “the majority of the simulation library models is
consistent with the data” and it is then explained that “Given the
uncertainties in the model – and our lack of knowledge of the source prior to
EHT2017 – it is remarkable that so many of the models are acceptable” [14],
pp. 5-15, p. 19; [15], p. 10, pp. 14-15, pp. 30-33; [4], p. 4.

What that meant was that the majority of the models were “true” models. The
ensuing steps therefore required narrowing down the range of the best-fit
models by imposing three constraints, the most important of which was the jet
power constraint. Based on measurements performed in 2012, 2015 and 2016, the
jet power $P_{jet}$ of the core of M87 was assumed to be large:
$P_{jet}>10^{42}ergs^{-1}$. The jet power constraint rejected the largest
number of models.

Although in the GRMHD models, “the most likely value from the stellar
absorption-line work $M_{BH}\approx 6.2\times 10^{9}M\odot$” was used, for
some GRMHD models, images were also generated with $M_{BH}\approx 3.5\times
10^{9}M\odot$ “to check that the analysis results are not predetermined by the
input black hole mass” [14], p. 5.

For a given magnetic field configuration: $P_{jet}=\frac{\dot{M}c^{2}}{10}$.
It was found by THEMIS-AIS that the SANE model with $a_{*}-0.94$ and
$R_{high}=1$ and with $M_{BH}\approx 3.5\times 10^{9}M\odot$ (and
$\frac{M_{Bh}}{d}\approx 2\mu as$) fitted the data.

But with this value of $M_{BH}$, $\dot{M}$ drops by a factor of two and so
$P_{jet}<10^{42}ergs^{-1}$. Consequently, the SANE model with $a_{*}-0.94$ and
$R_{high}=1$ was finally rejected. So, eventually the conclusion was: it is
unlikely that the 2017 data capture an $a_{*}-0.94$ and $R_{high}=1$ SANE
model, namely, it is unlikely that the 2017 EHT data capture a $M_{BH}\approx
3.5\times 10^{9}M\odot$ black hole.

Moreover, for SANE and MAD models that produced sufficiently powerful jets and
were consistent with the 2017 EHT data, $P_{jet}$ was found to be driven by
the extraction of black hole spin energy through the Blandford-Znajek
process.202020In 1969 Roger Penrose suggested that rotational energy can be
extracted from the rotating Kerr black hole [26], p. 1160. In 1977, Roger
Blandford and Roman Znajek extended Penrose’s “mechanical extraction of
energy” process to electromagnetic extraction of energy from a Kerr
supermassive black hole in a MHD environment [2], p. 434, p. 451. The Kerr
black hole is rotating like a conductor in a magnetic field and the total jet
power is:

$P_{jet}=\frac{k}{4\pi}\Phi_{BH}^{2}\Omega^{2},$ (6)

where $k\approx 0.045$ is a numerical constant which depends on the magnetic
field, $\Phi_{BH}$ is given by equation (2), $\Omega^{2}$ is the angular
velocity of the horizon:

$\Omega^{2}=\frac{a_{*}c}{2{r}_{+}}$ and $r_{+}$ is given by equation (3).

$P_{jet}$ increases quadratically with both the black hole spin $a_{*}$ and
the magnetic flux $\Phi_{BH}^{2}$, where the spin $-1<a_{*}<1$. Hence, GRMHD
models with $P_{jet}<10^{42}ergs^{-1}$ and zero spin $a_{*}=0$ are outright
rejected. For $a_{*}=0$ the bright section of the ring creates a nearly
symmetric ring, which is inconsistent with what is seen in the 2017 EHT
consensus image [14], pp. 9-10, p. 13-15; [5], pp. 2-3.

##  5 The Kerr black hole hypothesis and alternative hypotheses

The angular gravitational radius and ring diameter. The next stage of the
experiment was obtaining the angular gravitational radius $\theta_{g}$ by
three different methods – GRMHD model fitting, geometric model fitting and
image domain feature extraction; and estimating the ring diameter $\theta_{d}$
by the two latter methods:

1) _Geometric model fitting_ : Two kinds of geometric crescent models (called
_xs-ring_ and _xsringauss_) – collectively named, the _generalized crescent
model_ (GC) – were developed and compared directly with the 2017 EHT data. One
model was compared with data using the dynesty Python Bayesian dynamic NS
sampling code while the other was fitted to data by THEMIS. 212121NS sampling
is an Approximate Bayesian Computation method. In Bayesian statistics, we
start with Bayes’ theorem and update the prior probability distribution of the
model parameters upon receiving new data to obtain the posterior probability
distribution of the parameters. NS is designed to evaluate the Bayesian
evidence $\mathcal{Z}$ but as a by-product it can further sample the posterior
probability distribution. $\mathcal{Z}$ is calculated by sampling nested
points. $N$ live points are sampled from the prior space. At each iteration,
the live point with the lowest likelihood $\mathcal{L}_{i}$ among the live
points is found and a new live point is sampled. If
$\mathcal{L}_{i+1}\geq\mathcal{L}_{i}$, the old live point with
$\mathcal{L}_{i}$ is rejected but its values are stored to calculate
$\mathcal{Z}$. Dynamic NS (dynesty) has been developed to increase the
accuracy of nested sampling, to sort likelihoods more efficiently than NS and
speed up the process [15], p. 4, p. 6. Both THEMIS and dynesty produced the
following mean value for the ring diameter of the two geometric crescent
models: $\theta_{d}\approx 43\mu as$.

The GC model does not provide any scientific explanation for the 2017 EHT data
because no underlying mechanism based on physics is responsible for the
structures in it. The parameters of the GC model are therefore calibrated,
i.e. fitted, to the parameters of the GRMHD models. The crescent models are
associated with the emission surrounding the shadow of a black hole. If the
geometric crescent is formed by gravitational lensing, the angular diameter of
the photon ring obeys the equation:

$\theta_{d}=\alpha\theta_{g}.$ (7)

$\alpha$ represents the gravitational lensing factor and $\theta_{g}$ is
defined by equation (5). dynesty produced a ring diameter of
$\theta_{d}\approx 43.2\mu as$ for the _xs-ring_ model and THEMIS produced
$\theta_{d}\approx 43.4\mu as$ for the _xs-ringauss_ model. Those values were
then compared with the known value of $\theta_{g}$ that went into the GRMHD
simulations.222222This led to mean values of $\alpha$: $\alpha=11.55$ for the
_xs-ring_ model and $\alpha=11.50$ for the _xs-ringauss_ model. The two
$\theta_{d}$ measurements were combined with the two $\alpha$ values to arrive
at values of $\theta_{g}$ using equation (7): $\frac{43.2\mu as}{11.55}$ and
$\frac{43.4\mu as}{11.50}$. The final inferred value is
$\theta_{g}=3.77_{-0.40}^{+045}\mu as$ [15], pp. 10-13.

2) _GRMHD model fitting_ : GRMHD snapshot images were fit directly to data by
THEMIS and GENA (see Section 3). The combined value for the analysis performed
by both THEMIS and GENA is: $\theta_{g}=3.80_{-0.31}^{+039}\mu as$.

3) _Image domain feature extraction_ : The values of $\theta_{d}$ obtained by
image domain feature extraction (see end of Section 2) were converted to the
$\theta_{g}$ using the scaling factor $\alpha$ [Equation (7)], following a
similar procedure to the one used in the geometric model fitting. The
following value of $\theta_{g}$ was obtained:
$\theta_{g}=3.83_{-0.36}^{+042}\mu as$.

Fitting geometric models to data and extracting feature parameters in the
image domain both allowed quantifying the following properties of the object
at the center of M87: an angular ring diameter of $\theta_{d}\approx 42\pm
3\mu as$, an angular gravitational radius of $\theta_{g}\approx 3.8\pm 0.4\mu
as$, a deep central brightness depression (the fractional central brightness:
the ratio of the mean brightness interior to the ring to the mean brightness
around the ring), and $PA$. In the sixth letter it is concluded: “All of these
features support the interpretation that we are seeing emission from near the
event horizon that is gravitationally lensed into a crescent shape near the
photon ring” [15], pp. 20-21.

Prior measurements based on stellar dynamics had produced the value
$\theta_{g}=3.62_{-0.34}^{+060}\mu as$. This value is consistent with the
values calculated by the EHT project. On the other hand, gas dynamics
measurements led to a lower value: $\theta_{g}=2.05_{-0.16}^{+048}\mu as$.

It is remarked in the sixth letter that “All of the individual $\theta_{g}$
estimates use the GRMHD simulation library, either through directly fitting
GRMHD snapshots to the data […GRMHD model fitting] or through calibration of
diameters [$\theta_{g}$] resulting from geometric models or reconstructed
images”, i.e. from geometric crescent models or image domain feature
extraction. Hence “A degree of caution is therefore warranted. The
measurements rely on images generated from GRMHD simulations and should be
understood within that context” [13], pp. 27-30; [14], p. 2; [15], p. 1, pp.
4-21, p. 31, p. 39.

The mass of the black hole. In the first letter it is noted that “A basic
feature of black holes in GR is that their size scales linearly with mass”
[10], p. 9. The estimation of the angular ring diameter of $\theta_{d}\approx
42\pm 3\mu as$ should therefore allow for the determination of the mass
$M_{BH}$ of the core of M87.

In 1973, James Maxwell Bardeen found that the black hole casts a shadow on the
hot gas that surrounds it. The diameter of the apparent shadow $D_{sh}$ seen
by a distant observer depends on the gravitational lensing $\alpha$ around the
black hole. For a Schwarzschild black hole: $D_{sh}=2\sqrt{27}R_{g}\approx
10.39R_{g}$ where $\alpha=2\sqrt{27}$. However, in the vicinity of a rotating
Kerr black hole, “the effect of the frame dragging induced by the angular
momentum of the Kerr black hole is quite apparent” [1], pp. 230-233.232323The
Lense-Thirring effect due to the black hole rotation acts to compress the
shadow with respect to the rotation axis while the quadruple moment of the
rotating black hole causes an oblate shape of the shadow. The two effects
approximately cancel each other out and we are left with a nearly circular
shadow. Thus, for a Kerr black hole:

$D_{sh}<2\sqrt{27}R_{g}\approx 9.8R_{g},R_{sh}<\sqrt{27}R_{g}\approx
4.9R_{g}.$ (8)

In the GRMHD images: $D_{sh}=9.79M_{BH}$.

The radius $R_{g}$ is given by equation (4). Combining equation (8) with
equation (5) gives:

$\theta_{d}\approx\frac{2\sqrt{27}R_{g}}{d}\approx\frac{9.8R_{g}}{d}\equiv\frac{9.8GM_{BH}}{dc^{2}}=\alpha\theta_{g}.$
(9)

Inserting $\theta_{d}\approx 42\pm 3\mu as$ and $d=16.8\pm 0.8$ Mps into
equation (9) gives $M_{BH}$. It was shown that $\theta_{d}$ is consistent with
a mass of $M_{Bh}\approx 6.5\pm 0.7\times 10^{9}M\odot$.

Recall that measurements based on stellar dynamics had produced the value:
$M_{Bh}\approx 6.2\times 10^{9}M\odot$ [16]. This estimation is consistent
with the one calculated by the EHT project. On the other hand, gas dynamics
measurements had led to a lower value of $M_{Bh}\approx 3.5\times
10^{9}M\odot$ [32], see Section 4.

In the fifth letter it is stressed: “Although our working hypothesis has been
that M87 contains a Kerr black hole, it is interesting to consider whether or
not the data is also consistent with alternative models for the central
object” [14], p. 17.

One such alternative model is a black hole that is spinning more rapidly than
the Kerr bound, defined as follows: $a_{*}\leq M_{BH}$ or $J\leq M_{BH}^{2}$
($J\leq\frac{GM_{BH}^{2}}{c}$) see equations (1) and (3). Those black holes
are super-spinning black holes and are called superspinars. Their horizon
would disappear if $J=a_{*}$ and this would imply the existence of a naked
singularity and the violation of causality (spinning faster than the speed of
light). Computer simulations show that the shadows of superspinars are
significantly smaller compared with those of Kerr black holes [14], p. 18.

Hence, there exist solutions of the field equations of general relativity
which describe naked singularities not hidden by an event horizon. But the
possible existence of these exotic solutions causes severe problems to the
uniqueness theorems. The reason is that only if all singularities are
surrounded by event horizons, the Kerr black hole is a unique solution. So, in
1969 and 1976 Roger Penrose and Stephen Hawking phrased the censorship
hypothesis, which says that physics censors naked singularities by always
enshrouding them with a horizon [26], p. 1160, p. 1162; [21], p. 2461.

To verify the censorship hypothesis, we should demonstrate that the core of
M87 is not a naked singularity/superspinar. There also exist other horizonless
solutions of the field equations such as rotating wormholes. According to
equation (8), for a Kerr black hole: $R_{sh}\approx 4.9R_{g}$. A $6.5\times
10^{9}\odot$ rotating wormhole has half as big a shadow radius as a $6.5\times
10^{9}\odot$ black hole: $R_{sh}\approx 2.7R_{g}$. This is about the same size
as the shadow of the $3.5\times 10^{9}\odot$ black hole. Hence, for a
$3.5\times 10^{9}\odot$ black hole we would get a smaller shadow than for a
$6.5\times 10^{9}\odot$ black hole. A $6.5\times 10^{9}\odot$ naked
singularity (superspinar) has an even smaller shadow radius of $R_{sh}=R_{g}$.

The different shadow sizes ($R_{sh}$) were then overlaid on top of the April
11 2017 EHT fiducial images from each imaging pipeline (the consensus image).
The exotic possibilities were ruled out and one was left with a ring that
nearly perfectly matches that of a $6.5\times 10^{9}\odot$ black hole [10],
pp. 8-9; [14], p. 18; [23].

But there is a fly in the ointment because demonstrating that the core of M87
has an event horizon does not conclusively disprove the existence of naked
singularities elsewhere [28], p. 77.

In the first letter it is remarked that “However, other compact-object
candidates need to be analyzed with more care”. And in the fifth letter it is
again stressed: “Future observations and more detailed theoretical modeling,
combined with multiwavelength campaigns and polarimetric measurements, will
further constrain alternatives to Kerr black holes”. It is further noted that
“the comparisons carried out here must be considered preliminary.
Nevertheless, they show that the EHT2017 observations are not consistent with
several of the alternatives to Kerr black holes” [10], pp. 8-9; [14], p. 18.

##  6 The Kerr black hole is the most likely explanation of the evidence

A null hypothesis test, which consists of three ingredients, was subsequently
performed:

1) The dynamical measurements provide an accurate determination of $M_{BH}$.

2) The shadow in the 2017 EHT image is a black hole shadow.

3) The space-time of the black hole is described by the Kerr solution.

_Ingredient 1_ : An estimation was performed of the difference between the
measurement of $\theta_{g}$ performed by the EHT Collaboration and prior
measurements of $\theta_{dyn}$:

$\delta\equiv\frac{\theta_{g}}{\theta_{dyn}}-\frac{\theta_{g}}{\theta_{g}}=\frac{\theta_{g}}{\theta_{dyn}}-1.$
(10)

For gas dynamics: $\delta\approx 0.78$ and for stellar dynamics:
$\delta\approx-0.01$. In the sixth letter it is written that the fact that the
EHT measurement of $\theta_{g}$ is consistent with one of the measurements
“allows us to conclude that our null hypothesis has not been violated” [15],
pp. 22-23.

_Ingredient 2_ : Are black holes described entirely by their mass $M_{BH}$ and
spin $a_{*}$, or do they have hair (are they described by other parameters)?
To answer this question, we have to know $M_{BH}$ because the size of the
shadow is proportional to $M_{BH}$. Violations of the above no-hair theorem
generically change the shape and size of the shadow of the black hole. In
other words, for Kerr black holes of known $M_{BH}$, the size and shape of the
shadow remain nearly unchanged and shadows always appear nearly circular.
Detecting a shadow and extracting its characteristic properties offers a
chance to constrain the space-time metric. For a black hole of known mass
$M_{BH}$ and distance $d$ from Earth, identifying the presence of the shadow
and confirming that its size is in the narrow range
$\left(4.9-5.2\right)M_{BH}$ constitutes a statistical null hypothesis test of
the Kerr black hole [28], pp. 77-79; [29], p. 1; [15], p. 2; [23]. As said at
the end of section 5, different $R_{sh}$ were overlaid on top of the consensus
image. The ring of the 2017 EHT image nearly perfectly matches that of an
$R_{sh}\approx 4.9M_{BH}$ black hole whose mass is $M_{BH}\approx 6.5\pm
0.7\times 10^{9}M\odot$.

_Ingredient 3_ : In the fifth letter, it is explained: “we now adopt the
working hypothesis that M87 contains a turbulent, magnetized accretion flow
surrounding a Kerr black hole. To test this hypothesis quantitatively against
the EHT2017 data we have generated a Simulation Library of 3D time-dependent
ideal GRMHD models” [14], p. 3. And: “all [GRMHD] models assume a Kerr black
hole space-time, but there are alternatives” [14], p. 15. It is shown that
GRMHD models that assume a Kerr metric fit the 2017 EHT observations (see
Section 3) and it is concluded that the space-time of the black hole is
described by the Kerr metric.

Finally, in the fifth letter, three images are placed one next to the other: a
snapshot image based on a (spin-away) GRMHD model of a Kerr black hole, a
snapshot image based on a (spin-away) GRMHD model of a Kerr black hole
(restored with a FWHM Gaussian $20\mu as$ beam), and the 2017 EHT April 6
image. One can see that the two blurred images are almost the same [10], p. 6;
[14], p. 2, pp. 5-6, p. 8, p. 10, p. 15.

##  7 Testing the Kerr black hole hypothesis with gravitational waves

It is written in the first letter of the EHT Collaboration: ”Altogether, the
results derived here provide a new way to study compact-object spacetimes and
are complementary to the detection of gravitational waves from coalescing
stellar-mass black holes with LIGO/Virgo [10], p. 9. I shall now discuss the
detection of gravitational waves with Laser Interferometer Gravitational Wave
Observatory (hereafter LIGO)/Virgo interferometer.

A paper, “Testing the No-Hair Theorem with GW150914” [22], was published at
about the same time as the six letters published in _The Astrophysical Journal
Letters_ [10]-[15]. The paper presents an analysis of gravitational-wave data
from the first LIGO detection of the binary black-hole merger
GW150914.242424The gravitational-waves signals are surrounded by noise.
Gravitational-wave signals are extracted from the background noise using
statistical significance tests and ”templete waveforms” (model) fitting to
gravitational-wave data. The gravitational waves from the merger GW150914 were
so loud that they were visible in the data even with minimal data processing.

Two stellar-mass black holes that merge, form a single distorted compact
object that gradually settles to a final stationary form. Gravitational waves
are emitted throughout the entire process, at each moment carrying information
about the evolving compact object. There are three stages in the coalescence
of the two black holes:

1) _Inspiral_ : a long phase in which the two black holes are still quite far
one from each other but are slowly orbiting one another in a quasi-circular
orbit. Since we are dealing with weak gravitational fields, this phase allows
an analytic description using the post-Newtonian expansion (approximation).
During this phase, the amplitude and frequency of the gravitational waves are
increasing.

2) _Merger_ : the distance between the two black holes gets smaller, the orbit
gradually shrinks and the two bodies finally merge. Numerical relativity
(numerical simulations) is required. That is because the gravitational field
is so strong and also time-dependent. During this phase, the amplitude of the
gravitational waves gradually increases until it reaches a maximum.

3) _Ringdown_ : the newly created object releases its final gravitational wave
signals away into endless space. It wobbles and oscillates during which it
rings like a bell with characteristic frequencies and damping times determined
entirely by the mass and spin of the black hole. This stage is treated
analytically using perturbation theory. A linearly perturbed Kerr black hole
emits gravitational waves in the form of exponential damped sine waves, with
specific frequencies and decay rates determined exclusively by the black
hole’s mass and spin.

The ringdown phase consists of a superposition of quasi-normal modes. Each
quasi-normal mode has a characteristic complex angular frequency: the real
part is the angular frequency and the imaginary part is the inverse of the
damping time. These modes are distinguished by their longitudinal and
azimuthal indices, $l$ and $m$ respectively, as well as by their overtone
number $n$. Ringdown overtones are the quasi-normal modes with the fastest
decay rates. Each mode has a particular frequency and decay rate which are
functions of the Kerr black hole parameter spin and total mass of the black
hole that is being perturbed. Numerical simulations have demonstrated that the
fundamental mode $l=m=2$ seems to dominate the ringdown signal [3], p.
124018-18, p. 124018-21.

In 2016, the LIGO Scientific and Virgo Collaborations announced the first
joint detection of gravitational waves (GW150914) with both the LIGO and Virgo
detectors. In 2018, they reported of several quantitative tests made on the
gravitational-wave data from the detection of GW150914. The purpose of the
tests was to validate GR. The result of the tests indicates that the entire
GW150914 inspiral-merger-ringdown (IMR) waveform does not deviate from the
predictions of a binary black-hole merger in classical general relativity
[25], pp.221101-3-221101-5.

The detected gravitational-wave signal increases in frequency and amplitude in
about eight cycles from $35$ to $150$ Hz, where it reaches a peak amplitude.
After a time around $0.42s$, the amplitude drops rapidly, and the frequency
appears to stabilize. After the peak gravitational wave amplitude is reached,
the signal makes one to two additional cycles, continuing to rise in frequency
until reaching about $250$ Hz, while dropping sharply in amplitude.

The most plausible explanation for this empirical evolution is gravitational-
waves emission from two orbiting masses in the inspiral and merger phases. The
drop in amplitude is consistent with a Kerr black hole. Recall that for a Kerr
black hole, the ringdown is expected to have a damping time roughly equal to
the period of oscillation (inversely equal to the frequency). For a black hole
with spin $\chi=0.7$, a ringdown frequency of $\approx 260$
Hz$\left(\frac{65M\odot}{M}\right)$ and a damping time:
$4ms\left(\frac{M}{65M\odot}\right)$ are calculated. The LIGO Scientific and
Virgo Collaborations conclude: ”the signal in the data is fully consistent
with the final object being a Kerr black hole with a dimensionless spin
parameter $\chi=0.7$ and a mass $65M\odot$” [24], pp. 2-3, pp. 12-13.

The LIGO Scientific and VIRGO Collaborations explain that the initial behavior
of the gravitational-wave signal ”cannot be due to a perturbed system
returning back to stable equilibrium, since oscillations around equilibrium
are generically characterized by roughly constant frequencies and decaying
amplitudes”. On the other hand, the GW150914 data ”demonstrate very different
behavior. During the period when the gravitational wave frequency and
amplitude are increasing, orbital motion of two bodies is the only plausible
explanation: there, the only ’damping forces’ are provided by gravitational
wave emission, which brings the orbiting bodies closer (an ’inspiral’),
increasing the orbital frequency and amplifying the gravitational wave energy
output from the system” [24], p. 3.

It is concluded: ”our inspiral–merger–ringdown test shows no evidence of
discrepancies with the predictions of GR” [25], p.221101-5.

The LIGO Scientific and Virgo Collaborations performed a test that allowed
”for possible violations of GR” and concluded that the value of parameters was
usually found to represent GR. An additional test was performed. They asked:
Does GR fit the data better than alternative competing models? The following
model was compared with GR: a model based on theories of gravity mediated by a
graviton with a nonzero mass in which the gravitational waves travel at a
speed different than the speed of light. In GR, gravitons are massless and
travel at the speed of light. The presence of non-GR polarization states were
searched for. But it was difficult to distinguish between GR and non-GR models
on the basis of GW150914 data alone. Since the Hanford and Livingston LIGO
instruments have similar orientations, they are sensitive to a very similar
linear combination of the gravitational-wave polarizations, so it is difficult
to distinguish between the GR and non-GR states [25], p.221101-10.

It is concluded: ”With the exception of the graviton Compton wavelength and
the test for the presence of a non-GR polarization, we did not perform any
studies aimed at constraining parameters that might arise from specific
alternative theories” [25], p. 221101-8, pp. 221101-10-11.

Testing the black hole ring at the correct frequencies and damping times, it
ought to be possible to test the validity of the no-hair theorem, which states
that the remnant must be a Kerr black hole.

Already in 2004, a team of astrophysicists had asked: “can gravitational wave
observations provide a test of one of the fundamental predictions of general
relativity: the no-hair theorem?” They suggested “a definitive test of the
hypothesis that observations of damped, sinusoidal gravitational waves
originate from a black hole or, alternatively, that nature respects the
general relativistic no-hair theorem” [7], p. 787.

In 2017, three astrophysicists belonging to the LIGO Scientific and Virgo
Collaborations, Eric Thrane, Paul Lasky and Yuri Levin, wrote that “The recent
detections of gravitational waves from stellar-mass black hole mergers would
seem to suggest that a test of the no-hair theorem might be around the
corner”. A method was provided ”for testing the no-hair theorem using only
data from after the remnant black hole has settled into a perturbative state”.

The post-merger remnant must be allowed to settle into a perturbative, Kerr-
like state and this means that the ringdown frequencies and damping times
depend only on the mass and spin of the newly created black hole. In this way,
the no-hair theorem places stringent requirements on the asymptotic behavior
of perturbed black holes. The no-hair theorem concerns itself with linear
perturbations. But it was found that at no point in time is the post-merger
waveform precisely described by black hole perturbation theory. That is
because, there is always a contribution, however small, left over from the
merger. Furthermore, the ringdown signal becomes weaker as it settles into a
Kerr black hole.

The authors were confronted with the following problem: they could either
obtain higher signal-to-noise (SNR) ratio by adding louder signals from the
merger phase (the main gravitational wave signal) or, wait for the remnant
black hole to settle to the perturbative state where the no-hair theorem
applies. They chose the latter option because as louder signals were added, it
was no longer clear whether this improved the knowledge of the ringdown
frequency and damping times. So, they arrived at the conclusion that it may be
possible to test the no-hair theorem but the observed behavior could either be
attributed to a numerical relativity artifact, or to a residual non-linearity
of the signal from the merger decaying on a timescale comparable to the linear
ringdown signal [31], pp. 102004-1-2, p. 102004-5.

Matthew Giesler, Mark Scheel and Saul Teukolky have been working on the LIGO
data and gravitational waves. In May 2019, they submitted a paper with
Maximiliano Isi and Will Farr from the LIGO Scientific and Virgo
Collaborations in which they analyzed the ringdown gravitational-wave data.
They thought that a gestalt shift was needed. It does not mean that Thrane,
Lasky and Levin’s reservations are not sound. It only means that one team
thinks they have solved the problem whilst the other probably holds an
opposite view.

So, Geisler and his team have shown that including enough overtones of the
mode $l=2$ allows obtaining higher SNR. A GW150914-like signal was first
studied.252525A simulated output of a LIGO-like detector was studied in
response to the same gravitational waves as the ones from the asymptotic
remnant. That is, the $l=m=2$ mode of the signal was injected into simulated
Gaussian noise corresponding to the sensitivity of Advanced LIGO in its design
configuration. It was found that the inclusion of enough overtones associated
with the $l=m=2$ mode “provides a high-accuracy description of the ringdown”,
where the high SNR “can be exploited to significantly reduce the uncertainty
in the extracted remnant properties” [17], p. 041060-2, p. 041060-8, p.
041060-10.

Previously, overtones had been believed to be too faint to be detected and it
was believed that one had to wait at least ten years to test the no-hair
theorem. But as shown above, studies done by the team have shown that although
the overtones decay very quickly, the first overtones of GW150914 should be
loud enough to be detected.

Measuring the quasi-normal modes from gravitational-wave observations is
tantamount to measuring the ringdown spectrum. The ringdown spectrum is ”a
fingerprint that identifies a Kerr black hole” and the measurements of this
spectrum ”has been called black-hole spectroscopy” [22], p. 111102-1.

Isi et al. were “Assuming the remnant is a Kerr black hole”. That is,
“assuming first that quasinormal modes are as predicted for a Kerr black hole
within general relativity”, they checked this assumption [22], p. 111102-2.

Making use of overtones, Isi et al. extracted information about the GW150914
remnant using only post-inspiral data, starting at the peak of the signal. The
mass and spin of the remnant were measured using the overtones of the mode
$l=m=2$. The GW150914 ringdown using this mode and its first overtone were
analyzed. These are ringdown-only measurements. The ringdown-only measurements
of the remnant mass and spin magnitude were then compared to those obtained
from the analysis of the entire GW150914 inspiral-merger-ringdown (IMR)
waveform. It was concluded that evidence of the mode $l=m=2$ and at least one
overtone was found and, a $90\%$-credible measurement of the remnant mass and
spin was obtained in agreement with that inferred from the full waveform.

Measuring the frequencies of the fundamental and first overtone, information
was extracted about the GW150914 remnant and ”their consistency with the Kerr
hypothesis” was established. It was concluded that the GW150914 merger
”produced a Kerr black hole as described by general relativity” [22], p.
111102-5.

It is stressed at the end of Isi et al.’s paper that, as the precision of the
detectors improves, it can be expected that using overtones will provide
better tests of the no-hair theorem [22], p. 111102-1, p. 111102-3, p.
111102-5.

##  8 Inference to the best explanation

The first question asked by the EHT Collaboration is: Does M87’s core have a
horizon? Is the object at the center of the galaxy M87 a spinning Kerr black
hole as predicted by general relativity?

In order to answer this question, the EHT Collaboration first endorsed the
working hypothesis that the central object is indeed a black hole described by
the Kerr metric. They chose this hypothesis based on previous research and
observations of the galaxy M87. In 2015 a team of scientists belonging to the
EHT Collaboration made an experiment the goal of which was to compare the
merits of two rival hypotheses: a compact object with a surface and a black
hole with a horizon. The black hole hypothesis was selected as the best
explanation of the evidence.

The EHT Collaboration tested the Kerr black hole hypothesis and found that it
was consistent with the evidence. How did they test the Kerr black hole
hypothesis? GRMHD models based on the Kerr metric were used to extract
parameters from the 2017 EHT data (see Sections 3, 4 and 5) and then it was
shown that the data confirm the hypothesis that the compact object at the
center of M87 is a supermassive Kerr rotating black hole (see Section 6). The
sixth letter ends with the following statement: “Together, our results
strongly support the hypothesis that the central object in M87 is indeed a
Kerr black hole” [15], p. 23.

Ian Hacking once said: “This looks like a classic example of what
philosophers, following Gilbert Harman ([20]) have come to call inference to
the best explanation (IBE)”. He added: “there is an overall conjecture of the
sort that Popper would have called metaphysical” ([19], pp. 568-569). In our
case study, in the fifth letter it is written: “the results provided here are
consistent with the existence of astrophysical black holes” ([14], p. 19).
Hacking and Wesley Salmon are on the same page when it comes to supermassive
black holes.262626Salmon wrote: astrophysicists believe that there is a black
hole at the center of our galaxy. But he was not convinced of this particular
existence claim and he added that Hacking was “skeptical far more generally
about black holes”. From what emerges from Hacking’s papers, he is indeed
skeptical about black holes. They have argued that scientific existence claims
are always open to question. When one says that astrophysical black holes
exist, it adds nothing to our understanding and to the explanation. The fact
that the black hole hypothesis explains the observations does not provide
evidence that black holes exist. Existence claims with respect to black holes
are not part of the explanation [18], pp. 52-54; [30], p. 56, p. 58.

The Kerr black hole hypothesis has explanatory power in accounting for:
_ingredient 2_ , _ingredient 3_ (see Sections 6 and 3) and the GW150914
Ringdown. It is also the best explanation of the evidence for M87’s core and
for the GW150914 gravitational wave event. Does it mean that it is also true?
If truth means that no other hypothesis will explain the evidence as well as
the Kerr black hole hypothesis, then the answer is: we cannot say that it is
true. The reason is that the EHT Collaboration has written: “Future
observations and more detailed theoretical modeling, combined with
multiwavelength campaigns and polarimetric measurements, will further
constrain alternatives to Kerr black holes” [14], p. 18; see the end of
Section 5. And Isi et al. write: ”Future studies of black-hole ringdowns
relying on overtones could potentially allow us to identify black-hole
mimickers”, that is, exotic alternatives to the Kerr black hole hypothesis,
”and probe the applicability of the no-hair theorem with high precision, even
with existing detectors”. This would ”lead to more specific predictions from
general relativity” [22], p. 111102-5.

##  9 Underdetermination of theory by evidence

Doeleman has said: ”Now if we could make an image of this object, then we
could test Einstein’s theories of gravity in the one place where they might
break down in the universe” [6]. Can we also test Hawking radiation in the one
place where Einstein’s theory of gravity breaks down?

Hawking pointed out that evaporation violates the classical censorship
hypothesis. Hawking explained that if one tries to describe the process of a
black hole losing mass and eventually disappearing and evaporating by a
classical space-time metric, then “there is inevitably a naked singularity
when the black hole disappears. Even if the black hole does not evaporate
completely one can regard the emitted particles as having come from the
singularity inside the black hole and having tunneled out through the event
horizon on spacelike trajectories” [21], p. 2461; see [34], p. 23.

Since the rate of evaporation of a supermassive black hole is so slow and the
Hawking radiation is too faint to be observable, the 2017 EHT observations can
neither confirm nor refute Hawking’s evaporation. And since the evaporation
time of a supermassive black hole is no less than $10^{94}$ years, then
according to Hawking’s hypothesis at the time at which the EHT experiment was
performed in 2017, the compact object at the center of M87 could not be a
naked singularity.272727The evaporation time $t$ for the black hole, from the
point of view of an outside observer, is proportional to the cube of the mass
of the black hole: $t=\frac{5120\pi G^{2}M_{BH}^{3}}{\hbar c^{4}}$ [34], p.
18. And according to Hawking’s theory, it will have a horizon for many years
to come. Paradoxically, Hawking’s above hypothesis can explain at least as
well the evidence as the no-hair theorem. Although according to Hawking, a
supermassive black hole will gradually evaporate through Hawking radiation,
the rate of this evaporation is so slow that we cannot detect it. Collecting
additional and more accurate data from the object at the center of M87 will
not be enough to detect Hawking radiation.

That is exactly the point raised by Bas van Fraassen: “Is the best explanation
we have, likely to be true?” He answers this question in the negative saying
that there are many possible theories, perhaps never yet formulated, but that
can fit all the evidence so far. These theories can explain at least as well
the evidence as the best theory we have now. But with respect to statements
that go beyond the evidence we possess today these theories disagree with one
another. We consider them to be false because of underdetermination of theory
by evidence. Classical general relativity and the no-hair theorem are our best
explanation and they belong to the class of explanations that can fit all the
evidence to date, but they also disagree with certain quantum gravity
theories, theories with quantum corrections to classical general relativity
[9], pp. 160-161.

## References

  * [1] Bardeen, J. M. (1973). ”Timelike and Null Geodesics in the Kerr Metric.” In DeWitt C. and DeWitt B. S. (eds.). _Black Holes Les Astres Occlus._ New York: Gordon and Breach Science Publishers, pp. 216-239.
  * [2] Blandford, R. D. and Znajek, R. L. (1977). “Electromagnetic extraction of energy from Kerr black holes.” _Monthly Notices of the Royal Astronomical Society_ 179, pp. 433-456.
  * [3] Buonanno, A., Cook, G. B., and Pretorius, F. (2007). ”Inspiral, merger, and ring-down of equal-mass black-hole binaries.” _Physical Review D_ 75, pp. 124018-1-124018-42.
  * [4] Broderick, A. E., Gold, R. G., Karami, M., et al. (Event Horizon Telescope Collaboration) (2020). “Themis: A parameter estimation framework for the event horizon telescope.” _The Astrophysical Journal_ 897, pp. 1–38.
  * [5] Broderick, A. E., Narayan, R., Kormendy, J., Perlman, E. S., Rieke, M. J., and Doeleman, S. S. (2015). “The event horizon of M87.” _The Astrophysical Journal_ 805, pp. 1-9.
  * [6] Doeleman, S. S. (2020). “PSW 2421. The Event Horizon Telescope. First Ever Images of a Black Hole.” _The 89th Joseph Henry Lecture at the Philosophical Society of Washington Science_.
  * [7] Dreyer, O., Kelly, B., Krishnan, B. Finn, L. S., Garrison, D., and Lopez-Aleman, R. (2014). ”Black-hole spectroscopy: testing general through gravitational-wave observations.” _Classical and Quantum Gravity_ 21, pp. 787–803.
  * [8] Falcke, H., Melia, F., and Agol, E. (2000). “Viewing the Shadow of the Black Hole at the Galactic Center.” _The Astrophysical Journal_ 528, pp. L13–L16.
  * [9] van Fraassen, B. C. (1989). _Laws and Symmetry_. Oxford: Oxford University Press.
  * [10] Event Horizon Telescope Collaboration (2019). “First M87 event horizon telescope results. I. The shadow of the supermassive black hole.” _Astrophysical Journal Letters_ 875: L1, pp. 1-17.
  * [11] Event Horizon Telescope Collaboration (2019). “First M87 event horizon telescope results. II. Array and Instrumentation.” _Astrophysical Journal Letters_ 875: L2, pp. 1-28.
  * [12] Event Horizon Telescope Collaboration (2019). “First M87 event horizon telescope results. III. Data processing and calibration.” _Astrophysical Journal Letters_ 875: L3, pp. 1-32.
  * [13] Event Horizon Telescope Collaboration (2019). “First M87 event horizon telescope results. IV. Imaging the central supermassive black hole.” _Astrophysical Journal Letters_ 875: L4, pp. 1-52.
  * [14] Event Horizon Telescope Collaboration (2019). “First M87 event horizon telescope results. V. Physical origin of the asymmetric ring.” _Astrophysical Journal Letters_ 875: L5, pp. 1-31.
  * [15] Event Horizon Telescope Collaboration (2019). “First M87 event horizon telescope results. VI. The shadow and mass of the central black hole.” _Astrophysical Journal Letters_ 875: L6, pp. 1-44.
  * [16] Gebhardt, K., Adams, J., Richstone, D., Lauer, T. R., Faber, S. M., Gültekin, K., Murphy, J. and Tremaine, S. (2011). “The Black-Hole Mass in M87 from Gemini/NIFS Adaptive Optics Observations.” _The Astrophysical Journal_ 729, pp. 1-13.
  * [17] Giesler, M., Isi, M., Scheel, M. A., and Teukolsky, S. A. (2019). ”Black Hole Ringdown: The Importance of Overtones.” _Physical Review X_ 9, pp. 041060-1-041060-13.
  * [18] Hacking, I. (1983). _Representing and Intervening: Introductory. Topics in the Philosophy of Natural Science_. Cambridge: Cambridge University Press.
  * [19] Hacking, I. (1989). “Extragalactic Reality: The Case of Gravitational Lensing.” _Philosophy of Science_ 56, pp. 555-581.
  * [20] Harman, G. (1965). “The Inference to the best explanation.” _The Philosophical Review_ 74, pp. 88-95.
  * [21] Hawking, S. W. (1976). “Breakdown of predictability in gravitational collapse.” _Physical Review_ D 14, pp. 2460-2473.
  * [22] Isi, M., Giesler, M., Farr, W. M., Scheel, M. A., and Teukolsky, S. A. (2019). ”Testing the No-Hair Theorem with GW150914.” _Physical Review Letters_ 123, pp. 111102-1-111102-6.
  * [23] Johnson, M. D. (2020). “Photographing a Supermassive Black Hole with the Event Horizon Telescope.” _Lecture at Black hole Initiative seminar_ , Harvard university, April 12.
  * [24] LIGO Scientific and Virgo Collaborations (2017). ”The basic physics of the binary black hole merger GW150914.” _Annalen der Physik_ 529, pp. 1-17.
  * [25] LIGO Scientific and Virgo Collaborations (2018). ”Tests of General Relativity with GW150914.” _Physical Review Letters_ 116, pp. 221101-1-221101-19.
  * [26] Penrose, R. (1969). “Gravitational Collapse: The Role of General Relativity.” _Rivista del Nuovo Cimento, Numero Speziale_ 1, pp. 252-275; _General Relativity and Gravitation_ 34, 2002, pp. 1141-1165.
  * [27] Porth, O., Chatterjee, K., Narayan, R., et al. (Event Horizon Telescope Collaboration) (2019). “The Event Horizon General Relativistic Magnetohydrodynamic Code Comparison Project.” _Astrophysical Journal, Supplement Series_ 243, pp. 1-40.
  * [28] Psaltis, D. and Doeleman, S. (2015). “The Black Hole Test.” _Scientific American_ 313, pp. 74-79.
  * [29] Psaltis, D., Özel, F., Chan, C-K and Marrone, D. P. (2015). “A General Relativistic Null Hypothesis Test with Event Horizon Telescope Observations of the Black Hole Shadow in Sgr A*.” _The Astrophysical Journal_ 814, pp. 1-14.
  * [30] Salmon, W. C. (2005). _Reality and Rationality_. Dowe, P. and Salmon, M. H. (eds.). New York: Oxford University Press.
  * [31] Thrane, E. Lasky, P. D., and Levin, Y. (2017).”Challenges for testing the no-hair theorem with current and planned gravitational-wave detectors.” _Physical Review D_ 96, pp. 102004-1-102004-6.
  * [32] Walsh J. L., Barth A. J., Ho Luis C. and Sarzi, M. (2013). “The M87 Black Hole Mass from Gas dynamical Models of Space Telescope Imaging Spectrograph Observations.” _The Astrophysical Journal_ 770, pp. 1-11.
  * [33] Weinstein, G. (2020). “Coincidence and reproducibility in the EHT black hole experiment.” _Studies in History and Philosophy of Science_ , in press, pp. 1-16.
  * [34] Weinstein, Galina (2021). ”Demons in Black Hole Thermodynamics: Bekenstein and Hawking.” _ArXiv_ :2021.11209.v1 [physics.hist-ph] 22 Feb.