Effect of Electromagnetic Interaction on Galactic Center Flare Components

The present activity of Sgr A* is very low, and in comparison with active galactic nuclei (AGN), it can be generally characterized as extremely low-luminous (Genzel et al. 2010; Eckart et al. 2017), which stems from the comparison of its theoretical Eddington limit,

$$\begin{eqnarray}&&{L}_{\mathrm{Edd}}=5\times {10}^{44}\,\left(\displaystyle \frac{M}{4\times {10}^{6}\,{M}_{\odot }}\right)\,\mathrm{erg}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 1 }$$

and its 8 orders-of-magnitude smaller bolometric luminosity of $\sim {10}^{36}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ inferred from observations and explained by radiatively inefficient accretion flow models (RIAFs; Narayan et al. 1998; Blandford & Begelman 1999). The mass of Sgr A* in Equation (1) is scaled to the value of ∼4 × 10⁶ M_⊙ derived from the most recent S2 star observations by the Gravity Collaboration et al. (2018a; see also Boehle et al. 2016; Gillessen et al. 2017; Parsa et al. 2017, for comparison), which corresponds to the gravitational radius of ${R}_{{\rm{g}}}={GM}/{c}^{2}=5.9\times {10}^{11}\,\mathrm{cm}\sim {10}^{12}\,\mathrm{cm}$ that we apply in the further analysis.

The object Sgr A* is surrounded by ∼200 massive He I emission-line stars of spectral type OB, and it is thought to capture their wind material with an estimated rate of ${\dot{M}}_{{\rm{B}}}\approx {10}^{-5}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ at a Bondi radius of ${r}_{{\rm{B}}}=4^{\prime\prime} ({T}_{{\rm{a}}}/{10}^{7}\,{\rm{K}})\approx 0.16\,\mathrm{pc}=8.1\times {10}^{5}\,{R}_{{\rm{g}}}$ (Baganoff et al. 2003; Shcherbakov & Baganoff 2010; Ressler et al. 2019), where the gravitational pull of the SMBH prevails over that of the thermal gas pressure with temperature T_a. From the submillimeter Faraday rotation measurements within the inner r ≲ 200 R_g, it was inferred that Sgr A* accretes at least 2 orders of magnitude less than the Bondi rate, ${\dot{M}}_{\mathrm{acc}}\approx 2\times {10}^{-7}\mbox{--}2\times {10}^{-9}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ (Marrone et al. 2007); hence, most of the material captured at the Bondi radius is expelled and leaves the system as an outflow, which is also consistent with the RIAF solutions with a density profile in the power-law form $n(r)\propto {r}^{-p}$, where p ≲ 1 (Wang et al. 2013). The density profile of the hot RIAF flow is flatter than the density profile of the stationary spherical Bondi accretion, for which $n(r)\propto {r}^{-3/2}$. This flattening is caused by the presence of outflows (Yuan et al. 2012; Wang et al. 2013). The inflow–outflow RIAF models (disk–jet/wind or advection-dominated accretion flow (ADAF) and jet—jet-ADAF) can inhibit the accretion rate on smaller spatial scales by the transport of energy released during accretion to larger radii (Yuan et al. 2002; Shcherbakov & Baganoff 2010; Mościbrodzka et al. 2014; Chan et al. 2015; Ressler et al. 2017), which reduces the accretion rate to ≲1% of the Bondi rate, and the jet-ADAF models can generally capture the main features of the Sgr A* broadband spectrum. The extremely low luminosity of Sgr A* is thus best explained by the combination of a low accretion rate ${\dot{M}}_{\mathrm{acc}}$ and very low radiative efficiency of the accretion flow η_acc ≈ 5 × 10⁻⁶ (Yuan & Narayan 2014), which is 4 orders of magnitude below the standard 10% efficiency applicable to luminous AGN with significantly higher accretion rates.

In the radio/millimeter domain, the flux density generally increases with frequency with a rising spectral index from α = 0.1–0.4 to 0.76 at 2–3 mm⁷ and with a clear peak or bump close to 1 mm, which is referred to as the submillimeter bump (e.g., Falcke et al. 1998; Dexter et al. 2010; Bower et al. 2015). The submillimeter bump is produced by the optically thick synchrotron emission that originates from relativistic, thermal electrons (with a Lorentz factor of γ_e ∼ 10) in the innermost portions of the hot, thick ADAF (Narayan et al. 1995, 1998; Yuan et al. 2003). It marks the transition from the optically thick emission at lower frequencies to the optically thin emission at higher frequencies (Zylka et al. 1995; Serabyn et al. 1997; Falcke et al. 1998). Below 1 mm, the medium gets optically thin and the flux density gradually drops all the way to X-ray wavelengths, where the quiescent counterpart of Sgr A* was detected with the unabsorbed 2–10 keV luminosity of ${L}_{{\rm{x}}}\approx 2\times {10}^{33}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ produced by thermal bremsstrahlung from cooler electrons at larger distances close to the Bondi radius (Baganoff et al. 2003; Shcherbakov & Baganoff 2010), with no detected quiescent counterpart in the IR domain (see, however, the upper limits on the far-IR flux density based on the detected variability by von Fellenberg et al. 2018). Thanks to high-sensitivity GRAVITY observations in the K_s band (2.2 μm; The GRAVITY Collaboration et al. 2020), it was possible to detect a turnover in the flux density distribution of NIR flares with a median value of (1.1 ± 0.3) mJy. The flux density distribution in the NIR domain was found to have two states: the bulk of the emission can be described by a lognormal distribution with a median around 1.1 mJy, and on top of this quiescent emission, there are the sporadic flares at higher flux densities with a single power-law distribution. A single power-law or lognormal distribution cannot describe the flux density distribution as a whole.

While in the radio/millimeter domain, Sgr A* is mildly variable; it exhibits order-of-magnitude nonthermal high states or flares in the IR and X-ray domain a few times per day on a timescale of ∼1 hr (Baganoff et al. 2001; Genzel et al. 2003; Ghez et al. 2004; Eckart et al. 2006; Zamaninasab et al. 2010; Eckart et al. 2012; Witzel et al. 2012; Karssen et al. 2017; Gravity Collaboration et al. 2018b; Witzel et al. 2018), with the X-ray flares always being simultaneously associated with IR flares but not vice versa. The IR flares are linearly polarized with a polarization degree of 20% ± 15% and a rather stable polarization angle of 13° ± 15° (Shahzamanian et al. 2015), which likely reflects the overall stability of the disk–jet system of Sgr A*.

The K_s-band observations by GRAVITY@ESO (Gravity Collaboration et al. 2018b) brought the first direct evidence that flares are associated with the orbiting luminous mass or hot spots. The GRAVITY observations of hot spots close to the ISCO of Sgr A* have enabled the fitting of their orbital periods as well as orbital radii with the equatorial circular orbits of neutral test particles around rotating Kerr black holes of mass ∼4 million M_⊙.

The origin and nature of flares/hot spots still remains unclear. Despite many suggestions, including their connection to the tidal disruption of asteroids (Zubovas et al. 2012), they are most likely connected to dynamical changes in the hot, magnetized accretion flow. As transient phenomena, they could originate from magnetohydrodynamic instabilities or magnetic reconnection events, as is the case of X-ray flares on the Sun (Yuan et al. 2009) or has been discussed for M87 (Britzen et al. 2017). The model of ejected plasmoids during reconnection events is supported by their statistical properties; namely, the count rate versus flux density distribution can be fitted with the power law in the X-ray, IR, submillimeter, and radio domain, ${\rm{d}}N/{\rm{d}}E\propto {E}^{-\alpha }{\rm{d}}E$, which is consistent with the self-organized criticality phenomena of spatial dimension S = 3 (Witzel et al. 2012; Li et al. 2015; Subroweit et al. 2017; Witzel et al. 2018).

There are other mechanisms that lead to the power-law distribution of plasmoid properties. Namely, Uzdensky et al. (2010) showed analytically that the tearing (plasmoid) instability during magnetic reconnection leads to the power-law distribution of the magnetic flux ψ in plasmoids in high Lundquist number current sheets, f(ψ) ∝ ψ⁻², with an exponential decay in the tail of the distribution (Fermo et al. 2010), while Huang & Bhattacharjee (2012), using direct numerical simulations, showed that the slope is smaller, f(ψ) ∝ ψ⁻¹. In addition, magnetohydrodynamic turbulence driven by the magnetorotational instability can lead to a power-law distribution of dissipative events (see, e.g., Zhdankin et al. 2015). Zhdankin et al. (2015) found the probability distribution of the dissipated energy with an index of α = 1.75 ± 0.10 and the distribution of energy dissipation rates with a slope of α ∼ 2, which is two times less than the NIR, submillimeter, and radio distribution of flare flux densities, for which α = 4 (using the notation p(x) ∝ x^−α; see Witzel et al. 2012; Subroweit et al. 2017), but consistent with the energy distribution of X-ray flares, for which a slope of α = 1.65 ± 0.17 was recovered (Li et al. 2015). This is strikingly similar to the solar flare energy distribution with an index of α ∼ 1.8 (Hudson 1991). Neilsen et al. (2013) found a slope of ${1.9}_{-0.5}^{+0.4}$ for the X-ray flare peak rate distribution and 1.5 ± 0.2 for the fluence distribution, which is also consistent with the finding of Zhdankin et al. (2015) that the energy (fluence) distribution is shallower than the peak rate distribution. For the future, it will be necessary to verify whether the analytical and numerical studies of magnetohydrodynamic turbulence and instabilities are also applicable to the plasma in the strong gravity regime in 3D and whether the tearing and magnetohydrodynamic turbulence can lead to the power-law flux distribution of the flares that are sampled over a longer period of time.

To link the plasmoid model with the variability of Sgr A*, a plasma blob or plasmon that cools down via adiabatic expansion was applied to fit simultaneous multiwavelength flares. Yusef-Zadeh et al. (2008) found a time lag of 110 ± 17 minutes between X-ray and submillimeter 850 μm flares and ∼20–30 minutes between 7 and 13 mm flare peaks. The time delays are matched well by an initially optically thick cloud of synchrotron-emitting electrons that becomes optically thin toward consecutively lower frequencies as it expands and cools down adiabatically. The basic explanation of the delay between X-ray and submillimeter flare peak emissions is that at first, the emission is optically thick in the submillimeter domain, and later, it gets optically thin due to the blob expansion (van der Laan 1966). Furthermore, the model can reproduce well the asymmetric profile of the light curves (faster rising, slower fading) and the linear polarization degree of ∼1% at radio bands. The inferred parameters include the comoving expansion velocity of the blob, v_exp ∼ 0.003–0.1 c; a magnetic field in the range 10–70 G; and a particle spectral index of α = −1.5 ± 0.5. With this expansion velocity, plasma cannot escape from Sgr A*, unless a large bulk motion is present. These results have been confirmed independently by Kunneriath et al. (2010), Eckart et al. (2012), and Borkar et al. (2016). The NIR flares are explained via the synchrotron emission of localized, heated relativistic electrons, and the X-ray flares that are simultaneous with NIR flares are explained via three processes: (i) synchrotron emission of the same population of electrons as for NIR flares with a power-law distribution of energies, (ii) Compton upscattering of submillimeter seed photons by NIR-emitting electrons to X-ray energies (external inverse Compton), and (iii) NIR synchrotron photons upscattered to X-ray energies by the same population of heated electrons that produce NIR flares, which is referred to as the synchrotron self-Compton (SSC) mechanism (see Genzel et al. 2010, for a review).

Another model to explain the hot-spot phenomenon and the associated short-term variability in the Galactic center would be vortices and magnetic field flux tubes that could be sites of dissipation and collimated radiation (Abramowicz et al. 1992). Vortices as coherent structures are typical for any rotating fluid; hence, this scenario could also be applicable to the Galactic center hot flow.

The simplest explanation of the hot-spot nature is given by the discrete origin of the NIR flares. The single-state stochastic nature of Sgr A* variability (Meyer et al. 2014) suggests a clumpy mode of accretion. The hot spot would then correspond to confined islands or blobs of heated plasma that descend down the potential well toward the event horizon. This view is consistent with the class of magnetically arrested accretion flow models (MADs; Bisnovatyi-Kogan & Ruzmaikin 1974; Narayan et al. 2003; Igumenshchev 2008), in which the accretion flow becomes unstable due to the accumulation of the magnetic flux and fragments into magnetically confined blobs below the magnetospheric radius,

$$\begin{eqnarray}&&{R}_{{\rm{m}}}\sim \displaystyle \frac{8\pi {GM}\rho }{{B}_{\mathrm{pol}}^{2}}=\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&=181\,\left(\displaystyle \frac{M}{4\times {10}^{6}\,{M}_{\odot }}\right)\left(\displaystyle \frac{{n}_{\mathrm{acc}}}{{10}^{6}\,{\mathrm{cm}}^{-3}}\right){\left(\displaystyle \frac{{B}_{\mathrm{pol}}}{10{\rm{G}}}\right)}^{-2}\,{R}_{{\rm{g}}},\end{eqnarray} \tag{ 3 }$$

where the mass density is given by ${\rho }_{\mathrm{acc}}=\mu {m}_{{\rm{H}}}{n}_{\mathrm{acc}}$ (μ ∼ 0.5 for a fully ionized plasma), the poloidal magnetic field B_pol is scaled to 10 G, and the number density is scaled to ${10}^{6}\,{\mathrm{cm}}^{-3}$, according to the values obtained from the synchrotron emission models of the flares (Yusef-Zadeh et al. 2006, 2008; Eckart et al. 2012). For such an accretion flow model, see also Figure 1 for an illustration where an initially hot, diluted, and thick axisymmetric flow fragments into clumps below R_m. The clumpy structure of the flow is expected to dominate on A length scale of 100 R_g with a certain filling factor. The clumpy flow proceeds inwards diffusively through the poloidal magnetic field with the help of magnetic interchanges and reconnection events and the radial velocity is less than the freefall velocity (Narayan et al. 2003). The volume filling factor of the hot spots can be estimated ${f}_{{\rm{V}}}={V}_{\mathrm{hs}}/{V}_{\mathrm{acc}}\,={n}_{\mathrm{acc}}/{n}_{\mathrm{hs}}$, where V_hs is the volume of the hot spots, V_acc is the total volume of the accretion flow, n_acc is the mean number density of the accretion flow, and n_hs is the hot-spot number density. The hot-spot number density is thus expected to be larger than the mean density of the accretion flow, n_hs = n_acc/f_V, depending clearly on f_V. Typically, only one hot spot is present at the ISCO, according to NIR images and time series (two to three events per day; Gravity Collaboration et al. 2018b); therefore, within the ISCO volume, ${f}_{{\rm{V}}}\sim {\left({R}_{\mathrm{hs}}/{r}_{\mathrm{ISCO}}\right)}^{3}$, which for R_hs ∼ 1R_g and a nonrotating black hole leads to n_hs ∼ 216 n_acc. In this setup, the hot spot would clearly be an overdense blob with respect to the background medium.

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The relativistic motion of a plasma around a black hole in the presence of the ordered magnetic field component (orthogonal to the orbital plane) necessarily leads to the charge separation and the consequent growth of its net charge density. This charge increases due to the compensation of the electric field in the comoving frame induced by the motion of plasma in the external magnetic field. In the case where the rotating plasma is associated with a neutron star threaded by a magnetic field, the arising charge density is known as the Goldreich–Julian (GJ) charge density (Goldreich & Julian 1969). An analogous argument can be used to obtain a net charge density of the plasma of the flare components moving around the black hole, as will be described in Section 3.1.

On the other hand, a similar charging process occurs near the black hole due to the frame-dragging effect of the twisting of the magnetic field lines. In this scenario, both the black hole and the magnetosphere possess a nonnegligible electric charge (Ruffini & Wilson 1975). Damour et al. (1978) derived three different regions of charge particle trapping based on the interplay of electric and magnetic fields in the black hole magnetosphere. A magnetic field near a rotating black hole also plays a crucial role in the collimation of charged particles and thus also in the precollimation of astrophysical jets (Karas & Dovciak 1997).

For a black hole in this scenario, the charging mechanism was introduced by Wald (1974). Such an induced charge of Sgr A* has an upper limit of Q_BH ∼ 10¹⁵C (Zajaček et al. 2018), which is still quite weak to have a gravitational effect on the spacetime metric; however, its electrostatic counterpart is nonnegligible for the motion of charged matter. Similarly, in a nonrelativistic case, the net charge of a massive object arises due to its rotation in the magnetic field (Ruffini & Treves 1973). Computing Maxwell equations inside the magnetosphere (see, e.g., Ruffini & Wilson 1975), one can find that the total charge of the black hole magnetosphere is equal to the charge of the black hole with the opposite sign, i.e., Q_mag = −Q_BH. This condition holds for a large class of accretion models.

One of the interesting outcomes of the charge separation process in a plasma surrounding a black hole (and the consequent growth of the net charge densities of both the black hole and the magnetosphere) is that the black hole may act as a pulsar (Levin et al. 2018). For the dynamics of the flare components, furthermore, this leads to the inclusion of additional "electrostatic" interaction between the black hole and the hot spots if the black hole charge is not screened effectively.

In the Galactic center, a partially ordered magnetic field in the vicinity of Sgr A* has been estimated with a strength of 10–100 G. The NIR observations of horizontal polarization loops with timescales comparable to the orbital periods of the recently observed bright flares imply the prevalent orientation of magnetic field lines in a direction that is perpendicular to the orbital plane of the corresponding hot spots. This implies that the hot spots associated with flares orbiting at the relativistic orbits are expected to possess a net electric charge due to the charge separation in the Galactic center plasma.

In this paper, we focus on the possible interplay between gravitational and electromagnetic fields in the interpretation of the observational features of the flares. Orthogonal orbital orientation with respect to the magnetic field lines of the three most recent hot spots and the charge separation in plasma leads to the appearance of an external Lorentz force arising from interactions of the flare components with the magnetic field. Taking into account the error bars of the GRAVITY measurement arising mainly due to astrometric errors and incomplete orbital coverage, we put limits on the strength of the Lorentz force, electric charges of hot spots, and net charge densities.

Below, we consider hot-spot models with various charging mechanisms compared with observational data. It is worth noting that all obtained constraints on the charge values of hot spots have a comparable order of magnitude.

The paper is structured as follows. In Section 2 we introduce the basic equations describing the model and provide estimates on the magnetic field, black hole charge, size, and mass of the flare components. In Section 3 we study the charge separation in the plasma surrounding the black hole and estimate its net charge density. Applied to three recent flare components, we put tighter constraints on the hot spots' charge based on their dynamics with inclusion of the electromagnetic interaction and the synchrotron radiation from hot spots. We also calculate the shifts of the ISCO caused by the interplay between the hot-spot charge and external magnetic field and discuss the results. In Section 4 we study the possible inclination of the orbital plane and magnetic field lines with respect to the orientation of the black hole's spin and put constraints on the inclination angle. We discuss the main results and their consequences in Section 5 and give conclusions in Section 6.

Throughout the paper, we use the space-like signature (−, +, +, +) and the system of units in which c = 1 and G = 1. However, for the expressions with an astrophysical application and estimates, we use the units with the gravitational constant and the speed of light. Greek indices are taken to run from 0 to 3; Latin indices are related to the space components of the corresponding equations

Our modeling approach is primarily motivated by NIR observations in the K_s band (2.2 μm; Gravity Collaboration et al. 2018b), where three hot spots were detected at different epochs. These structures stay stable in terms of the luminosity for one orbital period or a large fraction of it. The small flux density changes can be attributed solely to the Doppler boosting along an orbit, which has a large inclination of ∼160° but is not face-on (see also Figure 1 for the inclination scheme). Flares that were found to be closer to edge-on orbits often exhibit a peak-shoulder structure in their light curves, which can be attributed to the combination of the gravitational lensing and Doppler boosting along their orbits (Eckart et al. 2017; Karssen et al. 2017). There is also no evidence of significant shearing along this orbit during one orbital period.

Keeping this in mind, we assume the following properties for the hot spot.

• (a)  
  The hot spot is a bound test and potentially charged mass moving on a circular orbit in the Kerr spacetime background, as well as in the global poloidal magnetic field.
• (b)  
  Given the lack of any shearing along the orbit, we also assume that the hot spot is a spherical mass, or at least that its shape does not change during one orbital timescale.
• (c)  
  We also assume that during the orbital timescale, the surrounding environment does not affect the hot-spot dynamics, or, in other words, these effects are negligible in comparison with the general relativistic effects and electromagnetic interaction.

Assumption (a) will be justified in more detail in Sections 2.2–2.5. Circular trajectories should be preferred, since the collisions of magnetically confined cloudlets in the MAD models on crossing orbits are highly dissipative in the central 100 R_g (see also Bao et al. 1994; Eckart et al. 2018, for the comparison of circular and elliptical hot-spot trajectories).

Concerning assumption (b), there is no observational information about the shape of the hot spot. However, given the stable intrinsic luminosity and no evidence for shearing, a stable spherical shape is a reasonable assumption. The shape of the hot spot may also be kept stable against magnetohydrodynamic instabilities by the tangled internal magnetic field (McCourt et al. 2015; Guillochon & McCourt 2017), which is also the property of MADs.

As a new feature in comparison with previous studies of hot-spot dynamics and radiative properties (Broderick & Loeb 2005, 2006a, 2006b; Meyer et al. 2006; Zamaninasab et al. 2008, 2010), we add the potential electromagnetic interaction. Any other effects from the surrounding medium or objects (stars) may have a certain effect, but it is observationally difficult to estimate them at the moment, as the hot spots are not observed beyond one orbital period. We leave the other potential magnetohydrodynamic effects for future studies when more data are available.

In addition, the accretion flow close to Sgr A* is diluted and potentially clumpy, as predicted by MADs (Narayan et al. 2003; Igumenshchev 2008), which we illustrate in Figure 1 and discuss in more detail in Sections 1.2 and 5. Therefore, it is likely that the hot spot is a dominating mass on a circular orbit close to the ISCO. This is also supported by the observations of discrete X-ray and NIR flares that point toward clumpy accretion. Given the general instability of the accretion flow below the magnetospheric radius (see Equation (3)), the flow continues inward in the form of magnetically confined blobs that diffuse through the poloidal magnetic field via reconnection events. In this way, a hot spot is a dominating mass at a given time and location close to the ISCO; hence, any effect from the surrounding medium is assumed to be negligible during one orbital timescale. The surrounding medium has the nature of a hot, diluted corona that is heated up by the released energy of the MAD flow (Narayan et al. 2003). In fact, the number density of the blob is expected to be larger than the mean accretion flow density by the inverse of the volume filling factor—${\left({r}_{\mathrm{ISCO}}/{R}_{\mathrm{hs}}\right)}^{3}\lt 216$—which we derive in Section 1.2.

In addition, even if the surrounding medium with a comparable number density were present around the hot spot, it would be comoving with the hot spot close to the ISCO. The magnetohydrodynamic drag force can be expressed as (Dursi & Pfrommer 2008; McCourt et al. 2015)

$$\begin{eqnarray}&&{F}_{\mathrm{drag}}\sim {\rho }_{{\rm{a}}}{v}_{\mathrm{rel}}^{2}{R}_{\mathrm{hs}}^{2}\,\left(1+\displaystyle \frac{{v}_{{\rm{A}}}^{2}}{{v}_{\mathrm{rel}}^{2}}\right),\end{eqnarray} \tag{ 4 }$$

where ρ_a is the ambient density, v_rel is the relative velocity between the hot spot and the ambient medium, R_hs is the hot-spot radius, and v_A is the Alfvén velocity. The drag force is therefore negligible for the case where the ambient medium is comoving with the hot spot, since v_rel ≈ 0. This justifies assumption (c).

It is likely that shearing and the associated departure from the quasi-spherical shape govern the hot-spot evolution after one orbital period when the hot-spot flux density falls beyond the detection limit. However, at the same time, the hot spot likely plunges from the ISCO toward and beyond the event horizon of Sgr A* on a timescale of ${t}_{{\rm{r}}{\rm{f}}}=12GM/{c}^{3}\,\approx 4$ minutes for the radial fall toward Sgr A* (in general, this depends on the initial angular momentum). The shearing and infall would manifest themselves by the gradually decreasing flux densities of the flare (see the simulated light curves of an infalling, shearing hot spot calculated by Dovčiak et al. 2004 and Zajacek 2017, the using KYSPOT code), which has not been detected so far (see the observed light curves in Figures 1 and 2 in Gravity Collaboration et al. 2018b). This so-far-unobserved regime is therefore beyond the scope of the current paper but may be of interest in our future studies.

In other words, the dynamical effects studied in this work focus on the transient hot-spot feature during its orbital timescale. Since this timescale, which can be calculated for the nonrotating black hole at the ISCO as

$$\begin{eqnarray}&&{P}_{\mathrm{hs}}=30.3\,\left(\displaystyle \frac{M}{4\times {10}^{6}\,{M}_{\odot }}\right)\,{\left(\displaystyle \frac{r}{6{R}_{{\rm{g}}}}\right)}^{3/2}\,\mathrm{minutes},\end{eqnarray} \tag{ 5 }$$

is much shorter than the viscous timescale of the hot and thick accretion flow around Sgr A* at larger scales of 100 R_g,

$$\begin{eqnarray}&&{t}_{\mathrm{visc}}\sim 2.3\left(\displaystyle \frac{M}{4\times {10}^{6}\,{M}_{\odot }}\right)\,{\left(\displaystyle \frac{h}{r}\right)}^{-2}\,{\left(\displaystyle \frac{r}{100{R}_{{\rm{g}}}}\right)}^{3/2}\,{\left(\displaystyle \frac{{\alpha }_{\mathrm{visc}}}{0.1}\right)}^{-1}\,\mathrm{days},\end{eqnarray} \tag{ 6 }$$

where we assumed a thick flow with a height-to-radius ratio of h/r ∼ 1 and a viscosity parameter of α_visc ∼ 0.1. Therefore, we can neglect the long-term behavior of the whole flow as such. These long-term effects will be studied in more detail in our upcoming studies.

Current estimates of the magnetic field around Sgr A*—at the scales of the ISCO—are consistent with its strength of the order of 10–100 G (Eckart et al. 2012). The time-variable flux density during high states—flares—is modeled using synchrotron components in the NIR domain (Witzel et al. 2012, 2018). To explain the emission mechanism of simultaneous X-ray and NIR flares (X-ray flares always have NIR counterparts, but not vice versa; Mossoux et al. 2016), a synchrotron—SSC model is often employed that requires relativistic electrons with a Lorentz factor of γ_e ∼ 10³ (Eckart et al. 2012), which is 3 orders of magnitude less than the pure synchrotron–synchrotron model. The time lag of t = 1.5 ± 0.5 h between X-ray/NIR and millimeter/submillimeter flares (Eckart et al. 2008) is successfully explained by comoving adiabatic expansion of plasma blobs with uniform expansion speeds of v_exp ∼ 0.005–0.017c (Yusef-Zadeh et al. 2008; Kunneriath et al. 2010 see also the model description in Section 1.2). The SSC model can also be used to estimate the magnetic field strength using $B\sim {\theta }_{\mathrm{ss}}^{4}{\nu }_{{\rm{m}}}^{5}{S}_{{\rm{m}}}^{-2}$, where θ_ss is the angular source size and S_m is the flux density at the turnover frequency ν_m. Typical values of the magnetic field during high states as derived from the SSC modeling are of the order of B ∼ 10–100 G (Kunneriath et al. 2010; Eckart et al. 2012), in general being variable by a factor of a few.

Another constraint can be derived from the Faraday rotation measurements and the accretion flow density and temperature profiles close to Sgr A* that are inferred from fitting the RIAF model to the X-ray observations of the hot flow. Based on the Faraday rotation measurements of the magnetar PSR J1745–2900, Eatough et al. (2013) put a lower limit of B ≳ 8 mG on the line-of-sight component of the magnetic field for the magnetar deprojected distance of r ≳ 0.12 pc from Sgr A*. They also confirmed that the ordered magnetic field is present at length scales of ∼0.1 pc, which is intermediate between the large-scale ordered magnetic field in the central molecular zone (Morris 2015) and the ordered field close to the ISCO of Sgr A* (Johnson et al. 2015; Gravity Collaboration et al. 2018b). The plasma magnetization parameter can be expressed as the ratio of its magnetic field energy density to its thermal pressure (thermal pressure of electrons), ${\beta }_{{\rm{p}}}={B}^{2}/8\pi {n}_{{\rm{p}}}{k}_{{\rm{B}}}{T}_{{\rm{p}}}$, where n_p and T_p are the plasma density and temperature, respectively. The value of β_p at the distance of the magnetar (r ∼ 0.1 pc) can be estimated based on the density and temperature as inferred for the Bondi radius, ${n}_{{\rm{p}}}\approx 26\,{\mathrm{cm}}^{-3}$ and T_p ≈ 1.5 × 10⁷ K (Baganoff et al. 2003). The value of the magnetic field is inferred from the Faraday rotation, B ∼ 8 mG (Eatough et al. 2013). Then the magnetization parameter is β_p ∼ 47.3. To estimate β_p at ISCO, we adopt the values of the density, temperature, and magnetic field from the plasmon model applied to the radio variability by Yusef-Zadeh et al. (2006), who got ${n}_{{\rm{p}}}\sim 6\times {10}^{5}\,{\mathrm{cm}}^{-3}$, T_p ∼ 10⁹ K, and B ∼ 10 G. Then the magnetization parameter at the ISCO can be estimated as β_p ∼ 48.1. Between the Bondi radius and the ISCO, the magnetization parameter may thus be considered constant within the uncertainties. Its value of β_p ∼ 50 also implies that the plasma in this region is magnetically dominated.

The X-ray spectroscopy measurements by the Chandra telescope (Wang et al. 2013) revealed an elongated extended emission structure with a radius of 1.5'' centered at Sgr A*. Based on the very weak Fe Kα line, a no-outflow scenario can be rejected. The radial density profile ${n}_{{\rm{p}}}\propto {r}^{-3/2+s}$ with s ≳ 0.6 best fits the continuum and emission lines in the 2–10 keV band. Using the upper limit on the mass accretion rate by Sgr A*, ${\dot{M}}_{\mathrm{SgrA}* }\sim 2\times {10}^{-7}\,{M}_{\odot }{\mathrm{yr}}^{-1}$ (Marrone et al. 2007), and the assumption of the inner radius of RIAF at r_i ∼ 200 R_g, Wang et al. (2013) got a radial temperature profile ${T}_{{\rm{p}}}\,\propto \,{r}^{-{\theta }_{T}}$ with θ_T ≳ 0.6. Using the thermal and magnetic pressure coupling via the magnetization parameter, ${P}_{\mathrm{mag}}\,={\beta }_{{\rm{p}}}{P}_{\mathrm{th}}$, and under the assumption of constant β_p, we obtain the power-law scaling of the magnetic field strength, $B\,\propto {r}^{-3/4+1/2(s-{\theta }_{T})}$, which for s ≈ θ_T (as inferred from X-ray spectra by Wang et al. 2013) simply becomes B ∝ r^−3/4. We can normalize the magnetic field profile using the line-of-sight magnetic field, as well as the magnetar distance from (Eatough et al. 2013)

$$\begin{eqnarray}&&B(r)\gtrsim 8\times {10}^{-3}\,{\left(\displaystyle \frac{r}{5.2\times {10}^{5}{R}_{{\rm{g}}}}\right)}^{-3/4}\,{\rm{G}},\end{eqnarray} \tag{ 7 }$$

where the distance was scaled to $r\simeq 0.1\,\mathrm{pc}\sim 5.2\times {10}^{5}\,{R}_{{\rm{g}}}$. The radiating plasma components are approximately at a distance of r = 10 R_g, which implies a magnetic field of $B(10\,{R}_{{\rm{g}}})\gtrsim 28\,{\rm{G}}$. This is consistent with the magnetic field as inferred from flare observations (Eckart et al. 2012). The Bondi flow (no outflow) with s = 0 and θ_T ∼ 1 would lead to a radial dependency B ∝ r^−5/4 and $B(10\,{R}_{{\rm{g}}})\gtrsim 6300\,{\rm{G}}$, which is 2 orders of magnitude larger than the flare value. This gives further support to the general RIAF model with an outflow, where ≲1% of the material captured at the Bondi radius is accreted by Sgr A*. The presence of an outflow then leads to the flattening of the density profile.

In Zajaček et al. (2018), we used the current observations of the hot phase of Sgr A* surroundings within the innermost arcsecond to place constraints on the electric charge of Sgr A*. The existence of a hot quasi-neutral, stationary plasma around Sgr A* leads to the existence of the equilibrium charge Q_eq, which stops the separation of lighter electrons from heavier protons in collisionless plasma. The equilibrium charge may be expressed as

$$\begin{eqnarray}&&{Q}_{\ \mathrm{eq}}^{{\eta }_{T}}=\displaystyle \frac{4\pi {\epsilon }_{0}G}{e}\,\left(\displaystyle \frac{{\eta }_{T}{m}_{{\rm{p}}}-{m}_{{\rm{e}}}}{1+{\eta }_{T}}\right)M,\end{eqnarray} \tag{ 8 }$$

where ${\eta }_{T}\equiv {T}_{{\rm{e}}}/{T}_{{\rm{p}}}\simeq 1\mbox{--}1/5$, i.e., the hot flow close to the black hole is characterized by different proton and electron temperatures, with the proton temperature up to five times larger than the electron temperature (Mościbrodzka et al. 2009; Dexter et al. 2010). For η_T = 1, we obtain ${Q}_{\mathrm{eq}}^{1}=3.1\times {10}^{8}\,C$, and for η_T = 1/5, the charge is lower by about a factor of three, ${Q}_{\mathrm{eq}}^{1/5}=1.02\times {10}^{8}\,C$. In some models, much smaller values of η_T are adopted. For example, Mościbrodzka et al. (2016) formally considered the values down to ${\eta }_{T}^{\min }=1/100$ for a highly magnetized accretion flow, which would lead to an equilibrium charge of ${Q}_{\mathrm{eq}}^{1/100}=5.85\times {10}^{6}$ C. On the other hand, at least in the accretion flow part, there are strong arguments for a considerable part of the heating going directly to electrons (see, e.g., Bisnovatyi-Kogan & Lovelace 1997; Marcel et al. 2018), which prevents such a small value of ${T}_{{\rm{e}}}/{T}_{{\rm{p}}}$.

In a more general case, the black hole rotation in an ordered, homogeneous magnetic field B_ext leads to the twisting of the magnetic field lines and the generation of an electric field associated with an induced Wald charge, ${Q}_{{\rm{W}}}=2{{aMB}}_{\mathrm{ext}}$ (Wald 1974), with respect to infinity, where a is a dimensionless spin parameter. The dimensionless spin parameter a is defined using the relation ${a}_{\mathrm{spin}}={aGM}/{c}^{2}$, with a = 1 standing for the maximum prograde rotation, a = −1 representing a maximally counterrotating black hole, and a = 0 being a nonrotating black hole. Considering the constraint on the spin, a ≤ M, the upper limit on the induced Wald charge for the Galactic center black holes is

$$\begin{eqnarray}&&{Q}_{{\rm{W}}}\leqslant 2.3\times {10}^{15}\,{\left(\displaystyle \frac{M}{4\times {10}^{6}{M}_{\odot }}\right)}^{2}\left(\displaystyle \frac{{B}_{\mathrm{ext}}}{10\,{\rm{G}}}\right)\,{\rm{C}}.\end{eqnarray} \tag{ 9 }$$

The expected black hole charge based on the realistic magnetohydrodynamic environment is in the range Q = (10⁸, 10¹⁵) C, which is at least 12 orders of magnitude below the extremal value,

$$\begin{eqnarray}&&{Q}_{\max }=6.9\times {10}^{26}\,\left(\displaystyle \frac{M}{4\times {10}^{6}\,{M}_{\odot }}\right)\sqrt{1-{a}^{2}}\,{\rm{C}}.\end{eqnarray} \tag{ 10 }$$

One should stress that the Galactic center black hole is not in a vacuum, which is assumed by the Wald solution. A force-free approximation may be more appropriate. However, Levin et al. (2018) showed that a black hole embedded in the force-free magnetosphere is expected to carry charge; hence, a black hole carrying a small charge seems more likely than a completely neutral black hole.

A charged black hole that rotates will itself generate a dipole magnetic field with a magnetic dipole moment m_d ∼ Q_WM. The dipole magnetic field strength is then given by ${B}_{{\rm{d}}}\sim {m}_{{\rm{d}}}/{r}^{3}=2{B}_{\mathrm{ext}}{{aM}}^{2}/{r}^{3}$, with the upper limit given by the maximally rotating black hole, a < M and r_ISCO = M, which gives ${B}_{{\rm{d}}}^{\max }(r={r}_{\mathrm{ISCO}})\lt 2{B}_{\mathrm{ext}}$.

The parameters of the flare components, such as size and mass, can be estimated using the results of current and previous flare studies from Sgr A* observed in millimeter, NIR, and X-ray parts of spectra (Eckart et al. 2012). Observed flares have strengths of 5.2 mJy in the NIR K band that is 40% of the S2 star having a corresponding strength of 13 mJy. A simple estimate of the length scale of the hot spot may be derived from the adiabatic expansion of the emitting sources observed at speeds of ∼0.01 c (Jones et al. 1974). Light-travel arguments give constraints on the size of the flare components as

$$\begin{eqnarray}&&{R}_{\mathrm{hs}}\approx {R}_{{\rm{g}}}\displaystyle \frac{{GM}}{{c}^{2}}\approx 6\times {10}^{11}\,\left(\displaystyle \frac{M}{4\times {10}^{6}{M}_{\odot }}\right)\,\mathrm{cm}.\end{eqnarray} \tag{ 11 }$$

Number densities of flare components can be estimated to be of the order of

$$\begin{eqnarray}&&{\rho }_{{\rm{N}}}\approx {10}^{7\pm 1}{\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 12 }$$

derived from both the synchrotron model and radio Faraday rotation of the polarization planes (Eckart et al. 2012; Yuan et al. 2003). For pure electron and electron–proton cases with a spherical hot spot, we get the following limits on the masses of the flare components:

$$\begin{eqnarray}&&{m}_{\mathrm{hs}}^{\min }\approx 8.7\times {10}^{14}{\rm{g}};\quad {m}_{\mathrm{hs}}^{\max }\approx 1.6\times {10}^{20}{\rm{g}}.\end{eqnarray} \tag{ 13 }$$

For the comparison, the solar mass is M_⊙ ≈ 2 × 10³³ g, and the typical masses of large asteroids are of the order of 10²³ g.

Another estimate comes from the SSC modeling of the simultaneous NIR and X-ray flares. The source properties of these flares may be constrained using the power-law energy distribution with an exponential cutoff, $N(\gamma )\,={N}_{0}{\gamma }^{-p}\exp \,(\,-\gamma /{\gamma }_{{\rm{c}}})$ (with p ∼ 2). In this model, NIR flares are produced via the synchrotron mechanism with electrons gyrating in the magnetic field, and the same electrons upscatter the emitted photons to higher energies, producing the X-ray emission. With the knowledge of the magnetic field strength and assuming that the hot spot is uniform and spherical, one obtains the radius and number density of the same order of magnitude (Melia 2007),

$$\begin{eqnarray}\begin{array}{rcl}{R}_{\mathrm{hs}} & \approx & 5.12\left(\displaystyle \frac{{L}_{\mathrm{syn}}}{{10}^{36}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right){\left(\displaystyle \frac{{L}_{\mathrm{SSC}}}{{10}^{35}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{-1/2}\\ & & \times {\left(\displaystyle \frac{{B}_{\mathrm{ext}}}{10{\rm{G}}}\right)}^{-1}\,{R}_{{\rm{g}}},\end{array}\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{\rm{N}}} & \approx & 1.15\times {10}^{6}{\left(\displaystyle \frac{{L}_{\mathrm{syn}}}{{10}^{36}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{-2}\,{\left(\displaystyle \frac{{L}_{\mathrm{SSC}}}{{10}^{35}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{3/2}\\ & & \times \left(\displaystyle \frac{{B}_{\mathrm{ext}}}{10\,{\rm{G}}}\right){\left(\displaystyle \frac{{\gamma }_{{\rm{c}}}}{100}\right)}^{-2}\,{\mathrm{cm}}^{-3},\end{array}\end{eqnarray} \tag{ 15 }$$

which, by the order of magnitude, is close to the estimates given by Equations (11) and (12). Here we define by L_syn and L_SSC the luminosities of the hot spot in the synchrotron and SSC regimes, respectively. This leads to hot-spot masses of ${m}_{{\rm{h}}{\rm{s}}}^{min}\sim 1.2\times {10}^{17}$ and ${m}_{\mathrm{hs}}^{\max }\sim 2.3\times {10}^{20}\,{\rm{g}}$ for the pure electron and proton limits, respectively.

In order to test the influence of electromagnetic interaction on the dynamics of the hot spot around Sgr A*, one can consider a simplified scenario of the motion of a charged test particle around a rotating black hole in the presence of a magnetic field. It is natural (and supported by the observed orthogonality of the magnetic field lines and the hot-spot orbital plane) to assume that the magnetic field shares the symmetries of the spacetime in the vicinity of a black hole. Therefore, using the stationarity and axial symmetry of the Kerr black hole spacetime, one can express the four-vector potential in terms of the time-like and space-like Killing vectors ξ^μ in the form

$$\begin{eqnarray}&&{A}^{\mu }={C}_{1}{\xi }_{(t)}^{\mu }+{C}_{2}{\xi }_{(\phi )}^{\mu },\end{eqnarray} \tag{ 16 }$$

where C₁ and C₂ are constants. The solution (Equation (16)) corresponding to the test field approximation was suggested by Wald (1974), and in the case of the asymptotically uniform magnetic field with strength B, the nonvanishing components of A_μ correspond to

$$\begin{eqnarray}&&{A}_{t}=\displaystyle \frac{B}{2}\left({g}_{t\phi }+2{{ag}}_{{tt}}\right),\quad {A}_{\phi }=\displaystyle \frac{B}{2}\,\left({g}_{\phi \phi }+2{{ag}}_{t\phi }\right).\end{eqnarray} \tag{ 17 }$$

This is also the historically first analytical solution of Maxwell equations in the Kerr spacetime background. If the field has an inclination angle with respect to the spin axis of a black hole, the solution for A_μ is given by Bicak & Janis (1985). A configuration of a magnetic field corresponding to the dipole type has been derived by Petterson (1974). In all axially symmetric electromagnetic field configurations, the energy and angular momentum of a test particle with charge q and mass m is modified according to (Tursunov et al. 2016)

$$\begin{eqnarray}&&-{ \mathcal E }\equiv -\displaystyle \frac{E}{m}={g}_{{tt}}\displaystyle \frac{{dt}}{d\tau }+{g}_{t\phi }\displaystyle \frac{d\phi }{d\tau }+\displaystyle \frac{q}{m}{A}_{t},\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{ \mathcal L }\equiv \displaystyle \frac{L}{m}={g}_{\phi \phi }\displaystyle \frac{d\phi }{d\tau }+{g}_{t\phi }\displaystyle \frac{{dt}}{d\tau }+\displaystyle \frac{q}{m}{A}_{\phi }.\end{eqnarray} \tag{ 19 }$$

The most general form of the equations of motion for charged particles in curved spacetime is given by the DeWitt–Brehme equation (DeWitt & Brehme 1960; Hobbs 1968), which can be written in the form

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{{Du}}^{\mu }}{d\tau } & = & \displaystyle \frac{q}{m}{F}_{\,\,\nu }^{\mu }{u}^{\nu }+\displaystyle \frac{2{q}^{2}}{3m}\left(\displaystyle \frac{{D}^{2}{u}^{\mu }}{d\tau }+{u}^{\mu }{u}_{\nu }\displaystyle \frac{{D}^{2}{u}^{\nu }}{d\tau }\right)\\ & & +\displaystyle \frac{{q}^{2}}{3m}\left({R}_{\,\,\lambda }^{\mu }{u}^{\lambda }+{R}_{\,\,\lambda }^{\nu }{u}_{\nu }{u}^{\lambda }{u}^{\mu }\right)+\displaystyle \frac{{q}^{2}}{m}\,{f}_{\,\mathrm{tail}}^{\mu \nu }\,{u}_{\nu },\end{array}\end{eqnarray} \tag{ 20 }$$

where ${R}_{\nu }^{\mu }$ is the Ricci tensor, ${F}_{\mu \nu }={A}_{\nu ,\mu }-{A}_{\mu ,\nu }$ is the Faraday tensor, D denotes a covariant derivative, and four-velocity ${u}^{\mu }={{dx}}^{\mu }/d\tau$ satisfies the condition ${u}^{\mu }{u}_{\mu }=-1$. The last term in Equation (20), known as the tail integral, reads

$$\begin{eqnarray}&&{f}_{\,\mathrm{tail}}^{\mu \nu }={\int }_{-\infty }^{\tau }{D}^{[\mu }{G}_{+\lambda ^{\prime} }^{\nu ]}\left(\tau ,\tau ^{\prime} \right){u}^{\lambda ^{\prime} }(\tau ^{\prime} )\,d\tau ^{\prime} ,\end{eqnarray} \tag{ 21 }$$

where ${G}_{+\lambda }^{\mu }$ is a retarded Green's function. A detailed analysis of this equation, together with numerical integration into an astrophysically relevant situation, can be found in Tursunov et al. (2018a, 2018b). In the realistic conditions, the leading forces acting on a charged test particle are the Lorentz force and the radiation reaction force, given by the first and second terms on the right-hand side of Equation (20). The terms containing Ricci tensors are irrelevant if the Kerr spacetime metric is assumed, while the tail term is negligible in comparison to the rest of the terms. Using the approach by Landau & Lifshitz (1975), we get the covariant dynamical equations of the motion of charged test particles in curved spacetime in the presence of an electromagnetic field,

$$\begin{eqnarray}&&\displaystyle \frac{{{Du}}^{\alpha }}{d\tau }=\displaystyle \frac{q}{m}{F}_{\,\,\beta }^{\alpha }{u}^{\beta }+\displaystyle \frac{2{q}^{3}}{3{m}^{2}}{f}_{R}^{\alpha },\end{eqnarray} \tag{ 22 }$$

where

$$\begin{eqnarray}&&{f}_{R}^{\alpha }=\displaystyle \frac{{{DF}}_{\,\,\beta }^{\alpha }}{{{dx}}^{\mu }}{u}^{\beta }{u}^{\mu }+\displaystyle \frac{q}{m}\left({F}_{\,\,\beta }^{\alpha }{F}_{\,\,\mu }^{\beta }+{F}_{\mu \nu }{F}_{\,\,\sigma }^{\nu }{u}^{\sigma }{u}^{\alpha }\right){u}^{\mu }.\end{eqnarray} \tag{ 23 }$$

Equation (22) with Equation (23) is the covariant form of the Landau–Lifshitz equation describing the dynamics of radiating charged particles. We will use this equation for constraints on the motion of hot spots.

It is usually assumed that a plasma surrounding astrophysical black holes is electrically neutral due to neutralization of charged plasma on relatively short timescales. Any oscillation of the net charge density in a plasma is supposed to disappear very quickly due to induction of a large electric field caused by charge imbalance. However, in the presence of an external magnetic field and when the plasma is moving at relativistic speeds, one can observe the charge separation effect in a magnetized plasma and consequently measure the net charge density. Applied to rotating neutron stars with magnetic fields, this special relativistic effect of plasma charging is known as the GJ charge density (Goldreich & Julian 1969). In fact, the motion of a plasma induces an electric field that, in the comoving frame of a plasma, should be neutralized, which leads to the appearance of the net charge in the rest frame. The GJ charge density is usually referred to for pulsar magnetospheres, although it is applicable in more general cases as well, as we will show below. For a black hole magnetosphere, the charging of the plasma was first described by Ruffini & Wilson (1975), who showed that the twisting of magnetic field lines due to rotation of the black hole induces an electric charge in both the black hole and surrounding magnetosphere with equal and opposite signs of the charge value.

3.1.1. Special Relativistic Case

Neglecting for now the general relativistic effects, Maxwell's equations read

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\boldsymbol{E}}=4\pi \rho ,\quad {\rm{\nabla }}\cdot {\boldsymbol{B}}=0,\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&{\rm{\nabla }}\times {\boldsymbol{E}}=-\displaystyle \frac{\partial {\boldsymbol{B}}}{\partial t},\quad {\rm{\nabla }}\times {\boldsymbol{B}}=4\pi {\boldsymbol{j}}+\displaystyle \frac{\partial {\boldsymbol{E}}}{\partial t},\end{eqnarray} \tag{ 25 }$$

where ρ and ${\boldsymbol{j}}$ are the charge and current densities. For a frame moving with a system with velocity v with respect to the rest frame, the Lorentz transformations lead to

$$\begin{eqnarray}&&{\boldsymbol{E}}^{\prime} =\gamma ({\boldsymbol{E}}+{\boldsymbol{v}}\times {\boldsymbol{B}})-\displaystyle \frac{{\gamma }^{2}}{\gamma +1}{\boldsymbol{v}}({\boldsymbol{v}}\cdot {\boldsymbol{E}}),\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&{\boldsymbol{B}}^{\prime} =\gamma ({\boldsymbol{B}}+{\boldsymbol{v}}\times {\boldsymbol{E}})-\displaystyle \frac{{\gamma }^{2}}{\gamma +1}{\boldsymbol{v}}({\boldsymbol{v}}\cdot {\boldsymbol{B}}),\end{eqnarray} \tag{ 27 }$$

where "primes" denote the quantities measured with respect to the inertial frame moving at the given moment together with the system and $\gamma ={\left(1-{v}^{2}\right)}^{-1/2}$. In a comoving frame of the system, the current density is connected with the electric field by Ohm's law, ${\boldsymbol{j}}^{\prime} =\sigma {\boldsymbol{E}}^{\prime}$, where σ is the conductivity of the medium. For an observer at rest, one gets Ohm's law in the form

$$\begin{eqnarray}&&{\boldsymbol{j}}=\gamma \sigma \left({\boldsymbol{E}}+{\boldsymbol{v}}\times {\boldsymbol{B}}-{\boldsymbol{v}}\cdot ({\boldsymbol{v}}\,{\boldsymbol{E}}\right)+\rho {\boldsymbol{v}}.\end{eqnarray} \tag{ 28 }$$

Let us now assume that the matter containing plasma is a perfect electrical conductor. This implies that the following relation holds:

$$\begin{eqnarray}&&{\boldsymbol{E}}=-{\boldsymbol{v}}\times {\boldsymbol{B}}.\end{eqnarray} \tag{ 29 }$$

From this, it follows that an external observer measures the induced electric field that arises in order to compensate for the electric field in the comoving frame of the system.

One can express the velocity in terms of an orbital angular velocity of the hot spot moving around the black hole in the equatorial plane, ${\boldsymbol{v}}={\boldsymbol{\Omega }}\times {\boldsymbol{R}}$. Substituting Equation (29) into the first equation of Equation (24) in terms of angular velocity and dividing to elementary charge e, we get the net charge number density in a plasma (number density of extra electrons or protons) in the form

$$\begin{eqnarray}&&{\rho }_{q}=\displaystyle \frac{1}{2\pi c}\displaystyle \frac{{\rm{\Omega }}\,{B}_{\perp }}{| e| },\end{eqnarray} \tag{ 30 }$$

where Ω is the orbital angular velocity of the hot spot and B_⊥ is the strength of the magnetic field orthogonal to the orbital plane. The orbital period of the hot spots at the distance of the ISCO (ISCO $\sim \,6{GM}/{c}^{2}$) of Sgr A* is $T\sim 45$ minutes, which corresponds to the angular velocity ${\rm{\Omega }}=2\pi /T\sim 2.33\times {10}^{-3}{{\rm{s}}}^{-1}$. The equipartition strength of the magnetic field at the ISCO scale can be assumed to be of the order of 10 G (Eckart et al. 2017). Thus, the number density of extra charged particles is

$$\begin{eqnarray}&&{\rho }_{q}\approx 2.57\times {10}^{-4}\left(\displaystyle \frac{B}{10{\rm{G}}}\right){\left(\displaystyle \frac{T}{45\mathrm{minutes}}\right)}^{-1}{\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 31 }$$

which is at least 10¹⁰ times less than the total number density in a plasma (Equation (12)). Assuming a spherical volume of radius $R\sim {R}_{{\rm{g}}}$, corresponding to the size estimate (Equation(11)), we get a net charge excess of the order of

$$\begin{eqnarray}&&| q| \approx 4\times {10}^{13}\,\left(\displaystyle \frac{B}{10{\rm{G}}}\right){\left(\displaystyle \frac{R}{{R}_{s}}\right)}^{3}\,{\rm{C}}.\end{eqnarray} \tag{ 32 }$$

One can see that the charge separation in a relativistic magnetized plasma can lead to the presence of a sufficient net charge in the hot spots that can considerably affect their motion. The dynamics of a charged hot spot is discussed in Section 3.2.

Let us now find the limits to the ratio of the Lorentz and gravitational forces acting on the hot spot at the ISCO scales, assuming that the hot spot has a charge given by Equation (32) and the magnetic field is orthogonal to the orbital plane with a strength of 10 G. For a hot spot with a mass in the range given by Equation (13), moving with a velocity $v\sim 0.3\,c$ around Sgr A* (as observed in recent GRAVITY flares) at a distance of $\sim 6{R}_{{\rm{g}}}$, we get the following limits:

$$\begin{eqnarray}&&{10}^{-5}\lt \displaystyle \frac{{F}_{\mathrm{Lor}.}}{{F}_{\mathrm{grav}.}}\lt 10.\end{eqnarray} \tag{ 33 }$$

We will give tighter constraints on this ratio in Section 3.2 by analyzing the period–radius relations of the hot-spot orbits (parameterizing the above ratio by the dimensionless parameter ${ \mathcal B }$) and comparing them with those of the three most recent flares.

3.1.2. General Relativistic Case

It is important to note that the general relativistic version of Equation (30) leads to a similar order estimate as Equations (31) and (32) unless a hot spot is moving in the very close vicinity of the event horizon. However, for completeness, we derive the net charge density of the flare component due to charge separation in a magnetized plasma in curved spacetime following the work of Bardeen et al. (1972), Ruffini & Wilson (1975), Thorne & MacDonald (1982), Muslimov & Tsygan (1992), and Izzard et al. (2004). For our purposes, the most convenient way to describe the electrodynamics of relativistic plasma around a black hole is to use the approach of 3 + 1 splitting of spacetime introduced by Thorne & MacDonald (1982) and further developed in Izzard et al. (2004). In curved spacetime, the Maxwell equations read in a similar way as in flat-space cases (Equations (24) and (25)), except that the operator ∇ is taken in 3D curved coordinates, implying a covariant derivative of absolute space. In tensor form, the covariant Maxwell equations read

$$\begin{eqnarray}&&{{\rm{\nabla }}}_{\nu }\,* {F}^{\mu \nu }=0,\quad {{\rm{\nabla }}}_{\nu }{F}^{\mu \nu }={J}^{\mu },\end{eqnarray} \tag{ 34 }$$

where ${F}^{\alpha \beta }$ and $* {F}^{\alpha \beta }$ are the Maxwell and Faraday tensors, respectively, and ${J}^{\mu }$ is the four-current. Splitting these equations into time and space components, we get

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\boldsymbol{B}}=0,\quad {\rm{\nabla }}\times {\boldsymbol{E}}=-\displaystyle \frac{\partial {\boldsymbol{B}}}{\partial t},\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\boldsymbol{D}}=4\pi \rho ,\quad {\rm{\nabla }}\times {\boldsymbol{H}}=4\pi {\boldsymbol{j}}+\displaystyle \frac{\partial {\boldsymbol{D}}}{\partial t}.\end{eqnarray} \tag{ 36 }$$

It should be noted that ${\boldsymbol{D}}$ and ${\boldsymbol{H}}$ coincide with ${\boldsymbol{E}}$ and ${\boldsymbol{B}}$ measured by a zero angular momentum observer (ZAMO), whose four-velocity in axially symmetric spacetime is defined by

$$\begin{eqnarray}&&{n}^{\mu }=({n}^{t},0,0,{n}^{\phi }),\end{eqnarray} \tag{ 37 }$$

where

$$\begin{eqnarray}&&{\left({n}^{t}\right)}^{2}=\displaystyle \frac{{g}_{\phi \phi }}{{g}_{t\phi }^{2}-{g}_{{tt}}{g}_{\phi \phi }},\quad {n}^{\phi }=-\displaystyle \frac{{g}_{t\phi }}{{g}_{\phi \phi }}\,{n}^{t}.\end{eqnarray} \tag{ 38 }$$

Applying the covariant derivative ∇ to Equation (36), we get the charge conservation law

$$\begin{eqnarray}&&{\partial }_{t}\rho +{\rm{\nabla }}\cdot {\boldsymbol{J}}=0.\end{eqnarray} \tag{ 39 }$$

Assuming that the magnetosphere of a black hole shares the background symmetry of the black hole, i.e., applying stationarity and axial symmetry, we get the effective charge density in the form

$$\begin{eqnarray}&&\rho =-\displaystyle \frac{1}{4\pi }{\rm{\nabla }}\cdot \left[\displaystyle \frac{1}{\alpha }\left(1-\displaystyle \frac{k}{{\eta }_{r}^{3}}\right){\boldsymbol{v}}\times {\boldsymbol{B}}\right],\end{eqnarray} \tag{ 40 }$$

$$\begin{eqnarray}&&{\boldsymbol{v}}={\boldsymbol{\Omega }}\times {\boldsymbol{r}},\quad k=\displaystyle \frac{{R}_{{\rm{g}}}\beta }{a},\quad {\eta }_{r}=\displaystyle \frac{a}{r},\quad \alpha =\sqrt{{g}_{{tt}}},\end{eqnarray} \tag{ 41 }$$

where a is the black hole's spin parameter, β is the moment of inertia of the plasma rotating around the black hole, and α is the lapse function.

It was argued by Komissarov (2004) that an electric field measured by ZAMO drives the electric current along the magnetic field lines, resulting in the separation of charges and the drop of the electrostatic potential, at least within the ergosphere. In this scenario, the black hole can act as the unipolar generator (Blandford & Znajek 1977), similar to the classical Faraday disk, which is based on the use of electromotive force $q{\boldsymbol{v}}\times {\boldsymbol{B}}$, resulting in the charge separation due to the voltage drop between the edge of the disk and its center. Further analysis led to the conclusion that any rotating compact object, like neutron stars or black holes, immersed into an external magnetic field and surrounded by plasma or an accretion disk generates a rotationally induced electric field at the object, as well as in the surrounding magnetosphere.

Let us start with a note on terminology. There have been a number of papers over the last three decades developing a phenomenological description of an electromagnetic signal from a hot spot orbiting near a black hole (see, e.g., Schnittman & Bertschinger 2004, Karas 2006, and references cited therein). This notion has been rather successful in predicting basic features that appear due to general relativity in light curves, spectra, and polarimetrical signal that are expected from a spatially localized source on a circular orbit or a plunging trajectory near the event horizon. Various physical representations have been invoked in order to understand the extended life span of the spot in the presence of strong tidal forces, where a plain blob of gas would disintegrate on a timescale shorter than the orbital period (which would prevent clear signatures from showing up and set the constraints of the black hole parameters); the term "spot" can thus represent a stable vortex (Abramowicz et al. 1992) or a wave pattern (Karas et al. 2001) within the accretion medium, it can be stabilized by the presence of a stellar object in the core (Cunningham & Bardeen 1973; Bao et al. 1994; Zajaček et al. 2014; Valencia-S. et al. 2015), or it can be confined by ambient magnetic pressure.

Although black holes do not support their own magnetic fields, large-scale organized fields are possible and even likely to occur due to external currents flowing in the accretion medium. Magnetic effects can visibly the influence the motion and radiation of an electrically charged spot. Even if a plasma blob is electrically neutral globally, the mechanism of charge separation produces an excess of charge that can prevail in certain regions.

The motion of a charged hot spot is described by Equation (22), which also includes, in addition to the geodesic term, the Lorentz and radiation reaction forces. In the case of a locally uniform magnetic field that satisfies the solution of Equation (17), the relative influence of magnetic and gravitational fields on the motion of the hot spot can be parameterized by the dimensionless parameter (Frolov & Shoom 2010; Tursunov et al. 2018a, 2016)

$$\begin{eqnarray}&&{ \mathcal B }=\displaystyle \frac{{q}_{\mathrm{hs}}\,G\,B\,M}{2\,{m}_{\mathrm{hs}}\,{c}^{4}},\end{eqnarray} \tag{ 42 }$$

where B is the strength of the magnetic field, M is the black hole mass, ${q}_{\mathrm{hs}}$ and ${m}_{\mathrm{hs}}$ are the charge and mass of the hot spot, and G and c are constants. The factor 1/2 is given for historical reasons. Hereafter, we call ${ \mathcal B }$ the magnetic parameter. The parameter ${ \mathcal B }$ is chosen in such a way that $2{ \mathcal B }v/c$ is the ratio of the Lorentz force to the gravitational force acting on the hot spot and constrained for a plasma surrounding Sgr A* in Equation (33).

Following Gravity Collaboration et al. (2018b), we assume that the magnetic field is orthogonal to the orbital plane of the hot spots that is placed at the equatorial plane of spinning Sgr A*. Depending on the orientation of the Lorentz force, the shift of the orbital frequency can occur in both directions for a given value of the orbital radii. Given the period–radius relations of components of three flares observed in 2018 July 22, May 27, and July 28, we solve the equations of motion for the charged hot spot numerically and put constraints on the magnetic parameter ${ \mathcal B }$ in Figure 2 as $-0.015\lt { \mathcal B }\lt 0.01$. One can also see in Figure 2 (right) that the positions of the centers of the observed flares on the period–radius plot are slightly lower than the theoretically predicted periods of neutral hot spots (red curve). The value of the magnetic parameter ${ \mathcal B }$ fitting the mean values (centers) of observed periods and radii of three flares is ${ \mathcal B }\sim -3\times {10}^{-3}$ (dashed middle curve). As discussed in Section 2.2, various measurements and estimates of magnetic field strength in the vicinity of Sgr A* suggest an equipartition magnetic field with a strength of $B\sim 10$ G (Eckart et al. 2017). This gives limits on the specific charge (charge-to-mass ratio) following Figure 2 in the range

$$\begin{eqnarray}&&\displaystyle \frac{| {q}_{\mathrm{hs}}| }{{m}_{\mathrm{hs}}}\lt {10}^{-3}\,{\left(\displaystyle \frac{B}{10{\rm{G}}}\right)}^{-1}\,{\left(\displaystyle \frac{{M}_{\mathrm{SgrA}* }}{4\times {10}^{6}{M}_{\odot }}\right)}^{-1}{\rm{C}}/{\rm{g}},\end{eqnarray} \tag{ 43 }$$

while the mean value of the specific charge fitting the observed mean periods and radii corresponds to

$$\begin{eqnarray}&&{\left.\displaystyle \frac{{q}_{\mathrm{hs}}}{{m}_{\mathrm{hs}}}\right|}_{\mathrm{mean}}\approx -3\times {10}^{-4}\,{\left(\displaystyle \frac{B}{10{\rm{G}}}\right)}^{-1}\,{\left(\displaystyle \frac{{M}_{\mathrm{SgrA}* }}{4\times {10}^{6}{M}_{\odot }}\right)}^{-1}{\rm{C}}/{\rm{g}}.\end{eqnarray} \tag{ 44 }$$

The same ratio for electrons is of order $e/{m}_{e}\sim -{10}^{8}$ C g^–1 and for protons $e/{m}_{p}\sim {10}^{5}$ C g^–1. Assuming that the constituents of the flare components are mainly protons and electrons, one can easily calculate the limiting ratio of the number of extra net charged particles to the number of neutral particles (proton–electron pairs) in the hot spot as

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{\mathrm{charged}}}{{N}_{\mathrm{neutral}}}=\displaystyle \frac{{q}_{\mathrm{hs}}}{e}\,\displaystyle \frac{{m}_{{\rm{p}}}+{m}_{{\rm{e}}}}{{m}_{\mathrm{hs}}}\lt {10}^{-8},\end{eqnarray} \tag{ 45 }$$

$$\begin{eqnarray}&&{\left.\displaystyle \frac{{N}_{\mathrm{charged}}}{{N}_{\mathrm{neutral}}}\right|}_{\mathrm{mean}}\approx 3\times {10}^{-9};\end{eqnarray} \tag{ 46 }$$

i.e., observed flare components have a net charge concentration corresponding to one extra charged particle to at least 10⁸ neutral pairs of protons and electrons. Since the mean value of the specific charge (Equation (44)) is negative, this corresponds to an extra electron in each $3\times {10}^{8}$ neutral pair. Based on the total number density obtained in Equations (12) and (15), one can estimate the number density of extra charged particles (${\rho }_{q}={\rho }_{N}\,{N}_{\mathrm{charged}}/{N}_{\mathrm{neutral}}$) as

$$\begin{eqnarray}&&{\rho }_{{\rm{q}}}\lt {10}^{-2}\,{\left(\displaystyle \frac{B}{10{\rm{G}}}\right)}^{-1}\,\left(\displaystyle \frac{{\rho }_{{\rm{N}}}}{{10}^{6}\,{\mathrm{cm}}^{-3}}\right)\,{\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 47 }$$

$$\begin{eqnarray}&&{\rho }_{{\rm{q}}}^{\mathrm{mean}}\approx 3\times {10}^{-3}\,{\left(\displaystyle \frac{B}{10{\rm{G}}}\right)}^{-1}\,\left(\displaystyle \frac{{\rho }_{{\rm{N}}}}{{10}^{6}\,{\mathrm{cm}}^{-3}}\right)\,{\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 48 }$$

that is, an order of magnitude larger than our earlier estimate (Equation (31)), which is based on the charge separation in a relativistic magnetized plasma. However, the discrepancy can be easily omitted if one assumes a slightly stronger magnetic field at the orbital location of the hot spots of the order of less than $\lesssim 100$ G. In that case, the two approaches will perfectly match.

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For the masses of the hot spots estimated in Equation (13), the limiting values for the charges of the hot spots are

$$\begin{eqnarray}&&| {q}_{\mathrm{hs}}| \lt {10}^{14.5\pm 2.5}\,{\left(\displaystyle \frac{B}{10{\rm{G}}}\right)}^{-1}\,\left(\displaystyle \frac{{m}_{\mathrm{hs}}}{{10}^{17.5\pm 2.5}{\rm{g}}}\right)\,{\rm{C}},\end{eqnarray} \tag{ 49 }$$

$$\begin{eqnarray}&&{q}_{\mathrm{hs}}^{\mathrm{mean}}\approx -{10}^{13.5\pm 2.5}\,{\left(\displaystyle \frac{B}{10{\rm{G}}}\right)}^{-1}\,\left(\displaystyle \frac{{m}_{\mathrm{hs}}}{{10}^{17.5\pm 2.5}{\rm{g}}}\right)\,{\rm{C}},\end{eqnarray} \tag{ 50 }$$

which does not contradict the value of Equation (32) estimated above. It is important to note that according to Figure 2, the spin parameter of the black hole does not play a crucial role in the fitting of the period–radius relation of circular orbits, while the magnetic parameter shifts the orbits and periods significantly, although one can conclude that the plasma containing the hot spot has a nonnegligible excess of net charge, which is shown by the above estimates.

Emission of flares observed mainly at X-ray wavelengths roughly appears every day, increasing the luminosity of Sgr A* up to 2 orders of magnitude (Yuan et al. 2003). On average, the flare state of Sgr A* corresponds to a luminosity of the order of 10³³ erg s⁻¹, although the brightest flares may reach a luminosity of ${10}^{35}\mbox{--}{10}^{36}$ erg s⁻¹ (Nowak et al. 2012). The flare activity states may last from a few minutes to hours. Usually, the shortest timescales correspond to the flare components at the closest distances from Sgr A*. Associating the nonthermal flare emission with the synchrotron radiation of a charged hot spot in a magnetic field surrounding Sgr A*, one can find other limits on the charge of the flare components and their emission timescales. For a magnetic field orthogonal to the orbital plane, the intensity of radiation in all directions of the hot spot orbiting the black hole in a fully relativistic approach is given by (Sokolov et al. 1978; Shoom 2015; Tursunov et al. 2018a)

$$\begin{eqnarray}&&L=\displaystyle \frac{2}{3}\displaystyle \frac{{q}_{\mathrm{hs}}^{4}{B}^{2}{v}^{2}{\gamma }^{2}}{{m}_{\mathrm{hs}}^{2}{c}^{3}}{\left(1-\displaystyle \frac{2{R}_{{\rm{g}}}}{{R}_{0}}\right)}^{3}\mathrm{erg}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 51 }$$

where v is the velocity of the hot spot in units of the speed of light, and R₀ is the orbital radius. Equalizing Equation (51) to 10³³ erg s⁻¹ for the orbit at the radius $R=6{R}_{{\rm{g}}}$ from Sgr A*, we get another limit for the charge of the hot spot as ${q}_{\mathrm{hs}}^{\min }\lt q\lt {q}_{\mathrm{hs}}^{\max }$, where

$$\begin{eqnarray}&&{q}_{\mathrm{hs}}^{\min }\approx -{10}^{13}\,{\left(\displaystyle \frac{B}{10{\rm{G}}}\right)}^{-\tfrac{1}{2}}\,{\left(\displaystyle \frac{{m}_{\mathrm{hs}}}{{10}^{14}{\rm{g}}}\right)}^{\tfrac{1}{2}}{\left(\displaystyle \frac{L}{{10}^{33}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{\tfrac{1}{4}}\,{\rm{C}},\end{eqnarray} \tag{ 52 }$$

$$\begin{eqnarray}&&{q}_{\mathrm{hs}}^{\max }\approx {10}^{16}\,{\left(\displaystyle \frac{B}{10{\rm{G}}}\right)}^{-\tfrac{1}{2}}\,{\left(\displaystyle \frac{{m}_{\mathrm{hs}}}{{10}^{20}{\rm{g}}}\right)}^{\tfrac{1}{2}}\,{\left(\displaystyle \frac{L}{{10}^{33}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{\tfrac{1}{4}}\,{\rm{C}},\end{eqnarray} \tag{ 53 }$$

whose orders of magnitude are very close to the limits (Equation (49)) given by the fitting of the period–radius data.

Given that the hot spot carries a small electric charge, one can also consider the possible influence on its motion of the unscreened charge of the black hole, which is discussed in Section 2.3. The realistic upper limit to the black hole charge is of the order of $\sim {10}^{15}{\rm{C}}$ (see Zajaček et al. 2018), which can arise due to magnetic field twist. In the case of a uniform magnetic field, the black hole possesses a Wald charge ${Q}_{{\rm{W}}}=2{G}^{2}{aMB}/{c}^{4}\approx {G}^{2}{M}^{2}B/(2{c}^{4})$. Using an approach similar to the magnetic field case (see Section 3.2 for details), we summarize the results of the fitting of the period–radius plots in Figure 3. For the sake of simplicity and in order to identify the pure contribution to the hot-spot dynamics due to the black hole's charge, we neglect the effects of the spin and magnetic field. Similar to the magnetic parameter (Equation (42)), we introduce the dimensionless parameter ${ \mathcal Q }$, reflecting the coulombic interaction between the hot spot and the black hole,

$$\begin{eqnarray}&&{ \mathcal Q }=\displaystyle \frac{{q}_{\mathrm{hs}}\,{Q}_{\mathrm{BH}}}{G\,{m}_{\mathrm{hs}}\,{M}_{\mathrm{SgrA}* }}.\end{eqnarray} \tag{ 54 }$$

Taking the charge of the black hole ${Q}_{\mathrm{BH}}\sim {10}^{15}$ C and constraining the charge parameter to $-0.5\lt { \mathcal Q }\lt 1$, with a mean value ${ \mathcal Q }=0.15$, we get the following constraints on the specific charge (charge-to-mass ratio) of the hot spot:

$$\begin{eqnarray}&&\displaystyle \frac{| {q}_{\mathrm{hs}}| }{{m}_{\mathrm{hs}}}\lt 2\times {10}^{-2}{\left(\displaystyle \frac{Q}{{10}^{15}{\rm{C}}}\right)}^{-1}{\left(\displaystyle \frac{{M}_{\mathrm{SgrA}* }}{4\times {10}^{6}{M}_{\odot }}\right)}^{-1}{\rm{C}}/{\rm{g}},\end{eqnarray} \tag{ 55 }$$

while the mean value of the specific charge fitting the observed mean periods and radii corresponds to

$$\begin{eqnarray}&&{\left.\displaystyle \frac{{q}_{\mathrm{hs}}}{{m}_{\mathrm{hs}}}\right|}_{\mathrm{mean}}\approx 7\times {10}^{-3}{\left(\displaystyle \frac{B}{10{\rm{G}}}\right)}^{-1}{\left(\displaystyle \frac{{M}_{\mathrm{SgrA}* }}{4\times {10}^{6}{M}_{\odot }}\right)}^{-1}{\rm{C}}/{\rm{g}}.\end{eqnarray} \tag{ 56 }$$

One can notice that the estimates in Equations (55) and (56) are only 1 order of magnitude larger than in the magnetic case (Equations (43) and (44)). Further analysis leads to the following constraints to the charge:

$$\begin{eqnarray}&&{q}_{\mathrm{hs}}^{\min }\approx -{10}^{12}\,\left(\displaystyle \frac{Q}{{Q}_{{\rm{W}}}(\sim {10}^{18}C)}\right)\left(\displaystyle \frac{{m}_{\mathrm{hs}}}{{10}^{17}{\rm{g}}}\right){\rm{C}},\end{eqnarray} \tag{ 57 }$$

$$\begin{eqnarray}&&{q}_{\mathrm{hs}}^{\max }\approx {10}^{18}\,\left(\displaystyle \frac{Q}{{Q}_{{\rm{W}}}(\sim {10}^{15}{\rm{C}})}\right)\left(\displaystyle \frac{{m}_{\mathrm{hs}}}{{10}^{20}{\rm{g}}}\right){\rm{C}},\end{eqnarray} \tag{ 58 }$$

which are close to the previous estimates by an order of magnitude.

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The location of the ISCO is among the few parameters that are strongly sensitive to the value of the black hole spin. The ISCO of a nonrotating black hole (a = 0) is located at a distance of $6{R}_{{\rm{g}}}$ from singularity. For rotating black holes, the ISCO of corotating matter shifts toward the black hole, coinciding with the event horizon at the extremal case (a = 1). For counterrotating matter, the ISCO shifts outward from the black hole, reaching up to $9{R}_{{\rm{g}}}$ in the extremal case. However, the inclusion of the interaction of the magnetic field with the charge of the accretion flow can shift the ISCO dramatically. The motion of relativistic plasma around a magnetized black hole puts limits on the ratio of the Lorentz force to the gravitation force acting at the ISCO scales, given by Equation (33). This implies that in general, for a plasma, the upper limit for the magnetic parameter ${ \mathcal B }$ defined in Equation (42) is $| { \mathcal B }| \lt 10$.

In Figure 4, we demonstrate the shift of the ISCO location toward the black hole by increasing the magnetic parameter ${ \mathcal B }$ in the case of uniform magnetic field configuration. The ISCO in the presence of a magnetic field has four branches, two for each of the corotating and counterrotating cases corresponding to the Larmor and anti-Larmor types of motion (Aliev & Özdemir 2002; Frolov & Shoom 2010; Tursunov et al. 2016). For the upper limit of the magnetic parameter $| { \mathcal B }| \approx 10$, the ISCO in the nonrotating black hole case shifts to the values ${r}_{\mathrm{ISCO}}\approx 2.1{R}_{{\rm{g}}}$ for ${ \mathcal B }\gt 0$ and ${r}_{\mathrm{ISCO}}\approx 4.3{R}_{{\rm{g}}}$ for ${ \mathcal B }\lt 0$. These values correspond to the ISCO of neutral matter moving around the rotating black hole with spin a = 0.93 and 0.48, respectively.

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A simple example demonstrated in Figure 4 shows that the effect of electromagnetic interaction on the matter surrounding the black hole can be of crucial importance, as it may lead to the discrepancy in the measurements of the spin of the SMBH. Since the magnetic field configuration in the Sgr A* environment can be more sophisticated, the problem requires further study.

Similar analyses of the location of the ISCO in the case of the Schwarzschild black hole with a small electric charge immersed in the external magnetic field were studied in Hackstein & Hackmann (2020).

We focused on the observed motion of three hot spots observed by GRAVITY (Gravity Collaboration et al. 2018b) that exhibit clockwise motion with inclinations $i=160^\circ \,\pm 10^\circ$. The polarization rotation implies a magnetic field oriented approximately parallel to the vector of the orbital plane, i.e., a poloidal configuration. On the other hand, the bright X-ray flares often exhibit a double-peak structure (Karssen et al. 2017; Ponti et al. 2017), which is fitted well by the combination of lensing and Doppler boosting of hot-spot emission due to its motion along the trajectory that is close to edge-on with respect to the observer (Eckart et al. 2018). Already, the first hot-spot models of Sgr A* flaring activity implied a departure from face-on orientation, with an inclination of i ≤ 145° (Meyer et al. 2006). Hence, the total range of inclinations is expected to be large, i = 90°–160°, i.e., from nearly edge-on to nearly face-on orbits. Future GRAVITY observations will likely provide better statistics to assess the inclination distribution.

A large range of hot-spot inclinations is generally expected in hot accretion flows (Yuan & Narayan 2014), whose half-thickness scales approximately as h ∼ r; i.e., the half-opening angle of the hot flow is close to 45°. The midplane of the flow is currently uncertain. Taking into account the inflow–outflow accretion flow model, in which most of the material is provided by the stellar winds of about 200 OB Wolf–Rayet stars in the central parsec with mass-loss rates of ${\dot{M}}_{{\rm{w}}}\approx {10}^{-5}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ and wind velocities of $\sim 1000\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (Cuadra et al. 2008; Shcherbakov & Baganoff 2010; Ressler et al. 2018; Calderón et al. 2020), the accretion flow midplane could be associated with the clockwise disk of massive OB stars (Levin & Beloborodov 2003), whose inclination is $i=122^\circ \pm 7^\circ$ (Genzel et al. 2010). Magnetohydrodynamic simulations of the inner accretion flow of Sgr  A* that is supplied by stellar winds confirm a broad range of angular momenta and a large thickness of the hot flow (Ressler et al. 2018). Therefore, the frequent occurrence of hot spots on clockwise orbits with various inclinations ranging from edge-on (≳90°) to nearly face-on (≲180°) could also be linked to the hot, thick flow that is supplied by the stellar winds of young stars concentrated at inclinations close to 120°. Recently, Murchikova et al. (2019) found observational evidence for the presence of a disklike structure with a radius of ∼0.004 pc based on the detection of a broad, double-peak 1.3 mm H30α line with ALMA. This colder disk has a temperature of ∼10⁴ K and a number density of 10⁵–${10}^{6}\,{\mathrm{cm}}^{-3}$, depending on its exact filling factor. Such a disklike structure embedded within the hot diluted plasma could be a result of the "disk" phase of the evolution of a system of hot Wolf–Rayet stars after ≳3000 yr (Calderón et al. 2020). Cuadra et al. (2008) calculated the mean value of the circularization radius ${R}_{\mathrm{circ}}\sim 0\buildrel{\prime\prime}\over{.} 05=0.002\,\mathrm{pc}$ of the gas that originates in the stellar winds, which is comparable to the radius of the colder disk of Murchikova et al. (2019). In addition, the statistics of NIR polarization data implies a rather stable geometrical orientation of the system. Shahzamanian et al. (2015) found typical polarization degrees of the order of 20% ± 10% and a preferred polarization angle of 13° ± 15°. This orientation is to first order consistent with that of the the primary He star clockwise disk (Levin & Beloborodov 2003).

Our analysis is generally applicable to the accretion flows where the magnetic field is dynamically significant, which seems to be the case for the Galactic center, according to our estimate of the magnetization parameter; see Section 2.2. This is especially the case for less dense hot flows, where the frozen-in magnetic field is dragged inward by the accreting gas and accumulates at the very center (Bisnovatyi-Kogan & Ruzmaikin 1974, 1976) with the dominant poloidal component, which at a certain point is strong enough to disrupt the axisymmetric flow at the magnetospheric radius. In Section 1.2, we estimated the magnetospheric radius R_m for the spherical flow; see Equation (3). For the axisymmetric flow, inside R_m, the flow is supported against gravity by the magnetic pressure, ${GM}{\rm{\Sigma }}/R\approx 2{B}_{{\rm{R}}}{B}_{{\rm{z}}}/4\pi$, where the surface density of the flow depends on the accretion rate $\dot{M}$ and the radial velocity, which is a factor of $\epsilon \approx 0.01-0.001$ smaller than the freefall velocity v_ff (Narayan et al. 2003), as ${\rm{\Sigma }}=\dot{M}/(2\pi R\epsilon {v}_{\mathrm{ff}})$. From this, assuming B_R ∼ B_z = B_pol, the relation for R_m is the following:

$$\begin{eqnarray}&&{R}_{{\rm{m}}}\approx {\left(\displaystyle \frac{\dot{M}\sqrt{{GM}}}{\epsilon {B}_{\mathrm{pol}}^{2}}\right)}^{2/5}.\end{eqnarray} \tag{ 65 }$$

Inside R_m, the accretion flow is broken up and continues to move inward at velocities much slower than the freefall velocity, v_R = v_ff. The MAD consists of magnetic islands and magnetic reconnection events that are expected to be frequent enough to give rise to the adiabatically expanding plasmoids or hot spots (Yuan et al. 2009; Eckart et al. 2012; Li et al. 2015), which can explain the observed flares and their multiwavelength properties, mainly the simultaneous NIR/X-ray flares and time-delayed submillimeter/millimeter/radio flares. In fact, the overall picture is consistent for the Galactic center environment, including the stellar and gaseous components. If we consider the disklike component of the accretion flow, which can be refueled by stellar winds (Murchikova et al. 2019; Calderón et al. 2020), with an accretion rate in the inner 100 R_g of $\dot{M}\approx {10}^{-8}\,{{\rm{M}}}_{\odot }\,{\mathrm{yr}}^{-1}$ (Marrone et al. 2007) and a poloidal magnetic field of ${B}_{\mathrm{pol}}\approx 10\,{\rm{G}}$, the estimate of R_m follows from Equation (65),

$$\begin{eqnarray}\begin{array}{rcl}{R}_{{\rm{m}}} & \sim & 78.5{\left(\displaystyle \frac{\dot{M}}{{10}^{-8}{{\rm{M}}}_{\odot }{\mathrm{yr}}^{-1}}\right)}^{2/5}\,{\left(\displaystyle \frac{M}{4\times {10}^{6}{M}_{\odot }}\right)}^{1/5}\\ & & \times {\left(\displaystyle \frac{\epsilon }{0.01}\right)}^{-2/5}{\left(\displaystyle \frac{{B}_{\mathrm{pol}}}{10{\rm{G}}}\right)}^{-4/5}{R}_{{\rm{g}}},\end{array}\end{eqnarray} \tag{ 66 }$$

which is of a comparable order of magnitude to the value derived for the spherical accretion; see Equation (3). So far, the observed flares or hot spots have been located within this radius (Karssen et al. 2017; Gravity Collaboration et al. 2018b). The disk plane could also influence the predominant orientation of the flare components as discussed earlier, albeit with a large scatter due to the thickness of the flow. The overall setup of the hot, thick flow with the inner MAD part is illustrated in Figure 1. However, the detailed numerical modeling of stellar wind feeding including the magnetic field is needed to test the self-consistency of this model, which is beyond the scope of this paper.

The analyses given in this paper could, in general, be applied to the case of the M87 black hole, for which the first black hole image has been obtained by the Event Horizon Telescope (EHT). The EHT image exhibits a bright, short crescent in the SSE–SWW position angle and a relatively compact hot spot at the SEE sector (Event Horizon Telescope Collaboration et al. 2019). Fitting the image and the orientation of the jet emission simultaneously with the general relativistic magnetohydrodynamic simulations makes it difficult to explain the emission in the SEE sector (Nalewajko et al. 2020), at least within strictly stationary and axially symmetric models. The presence of the charge in the accretion flow taking into account the electromagnetic interaction of the accretion disk of M87 could potentially help to resolve the problem, which requires further study. We note that the orbital timescale of the flare components in M87 is significantly longer. While for Sgr A*, we have a basic timescale at the ISCO of ${P}_{\mathrm{SgrA}* }\,=12\sqrt{6}\pi {GM}/{c}^{3}\sim 31.3\,\min$, for M87, we have an orbital timescale of ${P}_{{\rm{M}}87}=({M}_{{\rm{M}}87}/{M}_{\mathrm{SgrA}* }){P}_{\mathrm{SgrA}* }\approx 1570\times 31.3$ minutes ∼ 34 days. Therefore, if the flare component has a shorter timescale than the orbital timescale, it may be complicated to trace the flare components of M87 along their orbits, as was done for Sgr A*.

In our analysis, we considered the orbiting hot spot to arise in the accretion flow. Hence, the orbit is expected to be bounded or inspiralling with respect to the Sgr A* black hole. In case the hot spot had a large bulk motion, it could become a part of an outflow or the sheath of a helical jet (Ripperda et al. 2020). Several observations of faint jetlike structures in the vicinity of Sgr A* have been reported in the X-ray, radio, and IR domains, in general at different position angles (Eckart et al. 2006; Yusef-Zadeh et al. 2012; Li et al. 2013; Shahzamanian et al. 2015). Although the ADAF-jet model can explain the broadband characteristics of Sgr A* (Yuan et al. 2002), there is no clear observational evidence for hot spots to be jet plasmoids on outflowing trajectories. In fact, a significant linear component of the velocity is expected, which for large inclinations of i ≈ 160° should lead to much stronger Doppler boosting than observed. This can be shown from the general expectation that the jet velocity should be of the order of the local orbital velocity of the accretion material (de Gouveia dal Pino 2005), ${v}_{{\rm{j}}}\sim {v}_{\mathrm{orb}}(\mathrm{ISCO})\sim 0.5{\rm{c}}$. For a small angle to the line of sight, this would lead to a Doppler-boosting factor of $\delta \approx {[\gamma (1-\beta )]}^{-1}\approx \sqrt{3}$ and the overall Doppler term

$$\begin{eqnarray}&&\displaystyle \frac{{S}_{\mathrm{obs}}}{{S}_{\mathrm{HS}}}=D\equiv {\delta }^{3+\alpha }\approx 7.22\mbox{--}11.84,\end{eqnarray} \tag{ 67 }$$

where S_obs and S_HS are the flux densities measured in the observer's frame and the frame of the hot spot, respectively. The spectral index α, ${S}_{\nu }\,\propto \,{\nu }^{-\alpha }$, is 0.6 and 1.5 for the very bright and average bright flares, respectively (Witzel et al. 2018). This is clearly in contradiction to the observational constraints on the Doppler term of D < 2.5. In case the jet components had a significant ϕ-component of their velocity, the Doppler term would limit the forward speed to 0.20–0.24 c, which is comparable to the projection of a ϕ-component to the line of sight, ${v}_{\phi }\sin i=0.5\times \sin 20^\circ \approx 0.17$. Hence, the hot spots would be nearly stationary components of the jet. Although in some cases nearly stationary jet components are detected, e.g., in blazar jets (see, e.g., Britzen et al. 2018), their long-term kinematics evolves on a timescale of years, in comparison with the short-term hot-spot clockwise motion detected by the GRAVITY instrument (Gravity Collaboration et al. 2018b). Therefore, the assumption that hot spots arise in the accretion inflow rather than outflow is better justified, at least for a current set of orbiting hot spots.

Gravity Collaboration et al. (2020b) analyzed the astrometric positions of the three hot spots detected in 2018 with a general relativistic ray-tracing code. They included the effects of out-of-plane orbits, as well as the shearing of hot spots. They inferred a mean orbital radius of ∼9 R_g and an inclination of i ∼ 140°, and they constrained the hot-spot diameter to less than 5 R_g; i.e., the hot spots must be very compact emission sites.

Ripperda et al. (2020) studied the magnetic reconnection and plasmoid formation in current sheaths in the accretion flows using general relativistic magnetohydrodynamic simulations. They confirmed that plasmoids can form in the inner accretion flow parts between 5 and 10 Schwarzschild radii, regardless of the disk size and its magnetization. Plasmoids can further merge and grow to a macroscopic size of the order of a Schwarzschild radius (∼0.1 au) and get advected toward the black hole or become a part of the jet sheath.

Matsumoto et al. (2020) attempted to explain an apparent paradox in GRAVITY observations: the hot spots appear to traverse a loop on the sky in a time shorter than expected, which leads to a velocity of ∼0.3 c. The authors proposed that instead of material motions, in which the hot spot follows a gas parcel in the RIAF or along a geodesic, the radius–velocity discrepancy can be solved by the pattern motion at super-Keplerian speeds at larger radii. First, they proposed a magnetohydrodynamic perturbation at r ∼ 12.5 R_g, or, alternatively, the pattern could be created due to the interaction between the outflow and the inclined disk at a radius of ∼20 R_g.

Our work is complementary to the abovementioned studies in that it considers the model of a charged magnetosphere. In addition, the paradox of super-Keplerian speeds can be solved by the presence of the Lorentz force.