# Black holes and the nature of the event horizon

Svetlana Andrusenko  Physics Department, Bauman Moscow State Technical
University  MIREA — Russian Technological University  Daniil Krichevskiy
Institute for Theoretical Physics, University of Bern  Sternberg Astronomical
Institute, Lomonosov Moscow State University  Valentin Rudenko Corresponding
Author<EMAIL_ADDRESS>Sternberg Astronomical Institute, Lomonosov
Moscow State University  Faculty of Physics, Lomonosov Moscow State University

(2022)

###### Abstract

The problem of the event horizon in relativistic gravity is discussed.
Singular solutions in general relativity are well known. The Schwarschild
metric of a spherical mass is singular at zero ($r=0$) and at the event
horizon ($r=r_{g}$). Both features reflect the existence of the phenomenon of
collapse in general relativity for compact masses exceeding $3M_{\odot}$. A
material particle crossing the event horizon falls into a central singularity
according to the classical theory of general relativity. In the quantum theory
of gravity, there may be no central singularity. The physics of the event
horizon is currently being refined. A promising technique is the study of
gravitational waves (GW) accompanying the merger of binary black holes at the
ringdown stage. GW observations of quasi-normal modes of the newly formed
super dense remnant will help clarify the physics of the event horizon.

Keywords: general relativity, black holes, event horizon, gravitational waves

## Introduction

Formally, Newton’s discovery of the law of universal gravitation dates back to
1687 [1], although the laws of Kepler’s celestial mechanics (according to
Tycho Brahe’s observations) were established much earlier, in the interval
1609-1619 (for literary and historical details, see, for example, [2]). A
hundred years after Newton, in 1783, the English priest and amateur astronomer
John Mitchell suggested that there are astrophysical objects (bodies) for
which the speed of overcoming their gravitational attraction exceeds the speed
of light. Fifteen years later, in 1798, the French scientist P. Laplace also
expressed and substantiated, using Newton’s law of universal gravitation, the
idea of the existence of dark bodies in the Universe. At present, it is
generally accepted that two scientists were the predictors of dark stars: the
Englishman Mitchell and the Frenchman Laplace (More detailed information is
given in the work of Cherepashchuk A.M. [3]). However, in these
representations there was no hint of the gravitational collapse of stars, the
phenomenon of singularity, the effect of accretion of the surrounding matter
and the generation of X-ray luminosity. These details of the physical picture
of dark stars, or in modern terminology - Black Holes (BH), - came to science
after the creation of the general theory of relativity (GR) and the
accelerated development of tools for astronomical observations. The initial
dispute about the reality of the existence of BH was resolved in favor of GR
predictions about the behavior of matter in strong gravitational fields.
Experimental search for BHs of stellar masses $(3-100)M_{\odot}$ has been
carried out since the mid-60s of the 20th century, and giant BHs, in the
centers of galaxies and star clusters $(10^{6}-10^{9})M_{\odot}$, since the
mid-80s. Today these astrophysical objects occupy their equal place among the
classical population of the Universe: stars, planets, star clusters, etc.
Moreover, when studying the physics and evolution of the latter, it is
necessary to take into account the BHs influence. A new field of astrophysical
science has emerged – ‘demography of black holes’, which studies their birth,
growth and interaction with other objects of the Universe. At present, the
number of BHs of stellar and galactic masses is estimated as $10^{7}$ or
$0.1\%$ of the baryon mass of the Universe [3]. In parallel with the
observations (and stimulated by them), the BH theory was actively developed
(as an example, one can point to the well-known monographs [4] or [5]) both in
the framework of the classical BH and in its generalizations [6]. Note that in
a number of alternative theories of gravity, the BH phenomenon is completely
absent [7]. Many amazing properties of BH have been explained. However, due to
the complexity of the object, in particular rotating BH, a number of aspects
need further clarification. These include the non-trivial concept of ”BH event
horizon”. In this article, special attention is paid to this issue, in fact,
its discussion is the main goal of the work. The structure of the article is
as follows. Sections 1 and 2 reproduce the well-known evidence for the
inevitability of collapse (the singularity phenomenon) both due to the GR
mathematical structure and from the thermodynamical and statistical features
of the physics of the evolution of massive stars. Section 3 contains ideas for
searching for BH objects among X-ray sources detected by outside atmospheric
sensors. Such sources are often binary objects with a BH component, producing
X-rays due to the accretion of the surrounding matter onto the BH. This
section also describes the recent registration of gravitational radiation from
the merger of binary BHs. It has been suggested that the shape of the GW
signal should carry information about the conversion of two independent ”event
horizons” into a third one at the ”ring down” stage of the superdense remnant.
Section 4 discusses the latest advances in the registration of the BH shadow
at the centers of galaxies. Photos of the photon sphere are given allowing one
to see the silhouette of the event horizon.

##  1 Singular solutions in general relativity

The general theory of relativity (GR) fundamentally considers situations with
strong gravitation fields when the geometry differs significantly from the
Euclidean one, which entails the corresponding physical (observational)
manifestations (effects).

There are two principal implications of this kind:

  1. 1.

Black holes (BHs) (local geometry change)

  2. 2.

Cosmological singularities (the geometry of the Universe as a whole)

An example of a singular solution in GR is the Schwarzschild solution for a
spherically symmetric mass $M$:

$ds^{2}=\left(1-\frac{r_{g}}{r}\right)dt^{2}-\left(1-\frac{r_{g}}{r}\right)^{-1}dr^{2}-r^{2}\left(d\theta^{2}+sin^{2}\theta\
d\varphi^{2}\right),$ (1)

where $(r,\theta,\varphi)$ are spherical coordinates, $t$ is coordinate time
corresponding to proper time of an infinitely distant observer and
$r_{g}=2GM/c^{2}$ is Schwarzschild radius (or the gravitational radius)
defined with Newton’s constant $G$ and speed of light $c$.

What do these singularities at $r=0$ and $r=r_{g}$ mean? If we calculate the
square of the curvature tensor $R^{ikln}R_{ikln}$ which is an invariant (i.e.
doesn’t depend on the choice of coordinates) using (1) it turns out that it
has no singularities at $r=r_{g}$! This means that the Schwarzschild
singularity is a consequence of an unsuccessful choice of the coordinate
system.

###  1.1 Lemaitre’s metric

Let us consider the Lemaitre reference frame accompanying the particles (dust
grains) freely falling along the radial direction towards the center of the
field source [8]. In this system, the Schwarzschild metric looks like:

$ds^{2}=d\tau^{2}-\frac{1}{r}dR^{2}-r^{2}\left(d\theta^{2}+sin^{2}\theta\
d\varphi^{2}\right),$ (2)

where $R$ is the coordinate of the event (mark of the dust grain), the time
$\tau$ is proper time of a particle (taken from the clock of the particle).
The function $r\left(\tau,R\right)$ is given by the equations (for $r_{g}=1$)

$\begin{cases}\tau-R=\frac{2}{3}r^{3/2}&\text{(expansion)}\\\
R-\tau=\frac{2}{3}r^{3/2}&\text{(compression)}\end{cases}\longrightarrow
r=\left[3/2\left(R-\tau\right)\right]^{2/3}$ (3)

For $r=r_{g}=1$ the singularity is absent. Only the irremovable singularity
remains at $r=0\ (R=\tau)$.

Propagation of light signals ($ds^{2}=0$) in the Lemaitre metric along the
radial direction ($\left(d\theta^{2}+sin^{2}\theta\ d\varphi^{2}\right)=0$) is
described by the following equation:

$\begin{gathered}d\tau^{2}-\frac{1}{r}dR^{2}=0\Rightarrow,\\\
\frac{d\tau}{dR}=\pm\frac{1}{\sqrt{r}}\sim\pm\left[R-\tau\right]^{-1/3}.\end{gathered}$
(4)

It can be seen that the world line of a particle at rest lies inside the light
cone for $r>1$ and outside the light cone for $r<1$ [8].

Figure 1: Trajectories of particles in Lemaitre coordinates.

It follows from here that there are no particles at rest beyond the
gravitational radius; all particles tend to the center, i.e. collapse. Thus
gravitational collapse is one of the fundamental consequences of GR leading to
the BHs existences.

##  2 Thermodynamics of stars evolution

Gravitational collapse also follows from thermodynamic and statistical
concepts. Let’s consider the initial stage which precedes collapse: the star
is a ball of gas held in equilibrium by thermal pressure. Radiation comes at
the expense of gravitational energy, with constant compression and,
consequently, an increase in temperature in the center. At the advanced stage,
the temperature reaches the threshold of nuclear reactions. Then already
nuclear energy begins to compensate for the radiation. At the start of nuclear
combustion, the helium fusion reaction mainly takes place: $4p\longrightarrow\
^{4}He$. When the nuclear fuel runs out, the star turns into a white dwarf in
which the gravitational forces are balanced by the pressure of the cold Fermi
gas of electrons

$\epsilon_{F}\ll m_{e}c^{2}.$ (5)

###  2.1 Neutron star stage

When $\epsilon_{F}>\left(m_{n}-m_{p}\right)c^{2}$ neutronization occurs
$e^{-}+p\longrightarrow n+\nu_{e}$. A neutron star of mass $M$ and radus $R$
is formed which does not contain free electrons. A simple estimate of the
equilibrium condition can be done. The gravity pressure $P_{G}$

$P_{G}R^{2}\simeq\frac{aM^{2}}{R^{2}}$ (6)

has to be compensated by the neutron Fermi gas pressure $P_{F}$:

$P_{F}\sim\epsilon_{F}\sim\frac{1}{m_{n}}n^{5/3}\sim\frac{M^{5/3}}{m^{8/3}}\frac{1}{R^{5}},$
(7)

where $n$ is the concentration of the neutron gas. The equilibrium condition
$P_{G}\simeq P_{F}$ gives

$\frac{aM^{2}}{R^{4}}=\frac{M^{5/3}}{m^{8/3}}\frac{1}{R^{5}}\cdot const$ (8)

From the equilibrium condition, we see that the pressure of the neutron gas
increases with compression faster than the gravitational pressure. Equilibrium
radius $R_{0}$ is

$R_{0}\approx\frac{c}{4M^{1/3}m^{8/3}}=\frac{1}{\gamma
m}\left(\frac{m}{M}\right)^{1/3},$ (9)

with $\gamma=Gm^{2}=6\cdot 10^{-39}$.

If the stars are massive enough, then the neutron gas cannot be cold, it
becomes relativistic:

$P_{F}\sim n^{1/3}\sim mc,$ (10) $\left(\frac{M}{mR^{3}}\right)^{1/3}\sim mc$
(11)

Then the mass critical value $M_{cr}$ reads

$M_{cr}=\frac{m}{\gamma^{3/2}}\simeq 1.8\cdot 10^{33}$ (12)

When $M>M_{cr}$ the neutron star becomes unstable. For a relativistic neutron
gas the pressure is

$P_{n}=\epsilon_{F}\sim n^{1/3}n\longrightarrow
P_{n}\sim\frac{M^{4/3}}{m^{4/3}R^{4}}.$ (13)

At the same time, the gravitational mass and the negative pressure of gravity
read

$c^{2}M_{G}\sim\epsilon_{F}\sim n^{1/3}N,$ (14)
$M_{G}\sim\left(\frac{M}{mR^{3}}\right)^{1/3}\frac{M}{m}\longrightarrow
P_{G}=\frac{GM^{2}_{G}}{R^{4}}\sim\frac{1}{R^{6}},$ (15)

Thus, the force of gravitational pressure increases faster during contraction
than the pressure force of a relativistic Fermi gas, and the star must
contract uncontrollably.

###  2.2 X-ray sources as BH candidates

In GR motion under the gravitational radius leads to a fatal collapse. Stars
with masses $M>3M_{\odot}$ must collapse at the end of their evolution. What
happens at the singularity is unknown. Up to $10^{-33}$ cm, one can use the GR
equations, after - a quantum gravity comes into play.

The awareness of the inevitability of the gravitational collapse of massive
and rather compact stellar objects leads to the problem of their experimental
(observational) detection. Here are the following key questions. How to search
for BHs? What are their astrophysical manifestations? Outside the BH, the
observer should see the standard Schwarzschild field. If there is matter in
its vicinity in the form of gas or dust then it will be attracted and fall on
the BH. As shown in [9] non-spherical accretion must be accompanied by X-ray
emission. Due to the huge gravitational potential near the BH the speed of
matter in the inner parts of the accretion disk is close to the speed of
light. Mutual friction and collisions of gas flows in the disk heat up the
plasma to temperatures of tens of millions of degrees and form powerful
X-rays. The released energy can significantly (by an order of magnitude)
exceed the energy of thermonuclear reactions. Due to non-stationary processes
in the disk this X-ray emission is variable at times up to $10^{-3}$ s.

Such a picture is typical for a BH in a binary system with an optical star.
When entering the Roche lobe, the BH will pull the matter of the normal
component towards itself. Thus, the detection of BHs is associated with the
search for astrophysical objects with powerful X-ray luminosity.

##  3 Black holes in binary system

The search for and measurements of the parameters of X-ray sources with BHs
has been carried out (and continues) for several decades. Review articles with
morphological analysis and systematics of stellar-mass BHs were published, in
particular, by A.M. Cherepashchuk in Physics-Uspekhi journal. The main results
at the conceptual level are presented in the paper [3] cited above. It is
useful to present a number of illustrations and statistics from this work
below (Fig. 2 and 3).

Figure 2: Model of an X-ray binary system. Optical star in Roche lobe losing
its matter to form the accretion disk around the precessing BH component with
relativistic ejections jets.  Figure 3: Observational data for estimating the
mass of a BH in X-ray binary: (a) computer model of the binary system; (b)
X-ray spectrum of the binary system ; (c) radial velocity curve of the optical
component; (d) light curve of the system (along the y-axis are magnitudes of
the object; along the x-axis is the phase of orbital period).  Figure 4:
Diagram of a Schwarzschild BH in Euclidean space (neighborhood of the event
horizon). In the center is the singularity where the matter fell. The
singularity is surrounded by a spherical event horizon, with radius
$r_{g}=2GM/c^{2}$. Next is the photon sphere, in which the captured photons
move along closed trajectories, the radius of the photon sphere is $1.5r_{g}$.
Due to the strong curvature of light rays in the BH gravitational field, it
acts as a lens for itself. In this case, the size of the event horizon and the
photon sphere seem to be increased to an external observer.

###  3.1 Demographics of black holes

At the moment, thousands of X-ray binaries have been discovered from the
boards of orbital X-ray observatories in which NS and BH of stellar masses are
detected and studied. The masses of three dozen stellar BHs in X-ray binaries
have been measured, as well as the masses of about 70 NS. In this case, the BH
masses lie within $(4\div 16)M_{\odot}$ and NS masses - $(1\div 2)M_{\odot}$
and are concentrated near the value $\sim 1.4M_{\odot}$. It was also possible
to measure the angular momentum of rotation for more than a dozen BH of
stellar masses by comparing their X-ray spectra with theoretical spectra (the
model of thin accretion - Novikov and Thorne 1973 [10]). The normalized
torques of the BH lie in a wide range from $0.98$ to $0.12$, i.e. among
stellar BHs, there are both rapidly rotating and slowly rotating.

###  3.2 The physical riddle of the event horizon

Is this concept related to the presence of a material (observable) surface in
collapsars? The study of X-ray binaries showed that their relativistic NS and
BH objects differ not only in masses, but also in observational
manifestations. In all 70 cases, when a compact object with $M<3M_{\odot}$
shows signs of an observed surface, it is an NS: X-ray pulsar, radio-pulsar or
X-ray burster of the first type requiring the presence of a strong magnetic
field. On the contrary, none of the three dozen massive ($M>3M_{\odot}$)
compact objects (candidates for BHs) shows signs of an observable surface in
accordance with the GR prediction. Instead, there are only conceivable
boundaries - event horizons - light surfaces in spacetime (which depend on the
choice of an appropriate frame of reference). The statistics of relativistic
stellar mass objects is already sufficient to accept this significant result -
experimental evidence of the absence of observable surfaces in stellar BHs (or
the presence of an event horizon): New proof of this has recently been
obtained by detecting the gravitational radiation of merging binary BHs.

Four types of BHs are known as solutions to Einstein’s equations, in GR. Two
of them rotate: Kerr and Kerr-Newman BHs. It is believed that each such BH,
losing energy, quickly collapses and becomes a stable BH.

By the no-hair theorem, (except for quantum fluctuations) stable BHs can be
completely described at any time by these eleven numbers: mass - energy $M$,
momentum $P$ (three components), angular momentum $J$ (three components),
radius vector $x$ (three components), electric charge $Q$. These numbers
represent the lingering properties of an object, which can be determined from
a distance by studying its electromagnetic and gravitational fields. All other
changes in the BH will either go to infinity or be swallowed up by the BH.
This is because everything that happens inside (within the BH’s event horizon)
cannot affect events outside of it.

A falling observer with increasing acceleration hits a rotating BH in a finite
proper time. From the point of view of a distant observer at infinity, it
stops at the event horizon, approaching zero speed relative to a stationary
reference point in place, and at the same time rotates around the event
horizon infinitely often due to the drag effect of inertial frames of
reference.

Rotating BHs are formed by the gravitational collapse of massive rotating
stars or by the collapse of a cluster of stars or gas with a total angular
momentum other than zero. Since most stars rotate, it is assumed that most BHs
in nature are rotating BHs.

The rotating BHs besides the event horizon have also additional conceivable
surface, so called “ergosphere” (Fig. 5). Ergosphere is a region of space-time
near a rotating BH, located between the event horizon and the static limit.
Objects within the ergosphere inevitably rotate with the black hole due to the
Lense-Thirring effect. The characteristic size of a non-rotating BH is given
by its gravitational radius $r_{g}$, and the physical boundary of the BH is
the event horizon, a sphere of radius $r_{g}$. For a rotating BH, the event
horizon radius $r_{h}$ is less than $r_{g}$, and for an extremely fast
rotating BH (the so-called Kerr BH) $r_{h}=r_{g}/2$.

At the end of 2006, astronomers gave estimates of the rotation speed. A BH in
the Milky Way, GRS 1915+105, can rotate $1.150$ times per second (!), not yet
approaching the theoretical upper limit. A spinning BH can produce a large
amount of energy from its rotational energy. This happens through the Penrose
process (vacuum polarization with the birth of real particles and their
expansion into BHs and to infinity) in the ergosphere of a BH, a region
outside the event horizon. In this case, the rotating BH gradually shrinks
down to a Schwarzschild BH, the smallest form from which no energy can be
obtained. However, the spin rate of a Kerr BH will never reach zero.

Figure 5: Diagram of a Kerr BH. On the left is the event horizon surrounded by
an ergosphere in which matter and photons move rapidly entrained by the vortex
gravitational field of the BH (Lense-Thirring effect). On the right, a beam of
light passing near a BH swirls its motion.

###  3.3 GW from coalescing binaries

A priory the reliable forecast was to search GW radiation signal from the
coalescence of compact binary stars (BNS, BBH, NS/BH). For this type of events
the signal waveform was accurately modelled at the first stage (in spiral) and
last stage (ring down) of the process (Fig. 6). In principle it would allow to
use the matched filtering under detection (Fig. 9).

However much less known was the waveform in the merger phase. Here a new
interesting physics was assumed with the hope of obtaining an increased
amplitude of the emitted GWs. In this regard, a lot of effort was expended by
various university groups on the calculations of GW in the merger stage. The
very first experiments on detecting GW from merging binaries showed that these
expectations were untenable. The intensity of gravitational radiation did not
increase radically. It is now clear that the maximum amplitude of the GW
signal appears in the last orbital cycle of the spiral merger stage. Rate of
events depends on the source distance. A few events/year could be accessible
to the GW interferometers LIGO-Virgo.

Taking into account the experimental fact (the absence of features of the
metric amplitude at the stage of direct contact of the components), the
theoretical picture Fig.6 of the merger of a relativistic binary should be
presented as given in Fig.8.

Thus, after the inspiral phase, the ringdown phase immediately follows,
corresponding to the so-called. quasi-normal oscillation of the resulting BH.
The diagram in Fig.8 also gives insight that the stages of direct contact must
contain a ”collision of event horizons”

Figure 6: Phases of BH binary merger Figure 7: Gravitational wave amplitude of
a chirp signal. Figure 8: Modernized diagram of BH-binary coalescence (from
[11]).

GW emitted during the inspiral phase can be described by a chirp signal (see
Fig. 7 which is defined by so-called chirp mass parameter $\mathcal{M}$:

$\mathcal{M}=\frac{(m_{1}m_{2})^{3/5}}{(m_{1}+m_{2})^{1/5}}=\frac{c^{3}}{G}\left[\frac{5}{96}\pi^{-8/3}f^{-11}\dot{f}\right]^{3/5},$
(16)

where $m_{1}$ and $m_{2}$ are masses of the binary components, $f$ is the
chirp frequency and dot denotes the time derivative.

The first ever GW signal was detected in 2015 (GW150914) [11]). It was a
merger of two BHs (NS-NS, BH-NS options were rejected) from a distance of
$D\sim 400\ Mpc$. The duration of the signal was $\tau\approx 0.2\ s$. The
signal frequency increased during the inspiral $35\ Hz\rightarrow 150\ Hz$.
The chirp mass of the binary system was $\mathcal{M}\approx 30M_{\odot}$ with
$m_{1}+m_{2}\approx 70M_{\odot}$ and gravitaional radius $2GM/c^{2}\geq 210\
km$. The signal-to-noise ratio was $SNR\sim 24$ and the dimensionless
amplitude - $h\sim 2\times 10^{-21}$.

Figure 9: Event GW150914 (Sept. 14, 2015 at 09:50:45 UTC). All time series are
filtered with a 35–350 Hz band-pass filter and band-reject filters to remove
the strong instrumental spectral lines ; GW150914 arrived first at L1 and
$\sim 6.9\ ms$ late at H1; (for a visual comparison,the H1 data a real so
shown,shifted in time by this amount and inverted to account for the
detectors’ relative orientations); second row: solid lines show a numerical
relativity waveform for a system with parameters consistent with those
recovered from GW150914 third row: residuals after subtracting the numerical
relativity waveform from the filtered detector time series. bottom row: a
time-frequency representation of the strain data, showing the signal frequency
increasing over time. (from [11]).

###  3.4 Method of probing the event horizon for a black hole

One can study the event horizon nature using the GW observation from
coalescence binary BH systems. To prove the existence and study the properties
of the event horizon, it is necessary to observe the effects generated by its
changes. In case of a binary merger, the resulting BH with a new event horizon
is formed from two initial BHs with their own horizons. It has to be reflected
in the GW waveform at the ringdown stage. Experimentally the task is to detect
and study the quasi-normal mode oscillations of the newly formed remnant after
the merger i.e. in the process of damping this oscillations (ringdown phase)
[12]. The study of GW from BH quasi-normal modes is an important test for
quantum gravity theories [13].

Calculation of the merger dynamics requires numerical simulation (on modern
supercomputers) of GR non-linear differential equations. Today such a
calculation has been performed for many variants of the two BHs merger. At the
final stage, two initial BHs form one new BH with a larger mass, which is in
an excited state. Further, these excitations fade, emitting GW. Some of the
emitted GWs are absorbed by the BH itself, and some go to infinity, carrying
information about the specific nature of the event horizon. In fact, this is
information about the highly non-linear large-scale dynamics of the space-time
curvature. An experimental study of these GW signals will make it possible to
test the nonlinear equations of GR in all their completeness and adequacy to
nature. Apparently, this will provide decisive evidence for the existence of
an event horizon for BHs.

##  4 Supermassive black holes at the centers of galaxies

The first estimates of the masses of supermassive BHs in very active galactic
nuclei (quasars) were made by Zeldovich and Novikov in 1964 [14] under the
reasonable assumption that the luminosities are close to the critical
Eddington luminosity, at which the radiation pressure force balances the
gravitational force. It turned out that the masses of these BHs must be very
large - hundreds of millions of solar masses.

It is now clear that at the centers of most galaxies there are huge
supermassive BHs with masses $10^{6}-10^{10}M_{\odot}$. To date, reliable
methods for estimating the masses of supermassive BHs in the nuclei of
galaxies are based on studying the motion of various ”test bodies” in the
gravitational field of a central supermassive BH (stars, gaseous disks, gas
clouds, etc.) (Fig. 10).

Figure 10: Orbits of stars around the center of the Galaxy in the region
$r\sim 0.01\ pc$

In GR isolated BHs cannot be observed due to the infinitely large redshift of
photons traveling from the event horizon to a distant observer. However, the
dark shadow (silhouette) of a BH (object) can be seen against the background
of radiation from matter more distant along the line of the observer lensed by
the BH gravitational field. It is also stated there that the minimum size of
the shadow can be observed if the same BH is illuminated by photons from the
inner part of the accretion disk adjacent to the event horizon. In general, it
is clear that a remotely observed shadow pattern can be quite bizarre due to
the effects of gravitational lensing, the vortex gravitational field of the
ergosphere (the Lense-Thirring effect), as well as the Doppler frequency shift
of radiation from different zones of the accretion disk. In particular, recent
papers [15] in the framework of GR and [16] involving a generalization of
scalar fields are devoted to the analysis and calculations of the structure of
shadows.

The black holes predicted by the general theory of relativity are by their
nature (in principle) invisible objects. The event horizon of any black hole
is a surface formed by geodesic photon trajectories that do not go to spatial
infinity. Direct evidence was first presented in April 2019 by the
international collaboration EHT (Event Horizon Telescope) (Fig. 11). The image
of the shadow was made by linking 11 radio telescopes on four continents into
one huge radio interferometer has obtained the clearest image in history of a
supermassive BH in the M87 galaxy. The resolution of the resulting image
(several microarcseconds) turned out to be sufficient for the first time to
see the shadow of the BH itself, which goes along the event horizon in the
center of this object, as well as to estimate the size, plane and luminosity
of the accretion disk (substance fall) around it. For a mass of $6\times
10^{9}M_{\odot}$, the event horizon of this BH has exactly the predicted
angular size. The image qualitatively consists of a bright asymmetric ring,
interpreted as the glow of an accretion disk, and a dark central part,
interpreted as the observed silhouette of a BH.

The object at the center of M87 looks like a BH framed by a glowing crescent -
this is radiation from an accretion disk of matter falling into the hole. BHs
in M87 were observed by telescopes for four days - April 5, 6, 10 and 11, 2017
so much time in the observation window at eight points on the planet,
including the South Pole, there was good weather, which allowed scientists to
get a clean picture (Fig. 12) . The image of a BH shown by the EHT
collaboration was obtained using its model, in which the hole rotates
clockwise. The plasma disk around the hole is almost round - this means that
it is almost (but still at an angle) perpendicular to the picture plane in
which observers from the Earth look at M87.

Figure 11: The Event Horizon Telescope collaboration.  Figure 12: EHT images
of M87 on four different observing nights (from [17])

The closest supermassive BH to Earth, Sagittarius A*, is at the center of the
Milky Way, $26000$ ly away (while M87 is $55$ million ly away). But its radius
is close to the distance between the Sun and Pluto, and it is difficult to see
it due to its small angular diameter (its mass is $\sim 4\times
10^{6}M_{\odot}$ compared to the giant hole in the center of M87 with a mass
of $\sim 5\times 10^{9}M_{\odot}$). This is about how to take a picture of a
DVD disc lying on the surface of the Moon, or to see the Earth from Alpha
Centauri. Despite all the technical difficulties, scientists managed to obtain
images of the shadow of a black hole at the center of our galaxy (see Fig.
13), which was announced in 2022 [18].

Figure 13: EHT image of Sagittarius A* (from [19])

## Conclusion

The creation of detectors of gravitational radiation and its successful
detection from merging relativistic binary stars opened up a new unique
channel for obtaining astrophysical information about the surrounding
universe. The development of GW astronomy promises an acceleration in the rate
of knowledge of the surrounding world by mankind and the following technical
revolution in its development. The list of fundamental scientific problems
that the GW astronomy should help to solve already looks impressive. Here we
should note the conceptual issues of theoretical physics.

Are the detected GW those as predicted by Einstein’s theory? Is the GR theory
itself correct? Do the observed BHs correspond to the GR picture? Next, there
is a set of questions from astrophysics. What is the nature of observed gamma-
ray bursts? And also what is the structure of neutron stars and other compact
objects? How to clarify the process of gravitational collapse itself? Finally,
there are the inevitable questions of cosmology. What were the initial
conditions before the Big Bang? Nature of phase transitions (if any) in the
Early Universe ? What is the Dark Energy in reality? Obviously, this list is
not exhaustive.

The ability to register and measure the parameters of merging BH through the
gravitational-wave channel allows us to significantly advance our
understanding of the foundations of the Universe - in cosmology and physics of
the microworld. The masses of the merging components can be comparable, but
they can also have significant differences. These are events of moderate BH
falling onto supermassive BH at the centers of galaxies. In its spiral
trajectory, the test mass (smaller BH) will penetrate the zone of strong
gravitational field on the last orbits near the event horizon of the
supermassive BH. In this case, the structure of the emitted GW pulse should
contain unique information about the physics of this still unexplored area. It
will be possible to carry out such a research program using space GW
interferometers LISA or eLISA when they are created [20]. The sensitivity of
these cosmic detectors should be enough to register BH merger events at the
early stages of the development of the Universe, in the so-called epoch of
”cosmic noon” from $z\sim 3$ to $z\sim 1.5$ when the rate of star formation in
the Universe, the activity of quasars and galactic nuclei were the highest. In
”late cosmos”, at $z<1$, space interferometers will be able to continue
tracking binary BH merger events with an increased signal-to-noise ratio. It
is in this closer epoch that we can expect ”spiral signals” from the fall of
test BHs on the central supermassive BH.

Today, after the experimental confirmation of the existence of stellar-mass
black holes and giant black holes in the centers of galaxies, it seems that
GWs can help solve another intriguing mystery of philosophy and natural
science. What is a ”black hole event horizon”? How much mysticism and reality
is here? The way to solve, as the authors of this article think, lies through
the accumulation of observations of the shape of the GW radiation accompanying
the collisions of binary BHs at the final ringdown stage and its comparison
with the prediction given by calculations on damped oscillations of quasi-
normal BH modes.

##  5 Acknowledgements

The authors are grateful to their colleagues from Moscow State University,
professors Cherepashchuk A.M. and Galtsov D.V. for many helpful discussions.

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