[1,2,3]Reinhard Genzel

[1]Max-Planck Institute for Extraterrestrial Physics, Gießenbachstr. 1, 85748
Garching, Germany

2]Departments of Physics & Astronomy, University of California, Le Conte Hall,
Berkeley, CA, 94720, USA

3]Faculty of Physics, Ludwig Maximilian University, Munich, Germany

4]Department of Physics, TUM School of Natural Sciences, Technical University
of Munich, 85748 Garching, Germany

# Experimental studies of black holes: status and future prospects

<EMAIL_ADDRESS>Frank Eisenhauer<EMAIL_ADDRESS>Stefan Gillessen
<EMAIL_ADDRESS>*  [  [  [

###### Abstract

More than a century ago, Albert Einstein presented his general theory of
gravitation (GR) to the Prussian Academy of Sciences. One of the predictions
of the theory is that not only particles and objects with mass, but also the
quanta of light, photons, are tied to the curvature of space-time, and thus to
gravity. There must be a critical compactness, above which photons cannot
escape. These are black holes (henceforth BH). It took fifty years after the
theory was announced before possible candidate objects were identified by
observational astronomy. And another fifty years have passed, until we finally
have in hand detailed and credible experimental evidence that BHs of 10 to
$10^{10}$ times the mass of the Sun exist in the Universe. Three very
different experimental techniques, but all based on Michelson interferometry
or Fourier-inversion spatial interferometry have enabled the critical
experimental breakthroughs. It has now become possible to investigate the
space-time structure in the vicinity of the event horizons of BHs. We briefly
summarize these interferometric techniques, and discuss the spectacular recent
improvements achieved with all three techniques. Finally, we sketch where the
path of exploration and inquiry may go on in the next decades.

###### keywords:

Black holes, Galactic center, Interferometry, GRAVITY

###### Contents

  1. 1 Presto: Theoretical background
  2. 2 Vivace: X-ray binaries & quasars
  3. 3 Allegro: Testing the MBH paradigm in the Galactic Center with stellar orbits and radio emission
    1. 3.1 Initial statistical evidence for a compact central mass from gas and stellar motions
    2. 3.2 Sharper images and individual stellar orbits on solar system scales
    3. 3.3 Interferometry and detection of post-Newtonian orbital deviations
    4. 3.4 The Event Horizon Telescope and the detection of the ‘shadow’ as predicted by GR
  4. 4 Allegretto: GRAVITY measurements of BH masses in distant AGN and quasars
  5. 5 Allegro Molto: Experimental evidence for stellar BHs from gravitational waves
  6. 6 Rondo: Dark matter cusps
  7. 7 Coda Fortissimo: Future expectations for studying astrophysical BHs
  8. A Appendix: Instrumental techniques

##  1 Presto: Theoretical background

A ‘ _black hole_ ’ (e.g. Wheeler 215) conceptually is a region of space-time
where gravity is so strong that within its _event horizon_ neither particles
with mass, nor even electromagnetic radiation, can escape from it. Based on
Newton’s theory of gravity and assuming a corpuscular nature of light, Rev.
John Michell (in 1784) and Pierre-Simon Laplace (in 1795) were the first to
note that a sufficiently compact, massive star may have a surface escape
velocity exceeding the speed of light. Such an object would thus be ‘dark’ or
invisible. A proper mathematical treatment of this remarkable proposition had
to await Albert Einstein’s theory of General Relativity in 1915/1916 (Einstein
66, henceforth GR). Karl Schwarzschild’s (1916) first analytic solution of the
vacuum field equations in spherical symmetry revealed the unavoidable
existence of a characteristic _event horizon_ in the _metric_ of a mass M, the
_Schwarzschild radius_ $R_{s}=2\,GM/c^{2}=2\,R_{g}$ (with the gravitational
radius $R_{g}=GM/c^{2}$), within which no communication is possible with
external observers. It is a ‘one way door’. Radially inward moving observers,
after crossing the event horizon, cannot stop, nor reverse back out, but end
up in finite ‘Eigenzeit’ (proper time) at the center. All the mass/energy of a
BH is concentrated there in a central singularity.

[132] generalized this solution to spinning BHs. For the normalized spin
parameter $(0\leq\chi\leq 1)$ the event horizon becomes

$R_{\rm event\
horizon}=\frac{GM}{c^{2}}\times\left(1+(1-\chi^{2})^{1/2}\right).$ (1)

In 1965, Newman found the axisymmetric solution for a BH that is both rotating
and electrically charged. Israel, Carter, Robinson, Wheeler, Bekenstein and
Ruffini then formulated the so-called ‘ _no-hair theorem_ ’ (1967 –
1975)111See [124, 50, 194] and [29], stating that _a stationary BH solution is
completely described by the three parameters of the Kerr–Newman metric: mass,
angular momentum, and electric charge_. For the Kerr metric this means that
the quadrupole moment $Q_{2}$ of the BH is determined by the spin, namely
$Q_{2}/M=-\chi^{2}$. However, these solutions refer to configurations with
sufficiently high symmetry, so that Einstein’s equations can be solved
analytically. This led to a debate whether the conclusions obtained were
generally applicable. [177, 178] dropped the assumption of spherical symmetry,
and analyzed the problem topologically. Using the key concept of ‘ _trapped
surfaces_ ’ he showed that any arbitrarily shaped surface with a curvature
radius less than the Schwarzschild radius is a trapped surface. Any observer
is then inexorably pulled towards the center where time ends.

The distortion of the space-time outside the event horizon leads to a minimum
radius, where stable circular orbits are possible. For particles with mass
this _innermost stable, circular orbital radius_ (ISCO) is $6\,R_{g}=3\,R_{S}$
for $\chi=0$, and $R_{g}$ for $\chi=1$. For photons of no mass this innermost
stable orbital radius (called _photon orbit_) is $3\,R_{g}=1.5$ $R_{S}$ for
$\chi=0$, and $R_{g}$ for $\chi=1$ [24]. Finally, if a BH is irradiated by a
point source at large distance behind the BH, only photons with projected
radii $\geq 3\sqrt{3}R_{g}$ arrive at the distant observer in front of the BH.
Those inside form a ‘ _shadow_ ’ (a central depression of light, Bardeen et al
24, Luminet 143, Falcke et al 77) and do not reach the observer.

Work by Bardeen, Bekenstein, Carter, Christodoulou, Ruffini and Hawking in the
early 1970s222See [51, 52, 50, 25, 23, 119, 29] led to the formulation of _BH
thermodynamics_. These laws describe the behavior of a BH in close analogy to
the laws of classical thermodynamics, by relating mass to energy, area to
entropy, and surface gravity to temperature. The analogy was completed when
[119] showed that quantum field theory implies that BHs should emit particles
and photons like a black body with a temperature proportional to the surface
gravity of the BH, hence inversely proportionally to its mass. This predicted
effect is now known as _Hawking radiation_ ’. For the astrophysical BHs
discussed here, the Hawking radiation is out of reach of current detection
methods by many orders of magnitude.

From considerations of the _information content_ of BHs, there is significant
tension between the predictions of GR and general concepts of quantum theory
(e.g., Susskind 205, Maldacena 148, Bousso 40). It is likely that a proper
quantum theory of gravity will modify the concepts of GR on scales comparable
to or smaller than the _Planck length_ , $l_{\rm Pl}=\sqrt{\frac{\hslash
G}{c^{3}}}\sim 1.6\times 10^{-33}\,\mathrm{cm}$, remove the concept of the
central singularity, and potentially challenge the interpretation of the GR
event horizon [10]. If gravity is fundamentally a higher-dimensional
interaction, then the fundamental Planck length in 3D can be substantially
larger [18].

But are these bizarre objects of GR (and science fiction) actually realized in
Nature? The ultimate question discussed in the following is not just whether
the weak-ﬁeld gravity region near compact objects is qualitatively consistent
with the BH geometry of GR, but rather to quantify the limits of observations
(= experiments at a distance) in testing the existence of event horizons (cf.
Cardoso and Pani 47, and references therein). “How close” is a self-
gravitating object to a BH? One can introduce a _“closeness” parameter
$\epsilon$, such that $\epsilon\to 0$ corresponds to the BH limit_. For
example one can choose the compactness, such that for a spherically symmetric
space time

$\epsilon=1-\frac{2\,GM/c^{2}}{R}\,$ (2)

where $M$ is the object mass in the static case and $R$ is its radius (cf.
Cardoso and Pani 47). Likewise, one can introduce $\epsilon$ as a measure of
the violation of the no hair theorem above. Alternatives of the GR BH proposal
for compact astrophysical objects are ‘ _exotic compact objects_ ’ (’ECOs’,
Cardoso and Pani 47, Psaltis 182). These might be concentrations of heavy,
dark matter bosons or fermions, such as ‘ _boson stars_ ’ (Torres et al.
2000), or ‘ _fermion balls_ ’ [213, 209, 26], or ‘ _grava-stars_ ’ (stars
supported by negative vacuum pressure, e.g., Mazur and Mottola 152, Cardoso
and Pani 47), or ‘ _worm-holes’_ [165, 47].

##  2 Vivace: X-ray binaries & quasars

Astronomical evidence for the existence of BHs started to emerge sixty years
ago with the discovery of variable X-ray emitting binary stars in the Milky
Way [89, 88] on the one hand, and of distant, luminous ‘quasi-stellar-radio-
sources/objects’ (_quasars_ or _QSO_ s, Schmidt 199) on the other. Dynamical
mass determinations from Doppler spectroscopy of the visible primary star
established that the mass of the X-ray emitting secondary is sometimes
significantly larger than the maximum stable neutron star mass, $\sim$2.3
solar masses [153, 188, 176, 193]. The binary X-ray sources thus are excellent
candidates for stellar BHs (SBH, $\sim 8$–$20\,M_{\odot}$). If so they are
probably formed when a massive star explodes as a supernova at the end of its
fusion lifetime and the compact remnant collapses to an SBH.

The radio to X-ray luminosities of quasars often exceed by 3 to 4 orders of
magnitude the entire energy output of the Milky Way Galaxy. Furthermore, their
strong high-energy emission in the UV-, X-ray and $\gamma$-ray bands, as well
as their spectacular relativistic jets, can most plausibly be explained by
accretion of matter onto rotating (super)- massive BHs (henceforth (S)MBHs,
$10^{6}$–$10^{10}\,M_{\odot}$, e.g., Lynden-Bell 144, Shakura and Sunyaev 204,
Blandford and Znajek 38, Rees 187, Blandford 35, Yuan and Narayan 223,
Blandford et al 34). Between 5.7% (for a non-rotating Schwarzschild hole) and
42% (for a maximally rotating Kerr hole) of the rest energy of infalling
matter can, in principle, be converted to radiation outside the event horizon.
This efficiency is two orders of magnitude greater than nuclear fusion in
stars. To explain powerful QSOs by this mechanism, BH masses of $10^{8}$ to
$10^{10}$ solar masses, and accretion flows between 0.1 to tens of solar
masses per year are required. Often the accretion rate is expressed as
Eddington ratio, where a value of 1 corresponds to the situation that the
radiation pressure of the emission equals the gravitational pull of the MBH.

Quasars are located (without exception) in the nuclei of large, massive
galaxies (e.g., Osmer 175). Quasars represent the most extreme and spectacular
among the general nuclear activity of most galaxies [169]. There may also be
intermediate mass BHs (IMBHs, $10^{2}$–$10^{5}\,M_{\odot}$), for instance in
the cores of globular clusters or dwarf galaxies. Evidence for
$>10^{5}\,M_{\odot}$ MBHs in low mass galaxies is growing but the case for
IMBHs in globular clusters is still very much debated [114]. Finally, there
have been proposals that BHs with a wide mass spectrum might have been created
in the rapid cool-down phase after the Big Bang (e.g., Carr and Hawking 49,
Carr 48, Hasinger 118).

A conclusive experimental proof of the existence of a BH, as defined by GR,
requires the _determination of the gravitational potential_ _at or near the
scale of the event horizon_. This gravitational potential can be inferred from
spatially resolved measurements of the motions of test particles (interstellar
gas, stars, other BHs, or photons) in close trajectory around the BH [145], or
from gravitational waves emitted in the inspiral of a binary BH. [144] and
[145] proposed that (S)MBHs might be common in most galaxies (although in a
low state of accretion). If so, dynamical tests are feasible in nearby galaxy
nuclei, including the center of our Milky Way. Because of the small angular
radius of the event horizon (e.g. 10 micro-arcsec for the 4.3 million solar
mass MBH even in the ‘nearby’, 8.27 kpc, Galactic Center), achieving the
necessary instrumental resolution requires _extremely large telescopes (or
spatial interferometers) with exquisite sensitivity and spectral resolution_.

Over the past fifty years, increasingly solid _evidence for central ‘dark’
(i.e. non-stellar) mass concentrations_ has emerged for more than several
hundred galaxies in the local Universe (e.g., Magorrian et al 146, Gebhardt et
al 79, Ferrarese and Merritt 78, Kormendy 133, Gültekin et al 115, Fabian 74,
Kormendy and Ho 134, McConnell and Ma 154, Saglia et al 198, Greene et al 112,
113). The data come from optical/infrared imaging and spectroscopy on the
Hubble Space Telescope (HST, and most recently from the James Webb Telescope
(JWST)), from large ground-based telescopes, as well as from Very Long
Baseline radio Interferometry (VLBI). Further evidence comes from
relativistically broadened, redshifted iron K$\alpha$ line emission in nearby
Seyfert galaxies (e.g. Tanaka et al 206, Nandra et al 167, Fabian and Iwasawa
75), including the first statistical constraints on the BH spin distribution
[192].

In external galaxies a compelling case that such a dark mass concentration
cannot just be a dense nuclear cluster of white dwarfs, neutron stars and
perhaps stellar BHs, already emerged in the mid-1990s from spectacular VLBI
observations of the nucleus of NGC 4258. This is a mildly active galaxy at a
distance of 7 Mpc [161, 162]. The VLBI observations show that the galaxy
nucleus contains a thin, slightly warped disk of H2O masers (viewed almost
edge on), in beautiful Keplerian rotation around an unresolved mass of 40
million solar masses, as inferred from the maser motions. The maser motions
exceed 1000 km/s at the innermost edge of the disk of about 0.1pc. The
inferred density of this mass exceeds a few $10^{9}$ solar masses pc-3 and
thus cannot be a long-lived cluster of ‘dark’ astrophysical objects of the
type mentioned above [149]. [112] presented a survey of such H2O disk maser
MBHs. As we will discuss below, the Galactic Center provides a yet more
compelling case.

In the galaxies investigated, dark masses are found ranging from a few
$10^{4}$ to $10^{5}\,M_{\odot}$ in low mass systems [114], to
$10^{10+}\,M_{\odot}$ in very massive spheroidal/elliptical galaxies [134,
154]. For the ellipticals and for galaxies with ‘classical’ bulges [135],
there appears to be a fairly low scatter relationship between central mass and
bulge mass [117, 134, 154]. About 0.2–0.7% of the bulge mass is in the central
dark mass, increasing slowly with bulge mass, and strongly suggesting that
central dark mass and bulge have grown together over cosmological time scales
(Fig. 1).

Figure 1: Relationship between inferred central mass (from stellar and gas
dynamics) $M_{\bullet}$ and bulge mass in local Universe massive galaxies
(ellipticals, or disks with classical bulges). There clearly is a correlation
between these two components, with a best fit $\frac{M_{\bullet}}{M_{\rm
bulge}}=(4.9\pm 0.6)\times 10^{-3}\times\left(\frac{M_{\rm
bulge}}{10^{11}\,M_{\odot}}\right)^{0.14\pm 0.08}$ (adapted from Fig. 18 of
Kormendy and Ho 134).

##  3 Allegro: Testing the MBH paradigm in the Galactic Center with stellar
orbits and radio emission

The central light years of our Galaxy contain a dense and luminous star
cluster, as well as several components of neutral, ionized and extremely hot
gas (Fig. 2; Genzel and Townes 80, Genzel et al 81, Morris and Serabyn 163,
Melia and Falcke 156, Genzel et al 84, Morris et al 164). Compared to the
distant quasars, the Galactic Center is ‘just around the corner’ ($R_{0}=8.27$
kilo-parsecs (kpc), 27,000 light years). High resolution observations of the
Milky Way nucleus thus offer the unique opportunity of carrying out a
stringent test of the MBH-paradigm deep within its gravitational ‘sphere of
influence’ where gravity is dominated by the central mass ($R<1-3$ pc). Since
the center of the Milky Way is highly obscured by interstellar dust particles
in the plane of the Galactic disk, observations in the visible part of the
electromagnetic spectrum are not possible. The veil of dust, however, becomes
transparent at longer wavelengths (the infrared, microwave and radio bands),
as well as at shorter wavelengths (hard X-ray and $\gamma$-ray bands), where
observations of the Galactic Center thus become feasible [173].

###  3.1 Initial statistical evidence for a compact central mass from gas and
stellar motions

Starting in the late 1970s/1980s, observations of the Doppler motions of
ionized and neutral gas clouds in the central parsecs [222, 138, 203, 54], and
of stellar velocities [155, 136, 116] found the first evidence for a central
mass concentration of a few million solar masses, concentrated on or near the
compact radio source SgrA*. In the 1990s observations of stellar proper
motions with the telescopes of the European Southern Observatory (ESO) in
Chile [63, 82], and with the Keck telescopes on Mauna Kea [85] further
improved the statistical and systematic evidence. Yet in terms of the
compactness parameter $\epsilon$ introduced in the first paragraph, these
early measurements did not provide significant evidence that this mass
concentration must be a BH: $0\ll\epsilon\sim 1$–$10^{-5}$. It could instead
be a cluster of faint stars, neutron stars, or stellar BHs.

###  3.2 Sharper images and individual stellar orbits on solar system scales

Further progress required three new key elements. One is much higher angular
resolution and integral-field imaging spectroscopy (achieved with 8–10-m
telescopes, and aided by adaptive optics to reach the diffraction limit of
$\sim$50–60 milli-arcsec). These improvements were realized both in Chile
(ESO-VLT) and in Hawaii (Keck) between 2000 and 2005. The second are very long
duration observation campaigns ($>$1–2 decades) to observe not only stellar
velocities, but derive the full orbital parameters of individual stars as
precision tracers of the potential. The third was luck, namely to find stars
much closer to SgrA* than was theoretically expected (i.e., on solar system
scales. cf. Alexander 8, 9).

Figure 2: Summary of the different components: stars (old giants, red and blue
super-giants), cold (20–200 K) molecular/neutral gas and dust, ionized
($10^{4}$ K) and hot (1–$10\times 10^{6}$ K) gas, and their distributions on
sub-parsec, to 10 parsec scale in the Galactic Center (adapted from Genzel et
al 84). The cross in the center of the images marks the location of the
compact, non-thermal radio source SgrA*, probably a MBH of 4.3 million solar
masses [84]. Top right: Largest scales of the SgrA region, with the HII region
SgrA WEST and the supernova remnant SgrA EAST (presumably an explosion of one
or several massive O/Wolf–Rayet star(s) $\sim$20–40,000 years ago. Outside of
this region are two giant molecular clouds at ‘+20’ and ‘+50’ km/s LSR
velocity. Top left and center: zoom in onto SgrA WEST, which harbors the
center of a dense ($\rho_{*}>10^{6}\,M_{\odot}\,\mathrm{pc}^{-3}$) cluster of
old, and young, massive stars. The central 1.5-pc diameter region is filled
with ionized gas streamers (bottom left), hot X-ray emitting gas (bottom
center), and the most massive, recently formed O, Wolf–Rayet and B-stars
(bottom right). Winds and UV-radiation from these stars and the MBH have
created a lower density ‘cavity’ relatively devoid of dense molecular gas and
dust (average hydrogen density $n_{H}\sim 10^{3..4.5}\,\mathrm{cm}^{-3}$). The
central cavity in turn is surrounded by a rotating, clumpy ‘circum-nuclear’
ring of warm dust and dense, molecular gas (HCN and other high excitation gas
components are found here, and the molecular hydrogen density is $n_{\rm
H2}\sim 10^{5-6}\,\mathrm{cm}^{-3}$, Becklin et al 28, Ho 122). Gas is
streaming in and out of the central region in form of clumpy, tidally
disrupted ‘streamers’, such as the ‘northern’ and ‘eastern’ arms and the ‘bar’
(cf. Oort 173, Lo and Claussen 141, Genzel and Townes 80, Ho et al 123, Genzel
et al 81, Melia and Falcke 156, Genzel et al 84, Morris et al 164).

The most important scientific breakthrough started in 2002 when both the ESO-
VLT [201] and the Keck telescope [86] discovered that the star S2 (or S02 in
UCLA nomenclature) approached SgrA* to about 15 milli-arcsec ($\sim$15 light
hours or 1200 $R_{S}$), and sharply turned around SgrA* on a highly elliptical
orbit ($e=0.88$). By 2010, both the VLT- and the Keck-based groups were able
to derive orbits for about 10–20 stars remarkably close to SgrA* (top left
panel of Fig. 3, Ghez et al 87, Gillessen et al 90), followed in the next
decade by steady progress in the number and stars and quality of their derived
orbital parameters [39, 91]. Now the observations were able to exclude the
compact star cluster hypothesis, but a few speculative alternative
explanations to a MBH, such as ‘boson’ or ‘fermion’ stars (see Sects. 1 and 6)
still were possible. And in any case, there also remained the theoretical
possibility that GR was not applicable, since not yet tested in the MBH
regime.

###  3.3 Interferometry and detection of post-Newtonian orbital deviations

The next big breakthrough started in 2016/2017 with the completion and
continuous further improvements of the GRAVITY [94] and GRAVITY+ [107]
infrared interferometric beam combiner of the $4\times 8$-m telescopes of the
ESO-VLT (https://www.mpe.mpg.de/ir/gravity, Fig. 12 left panel, GRAVITY
Collaboration et al 106, Eisenhauer et al 67). GRAVITY(+) achieves a near-
infrared resolution of about 3 milli-arcsec, and an astrometric precision of
10–100 micro-arcseconds, about 10–20 times better than with individual
10m-class telescopes, and has polarization, spectroscopic, and wide and deep
imaging modes [94, 105, 111, 107, 110].

Figure 3: Tests of the MBH paradigm and GR in the Galactic Center using
individual stellar orbits Figure 4: Evidence for the concentration of the mass
of $4.3\times 10^{6}M_{\odot}$ (as determined from stellar orbits) within few
gravitational radii. Left: The flaring NIR emission of SgrA* revolves on a
time scale compatible with the Keplerian period at the observed radius. The
polarization of the NIR emission rotates on the same time scale. Middle: The
EHT-image of SgrA* shows a silhouette as expected for an MBH of $4.3\times
10^{6}M_{\odot}$. Right: The SED of SgrA*.

With the combined new instrumental capabilities, precision measurements of
$\sim$50 stellar orbits (central panel) of the so-called S-stars around SgrA*
determine the central mass and its distance to the Sun to be 4.300 ($\pm
0.011$ ($1\sigma$ statistical), $\pm 0.018$ ($1\sigma$ systematic)) $\times
10^{6}\,M_{\odot}$, and $R_{0}=8273$ ($\pm 7.5$, $\pm 15$) pc, centered on the
position of SgrA* [39, 91, 58, 98, 106, 111]. The stellar orbits set stringent
limits to any additional mass in the vicinity, of a few thousand solar masses
within a few $10^{4}\,R_{S}$ (bottom-right panel), as determined by the
orbital data of $\sim$15 stars [91, 58, 98, 106, 105, 109, 68, 218].

This conclusion is also supported the near-pericenter astrometric orbit of S2,
after subtracting the best fitting Newtonian orbit. While Newton’s theory
would thus expect these positions to lie in an inclined highly elliptical
orbit centered on SgrA*, GR in contrast predicts that the stellar motions
exhibit a prograde in-plane precession, the ‘ _Schwarzschild_ ’-precession,
with an advance of

$\Delta\Phi\text{(per orbit)}=3\pi f_{\rm SP}\frac{R_{S}}{a(1-e^{2})}\,.$

For the 16-year orbit of S2 ($a=0.125^{\prime\prime}$, or 1039 AU, $e=0.884$,
$R_{\rm peri}/R_{S}=a(1-e)/R_{S}=1200$) this precession is 12.1’ per orbit in
clockwise direction. In the framework of post-Newtonian (PPN) expansions
[127], the Schwarzschild precession is first order in $(v/c)^{2}$, or a few
times $10^{-4}$. GR has $f_{\rm SP}=1$, while Newton has $f_{\rm GR}=0$. The
top right panel of Fig. 3 shows the location of the averaged S2 residual data
obtained with GRAVITY (filled blue circles) and NACO (VLT adaptive optics
imaging, black cross), after removing the Newtonian best-fit orbit. In this
presentation the precession predicted by GR turns into the loopy red line. The
observed data points are in excellent agreement with the GR prediction (the
data in Fig. 3 give $f_{\rm SP}=1.2\pm 0.1$). This represents a confirmation
that GR is correct in the space-time environment of a 4 million solar mass
concentration; it also confirms that this mass concentration is compact and
not a centrally peaked, but spatially extended distribution [101, 106]. Most
of the mass is in a single concentrated object. Any nearby second
(intermediate mass) BH companion ($<10^{3}$ AU, or $10\,R_{\rm peri}$ of the
stars S2 and S29), can only have a mass of $<10^{3}\,M_{\odot}$ [109, 218]. In
PPN=1.5 ($\sim(v/c)^{3}$) GR has a second orbital precession, the
_Lense–Thirring_ precession per orbit is

$\Delta\Phi_{\rm
LT}=2\chi\times\left(\frac{R_{S}}{a(1-e^{2})}\right)^{1.5}\,,$

around the spin axis of the MBH. For the star S2, the Lense–Thirring
precession is about $0.0497\times\chi$ (arcminutes/orbit), for a spin
parameter $\chi(\leq 1)$, so at least 240 times smaller than the star’s
Schwarzschild precession, and out of reach of the current astrometry.

GR also has a PPN=1 order effect in line of sight velocity, the _gravitational
redshift_ , of 100 km/s for S2. The bottom left inset of Fig. 3 shows the
residual Doppler velocity of the star S2 as a function of time around the
peri-approach in 2018.4, relative to that predicted by a Newtonian orbit of
the same orbital parameters (horizontal grey line). The red curve are the
residuals curve predicted by GR. The actual observed residuals are the blue
filled circles. Our yield $f_{\rm GR}=0.97\pm 0.05$ (in addition to another
100 km/s redshift due to the transverse Doppler effect), again in excellent
agreement with GR ($f_{\rm GR}=1.0$, GRAVITY Collaboration et al 95, 101, Do
et al 58).

#### Further tests of General Relativity near a MBH.

[99] confirmed the Equivalence Principle in the orbit of S2 through a test of
the linear positional invariance. In that paper the redshift data are split
into spectroscopy of the hydrogen Br$\gamma$ line and the HeI 2.1$\upmu$m
line, and the gravitational redshift term is computed for the two data sets.
Einstein’s Equivalence Principle stipulates that in free fall the motion
should only depend on mass/energy, and not on composition. And indeed, [99]
set an upper limit of a few $10^{-2}$ to the fractional difference of the
gravitational redshift in hydrogen and helium. In another paper, [120, 121]
used the Galactic Center data to set limits on a hypothetical fifth force, and
variations in the fine structure constant. [130] analyzed the Schwarzschild
precession of S2 in the framework of Yukawa gravity theory, and set an upper
limit to the mass of the graviton, which is compatible with limits from aLIGO
gravitational-wave data.

#### Near event horizon motions and strong magnetic fields.

The near-IR emission from SgrA* itself is linearly polarized ($\sim$40%,
Eckart et al 64, 65, Genzel et al 84) and is synchrotron emission from very
hot gas in the accretion zone, like the radio emission. The near-IR emission
is constantly varying, with a red-noise power spectrum [57, 60, 61, 62, 219,
220]. The power spectrum is typically log-normal, but occasionally high
amplitude ’flares’ above occur over a few hours at $>$10–20 times the average
quiescent level [60, 62, 59, 84]. These flares exhibit ‘clockwise’ orbital
motion on a scale of 8–9 $R_{g}$, just outside the EHT ring [96, 108]. The
polarization direction also exhibits rotation at the same rate and in the same
direction as the astrometric motions. [108] conclude that the accretion zone
must be within a few tens of degrees of face-on. The polarization properties
clearly show that that the near-event horizon accretion zone is magnetically
dominated with a dominant poloidal field (Fig. 4: left panel, GRAVITY
Collaboration et al 96, 102, 100, 108, Wielgus et al 216). The accretion flow
in the Galactic Center thus is a variant of the ‘MAD’ flows ($B\sim 80$–100 G,
Yuan and Narayan 223, Bower et al 41, Dexter et al 56, Ressler et al 191).

The hot gas density in the accretion zone around SgrA* is comparably low,
$n_{e}\sim$ a few $10^{3}\,\mathrm{cm}^{-3}$ at a few $10^{3}\,R_{g}$ [92],
and $n_{e}\sim$ a few $10^{6}\,\mathrm{cm}^{-3}$ at $\geq 10\,R_{g}$ [150,
185]. The accretion flow in the Galactic Center thus is _radiative
inefficient_ and hot, since the density is too low to equilibrate the electron
and ionic accretion fluids [187, 186, 224, 223]. These properties are
consistent with strongly sub-Eddington accretion
($10^{-8\ldots-9}\,M_{\odot}\,\mathrm{yr}^{-1}$, Baganoff et al 21, Blandford
and Begelman 36, Gillessen et al 92 and references therein). If these motions
can be interpreted as Keplerian circular orbits of hot spots, the astrometric
data probe the potential on 8–10 $R_{g}$ scales (Broderick and Loeb 43, but
see Matsumoto et al 151). Combining the strong magnetic field and low density
it is then tempting to conclude that the near-face-on orientation of the
accretion flow as found by GRAVITY and ALMA reflects the angular momentum of
the accretion flow at large distances [189, 190]. And indeed the observed
orientation of the flow deduced from the infrared flares is consistent with
the angular momentum direction of the ‘clockwise’ disk of O/WR stars at
distances of 1–3” from SgrA* [108]. The winds from these stars currently
dominate the accretion flow onto SgrA* [190].

###  3.4 The Event Horizon Telescope and the detection of the ‘shadow’ as
predicted by GR

The central panel of Fig. 4 shows an image of the compact radio source SgrA*
obtained with the ‘Event Horizon Telescope’, an array of seven telescopes
across the globe, and measuring the 1.3-mm continuum radiation of SgrA*
(https://eventhorizontelescope.org/, see the more detailed discussion in Fig.
13). The EHT is the pinnacle (in terms of resolution and short wavelength
coverage) of the classical radio spatial interferometry Fourier inversion
technique (cf. Thompson et al 207). Once the collected data are calibrated and
analyzed, the EHT can reconstruct the 1.3mm brightness distribution with a
resolution of better than 20 micro-arcsec [72, 73]. As expected from theory
(Sect. 1), the image shows a bright ring with a central dip of diameter
$51.8\pm 2.3$ micro-arcsec. Given the mass and distance of SgrA* measured with
high precision from the stellar orbits [72, 73, 108], GR of a near-face-on
accretion zone of moderate to low spin predicts a shadow of diameter $52\pm
0.4$ micro-arcsec [126, 128], in excellent agreement with the EHT measurement
[72, 73] and a Kerr metric around a 4.3 million solar mass MBH.

The key finding thus is that the _near-event horizon-scale motions from
GRAVITY and the size of the ‘shadow’ of the EHT 1.3-mm emission at 6–9 $R_{g}$
are consistent with the GR ‘shadow’ computed from the prior of mass and
distance determined from the stellar orbits at a few $10^{3}\,R_{g}$._ This
indicates that any extended mass component within the peri-center motions of
the inner most stars is less than a few thousand solar masses [106, 218]. Both
the GRAVITY motions and the EHT size thus constrain the compactness parameter
$\epsilon$ to be 0.4–0.6, fully consistent with a single MBH in GR (bottom
right inset).

The EHT has also detected the predicted shadow in the massive central galaxy
of the Virgo galaxy cluster, M87 (or Virgo A) [69]. The distance of M87 is
16.8 Mpc, 2000 times further away than SgrA*. Since its mass of the central
SMBH is 1500 larger than that of SgrA*, $6.2$–$6.5\times 10^{9}\,M_{\odot}$,
the diameter of its shadow is $42\pm 3$ micro-arcsec, comparable to that of
SgrA*, and strengthening further the ‘shadow’ interpretation. Detailed
polarization images of M87 with the EHT that the near-event horizon magnetic
field structure is poloidal and the region is magnetically dominated, as in
SgrA* [70, 71].

#### Is there a relativistic jet emanating from SgrA*?

Given the strong magnetic field in the accretion zone, and the evidence for a
low value of the gas to magnetic pressure, $p(\rm gas)/p(B)\leq 1$ [96, 102,
103, 108, 216], one might expect a prominent relativistic radio jet from SgrA*
[76]. Moreover, if the BH spin were substantial, one would expect that the
[38] mechanism might be effective in accelerating the nuclear spin-driven
outflow. Yet, so far, no radio jet feature has been detected, even in
relatively high frequency 86 GHz-VLBI maps with exquisite sensitivity [125],
where foreground electron scattering should be less effective in smearing out
the jet feature. It is possible that the relative face-on orientation of the
central accretion zone would expect the spin to be along the line of sight and
thus hard to detect. Alternatively, the SgrA* spin might be low, on the
grounds that there is no evidence for strong accretion events in the last few
Myrs that could have spun up the MBH [84].

#### Lack of hard surface?

The right panel of Fig. 4 shows the observed radio to X-ray spectral energy
distribution of SgrA*, with red labels pointing out the origin of the data
(see Genzel et al 84, GRAVITY Collaboration et al 100, Falcke and Markoff 76,
Melia and Falcke 156). The radio to NIR emission exhibits substantial linear
polarization, pointing to non-thermal synchrotron emission from the hot
accretion flow, as well as possible jet driven outflows. The K- to mid-IR
emission is characterized by a semi-continuously variable, linearly polarized
source, with a near log-normal flux distribution over 1.5 decades [83, 64, 65,
60, 61, 62, 57, 219, 220, 100]. The X-ray emission is also highly variable
over two orders of magnitude and could come from synchrotron emission as well
[61, 180, 104], or, perhaps less likely, from Compton up-scattering of long
wavelength photons (Dodds-Eden et al 61, Witzel et al 221, see also Genzel et
al 84, Cardoso and Pani 47). [44] have used the relative weakness of the
steady infrared emission from the center of SgrA* ($<$ a few percent of the
total luminosity of a few $10^{36}\,\mathrm{erg\ s}^{-1}$) as an argument in
favor of the existence of an event horizon (but Abramowicz et al 6, Carballo-
Rubio et al 46). The basic argument is as follows. Assume matter were
accreting onto a hypothetical hard surface outside the gravitational radius,
but within the upper limit of $\sim 10R_{g}$ set by the VLBI images. When this
accreting matter hits the surface, it will shock, thermalize, and emit all its
remaining energy as black-body radiation of a few $10^{3}$ K in the IR range.
Such a thermal component is not observed in the steady spectral energy
distribution of SgrA* [84, 100], setting a stringent upper limit on the mass
accretion rate. In practice this limit is so low that even the low level of
observed quiescent radio/submm nonthermal emission requires an assumed
radiative efﬁciency of nearly 100%. This can be ruled out, which then leads to
the conclusion that the central object cannot have a hard surface but rather
must have an event horizon ($\epsilon\ll 1$). The caveat is that this
consideration does not include gravitational light bending. [142] make a
similar argument on the statistical lack of bright extragalactic tidal
disruption events in the Pan-STARRS1 survey.

##  4 Allegretto: GRAVITY measurements of BH masses in distant AGN and quasars

We discussed in Sect. 2 the discovery of quasars 60 years ago, interpreted to
be (S)MBHs ($10^{7}$ to $10^{10}\,M_{\odot})$ at large distances, and
accreting gas at large rates. Large samples of these active galactic nuclei
(AGN, of which quasars are the tip of the ice-berg) are now available, all the
way back to less than 1 Gyr after the Big Bang, in the Early Universe. In the
local Universe the mass of the central MBH and the mass of the central stellar
centroid/bulge are about 0.2–0.7% of their host galaxy (Fig. 1 right panel).
This correlation suggests that the evolution and growth of the two components
are correlated on cosmic time scales (Vestergaard et al 211, Vestergaard and
Osmer 210, Alexander and Hickox 7, Kormendy and Ho 134, McConnell and Ma 154,
Saglia et al 198, and references therein).

How can we test more quantitatively the correctness of the (S)MBH paradigm and
the (S)MBH-galaxy co-evolution? ‘Type I’ AGNs and quasars typically show $>$
few $10^{3}$ km/s line broadening of atomic emission lines by high-velocity
motion of gas near the center [168, 169]. If the line widths are due to
virialized motions caused by the central mass, and the size of regions for
which broad emission lines are observed (‘ _broad-line regions, BLR_ ’) it
then becomes possible to measure the masses of the central (non-stellar)
masses in individual AGNs.

One way to estimate the BLR sizes comes from measuring the delay in light
travel time between the variable brightness of the accretion disk continuum
and the emission lines, a method known as ‘ _reverberation mapping_ ‘ [37].
This method has been applied routinely to nearby AGNs (e.g. Kaspi et al 131,
Bentz et al 30, 31, 32, Peterson 179) but is somewhat limited because of the
necessary underlying assumptions on the source structure and geometry.

Moreover, ground-based reverberation mapping cannot be easily applied to the
very massive, large quasars because of the long time scales involved. Until
recently direct imaging of such BLRs has not been possible because of their
large distances and resulting small angular size (less than 100 micro-arcsec).
For the most distant SMBHs the approach has been to calibrate the relationship
between BLR size (inferred from reverberation) and optical luminosity, and
then apply the same relationship to higher $z$ SMBH where only line widths and
luminosities were available [211, 210].

The GRAVITY interferometric beam combiner of the $4\times 8$-m telescopes of
the ESO-VLT has changed this situation (Fig. 12, GRAVITY Collaboration et al
94, GRAVITY+ Collaboration et al 107). [97] were able to resolve the BLR of
the famous quasar 3C 273 ($z=0.16$, 550 Mpc distance, cf. Schmidt 199) at sub-
parsec level with interferometric spectro-astrometry. Figure 6 shows the
results. More recently, after incorporating new optics allowing much larger
offsets between phase reference star and science object, it has now become
possible to apply the same direct technique to faint, distant quasars, such as
the quasar J0920 ($z=2.32$, 17,700 Mpc distance, GRAVITY+ Collaboration et al
110; Fig. 5). At this time, there are about a dozen GRAVITY(+) measurements of
BLRs (bottom right of Fig. 6).

Looking ahead at the near-future, the enhanced capabilities of GRAVITY+ will
enable measuring super-massive BH masses and their evolution across the entire
cosmological evolution of galaxies, and answer the fundamental question
whether galaxies, and MBHs grew in lockstep, or whether one of them grew
faster and earlier. The high-$z$ detections with the new interferometric
technique of optically very luminous (i.e. high Eddington ratio accretion)
(S)MBHs, may have systematically smaller broad-line regions, and thus contain
a smaller (S)MBH mass, than lower Eddington rate AGNs measured in the local
Universe with the reverberation technique, and with the velocity-luminosity
relations at higher $z$ (bottom right of Fig. 6). Even small shifts as seen
here could have big impacts on our understanding of (S)MBH growth as these
relations are used to measure masses out to $z=10$ in usually high
luminosity/Eddington quasars and discriminate between black hole seed models.

Figure 5: Top: Schematic of the structures around a luminous, rapidly
accreting extragalactic active galactic nucleus
$(\dot{M}>0.01\times\dot{M}_{\max,\rm Eddignton})$, with a super-massive
$(>10^{8\ldots 10}\,M_{\odot})$ BH at its center (e.g. Osmer 175, Netzer 169).
The SMBH is surrounded by a hot accretion disk. Generalizing our current
GRAVITY results (Figure 3, GRAVITY Collaboration et al 97, GRAVITY+
Collaboration et al 110, and references therein) on its outer side are self-
gravitating ionized clouds, the BLR in virial equilibrium and rotating around
the SMBH. This central region in turn is surrounded by a dusty molecular
region (the “Torus”), and ionized clouds on 100 pc-10 kpc scale (the narrow-
line region, NLR). Image credit: Claudio Ricci. Bottom: GRAVITY spectro-
astrometry of the broad P$\alpha$ line in the $z=0.16$ Quasar 3C 273 [199].
The left panel shows the observed line profile and the inferred
interferometric phase gradient across the line, with a measurement accuracy of
about 1$\upmu$as (500 AU, or 0.0024 pc at 550 Mpc distance). The 2D spectro-
astrometry model (bottom middle panel) shows that this phase gradient extends
over 50 micro-arcsec, approximately perpendicular to the direction of the
known radio jet (black line). The model yields a MBH mass of $2.6\times
10^{8}\,M_{\odot}$, surrounded by a thick rotating gas disk of 0.18 parsec
(46$\upmu$as) diameter (GRAVITY Collaboration et al 97, cf. Netzer 170).
Figure 6: Top left: GRAVITY spectro-astrometry of the broad H$\alpha$ line in
the $z=2.32$ quasar J0920, with the new capabilities of ‘GRAVITY-wide’ to
fringe-track on a bright star up to 25” from the phase center [110]. The data
and modeling of this distant quasar (distance 17,700 Mpc, 2.89 Gyr after the
Big Bang) are qualitatively quite similar to those of the 33 times closer 3C
273, with a linear velocity gradient, indicating predominantly rotation in a
0.33 pc diameter (40 micro-arcsec) thick disk rotating around a $3.2\times
10^{8}\,M_{\odot}$ central mass (top right and bottom left). In contrast to
moderately luminous local AGN, 3C 273 and J0920, both optically very luminous
(i.e. high Eddington ratio accretion) MBHs, seem to have a smaller broad-line
region, and contain a smaller fraction of the overall baryonic mass of the
galaxy than predicted by scaling relations (bottom right).

##  5 Allegro Molto: Experimental evidence for stellar BHs from gravitational
waves

If two compact stellar masses orbit each other in a tight bound orbit, they
lose energy through the emission of gravitational waves. As a result, the
orbital semi-major axis shrinks. The inspiral rate is low initially at large
orbital radius, but then orbital speed, gravitational wave strain and
gravitational wave amplitude increase as the inspiral proceeds. Once the
binary has shrunk to the innermost stable orbit, plunge in to a SBH of the
combined mass (minus the mass-energy lost due to gravitational wave emission)
happens on a dynamical time scale of O (10milli-sec) or 200 Hz frequency (left
panel of Fig. 7). The gravitational strains that need to be detected for
successful of binary inspirals lead to pico-meter amplitudes, which can be
detected with second and third generation laser-powered Michelson
interferometers (https://www.ligo.caltech.edu/,
https://gcn.nasa.gov/missions/lvk).

Figure 7: Top: Inspiral, merger/plunge-in and ring-down of a SBH binary [1].
Bottom: ‘Spectroscopy’ of a SBH inspiral. Mid and right: The Kerr metric gives
a unique relation-ship between mass, and the orbital $l=2$ mode frequency
$\omega_{R}$ near the plunge in, and the imaginary frequency $\omega_{i}$
expressing the decay time of this mode, $t_{d}\sim 1/\omega_{i}$ [47]. Given
the short duration of a $2\times 30\,M_{\odot}$ inspiral, a single inspiral
like GW150914 does not yield enough SNR to determine these frequencies with
sufficient accuracy with the current aLIGO sensitivity, and a stacking of
about 30 such inspirals would be required (simulation by Brito et al 42). The
small yellow star is the true input value injected into the simulation. Figure
8:  Data of the first BH-binary inspiral, GW150914, as seen by the Hanford and
Livingston antennas of aLIGO, and the derived source properties —[1, 2].
Figure 9: Current status of the SBH-SBH and SBH-NS inspirals observed by aLIGO
and aLIGO+Virgo+KAGRA after GWTC-3 (Fig. 14, right panel, Abbott et al 1, 2,
4). Compilation of the inferred inspiral masses of all events seen at the end
of the GWTC-3 run with LIGO, Virgo and KAGRA [5]. Image courtesy of
https://www.ligo.caltech.edu/image/ligo20211107a.

The first spectacular detection was the gravitational-wave pattern from
GW150914, with the two gravitational-wave antennas of LIGO in the USA [1, 2].
The two initial (S)BHs had masses of about $35\,M_{\odot}$. Since 2015 and up
to the end of the GWTC-3 run, aLIGO, strengthened with the Virgo antenna in
Europe, and the KAGRA antenna in Japan, have detected 35 bona-fide mergers.
These turn out to be SBH-SBH and SBH-NS mergers, but no NS-NS mergers, as well
another $\sim$50 or so candidate detections [4]. This impressive harvest is
shown in the in Fig. 9. The data provide very strong, and arguably conclusive
evidence for the existence of SBHs.

Starting already with GW150914 (Fig. 8) it perhaps came as a surprise that the
SBH masses were larger than expected. According to standard stellar evolution
models, the typical SBH in the local Universe is expected to have a mass of
$\sim 10\,M_{\odot}$. While a number of such lower mass inspirals have by now
been seen, the relative fraction of $\sim 2\times 25$–$30\,M_{\odot}$ objects
is fairly, if not uncomfortably, large. Even more surprising is the case of
GW190521, with two SBHs of 85 and 66 $M_{\odot}$ combining to an ‘intermediate
mass BH’ (IMBH) of 142 $M_{\odot}$ [3]. The two initial SBHs of the binary are
both in, or near the so called ‘pulsational pair instability gap’, where the
creation of particle-antiparticle pairs during the supernova explosion there
prevents a stable SBH mass. While the large fraction of combined mass $\sim
50\,M_{\odot}$ SBH end states may be explainable by the instrumental
sensitivity bias of the current interferometers preferentially selecting
larger mass, larger amplitude systems, the case of GW190521 is truly
fascinating. Do repeated mergers in dense star clusters explain the large
masses, or is there a new, yet unknown channel of massive SBH and IMBH
creation?

#### Black-hole spectroscopy

It is clear that the largest information about the space-time and significant
tests of the no-hair theorem can in principle obtained in the last, near-event
horizon inspirals, before plunge in. In technical terms this ‘BH spectroscopy’
requires accurate determination of the near-photon orbit, orbital frequency
and its decay time (Brito et al 42, Berti et al 33, Cardoso and Pani 47; Fig.
7). The Kerr metric specifies a precise relation between the $l=2$ mode
‘normal’ frequency (at the photon orbit), and the decay time (Fig. 7). The
frequency analysis of the GW150914 ring down (the strongest and highest
quality inspiral as of today) is consistent with Fig. 7 in the $l=2$ mode, and
thus with the Kerr metric [2]. However, [42, bottom panels of Fig. 7] have
shown that a full test of the no-hair theorem requires also the determination
of the frequencies and decay times of the $l>2$ modes, which is not yet
possible with one inspiral. They estimate that about 30 inspirals need to be
combined to achieve that test. However, there is very good hope that the
ground-based interferometer of the next generation and the space
interferometer LISA will make dramatic improvements.

##  6 Rondo: Dark matter cusps

[213] and [166] proposed that the dark matter concentrations in galactic
nuclei, including QSOs, may not be Kerr BHs, but very compact ‘ _fermion
balls’_ (for example made of hypothetical, massive neutrinos) supported by
degeneracy pressure. The size of a fermion ball, and the maximum stable
‘Chandrasekhar’ mass (or its relativistic analog, the ‘Oppenheimer–Volkoff’
mass $M_{\rm OV}$, Oppenheimer and Snyder 174), increase the lighter the
fermion’s mass. In this scenario the largest observed central masses in
elliptical galaxies, a few $10^{10}\,M_{\odot}$, would approach the
Oppenheimer–Volkoff mass, resulting in an upper limit to the mass of the
constituent fermions to about $13\,\mathrm{keV}/c^{2}$, again with larger BH
masses requiring smaller neutrino masses. To still ‘fit’ within a given peri-
center distance the neutrinos would have to have a mass of [166]

$m_{\rm f}\sim 70\,\mathrm{keV\,c}^{-2}\left(\frac{R_{\rm
peri}}{10\,\text{light
hours}}\right)^{-3/8}\left(\frac{g}{2}\right)^{-1/4}\left(\frac{M_{\rm
OV}}{4.4\times 10^{6}\,M_{\odot}}\right)^{-1/8}$ (3)

where $g$ is the spin degeneracy factor of the fermion.

[196] and [15] have pointed out that a self-gravitating equilibrium
distribution of massive neutral fermions of given degeneracy g exhibits a
segregation into three physical regimes. There is an inner core of almost
constant density governed by degenerate quantum statistics. Surrounding this
core is an intermediate region with a sharply decreasing density distribution,
which in turn is surrounded by an extended plateau. Finally, there is an
outer, asymptotic halo where density scales as $\rho\propto R^{-2}$, i.e., a
classical Boltzmann regime (Fig. 10, adapted from Argüelles et al 15, 16. The
[196] model unifies dark matter on large, intergalactic and circum-galactic
scales, with the evidence of central massive concentrations discussed in
Sects. 2–5. The Boltzmann distribution would explain the ﬂat rotation curves
on $\gg$10 kpc scales, while baryons in form of gas and stars would explain
the mid-scales (parsec to 10+ kpc). Figure 10 shows the resulting density
distributions and rotation curve distributions as a function of radius, for
three different ‘fermions’, of $m_{\rm f}=$ 0.6, 48 and 345 keV. As pointed
out above, for a given fermion mass $m_{\rm f}$ in Eq. (3), $M_{\rm OV}$ gives
the critical stable mass, above which gravity will take over and the object
collapses (to a (S)MBH). Likewise, if there is substantial baryonic accretion
onto such a dark matter core, the central mass concentration would also
collapse to a (S)MBH [17].

Figure 10: Constraints on the contribution of various mass components of
baryonic and dark fermionic matter (fermion ball) to the central mass density
(left) and rotation velocity (right) at different radii $R$ (adapted from
Argüelles et al 15, 16).

Figure 10 summarizes the current constraints on the fermion ball, or fermion-
dark matter model in the Galactic Center. The outer flat rotation curve of the
Galaxy would be best matched by moderate mass, 10–50 keV fermions [15, 16,
27]. The nuclear stellar and gas velocities [106] are in principle also
consistent with a 40+ keV dark matter core [26, 27, 16]. However, the
excellent agreement of the measured prograde precession of S2 at $R_{\rm
peri}=5.7\times 10^{-4}\,\mathrm{pc}$ with that expected for a MBH [102, 106]
is barely consistent with such a lower mass dark matter particle. There is
also a modest inconsistency with the density of ionized baryonic mass
estimated at a few $10^{3}\,R_{g}$ [92].

If the infrared flare motions ($\sim 0.28c$ on the sky at 8–9 $R_{g}$, GRAVITY
Collaboration et al 96, 108) represent the Keplerian orbital speed, and the
EHT shadow diameter [72, 73] is a measure of the light bending by a central
mass, then the implied fermion mass would have to be $\sim$500 keV, obviously
very close to the MBH solution. Such a large fermion mass is disfavored for
the favored dark matter agent on large scales [16]. It would also be
inconsistent with the mass densities inferred from the BLR in AGN and quasars
(see Sect. 4).

Another proposed non-BH configuration is the ‘ _boson star_ ’ scenario
advanced by [208]. A wide range of boson star masses can theoretically be
imagined, including ones with masses of (S)MBHs, depending on the assumptions
about the specific boson particle masses and their self-interactions. Since
such an object consists of weakly interacting particles it is unclear how it
may have formed. The size of a boson star is only a few times $R_{S}$ of the
same mass (S)MBH, and is highly relativistic. It is clear from Fig. 10 that
even the astrometric observations of the IR flares are not sufficient to
distinguish a boson star from a MBH in the Galactic Center [195]. The motion
of test-particles crossing the boson star would allow to distinguish boson
stars from BHs [226], but are currently far out of reach. The EHT imagery is
more promising. Ray-tracing simulations by [212] and [172] suggest that the
mm-appearance of an accreting boson star can be close to that of a MBH,
although ‘ideal’ cases with a large shadow as seen in the Galactic Center, or
M87 [72, 73, 69] are rare. More detailed EHT observations are likely to
separate in more detail the time variable, from the stationary parts of the
shadow, thus helping to distinguish between the MBH and boson star cases.

Finally, Occam’s razor strongly disfavors the boson star interpretation for
rapidly accreting AGNs/quasars. A boson star obviously is unstable to collapse
to a BH if it experiences substantial baryonic accretion. While this is not
the case in the Galactic Center currently, such high Eddington ratio events
almost certainly happened in the evolution of most, and perhaps all, (S)MBHs.

Classical (particle-like) dark matter models predict that dark matter will
cluster around the massive black holes in galaxy centers [93, 197]. In
principle, such dark matter spikes could have high enough densities to affect
the motions of objects around the MBH [225, 137]. Yet, the detection of dark
matter in that way might be impossible, due to the expected background
population of dark objects such as white dwarfs, neutron stars and stellar
black holes, that should arrange into a spiky distribution centered on the MBH
[157, 14, 140], and that simply might outweigh the dark matter particles.
Current limits from GRAVITY observations stand at around $4000\,M_{\odot}$
[106].

##  7 Coda Fortissimo: Future expectations for studying astrophysical BHs

A century after the publication of Einstein’s field equations and
Schwarzschild’s first solution [66, 202], sixty years after the Kerr/Newman
(1963-65) solutions, and the discovery of X-ray binaries [89] and quasars
[199], BHs have come from theoretical speculation to near experimental
certainty. The recent progress in electromagnetic and gravitational wave
studies has been truly remarkable, the top experiments (VLT-GRAVITY-Keck, EHT,
aLIGO-Virgo-KAGRA) have been a tour de force of experimental physics, and the
experimental work and scientific results have been recognized with two Nobel
Prizes, and several Balzan, Shaw, Gruber and Breakthrough Prizes so far.

Yet let us stay realistic and humble. Measured on the expectation level of
critical theoretical colleagues, and certainly on the ultimate requirement for
establishing ‘scientific truth’, _the evidence we currently have is impressive
on all accounts but not (yet fully) conclusive_. We have constrained the
$\epsilon$-parameter to a few tenths, leaving in principle open the
possibility that the objects we have been studying are not BHs after all but
speculative ECOs, such as fermion or boson stars, gravastars, wormholes etc.
(see Sect. 6, and the detailed review of Cardoso and Pani 47). It is also
possible that GR is not the correct description of space time close to the
event horizon (cf. Cardoso and Pani 47 and references therein).

In Figure 11 below we have listed where we currently stand, and where we might
get to in the next decades, for proving the MBH paradigm in terms of the
compactness parameter, or equivalently the no-hair theorem test, with the
parameter $\epsilon=0$ for a Kerr hole.

Figure 11: Current status and future improvements in the quality of
experimental studies of the BH paradigm. The central table lists the
constraints (here in the compactness parameter $\epsilon$, where $\epsilon=0$
is a Kerr BH) achieved so far by the different techniques discussed in the
text are in white color, while the expected further improvements in the future
are in yellow. Current state of the art sets limits in $\epsilon$ of a few
tenths. The very faint stationary near-infrared emission of SgrA* can in
principle be interpreted as a strong evidence for the absence of a surface of
the source, and thus in favor of an event horizon. However, this argument
relies on the emission be isotropic and not strongly affected by gravitational
lensing. Detection with GRAVITY+ of the Lense–Thirring precession of a star
with $R_{\rm peri}<10\,\upmu{\rm as}$ would yield a spin determination of the
MBH in the Galactic Center, and together with other stars yield a limit of
$\epsilon\sim 0.1$. Higher quality measurements of the photon-ring ($n\geq 1$)
in SgrA* with ngEHT, or space VLBI (together with the priors from GRAVITY+
would reach $\epsilon<0.1$. The same limit could be reached with timing of a
Galactic Center pulsar within an arcsecond of SgrA*. Still better limits could
then come from a combination of all three techniques in the next ten years.
Detailed 100 m/s spectroscopy with MICADO@ELT in the 2030s of a late type star
in a close peri-approach to SgrA* might achieve $\epsilon\sim 0.03$. Finally
gravitational wave analysis of an inspiral of a stellar BH into a MBH (EMRI)
would reach $\epsilon\sim 0.0001$ with LISA data in 2+ decades.

It is clear that the best LIGO-Virgo-KAGRA SBH inspirals already set
$\epsilon$ significantly below unity, likewise so for the most recent
combination of stellar, flare and EHT-shadow data. If GRAVITY+ can measure a
high quality orbit of a star with a peri-center distance 2–4 times smaller
than S2, or if one can determine the pulse timing of a sufficiently nearby
pulsar, or the combination of GRAVITY+ and EHT (including possible upgrades in
performance), the next step is $\epsilon\leq 0.1$, allowing spin measurements
[214] and no-hair theorem tests (Falcke et al 77, Will 217, Psaltis and
Johannsen 183, Psaltis et al 184, Johannsen 127, Johnson et al 129, lower top
panel of Fig. 6). The enormously greater sensitivity of MICADO at the ESO-ELT
could push the spectroscopic measurement precision in 5–10 years to m/s level
[55]. For a late-type star near peri-center that could push to $\epsilon\sim
0.03$. Future extensions of the VLTI to a kilometer-wide interferometric array
could push the angular resolution to better than 100 micro-arcsecond and
astrometry to sub-micro-arcsec precision, thereby opening up the observation
of the scattering of S-stars by neutron stars and stellar black holes, and
spatially resolving the accretion zone and flaring activity at IR wavelengths.
An expanded ground-based mm-/submm-VLBI network, such as the ngEHT [19] could
test the space-time at the photon-orbit. The European ELT and GRAVITY+ will
also probe the formation and evolution of the first SMBHs, such as the JWST
source GN–z11 at $z=10.6$ [147, 200].

An important question is, how far the measurement accuracy of the small higher
order PPN-terms can be pushed before perturbations with other stars and SBHs
introduce ‘orbital chaos’ that diminishes or even wipes out the GR information
and significant tests of the no-hair theorem. [158] carried out a suite of
simulations and concluded that in the dense region around the MBH the
perturbations indeed are likely too strong to measure the Lense–Thirring
precession. However, the work of [158] only considered orbit averaged
quantities and assumed an overly high density of SBHs in the central region,
which the current limits on any extended mass around SgrA* now show to be far
too large. Moreover, we now know that many of the S-stars are on highly
elliptical orbits and spend little time in the high density region around the
MBH. The perturbative ‘chaos’ thus is dominated by single star-star and star-
SBH events, when the three are near peri-approach. [181] have studied this
‘punctuated chaos’ with high quality N-body integrations of the known S-star
orbits. When two S-stars come very close (a few tens of AU) during peri of
100-1000 AU, they indeed experience exponential growth of the orbital
deviations in their separation in parameter space. Still the average
exponential growth time, the Lyapunov time, is 460 years. This is encouraging
if indeed faint stars of $<$100 AU peri-distance occur that in principle allow
measurements of the Lense–Thirring and quadrupole terms [214].

On a time scale of twenty years, the gravitational-wave mission LISA of ESA
should deliver the ultimate test. This space interferometer with three
satellites forming a laser interferometer of 2.5 million km length is
sensitive to gravitational waves with 4 to 5 orders lower frequency, and thus
correspondingly larger masses [13, 53]. LISA will be able to observe the
inspirals of MBHs across the entire Universe [22]. The inspiral of a SBH into
a MBH (an extreme mass ratio inspiral, EMRI: Amaro-Seoane et al 12, Amaro-
Seoane 11) should deliver enough SNR in the inspiral before plunge, to obtain
a high-quality measurements of the fundamental ($l=2$) quasi-normal mode of
the MBH at the photon orbit, and get to $\epsilon\sim 10^{-4}\ldots 10^{-3}$
[45, 20, 47]. This would provide the ultimate culmination of this exciting
journey, which Albert Einstein started more than a century ago.

Acknowledgements The authors are grateful to the numerous contributions from
GRAVITY team members in Germany, France, and Portugal, as well as the generous
support from ESO and their staff, including the entire Paranal Observatory.
None of the relevant research discussed in this review would have been
possible without this fantastic team spirit and the key technical and
scientific inputs from all our colleagues. We are grateful to Vitor Cardoso,
Thibault Damour, Hagai Netzer, Diogo Ribeiro, Taro Shimizu, and our Editor,
P.T.P. Ho, for helpful comments on the text.

##  Appendix A Appendix: Instrumental techniques

All three experimental approaches discussed in this review apply variations of
the interference of light in a two-beam Michelson interferometer [160].

#### GRAVITY.

Figure 12 summarizes the essentials of the stellar interferometry at 2$\upmu$m
(K-band) with the GRAVITY(+) beam combiner at the ESO VLTI (GRAVITY
Collaboration et al 94, Eisenhauer et al 67,
https://www.mpe.mpg.de/ir/gravity, https://www.mpe.mpg.de/ir/gravityplus).
GRAVITY coherently combines the light from the four 8m UT or 1.8m AT
telescopes (left bottom). Each telescope is equipped with adaptive optics to
provide a diffraction-limited beam, which is then transported through mirrored
delay lines to the cryogenic beam combiner instrument (right top). The
instrument provides two beam combiners, one for fringe tracking, the other
optimized for long exposure, high spectral resolution interferometry of the
science target. The optical path length within the observatory is controlled
via several laser metrology systems, delay lines, and differential delay lines
(top).

Figure 12: GRAVITY. See text. Image credits: ESO/GRAVITY(+) collaborations.

#### Event Horizon Telescope.

The Event Horizon Telescope (EHT) is a further development of the technique of
intercontinental, heterodyne stellar interferometry [207] pushed to the
highest microwave frequencies (230/345 GHz), at which the Earth Atmosphere is
still transparent. The Very Large Baseline Interferometry (Event Horizon
Telescope Collaboration et al 72, https://eventhorizontelescope.org/,
https://blackholecam.org/) links radio telescopes across the globe to create
an Earth sized interferometer (left panel in Fig. 13). Both techniques –
optical/IR and radio interferometery – synthesize a virtual telescope of
diameter $B_{\max}$ (the maximum separation between two telescopes in the
array) with an angular resolution $\lambda/B_{\max}$. They measure the Fourier
components of the image at the projected separation of the telescopes (right).

Figure 13: Event Horizon Telescope. See text. Image credits: EHT
collaboration.

#### Ground-based gravitational-wave interferometry.

Figure 14 summarizes the essentials of ground-based gravitational wave, laser
interferometry, and in particular the LIGO, Virgo, KAGRA gravitational-wave
observatories (Abbott et al 1, https://www.ligo.caltech.edu/,
https://gcn.nasa.gov/missions/lvk). The experiments measure the small
distortions of space-time when gravitational waves pass through the two arms
of the interferometer (right). In this case one of the two arms will be
stretched, while the other arm shortened. Ground-based gravitational-wave
detectors are most sensitive at frequency of 20 to several hundred Hz (inset),
which matches the orbital frequencies of stellar black-hole mergers.

Figure 14: Ground-based gravitational-wave interferometry. See text. Image
credits: LIGO/Virgo/KAGRA collaborations.

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