1 INTRODUCTION

After WWII an importance of development of theoretical and experimental aspects of general relativity (GR) started to be evident and the conference dedicated to the Jubilee of Relativity Theory was organized by W. Pauli in Bern in 1955Footnote 1. In 1957 the next conference (GR1) was organized by B. De Witt in Chapel Hill [2] where he was the director of Institute of Field Theory at University of North Carolina. The goal of this conference was to provide a good platform where experimentalists could met theorists to boost research in gravitational physics. Intensive discussions of opportunity to detect gravitational waves were started by R. Feynman, H. Bondi, J. Weber at this conference and now we could see that this brain storm was very efficient.

In 1962 GR3 conference was organized in Warsaw and Jablonna (it was the first conference where a large group of Western scientists met with scientists from Eastern Europe (the conference was organized in times of Cold War and the Iron Curtain was very high). Among 114 participants 33 participants were from Eastern countries. The chairman was L. Infeld. Many outstanding scientists attended the conference including P.A. M. Dirac (who got his Nobel prize many years before the conference), and many others (including R. Feynman, S. Chandrasekhar, V.L. Ginzburg, P. Higgs, R. Penrose) who got the Nobel prizes later. This fact illustrates a high level of the conference organization and it is also the evidence that in 1962 GR studies were among the hottest topics in physics in spite of Feynman’s criticism of GR conferences [3, 4]. A representative group of Soviet scientists (11 people) attended the activity and two of them (V. Ginzburg and V.A. Fock) presented plenary talks at the meeting. The title of Ginzburg’s talk was “Experimental Verification of General Relativity Theory” [5] (it was the only talk devoted to observational tests of GR [6]). Later, Ginzburg wrote a review about the most important and interesting problems in physics and astrophysics and many versions of the review were published as journal articles, chapters of Ginzburg’s books and separate booklets (see, versions of the review published in Physics Uspekhi [7, 8]). Ginzburg called the articles as realizations of the Project on Physical Minimum and he claimed that a basic knowledge of these issues is necessary to increase a qualification of young physicists and astrophysicistsFootnote 2. These Ginzburg’s articles provided a high impact on scientific community in Russia and abroad. Astrophysical black holes are directly connected with the following problems from the Ginzburg’s list: 21. Experimental verification of general theory of relativity; 22. Gravitational waves and their detection; 25. Black holes. Cosmic strings (?);Footnote 3 26. Quasars and galactic nuclei. Formation of galaxies; 27. The problem of dark matter (hidden mass) and its detection.

Therefore, international conferences on GR played the extremely important role to settle and solve many theoretical and experimental aspects of gravity and to start intensive studies of theoretical and experimental aspects of gravitational wave detectionFootnote 4 because even the question about an existence of gravitational waves was a subject of doubts among many researchers including A. Einstein and many others and an existence of gravitational waves started to much more clear in a result of efforts of many scientists. For example, in 1936 Einstein and Rosen submitted under the title “Do gravitational waves exist?” in Physical Review and the Einstein’s answer for the title question was “No” as it was noted in [9, 10]. Before the submission, Einstein wrote a letter to M. Born in letter “Together with a young collaborator, I arrived at the interesting result that gravitational waves do not exist through they had assumed certainly to the first approximation” [11]. Einstein received a negative referee’s report from the Chief Editor of Physical Review (it was John Tate at this time). Einstein was very angry and he wrote to Tate: “Dear Sir, We (Mr Rosen and I) had sent you our manuscript for publication and had not authorized you to show it to specialists before it is printed… On the basis of this incident I prefer to publish the paper elsewhere…” [10]. After conversations with H.P. Robertson and L. Infeld (who were in Princeton in 1936) Einstein revised his conclusion and submitted the joint paper with Rosen in Journal of Franklin Institute [12] and at the end the paper Einstein wrote “The second part of this paper altered by me after departure of Mr. Rosen for Russia since we had interpreted our formula erroneously. I wish to thank my colleague Professor Robertson for his friendly assistance in the clarification of the original error”. As it was noted in [10] Robertson was the referee of the paper by Einstein and Rosen submitted in Physical Review.

2 DIRECT WAYS TO EVALUATE GRAVITATIONAL POTENTIAL NEAR SUPERMASSIVE BLACK HOLES

The most natural way to evaluate a gravitational potential is a consideration of test body trajectories in the potential and comparison of theoretical orbits calculated in the framework of a selected model with observational results. For solar system potential this way was passed due to efforts of scientific giants like Tycho Brahe, J. Kepler, R. Hooke and I. Newton. Really, Tycho Brahe collected observational data about trajectories of planets, Kepler formulated laws for planet motions, Hooke wrote a letter to Newton [13, 14] and later Newton proved that really this gravity law could explain Kepler’s laws for solar system and established his gravity law for all celestial objects in the Universe. Later, Rutherford used trajectories of \(\alpha \)‑particles to investigate a structure of atoms. Similarly, we could use trajectories of bright stars, gas clouds to test gravitational field in centers of galaxies including our Galactic Center.

3 BRIGHT STARS AS TEST BODIES IN CENTERS OF GALAXIES

Two groups of astronomers monitor bright stars near the Galactic Center (GC) for decades (see early publications on the subject in [15, 16]). Currently the group led by A. Ghez uses the Keck twin telescope in Hawaii while the European group led by R. Genzel uses VLT facilities in Chile (in the last years, four VLT telescopes with 8 m diameter mirrors can form the interferometer which is called GRAVITY). Recently, relativistic predictions evaluated in the framework of the first post-Newtonian approximation about gravitational redshifts for S2 star near its pericenter passage in May 2018 have been confirmed in observations (these conclusions were done by both Keck and GRAVITY collaborations) [17–19]. The GRAVITY collaboration reported about a discovery of hot spot motions near the innermost stable circular orbit of supermassive black hole at the GC [20] and these achievements give an opportunity to investigate GR predictions in a strong gravitational field limit. In 2020 the GRAVITY collaboration found that the Schwarzschild precession corresponds to its relativistic estimates done in the first post-Newtonian approximation [21]. These observational results support the Newton’s declaration that the gravity law is universal elsewhere including the Solar system and the Galactic Center. This conclusion is important since in last years theorists proposed a number of alternative theories of gravity and gravity laws may be different in different astronomical systems.

4 SCHWARZSCHILD PRECESSION AS A DISTANCE MEASURE FOR GC MODEL

Observations of bright stars at GC give an opportunity to choose the most suitable model for gravitational potential. If in the first approximation we can use a Newtonian approximation for gravity and we can assume that a gravitational potential is spherically symmetrical, so we can test any theoretical model in a spherical shell where the orbits of observed stars are located if observation accurate enough even in the case if observers monitor a small piece of an entire trajectory of a star. However, if we wish to rule out a set of models which do not correspond to observations we could theoretically evaluate the Schwarzschild precession for monitored stars for more than one period and after that we could reject models (or gravity theories) which are nor consistent with observations. Using such an approach we constrain GC models with dark matter based on properties of the S2 star trajectory [22].

5 ALTERNATIVE THEORIES OF GRAVITY

As the first case to constrain alternative theories of gravity we choose a version of \(f(R)\) theory of gravity [23–25] where \(f(R) = {{R}^{n}}\) (it is clear that for \(n = 1\) the theory coincides with GR) and this approach was proposed in [26]. Later it was found that \({{R}^{n}}\) theory can explain an accelerated expansion of the Universe [27] and flat rotation curves for spiral galaxies [28] or in other words for some cases the considered gravity change could fit the accelerated expansion of the Universe due to dark energy (or \(\Lambda \)-term) and dark matter phenomena (naturally to explain DM and DE phenomena \(n\) must be significantly different from unity, since really, to fit supernovae type Ia data which was used to discover the accelerated expansion of the Universe \(n\) parameter must be around 3 [27], while to fit rotation curves for spiral galaxies we have to choose \(n \approx 3.5\) [28]). However, Solar system data are not consistent with so big \(n\) numbers since we found that \(n\) must be very close to 1 [29]. From a comparison of theoretical estimates and observational data for the S2 star trajectory we concluded that \(n\) parameter must be very close to 1 otherwise the precession is much greater than its observed quantity [30]. Later we assumed a presence of \({{R}^{n}}\) gravity and a bulk concentration of matter in GC we also obtained even more strict constraints on \(n\) parameter [31] since both \({{R}^{n}}\) (for \(n > 1\)) and extended mass distribution cause retrograde orbital precession. Evaluating the Schwarzschild precession in the framework of Yukawa gravity and taking into account an absence of the precession for S2 star orbit we constrained Yukawa gravity parameters in [32] (it was useful result since there is a class of extended theories of gravity which have an Yukawa gravity as a weak gravitational field limit [33]). Constraints on parameters of extended gravity theory from observations of the S2 trajectory are presented in [34].

6 THEORIES OF MASSIVE GRAVITY

The first version of gravity theory where graviton is massive was proposed by Fierz and Pauli [35], however, later pathologies have been discovered in this theory, in particular, a presence of ghosts have found [36]. A version of massive theory of gravity have been developed by A.A. Logunov and his group in the framework of relativistic theory of gravity and using this approach astrophysical consequences were discussed, in particular, graviton mass have been constrained [37, 38] (see references therein as well) and the graviton mass constraint \({{m}_{{\text{g}}}} < 9 \times {{10}^{{ - 34}}} \) eV obtained in [37] is still the strictest according to the last PDG review [39] (however, we have to note that it very hard to control systematic errors in different estimates of graviton mass bounds).

Several years ago C. de Rham and her co-authors found a way to construct a massive theory of gravity without ghosts [40] (see also a more extended review on the subject [41]). In the first publication on gravitational wave discovery [42] the LIGO and Virgo collaborations reported about detections of gravitational waves from binary black hole system and the authors considered a massive theory of gravity and found a constraint on graviton mass \({{m}_{{\text{g}}}} < 1.2 \times {{10}^{{ - 22}}} \) eV. Assuming that massive gravity is valid at GC and considering the S2 star trajectory observed by Keck and VLT telescopes we obtained a graviton mass constraint \({{m}_{{\text{g}}}} < 2.9 \times {{10}^{{ - 21}}} \) eV [43]. If we suppose that GR estimates concerning the orbital precessions of bright stars will be confirmed by future observations we showed that the current graviton mass estimate could be significantly improved at a level around \(5 \times {{10}^{{ - 23}}}\) eV [44] which is comparable with constraints found by LIGO–Virgo collaborations.

7 IS IT POSSIBLE TO SUBSTITUTE SUPERMASSIVE BLACK HOLE WITH DARK MATTER CLOUD IN GC?

It was generally adopted that there supermassive black holes (SMBHs) in galactic nuclei (including our GC). However, other models are also proposed. For instance, it was suggested to substitute supermassive black hole in GC by dense core and diluted halo produced by dark matter [45] (it is now called RAR-model, since Ruffini, Argüelles and Rueda were the authors in [45]) it was also declared [46] that this model provided a better fit of trajectories of bright stars in comparison with the conventional model where SMBH is a key component. However, if we adopt RAR-model for GC we have a harmonic potential for the central core of dark matter and trajectories of bright stars are elliptical where GC coincides with centers of ellipses while in reality GC coincides with foci of observed trajectories of bright stars [47, 48]. We should also mention that the RAR model for GC is not consistent the shadow reconstruction in Sgr A* [55] which was reproduced by the Event Horizon Telescope Collaboration.

8 SHADOWS AS BLACK HOLE FINGERPRINTS

Thought experiments were popular at the dawn of general relativity and quantum mechanics development. Around 50 yr ago when people started to analyze consequences of an existence of astrophysical black holes James Maxwell Bardeen proposed to consider a bright screen behind an astrophysical rotating black hole assuming that photons propagated along geodesics (without scattering) and in this case he concluded that a virtual observer could detect a small spot in the sky [50] (and later this spot was called shadow). However, in these times people did not discuss the Bardeen’s consideration as a GR test or a test for BH existence in an observed astronomical object, since first there is no a bright screen behind a selected black hole, second, for known black hole candidates sizes of these shadows are extremely small to be detected. Later, it was understood that secondary images should be concentrated near shadows [51] and shapes and sizes of shadows could be reconstructed from bright structure distributions around shadows and we declared that the shadow for Sgr A* could be reconstructed from global (or and ground—space) VLBI observations in mm or/and sub-mm bands (or X-ray band) [52] (simplifying our proposal we said that for theorists black holes are vacuum solutions of Einstein equations while for observers black holes are small spots (shadows) in the sky). These spectral bands were chosen since H. Falcke at al. showed in numerical simulations [53] that for 1 cm or longer wave lengths scatter of photons on electrons could spoil bright images around shadows while for 1.3 mm wave lengths or shorter shadows could be detectable. Consequent studies confirmed our predictions in [52] since the Event Horizon Telescope (EHT) Collaboration reconstructed the shadow around Sgr A* observed in April 2017 at 1.3 mm wavelength [55] (earlier, the EHT reported about shadow reconstruction for M87* [54). In spite of great differences in masses and distances for Sgr A* and M87* their shadow diameters are comparable since as it was found we have \(52 \mu as\) for Sgr A* and \(42 \mu as\) for M87*. We showed that a black hole spin could be evaluated from an analysis of shadow shape [52].

A cosmic plasma is quasi-neutral it is natural to expect that astrophysical black hole has a very small electric charge. In spite of these expectations we derived an analytical expression for a shadow size as a function of charge [56] (we followed an approach used earlier in [57, 58]). It means that photons could measure a black hole charge since a charge changes the Schwarzschild metric with the Reissner–Nordström one. We also should to note that Reissner–Nordström metric is a solution in Randall–Sundrum gravity theory with an extra dimension [59]. Really, this solution looks like Reissner–Nordström metric but it is a generalization of this solution since parameter \({{q}^{2}}\) may be negative (\(q\) is a black hole charge) and Dadhich et al. called it a Reissner–Nordström metric with a tidal charge since this additional parameter was caused by an existence of an extra dimension [59]. Later, it was proposed to adopt a Reissner–Nordström metric with a tidal charge for the GC [60], however, it was shown that a significant negative tidal charge is inconsistent with current estimates of a shadow size in Sgr A* [61].

Earlier we found allowed intervals for tidal charges based on EHT estimates of shadow sizes in M87* [54] and Sgr A* [55]. We will remind expression for a Reissner–Nordström black hole with a tidal charge in natural units (\(G = c = 1\)) in a form

$$\begin{gathered} d{{s}^{2}} = - \left( {1 - \frac{{2M}}{r} + \frac{{{{\mathcal{Q}}^{2}}}}{{{{r}^{2}}}}} \right)d{{t}^{2}} \\ + \,\,{{\left( {1 - \frac{{2M}}{r} + \frac{{{{\mathcal{Q}}^{2}}}}{{{{r}^{2}}}}} \right)}^{{ - 1}}}d{{r}^{2}} \\ + \,\,{{r}^{2}}(d{{\theta }^{2}} + {{\sin }^{2}}\theta d{{\phi }^{2}}), \\ \end{gathered} $$
(1)

where \(M\) is a black hole mass, \(\mathcal{Q}\) is its charge. Constants \(E\) and \(L\) are connected with photon and they are describe photon geodesics, namely \(E\) is photon’s energy, \(L\) is its angular momentum. If we introduce normalized radial coordinate, impact parameter and charge \(\hat {r} = {r \mathord{\left/ {\vphantom {r M}} \right. \kern-0em} M},\) \(\xi = {L \mathord{\left/ {\vphantom {L {(ME)}}} \right. \kern-0em} {(ME)}}\), \(\hat {\mathcal{Q}} = {\mathcal{Q} \mathord{\left/ {\vphantom {\mathcal{Q} M}} \right. \kern-0em} M}.\) We introduce also variables \(l = {{\xi }^{2}},\,\,q = {{\hat {\mathcal{Q}}}^{2}}\), then critical impact parameter corresponding to shadow radius [62]

$${{l}_{{{\text{cr}}}}} = \frac{{(8{{q}^{2}} - 36q + 27) + \sqrt D }}{{2(1 - q)}},$$
(2)

where \(D = - 512{{\left( {q - \frac{9}{8}} \right)}^{3}}.\) As we noted earlier, parameter \(q\) may be negative for a Reissner–Nordström black hole with a tidal charge (or for Horndeski scalar-tensor theory of gravity [63, 64]).

The EHT Collaboration evaluated the shadow radius in M87* and estimated parameters of several spherically symmetric metrics which may be considered as alternatives for Schwarzschild metric in M87* [65]. In [66] we generalizes results [65] for a Reissner–Nordström black hole with a tidal charge assuming similarly to [65], that angular diameter of a shadow in M87* \({{\theta }_{{{\text{sh M87*}}}}} \approx 3\sqrt 3 (1 \pm 0.17){\kern 1pt} {{\theta }_{{{\text{g M87*}}}}}\), at confidence level around 68% or \({{\theta }_{{{\text{sh M87*}}}}} \in [4.31,6.08]{{\theta }_{{{\text{g M87*}}}}}\), where \({{\theta }_{{{\text{g M87*}}}}} \approx 8.1 \mu as\), since \({{\theta }_{{{\text{g M87*}}}}} = {{2{{M}_{{{\text{M87*}}}}}} \mathord{\left/ {\vphantom {{2{{M}_{{{\text{M87*}}}}}} {{{D}_{{{\text{M87*}}}}}}}} \right. \kern-0em} {{{D}_{{{\text{M87*}}}}}}}\) (\({{M}_{{{\text{M87*}}}}} = 6.5 \times {{10}^{9}}{{M}_{ \odot }}\) and \({{D}_{{{\text{M87*}}}}} = 17\) Mpc, we found \(q \in [ - 1.22,0.814]\) from Eq. (2). In this case an upper limit for \(q\) parameter (\({{q}_{{{\text{upp}}}}} = 0.814\)) corresponds to an upper parameter \({{\mathcal{Q}}_{{{\text{upp}}}}} = \sqrt {{{q}_{{{\text{upp}}}}}} \approx 0.902\), which corresponds to quantity calculated numerically and shown in Fig. 2 in [65].

Similarly to our previous estimates for tidal charge in M87* in [67] we estimated a tidal charge for the black hole in GC. We used estimates of shadow radius in GC from [55]. Following these studies, we assume that the shadow diameter in GC is \({{\theta }_{{{\text{sh M87*}}}}} \approx (51.8 \pm 2.3) \mu as\) at C. L. 68% and in this case we obtain constraints for a tidal charge \( - 0.27 < q < 0.25\) at the same confidence level.

These results may be used for analytical estimates of charge for Kazakov–Solodukhin (KS) black hole. Really Kazakov and Solodukhin considered a Schwarzschild black hole perturbed by quantum fluctuations [68]. We should note that black hole with a negative tidal charge (or scalar-tensor charge in Horndeski gravity) could treated as a good approximation for KS black hole for a small KS charge, really according to Eq. (3.21) in [68] we have

$$g(r) = - \frac{{2M}}{r} + \frac{1}{r}{{\left( {{{r}^{2}} - q_{{{\text{KS}}}}^{2}} \right)}^{{1/2}}} \approx 1 - \frac{{2M}}{r} - \frac{{q_{{{\text{KS}}}}^{2}}}{{{{r}^{2}}}},$$
(3)

where \({{q}_{{{\text{KS}}}}}\) is a KS charge. For small parameter \({{q}_{{{\text{KS}}}}}\) approximation we could use previous estimates for a KS charge in Sgr A* \({{({{q}_{{{\text{KS}}}}})}^{2}} < 0.27\) (\(({{q}_{{{\text{KS}}}}}) < 0.52\)). As we see in Fig. 2 in [65] the shadow radius is growing as \({{q}_{{{\text{KS}}}}}\) is growing and it corresponds to the shadow diameter dependence of a tidal charge given in Eq. (2).

9 CONCLUSIONS

Observations of bright stars near the GC confirmed predictions of GR in the first post-Newtonian approximation for gravitational redshift for S2 star trajectory near its pericenter passage in May 2018. The GRAVITY collaboration found that the Schwarzschild precession for S2 star corresponds to GR predictions. Several alternative theories of gravity were constrained with observations of bright stars. Reconstructions of shadows in M87* and Sgr A* give an opportunity to check GR predictions in these objects and to constrain parameters of alternative models for these objects [69, 70]. In 2020 the GRAVITY collaboration found that the Schwarzschild precession of S2 star corresponds to GR estimates [71]. Recently, constraints on Yukawa gravity parameters have been found from these observational results [72] (previous estimates of Yukawa gravity parameters were done in [73]). An improvement of graviton mass constraints was obtained in [74]. Possible models for GC and M87* were discussed in [75]. A development of shadow from a theoretical concept in GR test was reviewed in [76].