Gravitational redshift/blueshift of light emitted by geodesic test particles, frame-dragging and pericentre-shift effects, in the Kerr-Newman-de Sitter and Kerr-Newman black hole geometries

We investigate the redshift and blueshift of light emitted by timelike geodesic particles in orbits around a Kerr-Newman-(anti) de Sitter (KN(a)dS) black hole. Specifically we compute the redshift and blueshift of photons that are emitted by geodesic massive particles and travel along null geodesics towards a distant observer-located at a finite distance from the KN(a)dS black hole. For this purpose we use the Killing-vector formalism and the associated first integrals-constants of motion. We consider in detail stable timelike equatorial circular orbits of stars and express their corresponding redshift/blueshift in terms of the metric physical black hole parameters (angular momentum per unit mass, mass, electric charge and the cosmological constant) and the orbital radii of both the emitter star and the distant observer. These radii are linked through the constants of motion along the null geodesics followed by the photons since their emission until their detection and as a result we get closed form analytic expressions for the orbital radius of the observer in terms of the emitter radius, and the black hole parameters. In addition, we compute exact analytic expressions for the frame dragging of timelike spherical orbits in the KN(a)dS spacetime in terms of multivariable generalised hypergeometric functions of Lauricella and Appell. Last but not least we derive a very elegant and novel exact formula for the periapsis advance for a test particle in a non-spherical polar orbit in KNdS black hole spacetime in terms of Jacobi's elliptic function sn and Lauricella's hypergeometric function $F_D$.


Introduction
General relativity (GR) has triumphed all experimental tests so far which cover a wide range of field strengths and physical scales that include: those in large scale cosmology [17], [18], the prediction of solar system effects like the perihelion precession of Mercury with a very high precision [1], [19], the recent discovery of gravitational waves in Nature [34], [35], as well as the observation of the shadow of the M87 black hole [24], see also [5].
The orbits of short period stars in the central arcsecond (S-stars) of the Milky Way Galaxy provide the best current evidence for the existence of supermassive black holes [2], [3].
In a series of papers we solved exactly timelike and null geodesics in Kerr and Kerr-(anti) de Sitter black hole spacetimes [6], [7], [20], and null geodesics and the gravitational lens equations in electrically charged rotating black holes in [8].
We also computed in [6] elegant closed form analytic solutions for the general relativistic effects of periapsis advance, Lense-Thirring precession, orbital and Lense-Thirring periods and applied our solutions for calculating these GR-effects for the observed orbits of S-stars. The shadow of the Kerr and charged Kerr black holes were computed in [9], [7] and [8] respectively.
One of the targets of observational astronomers of the galactic centre is to measure the gravitational redshift predicted by the theory of general relativity [10]. In the Schwarzschild spacetime geometry the ratio of the frequencies measured by two stationary clocks at the radial positions r 1 and r 2 is given by [15]: where G is the gravitational constant and M is the mass of the black hole.
Recently, the redshift/blueshift of photons emitted by test particles in timelike circular equatorial orbits in Kerr spacetime were investigated in [30]. It is the purpose of this paper to extend our previous results on relativistic observables and compute for the first time the redshift and blueshift of light emitted by timelike geodesic particles in orbits around the Kerr-Newman-(anti) de Sitter (KN(a)dS) black hole. In addition, we derive new exact analytic expressions for the pericentre-shift and frame-dragging for non-spherical nonequatorial (polar) timelike KNdS and KN black hole orbits. Moreover, we derive novel exact expressions for the frame dragging effect for particles in spherical, non-equatorial orbits in KNdS and KN black hole geometries. These results will be of interest to the observational astronomers of the Galactic centre [2], [3] whose aim is to measure experimentally, the relativistic effects predicted by the theory of General Relativity [29] for the observed orbits of short-period stars-the so called S-stars in our Galactic centre. During 2018, the close proximity of the star S2 (S02) to the supermassive Galactic centre black hole allowed the first measurements of the relativistic redshift observable by the GRAVITY collaboration [11] and the UCLA Galactic centre group whose astrometric measurements were obtained at the W.M. Keck Observatory [12] 1 .
One of the most fundamental exact non-vacuum solutions of the gravitational field equations of general relativity is the Kerr-Newman black hole [14]. The Kerr-Newman (KN) exact solution describes the curved spacetime geometry surrounding a charged, rotating black hole and it solves the coupled system of differential equations for the gravitational and electromagnetic fields [14] (see also [15]).
The KN exact solution generalised the Kerr solution [16], which describes the curved spacetime geometry around a rotating black hole, to include a net electric charge carried by the black hole.
Taking into account the contribution from the cosmological constant Λ, the generalisation of the Kerr-Newman solution is described by the Kerr-Newman de Sitter (KNdS) metric element which in Boyer-Lindquist (BL) coordinates is given by [36], [37], [39], [38] (in units where G = 1 and c = 1): ∆ θ := 1 + a 2 Λ 3 cos 2 θ, Ξ := 1 + a 2 Λ 3 , where a, M, e, denote the Kerr parameter, mass and electric charge of the black hole, respectively. The KN(a)dS metric is the most general exact stationary black hole solution of the Einstein-Maxwell system of differential equations. This is accompanied by a non-zero electromagnetic field F = dA, where the vector potential is [40], [39]: The KN(a)dS dynamical system of geodesics is a completely integrable system 2 as was shown in [37], [36], [40], [20] and the geodesic differential equations take the form: where Null geodesics are derived by setting µ = 0. The proper time τ and the affine parameter λ are connected by the relation τ = µλ. In the following we use geometrised units, G = c = 1, unless it is stipulated otherwise. The first integrals of motion E and L are related to the isometries of the KNdS metric while Q (Carter's constant) is the hidden integral of motion that results from the separation of variables of the Hamilton-Jacobi equation. The material of the paper is organised as follows: In Sec. 2 we consider the Killing vector formalism and the corresponding conserved quantities in Kerr-Newman-(anti) de Sitter spacetime. In Sec.3 we consider equatorial circular geodesics in KN(a)dS spacetime and derive novel expressions for the specific energy and specific angular momentum for test particles moving in such orbits, see equations (26) and (27). Typical behaviour of these functions is displayed in Fig.1-Fig.17. In Sec. 4 we provide general expressions for the redshift/blueshift that emitted photons by massive particles experience while travelling along null geodesics towards an observer located far away from their source by making use of the Killing vector formalism. In Sec.4.1 we derive novel exact analytic expressions for the redshift/blueshift of photons for circular and equatorial emitter/detector orbits around the Kerr-Newman-(anti) de Sitter black hole-see equations (56) and (57) respectively. In the procedure we take into account the bending of light due to the field of the Kerr-Newman-(anti)de Sitter black hole at the moment of detection by the observer. In Sec.5 we study non-equatorial orbits in rotating charged black hole spacetimes. Specifically, we compute in closed analytic form the frame-dragging for test particles in timelike spherical orbits in the Kerr-Newman and Kerr-Newman-de Sitter black hole spacetimes-equations (72),theorem 4 and (80) respectively. The former equation (KN case) involves the ordinary Gauß hypergeometric function and Appell's F 1 two-variable hypergeometric function, while the latter (KNdS case) is expressed in terms of Lauricella's F D and Appell's F 1 generalised multivariate hypergeometric functions [32]. In Sec.5.5 & 5.6 we derive new closed form analytic expressions for the periapsis advance for test particles in non-spherical polar orbits in KN and KNdS spacetimes respectively. In the latter case we derive a novel, very elegant, exact formula in terms of Jacobi's elliptic function sn and Lauricella's hypergeometric function F D of three variables-see equation (108).

Particle orbits and Killing vectors formalism in Kerr-Newman-(anti)de Sitter spacetime
From the condition for the invariance of the metric tensor: under the infinitesimal transformation: it follows that whenever the metric is independent of some coordinate a constant vector in the direction of that coordinate is a Killing vector . Thus the generic metric: possesses two commuting Killing vector fields: ψ µ = (0, 0, 0, 1) rotational Killing vector.
According to Noether's theorem to every continuous symmetry of a physical system corresponds a conservation law. In a general curved spacetime, we can formulate the conservation laws for the motion of a particle on the basis of Killing vectors. We can prove that if ξ ν is a Killing vector, then for a particle moving along a geodesic, the scalar product of this Killing vector and the momentum P ν = µ dx ν dτ of the particle is a constant [15]: Due to the existence of these Killing vector fields (17), (18) there are two conserved quantities the total energy and the angular momentum per unit mass at rest of the test particle 3 : Thus, the photon's emitter is a probe massive test particle which geodesically moves around a rotating electrically charged cosmological black hole in the spacetime with a four-velocity: The conservation law (19) also applies to photon moving in the curved spacetime. Thus, if the spacetime geometry is time independent, the photon energy P 0 is constant. In section 4.1 we will extract the redshift/blueshift of photons from this conservation law.

Equatorial circular orbits in Kerr-Newman spacetime with a cosmological constant
It is convenient to introduce a dimensionless cosmological parameter: and set M = 1. For equatorial orbits Carter's constant Q vanishes. For the following discussion, it is useful to introduce new constants of motion, the specific energy and specific angular momentum: This is equivalent to setting µ = 1. Thus for reasons of notational simplicity we omit the caret for the specific energy and specific angular momentum in what follows. Equatorial circular orbits correspond to local extrema of the effective potential. Equivalently, these orbits are given by the conditions R ′ (r) = 0, dR ′ /dr = 0, which have to be solved simultaneously. Following this procedure, we obtain the following novel equations for the specific energy and the specific angular momentum of test particles moving along equatorial circular orbits in KN(a)dS spacetime: The reality conditions connected with equations (26) and (27) are given by the inequalities: and 1 − Λ ′ r 3 ≥ e 2 /r.
In the figures 1-17, for concrete values of the electric charge and the cosmological parameter we present the radial dependence of the specific energy and specific angular momentum for various values of the black hole's spin. For the cosmological parameter Λ ′ we choose the values Λ ′ = 10 −5 , 10 −4 , 10 −3 as well as their negative counterparts. For stellar mass black holes, and positive cosmological constant this corresponds to Λ ∼ 10 −15 cm −2 − 10 −13 cm −2 . For supermassive black holes such as at the centre of Galaxy M87 with mass M M87 BH = 6.7 × 10 9 solar masses [24] the value of Λ ′ = 10 −5 corresponds to the value for the cosmological constant: Λ = 3.06 × 10 −35 cm −2 .

Gravitational redshift-blueshift of emitted photons
In this section we will provide general expressions for the redshift/blueshift that emitted photons by massive particles experience while travelling along null geodesics towards an observed located far away from their source.                 In general, the frequency of a photon measured by an observer with proper velocity U µ A at the spacetime point P A reads [15], [30]: where the index A refers to the emission (e) and/or detection (d) at the corresponding point P A . The emission frequency is defined as follows: Likewise the detected frequency is given by the expression: In producing (32),(33) we used the expressions for U t and U φ in terms of the metric components and the conserved quantities E, L: Thus, This is the most general expression for the redshift/blueshift that light signals emitted by massive test particles experience in their path along null geodesics towards a distant observer (ideally located near the cosmological horizon in particular or at spatial infinity assuming a zero cosmological constant).

The redshift/blueshift of photons for circular and equatorial emitter/detector orbits around the Kerr-Newman-(anti) de Sitter black hole
For equatorial circular orbits U r = U θ = 0 thus Following the procedure for the Kerr black hole in [30], we consider the kinematic redshift of photons either side of the line of sight that links the Kerr-Newman-de Sitter black hole and the observer, and subtract from Eq.(37) the central value z c . Then we obtain: Let us now consider photons with 4-momentum vector k µ = (k t , k r , k θ , k φ ) which move along null geodesics k µ k µ = 0 outside the event horizon of the Kerr-Newman-de Sitter black hole, which explicitly can be expressed as We must take into account the bending of light from the rotating and charged Kerr-Newman-(anti) de Sitter black hole. From (38) it follows that the apparent impact parameter must be maximised. The apparent impact factor Φ γ ≡ L γ /E γ can be obtained from the expression k µ k µ = 0 4 as follows: Solving the quadratic equation we obtain: where we got two values, Φ + γ and Φ − γ (either evaluated at the emitter or detector position, since this quantity is preserved along the null geodesic photon orbits, i.e., Φ e = Φ d ) that give rise to two different shifts respectively, z 1 and z 2 of the emitted photons corresponding to a receding and to an approaching object with respect to a far away positioned observer: 4 Taking into account that k r = 0 and k θ = 0.
In general the two values z 1 and z 2 differ from each other due to light bending experienced by the emitted photons and the differential rotation experienced by the detector as encoded in U φ d and U t d components of the four-velocity 5 . In order to get a closed analytic expression for the gravitational redshift/blueshift experienced by the emitted photons we shall express the required quantities in terms of the Kerr-Newman-(anti) de Sitter metric. Thus, the U φ and U t components of the four-velocity for circular equatorial orbits read: Substituting the expressions (26)-(27) for E ± and L ± into U t (r, π/2), U φ (r, π/2) we finally obtain remarkable novel expressions for these four-velocity components in Kerr-Newman-(anti) de Sitter spacetime: We now compute the angular velocity Ω: In terms of the angular velocities the quantities z 1 , z 2 read as follows: . (53) Thus for the Kerr-Newman-(anti) de Sitter black hole we can write for the redshift and blueshift, respectively: where now r e and r d stand for the radius of the emitter's and detector's orbits, respectively. These elegant and novel expressions can be written in terms of the physical parameters of the Kerr-Newman-(anti) de Sitter black hole and the detector radius, r d , as follows: where we define: and we have made use of the relation Φ e = Φ d . The remarkable closed form analytic expressions for the frequency shifts we obtained in eqns.(56)-(57), constitute a new result in the theory of General Relativity, in which all the physical parameters of the exact theory enter on an equal footing.
In the particular case when the detector is located far away from the source and the condition is fulfilled: r d ≫ M ≥ a, the redshift and blueshift respectively take the form: .
Thus for instance, expressed in terms of the new variable : where α = a 2 (1 − E 2 ), β = L 2 + Q. The range of z for which the motion takes place includes the equatorial value, z = 0: We will prove first the following exact result: Proof. We compute first the integral: Applying the transformation z = z − + ξ 2 (z j − z − ) in (67) we obtain where x ≡ ξ 2 . Setting z j = 0 yields: For producing the result in the last line of equation (69), we used the following transformation property of Appell's hypergeometric function F 1 : On the other hand we compute analytically the second integral that contributes to frame-dragging and we obtain: We thus obtain the following result in closed analytic form for the amount of frame-dragging that a timelike spherical orbit in Kerr-Newman spacetime undergoes.
Theorem 4 As θ goes through a quarter of a complete oscillation we obtain the change in azimuth φ, ∆φ GTR :

Periods
Squaring the geodesic differential equation for the polar variable (11) (for Λ = 0), multipying by the term cos 2 θ sin 2 θ, and making the change to the variable z, yields the following differential equation for the proper polar period: Finally our closed form analytic computation for the proper polar period yields: Proof. We compute first: We write: Applying the change of variables:z = z − + ξ 2 (z j − z − ) we compute the term: Setting z j = 0 in the expression which involves Appell's hypergeometric function F 1 yields:

Spherical orbits in Kerr-Newman-(anti) de Sitter spacetime
From (8) and (11) we derive the equation : Using the variable z we obtain the following novel exact result in closed analytic form for the amount of frame-dragging that a timelike spherical orbit in Kerr-Newman-(anti)de Sitter spacetime undergoes. As θ goes through a quarter of a complete oscillation we obtain the change in azimuth φ, ∆φ GTR in terms of Lauricella's F D and Appell's F 1 multivariable generalised hypergeometric functions: where we define: In our calculations we used the following property for the values of Lauricella's multivariate function F D :

Frame-dragging effect for polar non-spherical bound orbits in Kerr-Newman spacetime
Polar spherical orbits are characterised by the vanishing of the angular momentum of the particle, i.e. L = 0. We further assume in this section that Λ = 0. The relevant differential equation for the calculation of frame-dragging is: The quartic radial polynomial R is obtained from R ′ in (12) for Λ = L = 0.
Using the partial fractions technique we integrate from the periastron distance r P to the apoastron distance r A : Applying the transformation: and organizing the roots of the radial polynomial and the radii of the event and Cauchy horizon in the ascending order of magnitude: with the correspondence α ρ = α µ = α, α σ = α µ+1 = β, α ν = α µ+2 = γ, α i = α µ−i , i = 1, 2, 3, α µ−1 = a µ−2 = r ± , α µ−3 = δ we compute the exact analytic result in terms of Appell's hypergeometric function F 1 : where the partial fraction expansion parameters are given by: The variables of the hypergeometric functions are given in terms of the roots of the quartic and the radii of the horizons by the expressions: while

Exact calculation of the orbital period in non-spherical polar Kerr-Newman geodesics
In this section we will compute a novel exact formula for the orbital period for a test particle in a non-spherical polar Kerr-Newman geodesic. The relevant differential equation is: and we integrate from periapsis to apoapsis and back to periapsis. Indeed, our analytic computation yields: Parameters for the star S2 15.15 6.25 × 10 6 Table 1: Lense-Thirring precession for the star S2 in the central arcsecond of the galactic centre, using the exact formula (86) . We assume a central galactic Kerr-Newman black hole with mass M BH = 4.06 × 10 6 M ⊙ and that the orbit of S2 star is a timelike non-spherical polar Kerr-Newman geodesic. The computation of the orbital period of the star S2 was performed using the exact result in eqn.(91).
and the moduli (variables) of the hypergeometric function of Appell are given by: For zero electric charge, e = 0, Eqn.(91) reduces correctly to Eqn. (33) in [6] for the case of a Kerr black hole. The Lense-Thirring period for a non-spherical polar timelike geodesic in Kerr-Newman BH geometry, is defined in terms of the Lense-Thirring precession Eq. (86) and its orbital period Eq. (91) as follows: We now proceed to calculate using our exact analytic solutions and assuming a central galactic Kerr-Newman black hole, the Lense-Thirring effect and the corresponding Lense-Thirring period for the observed stars S2,S14 for various values of the Kerr parameter and the electric charge of the central black holesee Tables 1-2. We observe that the contribution of the electric charge on the frame-dragging precession is small.

Periapsis advance for non-spherical polar timelike Kerr-Newman orbits
The purpose of this section is twofold. First, we apply closed form analytic expressions for the periapsis advance for non-constant radius orbits in Kerr-Newman spacetime, for the computation of this relativistic effect for the observed S-star orbits in the central arcsecond of SgrA* supermassive black hole.
Parameters for the star S14 37.88 7.38 × 10 6 Table 2: Lense-Thirring precession for the star S14 in the central arcsecond of the galactic centre, using the exact formula (86) . We assume a central galactic Kerr-Newman black hole with mass M BH = 4.06 × 10 6 M ⊙ and that the orbit of S14 star is a timelike non-spherical polar Kerr-Newman geodesic. The computation of the orbital period of the star S14 was performed using the exact result in eqn.(91).
Secondly, this computation will provide us with realistic values for the first integrals of motion associated with non-circular orbits in KN and KN-(a)dSspacetime. In principle, the latter values can be used as input in our analytic expressions for the redshift/blueshift experienced by photons emitted by test particles such as S-stars.
In [6] we computed a closed-form analytic expression for the periapsis advance that a non-spherical polar timelike orbit undergoes in Kerr spacetime in terms of Abel-Jacobi's amplitude function: The functional form of the solution will remain the same by incorporating the electric charge of the black hole, however the roots α, β, γ, δ will now be solutions of the quartic polynomial: and thus they will differ from those of [6]. We compute with the aid of (96) the periapsis advance for the stars S2 and S14 assuming that they orbit in a timelike non-spherical polar Kerr-Newman geodesic. Our results are displayed in Tables 3 and 4. We also computed with the aid of the exact formula Eqn. (103) in [8], the pericentre-shift for the stars S2 and S14 for various values for the spin and charge of the central black hole.. By doing this exercise, we gain an appreciation of the effect of the electric charge of the rotating galactic black hole (we assume that the KN solution describes the curved spacetime geometry around SgrA*) on this observable. We also assume that the angular momentum axis of the orbit is co-aligned with the spin axis of the black hole and that the S−stars can be treated as neutral test particles i.e. their orbits are timelike non-circular equatorial Kerr-Newman geodesics, Our results are displayed in Tables 5-8   is evident in this case that the value of electric charge plays a significant role in the value of the pericentre-shift 8 .
A few comments are in order. The values of the hypothetical electric charge of the central Kerr-Newman black hole have been chosen so that the surrounding spacetime represents a black hole, i.e. the singularity surrounded by the horizon, the electric charge and angular momentum J must be restricted by the relation: where in the last inequality a ′ = a GM/c 2 denotes a dimensionless Kerr parameter. Concerning the tentative values for the electric charge e we used in applying our exact solutions for the case of SgrA* black hole we note that their likelihood is debatable: There is an expectation that the electric charge trapped in the galactic nucleous will not likely reach so high values as the ones close to the extremal values predicted in (100) that allow the avoidance of a naked singularity. However, more precise statements on the electric charge's magnitude of the   Table 6: Periastron precession for the star S14 in the central arcsecond of the galactic centre, using the exact formula Eqn. (103) in [8] . We assume a central galactic Kerr-Newman black hole with mass M BH = 4.06 × 10 6 M ⊙ and that the orbit of the star S14 is a timelike non-circular equatorial Kerr-Newman geodesic.
galactic black hole or its upper bound will only be reached once the relativistic effects predicted in this work are measured and a comparison of the theory we developed with experimental data will take place.

Periapsis advance for non-spherical polar timelike Kerr-Newman -de Sitter orbits
In this section we are going to derive a new closed form expression for the pericentre-shift of a test particle in a timelike non-spherical polar Kerr-Newman- Inverting the elliptic integral for z we obtain: Equivalently the change in latitude after a complete radial oscillation leads to the following exact novel expression for the periastron advance for a test particle in an non-spherical polar Kerr-Newman-de Sitter orbit: where: Also the Jacobi modulus κ of the Jacobi's sinus amplitudinous elliptic function in formula (108), for the periapsis advance that a non-spherical polar orbit undergoes in the Kerr-Newman-de Sitter spacetime, is given in terms of the roots of the angular elliptic integral by: The roots z Λ , z + , z − appearing in (106) are roots of the polynomial equation: after setting 10 L = 0. 10 We have the correspondence α 1 = z + , β 1 = z − , δ 1 = z Λ .

Computation of first integrals for spherical timelike geodesics in Kerr-Newman spacetime
For zero electric charge e = 0, Equations (118) and (119) reduce correctly to the corresponding equations for the first integrals of motion in Kerr spacetime [42]:

The apparent impact factor for more general orbits
The apparent impact parameter Φ for the Kerr-Newman-(anti) de Sitter black hole can also be computed in the case in which the considered orbits depart from the equatorial plane and therefore θ = π/2. Again, we compute this quantity from the k µ k µ = 0 relation just taking into account its maximum character, i.e., that k r = 0. Our calculation yields: (122) Our exact expression (122) for the apparent impact parameter in the KN(a)dS spacetime, for zero cosmological constant (Λ = 0) and zero electric charge (e = 0), reduces to eqn.(59) (the apparent impact parameter for the Kerr black hole) in [30]. Also for zero value for Carter's constant Q γ equation (122) reduces to eqn.(44).

Conclusions
In this work using the Killing-vector formalism and the associated first integrals we computed the redshift and blueshift of photons that are emitted by geodesic massive particles and travel along null geodesics towards a distant observerlocated at a finite distance from the KN(a)dS black hole. As a concrete example we calculated analytically the redshift and blueshift experienced by photons emitted by massive objects orbiting the Kerr-Newman-(anti) de Sitter black hole in equatorial and circular orbits, and following null geodesics towards a distant observer.
In addition and extending previous results in the literature we calculated in closed analytic form firstly, the frame-dragging that experience test particles in non-equatorial spherical timelike orbits in KN and KNdS spacetimes in terms of generalised hypergeometric functions of Appell and Lauricella. Secondly, we computed in closed analytic the periapsis advance for timelike non-spherical polar orbits in Kerr-Newman and Kerr-Newman de Sitter spacetimes. In the Kerr-Newman case, the pericentre-shift is expressed in terms of Jacobi's amplitude function and Gauß hypergeometric function, while in the Kerr-Newman-de Sitter the periapsis-shift is expressed in an elegant way in terms of Jacobi's sinus amplitudinus elliptic function sn and Lauricella's hypergeometric function F D with three-variables.
We also computed the first integrals of motion for non-equatorial Kerr-Newman geodesics of constant radius. These expressions together with the analytic equation for the apparent impact factor we derived in this work-eqn (122), can be used to derive close form expressions for the redshift/blueshift of the emitted photons for such non-equatorial orbits in Kerr-Newman and Kerr-Newman (anti) de Sitter spacetimes. This will be a task for the future. Such a future endeavour will also involve the computation of the redshift/blueshift of the emitted photons for realistic values of the first integrals of motion associated with the observed orbits of S-stars (the emitters) such as those of Section 5.5, especially when the first measurements of the pericentre-shift of S2 will take place. The ultimate aim of course is to determine in a consistent way the parameters of the supermassive black hole that resides at the Galactic centre region SgrA*.
It will also be interesting to investigate the effect of a massive scalar field on the orbit of S2 star and in particular on its redshift and periapsis advance by combining the results of this work and the exact solutions of the Klein-Gordon-Fock (KGF) equation on the KN(a)dS and KN black hole backgrounds in terms of Heun functions produced in [43] (see also [44]) . This research will be the theme of a future publication 11 .
The fruitful synergy of theory and experiment in this fascinating research field will lead to the identification of the resident of the Milky Way's Galactic centre region and will provide an important test of General Relativity at the strong field regime.