Gravitational Redshift/Blueshift of Light Emitted by Geodesic Test Particles

Eur. Phys. J. C (2021) 81:147 https://doi.org/10.1140/epjc/s10052-021-08911-5

Regular Article - Theoretical Physics

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

G. V. Kraniotisa Section of Theoretical Physics, Physics Department, University of Ioannina, 451 10 Ioannina, Greece

Received: 22 January 2020 / Accepted: 22 January 2021 © The Author(s) 2021

Abstract We investigate the redshift and blueshift of light 1 Introduction emitted by timelike geodesic particles in orbits around a Kerr–Newman–(anti) de Sitter (KN(a)dS) black hole. Specif- General relativity (GR) [1] has triumphed all experimental ically we compute the redshift and blueshift of photons that tests so far which cover a wide range of field strengths and are emitted by geodesic massive particles and travel along physical scales that include: those in large scale cosmology null geodesics towards a distant observer-located at a finite [2–4], the prediction of solar system effects like the perihe- distance from the KN(a)dS black hole. For this purpose lion precession of Mercury with a very high precision [1,5], we use the killing-vector formalism and the associated first the recent discovery of gravitational waves in Nature [6–10], integrals-constants of motion. We consider in detail stable as well as the observation of the shadow of the M87 black timelike equatorial circular orbits of stars and express their hole [11], see also [12]. corresponding redshift/blueshift in terms of the metric physi- The orbits of short period stars in the central arcsecond cal black hole parameters (angular momentum per unit mass, (S-stars) of the Milky Way Galaxy provide the best current mass, electric charge and the cosmological constant) and the evidence for the existence of supermassive black holes, in orbital radii of both the emitter star and the distant observer. particular for the galactic centre SgrA* supermassive black These radii are linked through the constants of motion along hole [13–17]. the null geodesics followed by the photons since their emis- One of the most fundamental exact non-vacuum solutions sion until their detection and as a result we get closed form of the gravitational field equations of general relativity is analytic expressions for the orbital radius of the observer in the Kerr–Newman black hole [18]. The Kerr–Newman (KN) terms of the emitter radius, and the black hole parameters. exact solution describes the curved spacetime geometry sur- In addition, we compute exact analytic expressions for the rounding a charged, rotating black hole and it solves the cou- frame dragging of timelike spherical orbits in the KN(a)dS pled system of differential equations for the gravitational and spacetime in terms of multivariable generalised hypergeo- electromagnetic fields [18](seealso[19]). metric functions of Lauricella and Appell. We apply our exact The KN exact solution generalised the Kerr solution [20], solutions of timelike non-spherical polar KN geodesics for which describes the curved spacetime geometry around a the computation of frame-dragging, pericentre-shift, orbital rotating black hole, to include a net electric charge carried period for the orbits of S2 and S14 stars within the 1 of by the black hole. SgrA*. We solve the conditions for timelike spherical orbits A wide variety of astronomical and cosmological obser- in KN(a)dS and KN spacetimes. We present new, elegant vations in the last two decades, including high-redshift type compact forms for the parameters of these orbits. Last but Ia supernovae, cosmic microwave background radiation and not least we derive a very elegant and novel exact formula large scale structure indicate convincingly an accelerating for the periapsis advance for a test particle in a non-spherical expansion of the Universe [2,3,21,22]. Such observational polar orbit in KNdS black hole spacetime in terms of Jacobi’s data can be explained by a positive cosmological constant elliptic function sn and Lauricella’s hypergeometric function (>0) with a magnitude ∼ 10−56cm−2 [4]. On the other FD. hand, we note the significance of a negative cosmological constant in the anti-de Sitter/conformal field theories corre- a e-mail: [email protected] (corresponding author)

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dr √ spondence [23–25]. Thus, solutions of the Einstein-Maxwell ρ2 =± R, (9) equations with a cosmological constant deserve attention. dλ dθ √ A more realistic description of the spacetime geometry ρ2 =± , (10) surrounding a black hole should include the cosmological dλ constant [5,26–34]. where Taking into account the contribution from the cosmological constant , the generalisation of the Kerr–Newman solu- 2 tion is described by the Kerr–Newman de Sitter (KNdS)met- R := 2 (r 2 + a2)E − aL ric element which in Boye-r-Lindquist (BL) coordinates is − KN(μ2r 2 + Q + 2(L − aE)2), (11) given by [35–38] (in units where G = 1 and c = 1): r 2 2 2 2 2 := Q + (L − aE) − μ a cos θ θ KN ρ2 ρ2 ds2 = r (dt − a sin2 θdφ)2 − dr 2 − dθ 2 2 θ − 2 2ρ2 KN 2 aE sin L r θ − . (12) sin2 θ θ sin2 θ 2 − adt − (r 2 + a2)dφ (1) 2ρ2 Null geodesics are derived by setting μ = 0. The proper time 2 2 τ λ a 2 a and the affine parameter are connected by the relation θ := 1 + cos θ, := 1 + , (2) 3 3 τ = μλ. In the following we use geometrised units, G = c = 1, unless it is stipulated otherwise. The first integrals of KN := 1 − r 2 r 2 + a2 − 2Mr + e2, (3) r 3 motion E and L are related to the isometries of the KNdS ρ2 = r 2 + a2 cos2 θ, (4) metric while Q (Carter’s constant) is the hidden integral of motion that results from the separation of variables of the Hamilton-Jacobi equation. where a, M, e, denote the Kerr parameter, mass and electric Although it is not the purpose of this paper to discuss how charge of the black hole, respectively. The KN(a)dS metric is a net electric charge is accumulated inside the horizon of the the most general exact stationary black hole solution of the black hole, we briefly mention recent attempts which address Einstein–Maxwell system of differential equations. This is the issue of the formation of charged black holes. Indeed, we accompanied by a non-zero electromagnetic field F = dA, note at this point, that the authors in [40], have studied the where the vector potential is [38,39]: effect of electric charge in compact stars assuming that the er A =− (dt − a sin2 θdφ). (5) charge distribution is proportional to the mass density. They (r 2 + a2 cos2 θ) found solutions with a large positive net electric charge. From Properties of the Kerr–Newman spacetimes with a non- the local effect of the forces experienced on a single charged zero cosmological constant are appropriately described by particle, they concluded that each individual charged parti- their geodesic structure which determines the motion of test cle is quickly ejected from the star. This is in turn produces a particles and photons. The KN(a)dS dynamical system of huge force imbalance, in which the gravitational force over- geodesics is a completely integrable system1 as was shown whelms the repulsive Coulomb and fluid pressure forces. In in [26,35,36,39] and the geodesic differential equations take such a scenario the star collapses to form a charged black hole the form: before all the charge leaves the system [40]. A mechanism for generating charge asymmetry that may be linked to the dr dθ formation of a charged black hole has been suggested in [41]. √ = √ , (6) R Besides these theoretical considerations for the formation of φ 2 charged black holes, recent observations of structures near ρ2 d =− 2 θ − aE sin L SgrA* by the GRAVITY experiment, indicate possible pres- dλ θ sin2 θ ence of a small electric charge of central supermassive black a2 + (r 2 + a2)E − aL , (7) hole [42,43]. Accretion disk physics around magnetised Kerr KN r black holes under the influence of cosmic repulsion is exten- dt 2(r 2 + a2)[(r 2 + a2)E − aL] sively discussed in the review [44] 2. Therefore, it is quite cρ2 = λ KN d r a2(aE sin2 θ − L) 2 We also mention that supermassive black holes as possible sources − , (8) θ of ultahigh-energy cosmic rays have been suggested in [45], where it has been shown that large values of the Lorentz γ factor of an escap- ing ultrahigh-energy particle from the inner regions of the black hole 1 This is proven by solving the relativistic Hamilton–Jacobi equation accretion disk may occur only in the presence of the induced charge of by the method of separation of variables. the black hole. 123 Eur. Phys. J. C (2021) 81:147 Page 3 of 37 147 interesting to study the combined effect of the cosmological As we mentioned a very important motivation for our constant and electromagnetic fields on the black hole astro- present work is to investigate frequency redshift/blueshift of physics. light emitted by timelike geodesic particles in orbits around Most of previous studies of the black hole backgrounds a Kerr–Newman–(anti) de Sitter black hole using the Killing with >0 were mainly concentrated on uncharged vector formalism. Schwarzschild–de Sitter and Kerr–de Sitter spacetimes [5, Indeed, one of the targets of observational astronomers 26,27,29,46–49]. Particle motion and shadow of rotating of the galactic centre is to measure the gravitational red- spacetimes have been studied for the Kerr case in [50]for shift predicted by the theory of general relativity [54]. In the the braneworld Kerr–Newman case with a tidal charge in Schwarzschild spacetime geometry the ratio of the frequen- [51,52], and for the Kerr–de Sitter case in [53]. In a series of cies measured by two stationary clocks at the radial positions papers we solved exactly timelike and null geodesics in Kerr r1 and r2 is given by [19]: and Kerr–(anti) de Sitter black hole spacetimes [26,27,46], √ ν1 1 − 2GM/r2 and null geodesics and the gravitational lens equations in = √ , (13) ν 1 − 2GM/r electrically charged rotating black holes in [28]. We also 2 1 computed in [46] elegant closed form analytic solutions for where G is the gravitational constant and M is the mass of the general relativistic effects of periapsis advance, Lense– the black hole. Recently, the redshift/blueshift of photons Thirring precession, orbital and Lense–Thirring periods and emitted by test particles in timelike circular equatorial orbits applied our solutions for calculating these GR-effects for in Kerr spacetime were investigated in [55]. the observed orbits of S-stars. The shadow of the Kerr and The analytic computation we perform in this work for charged Kerr black holes were computed in [27,47] and [28] the first time, for the redshift and blueshift of light emit- respectively. ted by timelike geodesic particles in orbits around the Kerr– It is the main purpose of this work to derive new exact ana- Newman–(anti) de Sitter (KN(a)dS) black hole, extend our lytic solutions for timelike geodesic equations in the Kerr– previous results on relativistic observables. The theory we Newman and Kerr–Newman–de Sitter spacetimes as well as develop enriches our arsenal for studying the combined effect to develop a theory of gravitational redshift/blueshift of pho- of the cosmological constant and electromagnetic fields on tons emitted by geodesic particles in timelike orbits around a the black hole astrophysics and can serve as a method to Kerr–Newman–(anti) de Sitter black hole. We derive closed infer important information about black hole observables form analytic solutions for both spherical orbits as well as from photon frequency shifts. In addition, we derive new for non-spherical orbits. A special class of orbits around a exact analytic expressions for the pericentre-shift and frame- charged rotating black hole that we investigate in this paper dragging for non-spherical non-equatorial (polar) timelike are spherical orbits i.e. orbits with constant radii that are not KNdS and KN black hole orbits. Moreover, we derive novel necessarily confined to the equatorial plane. We solve for the exact expressions for the frame dragging effect for particles in first time the conditions for spherical timelike orbits in Kerr– spherical, non-equatorial orbits in KNdS and KN black hole Newman and Kerr–Newman–(anti) de Sitter spacetimes. As geometries. These results could be of interest to the obser- a result of this procedure we derive elegant, compact forms vational astronomers of the Galactic centre [13,14,16,17] for the parameters of these orbits which are associated with whose aim is to measure experimentally, the relativistic the Killing vectors of these spacetimes. For such spherical effects predicted by the theory of General Relativity [28,56– orbits we derive closed form analytic expressions in terms of 68] for the observed orbits of short-period stars-the so called special functions (multivariable hypergeometric functions of S-stars in our Galactic centre. During 2018, the close prox- Appell–Lauricella) for frame dragging precession. We per- imity of the star S2 (S02) to the supermassive Galactic centre form an analysis of the parameter space for such spherical black hole allowed the first measurements of the relativistic orbits around a Kerr–Newman black hole and present many redshift observable by the GRAVITY collaboration [69] and examples of such orbits. We also derive the exact orbital the UCLA Galactic centre group whose astrometric measure- solution for non-spherical timelike geodesics around a Kerr– ments were obtained at the W.M. Keck Observatory [70].3 Newman black hole, that cross the symmetry axis of the black The material of the paper is organised as follows: In hole, in terms of elliptic functions. Moreover, for such orbits Sect. 2 we consider the killing vector formalism and the cor- we derive closed form analytic expressions for relativistic responding conserved quantities in Kerr–Newman–(anti) de observables that include the pericentre shift, frame-dragging Sitter spacetime. In Sect. 3 we consider equatorial circular precession and the orbital period. We extend our analytic computations by deriving a very elegant analytic solution for 3 Observational work is ongoing towards the detection of the periastron the periapsis advance for a test particle in a non-spherical shift of the star S2 and the discovery of putative closer stars-in the central milliarcsecond of SgrA* supermassive black hole [71], which could polar orbit around a Kerr–Newman–de Sitter black hole. allow an astrometric measurement of the black hole spin as envisaged e.g. in [46]. 123 147 Page 4 of 37 Eur. Phys. J. C (2021) 81:147

Fig. 1 Speciﬁc energy E+ for e = 0.11, = 0.001 for different Fig. 3 Speciﬁc energy E+ for e = 0.11, = 0.00001 for different values for the Kerr parameter values for the Kerr parameter

Fig. 2 Specific energy E+ for e = 0.11, = 0.0001 for different Fig. 4 Radial profile for specific energy E+ of test particles moving values for the Kerr parameter on equatorial circular orbits in KN(a)dS black hole with Kerr parameter a = 0.52, and various values for the cosmological constant and the black hole’s electric charge. The case of the Kerr black hole e = = 0is geodesics in KN(a)dS spacetime and derive novel expres- displayed sions for the specific energy and specific angular momentum for test particles moving in such orbits, see Eqs. (26) and (27). Typical behaviour of these functions is displayed in Figs. 1, the moment of detection by the observer. We derive the cor- 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, responding frequency shifts for circular equatorial orbits in 20 and 21. Furthermore, we investigate the stability of such Kerr–de Sitter spacetime in Sect. 4.1.1. We also examine the timelike circular equatorial geodesics in Kerr–Newman–de particular case when the detector is located far away from Sitter spacetime and derive a new condition that restricts their the source. In Sect. 5 we study non-equatorial orbits in rotat- radii, namely the inequality (34). In Sect. 4 we provide gen- ing charged black hole spacetimes. Specifically, we compute eral expressions for the redshift/blueshift that emitted pho- in closed analytic form the frame-dragging for test particles tons by massive particles experience while travelling along in timelike spherical orbits in the Kerr–Newman and Kerr– null geodesics towards an observer located far away from Newman–de Sitter black hole spacetimes-equations (95), their source by making use of the Killing vector formalism. Theorem 4 and (141) respectively. The former equation (KN In Sect. 4.1 we derive novel exact analytic expressions for case) involves the ordinary Gauß hypergeometric function the redshift/blueshift of photons for circular and equatorial and Appell’s F1 two-variable hypergeometric function, while emitter/detector orbits around the Kerr–Newman–(anti) de the latter (KNdS case) is expressed in terms of Lauricella’s Sitter black hole-see Eqs. (69) and (70) respectively. In the FD and Appell’s F1 generalised multivariate hypergeomet- procedure we take into account the bending of light due to ric functions [72,73]. We solve the conditions for spheri- the field of the Kerr–Newman–(anti)de Sitter black hole at cal orbits in Kerr Newman spacetime and derive new ele- 123 Eur. Phys. J. C (2021) 81:147 Page 5 of 37 147

Fig. 5 Radial profile for specific energy E+ of test particles moving Fig. 8 Specific energy E− for e = 0.11, = 0.00001 for different on equatorial circular orbits in KN(a)dS black hole with Kerr parameter values for the Kerr parameter a = 0.9939, and various values for the cosmological constant and the black hole’s electric charge. The case of the Kerr black hole e = = 0 is displayed

Fig. 9 Speciﬁc angular momentum L+ for e = 0.11, = 0.001 for different values for the Kerr parameter

Fig. 6 Speciﬁc energy E− for e = 0.11, = 0.001 for different values for the Kerr parameter

Fig. 10 Speciﬁc angular momentum L+ for e = 0.11, = 0.0001 for different values for the Kerr parameter

Fig. 7 Speciﬁc energy E− for e = 0.11, = 0.0001 for different values for the Kerr parameter

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− Fig. 11 Specific angular momentum L+ for e = 0.11, = 10 5 for Fig. 13 Radial profile for specific angular momentum L+ of test par- different values for the Kerr parameter ticles moving on equatorial circular orbits in KN(a)dS black hole with Kerr parameter a = 0.9939, and various values for the cosmological constant and the black hole’s electric charge. The case of the Kerr black hole e = = 0 is displayed

Fig. 12 Radial profile for specific angular momentum L+ of test particles moving on equatorial circular orbits in KN(a)dS black hole with Kerr parameter a = 0.52, and various values for the cosmological con- = . , = −3 stant and the black hole’s electric charge. The case of the Kerr black Fig. 14 Specific angular momentum L− for e 0 11 10 for hole e = = 0 is displayed different values for the Kerr parameter gant compact forms for the particle’s energy and angular momentum about the φ-axis, Theorem 5, and relations (101), (102). We investigated in some detail the ranges of r and Carter’s constant Q for which the solutions (101), (102)are valid. Having solved analytically the geodesic equations and armed with Theorems 4 and 5, we present several examples of timelike spherical orbits around a Kerr–Newman black hole in Tables 1 and 2. In Sect. 5.1.3 we investigate an important subset of spherical orbits: polar spherical orbits i.e. timelike geodesics with constant coordinate radii crossing the symmetry axis of the Kerr–Newman spacetime. We perform an effective potential analysis and determine sim- plified forms for the physical parameters of such orbits: rela- − tions (123) and (124). We solve analytically the geodesic Fig. 15 Specific angular momentum L− for e = 0.11, = 10 4 for equations and derive closed analytic form expression for the different values for the Kerr parameter Lense–Thirring precession, Eq. (126). Moreover we present 123 Eur. Phys. J. C (2021) 81:147 Page 7 of 37 147

−5 Fig. 16 Specific angular momentum L− for e = 0.11, = 10 for Fig. 18 Specific momentum L+ for e = 0.11, = different values for the Kerr parameter −0.001 for different values for the Kerr parameter. The radii of the event horizons for the specific spins of the black hole (r+, a) are:(1.59128,0.7939),(1.78524,0.6), (1.83972,0.52),(1.9516,0.26),(1.96582,0.2)

Fig. 17 Specific energy E+ for e = 0.11, =−0.001 for different values for the Kerr parameter. The radii of the event horizons for the specific spins of the black hole (r+, a) = . , =− . are:(1.59128,0.7939),(1.78524,0.6),(1.83972,0.52),(1.9516,0.26), Fig. 19 Specific angular momentum L− for e 0 6 0 01 for (1.96582,0.2) different values for the Kerr parameter several examples of such orbits in Table 3. Remarkably, In Theorem 7 we have solved the conditions for timelike spherical orbits for the constants of motion E, L that are associated with the two Killing vectors of the Kerr–Newman–(anti) de Sitter black hole spacetime. The original elegant relations we derive, Eqs. (144), (145), represent the most general forms for these parameters of timelike spherical geodesics around the fundamental Kerr–Newman(anti)de Sitter black hole: they have as limits eqns (26), (27) and eqns (101), (102). In The- orem 8 of Sect. 5.3.2 we derive a closed form analytic solution for frame dragging for a timelike polar spherical orbit in Kerr–Newman de Sitter spacetime, Eq. (157). The solution is written in terms of Lauricella’s hypergeometric function F D Fig. 20 Specific energy E− for e = 0.11, =−0.001 for different and Appell’s F1. In Sects. 5.4 and 5.4.1 we derive new exact values for the Kerr parameter solutions for the Lense–Thirring precession and the orbital period for a test particle in a polar non-spherical orbit around 123 147 Page 8 of 37 Eur. Phys. J. C (2021) 81:147

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 ξν is a Killing vector, then for a particle moving along a geodesic, the scalar product of this ν = μ dxν Killing vector and the momentum P dτ of the particle is a constant [19]: ν ξν P = constant (19)

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 4 : Fig. 21 Specific angular momentum L− for e = 0.11, =−0.001 for different values for the Kerr parameter ˜ E μ ν t φ E = = gμνξ U = g U + g φU , μ tt t (20) a Kerr—Newman black hole, Eqs. (167) and (172) respec- ˜ L μ ν t φ tively. In Sects. 5.5 and 5.6 we derive new closed form ana- L = =−gμνψ U =−gφ U − gφφU . μ t (21) lytic expressions for the periapsis advance for test particles in non-spherical polar orbits in KN and KNdS spacetimes Thus, the photon’s emitter is a probe massive test parti- respectively. In the latter case we derive a novel, very ele- cle which geodesically moves around a rotating electrically gant, exact formula in terms of Jacobi’s elliptic function sn charged cosmological black hole in the spacetime with a four- and Lauricella’s hypergeometric function F of three vari- D velocity: ables – see Eq. (202). The exact results of Sects, 5.4, 5.4.1, 5.5, are applied for the computation of the relativistic effects μ = ( t , r , θ , φ) . Ue U U U U e (22) for frame-dragging, pericentre shift and the orbital periods for the observed orbits of stars S2 and S14 in the central The conservation law (19) also applies to photon moving arcsecond of Milky way, assuming that the galactic centre in the curved spacetime. Thus, if the spacetime geometry supermassive black hole is a Kerr–Newman black hole. is time independent, the photon energy P0 is constant. In Sect. 4.1 we will extract the redshift/blueshift of photons from this conservation law. 2 Particle orbits and killing vectors formalism in Kerr–Newman–(anti)de Sitter spacetime 3 Equatorial circular orbits in Kerr–Newman From the condition for the invariance of the metric tensor: spacetime with a cosmological constant α β α ∂gμν ∂ξ ∂ξ 0 = ξ + gαν + gμβ , (14) ∂xα ∂xμ ∂xν It is convenient to introduce a dimensionless cosmological parameter: under the infinitesimal transformation: α α α 1 x = x + ξ (x), with → 0 (15) = M2, (23) 3 it follows that whenever the metric is independent of some and set M = 1. For equatorial orbits Carter’s constant Q van- coordinate a constant vector in the direction of that coordinate ishes. For the following discussion, it is useful to introduce is a Killing vector . Thus the generic metric:

2 2 2 2 2 4 ds = gttdt + 2gtφdtdφ + gφφdφ + grrdr + gθθdθ , As we mentioned already in the introduction, the charged Kerr solution possesses another hidden constant, Carter’s constant Q. Alterna- (16) tively, the complete integrability of the geodesic equations in KN(a)dS spacetime can be understood as follows: The Kerr–Newman family of possesses two commuting Killing vector ﬁelds: spacetimes possesses in addition to the two Killing vectors a Killing tensor ﬁeld Kαβ . This tensor can be expressed in terms of null tetrads μ ξ = (1, 0, 0, 0) timelike Killing vector, (17) (e.g. see eqn (7) in [74]) which implies the existence of a constant = μ ν ψμ = ( , , , ) of motion K KμνU U . This is related to Carter’s constant by: 0 0 0 1 rotational Killing vector. (18) K ≡ Q + (L − aE)2. See also [75,76]. 123 Eur. Phys. J. C (2021) 81:147 Page 9 of 37 147

new constants of motion, the specific energy and specific 1 − r 3 ≥ e2/r. (30) angular momentum: For zero electric charge e = 0 equations (26), (27) reduce E correctly to those in Kerr-anti de Sitter (KadS) spacetimes Eˆ ≡ , μ (24) [48]. For zero electric charge and zero cosmological constant = = L ( e 0) equations (26), (27) reduce correctly to the Lˆ ≡ . μ (25) corresponding ones in Kerr spacetime [84]. Relations (26), (27) have also been discovered independently in [85]. This is equivalent to setting μ = 1. Thus for reasons of In the Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, notational simplicity we omit the caret for the specific energy 16, 17, 18, 19, 20 and 21, for concrete values of the elec- and specific angular momentum in what follows. tric charge and the cosmological parameter we present the Equatorial circular orbits correspond to local extrema of radial dependence of the specific energy and specific angu- the effective potential. Equivalently, these orbits are given lar momentum for various values of the black hole’s spin. by the conditions R(r) = 0, dR/dr = 0,whichhavetobe For the cosmological parameter we choose the values = −5, −4, −3 solved simultaneously. Following this procedure, we obtain 10 10 10 as well as their negative counter- the following novel equations for the specific energy and the parts. For stellar mass black holes, and positive cosmological ∼ −15 −2 − −13 −2 specific angular momentum of test particles moving along constant this corresponds to 10 cm 10 cm . equatorial circular orbits in KN(a)dS spacetime: For supermassive black holes such as at the centre of Galaxy M87 = . × 9 M87 with mass MBH 6 7 10 solar masses [11]the − E±(r; , a, e) = 5 value of 10 corresponds to the value for the cosmological constant: = 3.06 × 10−35cm−2. The physical e2 + r(r − 2) − r 2(r 2 + a2) ± a r 4 1 − − e2 r3 relevance of the graphs in Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, = , (26) 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 and 21 lies in the fol- r 2e2 + r(r − 3) − a2r 2 ± 2a r 4 1 − − e2 r3 lowing fact: the local extrema of the radial profiles of the L±(r; , a, e) specific energy and specific angular momentum of particles

on equatorial circular orbits given by the relations (26) and ±(r 2 + a2) r 4 1 − − e2 − 2ar − ar2(r 2 + a2) + ae2 r3 (27) correspond to the marginally stable orbits that we shall = discuss later. The apparent singularity that appears in some r 2e2 + r(r − 3) − a2r 2 ± 2a r 4 1 − − e2 r3 graphs for negative cosmological constant e.g. in Figs. 17, (27) 18 is due to fact that the quantity inside the larger root in the denominator of the radial profiles in relations (26), (27), and The upper sign in (27)–(26) corresponds to the parallel ori- for a range of radii values, becomes negative and therefore entation of particle’s angular momentum L and black hole the constants of motion become complex numbers. In Figs. 5, spin a (corotation-prograde motion), the lower to the antipar- 4, 12, 13 we plot the radial profiles for the direct constants allel one (counter-rotation or retrograde motion). The reality of motion for the KN(a)dS black hole together with the pro- conditions connected with Eqs. (26) and (27)aregivenby files that correspond to the Kerr case (e = = 0) in order the inequalities: appreciate the electric charge and cosmological constant con- tributions. Horizons of the KN(a)dS geometries are given by the con- 1 2e2 + r(r − 3) − a2r 2 ± 2a r 4 − − e2 ≥ 0 dition KN = 0, which determines pseudosingularities of r 3 r the spacetime interval (1). This condition yields the relations: (28) 2 2 2 r + a + e − 2r 2 2 = ( , , ) ≡ . 2e r − 3 1 e h r a e (31) ⇔ + − a2 ± 2a − − ≥ 0, (29) r 2(r 2 + a2) r 2 r r 3 r 4 Figures 22, 23 exhibit typical behaviour of these functions for positive and negative cosmological constant respectively. and5 For the KNdS black hole spacetime there are three hori-

5 We note that for zero electric charge and >0, inequality (30) deﬁnes the concept of static radius rs at which, the gravitational attrac- Footnote 5 continued tion caused by the central mass is just compensated by the cosmic also for the polytropic structures that could model dark matter halos, repulsion due to cosmological constant. It has been shown that the because it represents the upper limit on their extension, as shown in static radius gives an upper limit on an extension of disk-like structures [79]. Moreover, it is relevant also for motion around galaxies, as dis- around black holes, coinciding with dimensions of large galaxies with cussede.g.in[80–82]. On the other hand, see [83] for arguments about central supermassive black holes [77,78]. The static radius is relevant the more likely detection of in a galaxy which is in the Hubble ﬂow. 123 147 Page 10 of 37 Eur. Phys. J. C (2021) 81:147

Fig. 22 Horizons of the Kerr–Newman–de Sitter spacetimes deﬁned Fig. 23 Horizons of the Kerr–Newman–anti de Sitter spacetimes ( ; , ) ( ; , ) by the function h r a e illustrated for different values of the rota- deﬁned by the function h r a e illustrated for different values of tional (Kerr) parameter a and the electric charge of the black hole the rotational (Kerr) parameter a and the electric charge of the black hole zons (event, apparent or Cauchy, cosmological) while for the + r 2(6 − r + r 3(−15 + 4r) ) + 4e4 + 3e2r(−3 + 4r 3 ) KNadS spacetime there are two horizons (event, apparent). + a2(−4e2 + r[3 + r 2 (1 − 4r 3 )])

3.1 Stability of circular equatorial orbits in 2 1 e Kerr–Newman–de Sitter spacetime × 2 2e2 + r − 3 + r ± 2ar − − − a2r r 3 r 4

The loci of the stable equatorial circular orbits are determined 2 2 1 e × 2e2 + r −3 + r ± 2ar − − − a2r ≥ 0. by the inequality condition: r 3 r 4 (33) d2 R ≥ 0, (32) dr 2 Due to reality conditions (28) we find that the radii of the stable orbits in Kerr–Newman—de Sitter spacetime are which has to be satisfied simultaneously with the conditions restricted by the inequality: R(r) = 0 and dR/dr = 0 determining the specific energy 2 3/2 and specific angular momentum of the constant radius orbits. 1 e ∓ 8ar6 − − Using relations (26)–(27) we find that: r 3 r 4 + 2( − + 3(− + )) + 4 + 2 (− + 3) r 6 r r 15 4r 4e 3e r 3 4r 2 2 3/2 d R 1 e 2 2 2 3 ≥ 0 ⇔ ∓8ar6 − − + a (−4e + r[3 + r (1 − 4r )]) ≥ 0. (34) dr 2 r 3 r 4 The marginally stable orbits can be obtained by solving the quadratic equation (embedded in (34)) for the rotational parameter a procedure that yields the relation:

2 = 2 ≡ ( − 3 − 2/ )3 2 − [− 2 + ( + 2( − 3))]{ 2( − + 3(− + )) a ams(1,2) 128r 1 r e r r 4 4e r 3 r 1 4r r 6 r r 15 4r √ / + 4e4 + 3e2r(−3 + 4r 3 )}±32[r(1 − r 3 − e2/r)3]1 2r r 3 × (1 − 4r 3) ⎫ ⎬ 3e2 × −2 + 3r − r 2(6 + 10r − 15r 3) − 4e4 + (1 + 4r 3) − 4e2 + 9e2r − 12r 4e2 r ⎭

− − 2 × 4 1 −4e2 + r(3 + r 2 (1 − 4r 3 )) . (35)

123 Eur. Phys. J. C (2021) 81:147 Page 11 of 37 147

From (35) we deduce the following reality conditions : where the index A refers to the emission (e) and/or detection (d) at the corresponding point P . 1 A ≤ (r) ≡ , (36) The emission frequency is deﬁned as follows: ms 4r 3 and μ ωe = kμU 2 3 4 t r θ φ − 2 + 3r − r (6 + 10r − 15 r ) − 4e = kt U + kr U + kθ U + kφU 2 t φ r r θ θ 3e = (k E − k L + g k U + gθθk U )| . (42) + (1 + 4r 3 ) − 4e2 + 9 e2r − 12r 4 e2 ≥ 0 rr e r (37) Likewise the detected frequency is given by the expression: ⇔ 215r 5 + μ ω =+kμU −r 2(6 + 10r) − 4e4 + 12e2r 2 + 9e2r − 12r 4e2 d t φ r r θ θ = (Ek − Lk + grrk U + gθθk U )|d . (43) 3e2 − 2 + 3r + − 4e2 ≥ 0. (38) r In producing (42), (43) we used the expressions for U t and U φ in terms of the metric components and the conserved From (38) and the fact that 15r 5 > 0 we derive two more quantities E, L: conditions: ≤ or ≥ , (39) −Egφφ − Lg φ ms− ms+ U t = t , (44) 2 − gtφ gttgφφ where ms± are the two roots of the quadratic equation (38). A detailed discussion of stability of the geodesic circular φ gtt L + gφt E U = . (45) 2 − equatorial motion in the KNdS spacetimes is presented in gtφ gttgφφ [85]. Thus, the frequency shift associated to the emission and 3.1.1 Stability of equatorial circular geodesics in Kerr-de detection of photons is given by either of the following rela- Sitter spacetime tions:

= ωe Our results in inequality (34), for zero electric charge, e 0, 1 + z = give a condition6 that agrees with the results in [49] and ωd φ θ θ restricts the radii of the stable orbits in Kerr–de Sitter space- ( t − + r r + θθ )| = k E k L grrk U g k U e time: ( t − φ + r r + θ θ )| Ek Lk grrk U gθθk U d t φ r r θ θ 1 (Eγ U − Lγ U + grr K U + gθθ K U )|e ± ar2 − (− + r 3) = 8 3 1 t φ r r θ θ r (Eγ U − Lγ U + grr K U + gθθ K U )|d +r(6 − r + r 3(−15 + 4r) ) (46) +a2(3 + r 2 (1 − 4r 3 )) ≥ 0. (40) This is the most general expression for the redshift/blueshift that light signals emitted by massive test particles experience 4 Gravitational redshift-blueshift of emitted photons in their path along null geodesics towards a distant observer (ideally located near the cosmological horizon in particular In this section we will provide general expressions for the or at spatial inﬁnity assuming a zero cosmological constant). redshift/blueshift that emitted photons by massive particles experience while travelling along null geodesics towards an 4.1 The redshift/blueshift of photons for circular and observed located far away from their source. equatorial emitter/detector orbits around the In general, the frequency of a photon measured by an Kerr–Newman–(anti) de Sitter black hole μ observer with proper velocity U at the spacetime point PA A θ reads [19,55]: For equatorial circular orbits Ur = U = 0 thus μ t φ t φ ω = kμU | , (41) (Eγ U − Lγ U )| U − U | A A PA + = e = e 1 z t φ t φ (Eγ U − Lγ U )|d U − U |d 6 = = φ φ We observe that for e 0, (34) reduces correctly to the equation U t − U U t − U that determines the radii of marginally stable orbits in Kerr spacetime: = e e e = e e , √ φ φ (47) r 2 − 6Mr − 3a2 ∓ 8a Mr = 0forM = 1, [84]. t − t − Ud dUd Ud Ud 123 147 Page 12 of 37 Eur. Phys. J. C (2021) 81:147

U t = γ / γ 7 = , + = e where L E . For 0 1 zc t . Following 2(aEγ sin2 θ−Lγ )2 Ud Qγ + ( γ − γ )2 − θ θ L aE θ 2 θ the procedure for the Kerr black hole in [55], we consider the (k )2 = sin , (52) ρ4 kinematic redshift of photons either side of the line of sight 2[(r2 + a2)Eγ − aLγ ]2 − KN(Qγ + 2(Lγ − aEγ )2) that links the Kerr–Newman–de Sitter black hole and the (kr )2 = r . ρ4 observer, and subtract from Eq. (47) the central value zc.We (53) note that zc corresponds to a gravitational frequency shift of a photon emitted by a static particle located in a radius equal We must take into account the bending of light from the rotat- to the circular orbit radius and on the signal line going from ing and charged Kerr–Newman–(anti) de Sitter black hole. the centre of coordinates to the far detector [55,86]. Then we In other words we need to find the apparent impact param- obtain: eter for every orbit, i.e. to find the map, = (r) as a function of the radius r of the circular orbit associated with t − φ t Ue eUe Ue zkin ≡ z − zc = − the emitter. The criteria employed in [55] to construct this t − φ U t Ud dUd d mapping is to choose the maximum value of z at a fixed dis- φ φ U U t − U U t tance from the observed centre of the source at a fixed and = d e e d . φ (48) that the apparent impact parameter must also be maximised. U t U t − U U t d d d d The apparent impact factor γ ≡ Lγ /Eγ can be obtained μ from the expression k kμ = 0 8 as follows: We further comment on the different gravitational frequency shifts of photons included in (47) and (48). In Eq. (47), the μ t φ k kμ = 0 ⇔ k k + k kφ = 0 redshift/blueshift is indeed gravitational, but it includes an t equivalent Doppler effect (redshift/blueshift) as the emitter Eγ gφφ + gtφ Lγ ⇔ (−Eγ ) moves towards/away from the observer along the timelike 2 − gtφ gφφgtt circular orbit and, additionally, two gravitational effects: a −Lγ gtt − Eγ gφt gravitational redshift for the photon emitted by a static par- + Lγ = 0 2 − ticle and a redshift/blueshift due to the rotation of the space gtφ gφφgtt time (as is the case for the Kerr–Newman–de Sitter black 2 ⇔ gφφ + 2gtφγ + γ gtt = 0(54) hole spacetime). Let us now consider photons with 4-momentum vector Solving the quadratic equation we obtain: kμ = (kt , kr , kθ , kφ) which move along null geodesics μ k kμ = 0 outside the event horizon of the Kerr–Newman–de Sitter black hole, which explicitly can be expressed as

t 2 t φ φ 2 r 2 θ 2 0 = gtt(k ) + 2gtφ(k k ) + gφφ(k ) + grr(k ) + gθθ(k ) . (49) 2θ (r 2 + a2)[(r 2 + a2)Eγ − aLγ ]−a2KN(aEγ sin2 θ − Lγ ) kt = r KN ρ2 r θ Eγ [2θ (r 2 + a2)2 − a2 sin2 θ2KN]+Lγ [−a2θ (r 2 + a2) + a2KN] = r r , (50) KN ρ2 r θ −2KN(aEγ sin2 θ − Lγ ) + a2θ sin2 θ[(r 2 + a2)Eγ − aLγ ] kφ = r KN ρ2 2 θ r θ sin [−2KNa sin2 θ + a2θ sin2 θ(r 2 + a2)]Eγ + Lγ [2KN − a22θ sin2 θ] = r r , (51) KN ρ2 2 θ r θ sin

7 Since the constants of motion Eγ and Lγ are preserved along the null geodesics followed by the photons from emission till detection, we have e = d = , i.e. this quantity is also constant along the whole photons trajectory. 8 Taking into account that kr = 0andkθ = 0. 123 Eur. Phys. J. C (2021) 81:147 Page 13 of 37 147

Substituting the expressions (26)–(27)forE± and L± into t ( ,π/ ), φ( ,π/ ) −gφ ± g2 − gφφg U r 2 U r 2 we finally obtain remarkable novel ± t tφ tt γ = expressions for these four-velocity components in Kerr– g tt Newman–(anti) de Sitter spacetime: a(KN − (r 2 + a2)) ± r 2 KN = r r , (55) KN − 2 t r a U (r,π/2) + − where we got two values, and (either evaluated at (r2 ± a −e2 + r4 1 − )2 γ γ r3 = , the emitter or detector position, since this quantity is pre- (60) served along the null geodesic photon orbits, i.e., = ) r 2e2 + r(r − 3) − r2a2 ± 2a −e2 + r4 1 − e d r3 that give rise to two different shifts respectively, z1 and z2 φ( ,π/ ) of the emitted photons corresponding to a receding and to U r 2 an approaching object with respect to a far away positioned ± −e2 + r4 1 − 2 r3 observer: = . (61) 2 + ( − ) − 2 2 ± − 2 + 4 1 − − φ φ φ φ r 2e r r 3 r a 2a e r 3 U U t − −U U t −U U t − −U U t r = d d e e e d = e d e e e d , z1 − φ − φ U t (U t − U ) U t (U t − U ) d d d d d d e d (56) We now compute the angular velocity : + φ φ φ φ U U t − +U U t +U U t − +U U t dφ 1 = d d e e e d = e d e e e d . ≡ = . z2 φ φ 3/2 (62) t ( t − + ) t ( t − + ) dt a ± r Ud Ud d Ud Ud Ud e Ud 2 1−r3− e (57) r In terms of the angular velocities the quantities z , z read In general the two values z and z differ from each other due 1 2 1 2 as follows: to light bending experienced by the emitted photons and the − φ − differential rotation experienced by the detector as encoded U t − −U U t [ − − ] φ t 9 d d e e e e d d e e in U and U components of the four-velocity . z1 = = − d d t − − φ U t (1 − ) In order to get a closed analytic expression for the grav- Ud d Ud d d d t − itational redshift/blueshift experienced by the emitted pho- U [d − e] = e e , (63) tons we shall express the required quantities in terms of the t ( − − ) Ud 1 e d φ t Kerr–Newman–(anti) de Sitter metric. Thus, the U and U + t − + φ + t − + φ d dUe e Ue d dUe e Ue components of the four-velocity for circular equatorial orbits z2 = = + t − + φ U t (1 − ) read: Ud d Ud d d d t [+ − + ] t +[ − ] Ue d d e e Ue e d e t = = . (64) U (r,π/2) t ( − + ) t ( − + ) Ud 1 d d Ud 1 e d −(KN 2 − ( 2 + 2)2) − (− (KN − ( 2 + 2))) = r a r a E L a r r a , 2 2KN r r Thus for the Kerr–Newman–(anti) de Sitter black hole we 4 can write for the redshift and blueshift, respectively: (58) √ φ − − − [ 3/2 ± G ] ( ,θ = π/ ) d d e e re a e U r 2 zred = − − / / √ / √ 1 d d 3 4 2e2 + 3 2 − − 3 2 2 ± G 2 KN 2 2 KN 2 2 re 1/2 re 3 re re a 2a e ( − a )L + E (−a( − (r + a ))) re = r r . 2 KN / / √ / √ r 3 4 2e2 3 2 3 2 2 r r / + r − 3 rd − r a ± 2a Gd d r1 2 d d (59) × d √ , (65) 3/2 ± G rd a d √ + − + [ 3/2 ± G ] 9 The second term in the denominator in Eqs. (56)–(57) encodes the d d e e re a e zblue = + − / / √ / √ contribution of the movement of the detector’s frame [55]. If this quan- 1 d d 3 4 2e2 + 3 2 − − 3 2 2 ± G re 1/2 re 3 re re a 2a e t re tity is negligible in comparison to the contribution steming from the Ud φ t / / √ / √ component (U U ) then the detector can be considered static at spa- 3 4 2e2 3 2 3 2 2 d d r / + r − 3 rd − r a ± 2a Gd d r1 2 d d tial infinity for the case of Kerr black hole. There are no static observers d × √ , (66) in infinity of the Kerr–de Sitter (KdS) spacetimes [53]. Indeed, in the 3/2 ± G rd a d KdS black hole spacetimes the free static observers are expected near the static radius representing the outermost region of gravitationally bound systems in Universe with accelerated cosmic expansion-see discussion in [53]. 123 147 Page 14 of 37 Eur. Phys. J. C (2021) 81:147 where now re and rd stand for the radius of the emitter’s and where we define: detector’s orbits, respectively. We also define KN 2 2 2 2 (re) := (1 − r )(r + a ) − 2re + e (71) r e e 1 and we have made use of the relation e = d .Theremark- G ≡−e2/r + r 3 − , (67) d d d 3 able closed form analytic expressions for the frequency shifts rd we obtained in Eqs. (69)–(70), constitute a new result in the 1 G ≡−e2/r + r 3 − theory of General Relativity, in which all the physical param- e e e 3 (68) re eters of the exact theory enter on an equal footing. 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, rd , as follows:

√ 3/4 2 3/2 √ 3/2 √ √ 2e + − − 2 ± G 3/2 3/2 rd 1/2 rd 3 rd rd a 2a d (− 2( 2 + 2) − + 2) − 2 KN( ) (±[ G − G ]) r a re re a 2re e re r re re d rd e = d √ √ , zred √ 3/2 3/2 (69) / 2 / √ / ( ± G )[(KN( ) − 2) + ( 2 + 2 KN( ))(± G )] 3 4 2e + 3 2 − − 3 2 2 ± G r a d r re a r are re r re d re 1/2 re 3 re re a 2a e d d re

√ 3/4 2 3/2 √ 3/2 √ √ 2e + − − 2 ± G 3/2 3/2 rd 1/2 rd 3 rd rd a 2a d (− 2( 2 + 2) − + 2) + 2 KN( ) (±[ G − G ]) r a re re a 2re e re r re re d rd e = d √ √ , zblue √ 3/2 3/2 (70) / 2 / √ / ( ± G )[(KN( ) − 2) + ( 2 − 2 KN( ))(± G )] 3 4 2e + 3 2 − − 3 2 2 ± G r a d r re a r are re r re d re 1/2 re 3 re re a 2a e d d re

In the particular case when the detector is located far away from the source and the detector radius rd is much larger than the black hole parameters ( its mass,spin and electric charge), the frequency shifts (69)–(70) become:

√ √ 2 1 − a2 a(r 2(r 2 + a2) + 2r − e2) + r 2 (r ) ± 1 − r 3 − e e e e e r e e re z = , (72) red / / √ / 3 4 2e2 + 3 2 − − 3 2 2 ± − e2 + 3 1 − [ ( ) − 2] re 1/2 re 3 re re a 2a re 3 r re a r re re e √ √ 2 1 − a2 a(r 2(r 2 + a2) + 2r − e2) − r 2 (r ) ± 1 − r 3 − e e e e e r e e re z = . (73) blue / / √ / 3 4 2e2 + 3 2 − − 3 2 2 ± − e2 + 3 1 − [ ( ) − 2] re 1/2 re 3 re re a 2a r re 3 r re a re e re

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4.1.1 Redshift/blueshift for circular equatorial orbits in Kerr–de Sitter spacetime

For zero electric charge, e = 0, Eqs. (69)–(70) reduce to: / / √ / 3 4 3 2 3 2 2 3 1 r r − 3 rd − r a ± 2a r − √ / / d d d d r 3 [ (− 2( 2 + 2) − ) − 2 ( )](±[ 3 2 − 3 − 3 2 − 3]) d a re re a 2re re r re re 1 rd rd 1 re zred = √ , (74) 3/2 3/2 / / √ / (r ± a 1 − r 3)[( (r ) − a2)r + (ar2 + r 2 (r ))(± 1 − r 3)] 3 4 3 2 − − 3 2 2 ± 3 1 − d d r e d e e r e d re re 3 re re a 2a re 3 re / / √ / 3 4 3 2 3 2 2 3 1 r r − 3 rd − r a ± 2a r − √ / / d d d d r 3 [ (− 2( 2 + 2) − ) + 2 ( )](±[ 3 2 − 3 − 3 2 − 3]) d a re re a 2re re r re re 1 rd rd 1 re zblue = √ . (75) 3/2 3/2 / / √ / (r ± a 1 − r 3)[( (r ) − a2)r + (ar2 − r 2 (r ))(± 1 − r 3)] 3 4 3 2 − − 3 2 2 ± 3 1 − d d r e d e e r e d re re 3 re re a 2a re 3 re

In Eqs. (74), (75) we deﬁne: The frequency shifts (77)–(78) are plotted in Figs. 24 and ( ) := ( − 2)( 2 + 2) − . r re 1 re re a 2re (76) 25 for different values of the spin of the central black hole and for positive cosmological constant for the corotating case. As In the particular case when the detector is located far away the radius increases, zred →−zblue. from the source and the detector radius rd is much larger than the black hole parameters , the frequency shifts (69)–(70) become: 5 More general orbits for rotating charged black holes

√ √ − 2 ( 2( 2 + 2) + ) + 2 ( ) ± − 3 5.1 Spherical orbits in Kerr–Newman spacetime 1 a a re re a 2re re r re 1 re z = , red / / √ / 3 4 3 2 − − 3 2 2 ± 3 1 − [ ( ) − 2] re re 3 re re a 2a re 3 r re a Depending on whether or not the coordinate radius r is con- re (77) stant along a given timelike geodesic, the corresponding √ √ − 2 ( 2( 2 + 2) + ) − 2 ( ) ± − 3 particle orbit is characterised as spherical or nonspherical, 1 a a re re a 2re re r re 1 re z = . blue respectively. In this subsection we will focus on spherical / / √ / 3 4 3 2 − − 3 2 2 ± 3 1 − [ ( ) − 2] re re 3 re re a 2a re 3 r re a non-equatorial orbits. re (78) 5.1.1 Frame-dragging for timelike spherical orbits

Assuming = 0 we derive from (7) and (10) the following equation:

aP − + / 2 θ dφ KN aE L sin =− . (79) θ 2 2 θ d Q − L cos + a2 cos2 θ(E2 − 1) sin2 θ

In (79) KN := r 2 + a2 + e2 − 2Mr.10 Also P = E(r 2 + a2) − La. Using the variable z = cos2 θ, − 1 √dz √ 1 = 2 z 1−z sgn(π/2−θ)dθ we will determine for the ﬁrst time in closed analytic form the amount of frame-dragging for timelike spherical orbits in the Kerr–Newman spacetime.11

10 When we set M = 1, KN = r 2 + a2 + e2 − 2r. 11 We should mention at this point the extreme black hole solutions for spherical timelike non-polar geodesics in Kerr–Newman spacetime Fig. 24 The functions zred, zblue as functions of radius re. The spin of obtained in [87] in terms of formal integrals. Latitudinal motion in the the Kerr–Newman–de Sitter black hole was chosen as a = 0.52 and the Kerr and KN spacetimes has been studied in [88] and complemented in −33 dimensionless cosmological parameter as = 10 [89]. 123 147 Page 16 of 37 Eur. Phys. J. C (2021) 81:147 1 α− γ −α− −β −β u 1(1 − u) 1(1 − ux) (1 − uy) du 0 (α)(γ − α) = F (α,β,β,γ,x, y). (γ) 1 (83) Also (a) denotes Euler’s gamma function.

Proof We compute ﬁrst the integral: z− L 1 dz 1 1 − √ √ . − 2 ( − )( − ) z j 1 z 2 z |a| 1 − E z z+ z z− (84)

2 Applying the transformation z = z− + ξ (z j − z−) in (84) we obtain Fig. 25 The functions zred, zblue as functions of radius re. The spin of z− the Kerr–Newman–de Sitter black hole was chosen as a = 0.9939 and L 1 dz 1 1 − − √ √ = 33 − 2 ( − )( − ) the dimensionless cosmological parameter as 10 z j 1 z 2 z |a| 1 − E z z+ z z− −L z− − z 1 = √ j √ √ √ Thus for instance, expressed in terms of the new variable: 2|a| 1 − E2 (1 − z−) z− z− − z+ z j − z− 1 x × d 1 √1 1 θ z j −z− z−−z z−−z L d L 1 dz 1 0 [1 − x ] − j x − j √ = − √ 1−z− 1 x z−−z+ 1 x z− 2 θ 1 − z 2 z α 2 − (α + β) + sin z z Q −L z− − z 1 = √ j √ √ √ L 1 dz 2|a| 1 − E2 (1 − z−) z− z− − z+ z j − z− = − √ 1 − z 2 z 1 (1) × 2 1 1 3 × √ , (80) 2 2 ( − )( − ) |a| 1 − E z z+ z z− 1 1 1 3 z j − z− z− − z j z− − z j × FD , 1, , , , , , , (85) 2 2 2 2 1 − z− z− z− − z+ where α = a2(1 − E2), β = L2 + Q. The range of z for where x ≡ ξ 2.InEq.(85) F denotes Lauricella’s fourth which the motion takes place includes the equatorial value, D multivariable hypergeometric function (here is a function of z = 0: three variables). The general multivariable Lauricella’s function F is defined as follows: 0 ≤ z ≤ z−. (81) D !∞ (α) +··· (β ) ···(β ) F (α, β,γ,z) = n1 nm 1 n1 m nm zn1 ···znm D (γ ) ( ) ···( ) 1 m (86) The integral of (79) will be split into two parts: Propositions 1 n1+···+nm 1 n1 1 nm n1,n2,...,nm =0 and 3. We prove first the exact result of Proposition 1: where Proposition 1 z = (z1,...,zm), β = (β ,...,β ). z− 1 m (87) L 1 dz 1 1 − √ √ (α) = (α, ) 0 1 − z 2 z |a| 1 − E2 (z − z+)(z − z−) The Pochhammer symbol m m is defined by π L −1 −1/2 (α + m) =− √ (1 − z−) (z+ − z−) (α) = |a| 1 − E2 2 m (α) 1 1 z− z− , m = × F , 1, , 1, , = 1 if 0 1 − − α(α + ) ···(α + − ) = , , (88) 2 2 z− 1 z− z+ 1 m 1 if m 1 2 3 L π −1/2 1 1 z− =− √ z+ F1 , , 1, 1, , z− , (82) The series (86) admits the following integral representation: |a| 1 − E2 2 2 2 z+ 1 (γ) α− γ −α− F (α, β,γ,z)= t 1(1−t) 1(1− D (α,β,β ,γ, , ) (α)(γ − α) 0 where F1 x y denotes the first Appell’s hyper- −β −β ) 1 ···( − ) m geometric function of two variables (x and y), which admits z1t 1 zmt dt the integral representation: (89) 123 Eur. Phys. J. C (2021) 81:147 Page 17 of 37 147

Table 1 Properties including frame-dragging (Lense–Thirring precession) of spherical timelike Kerr–Newman (prograde) orbits, applying the exact analytic formula (95) and the ﬁrst branch of solutions for the constants of motion in Eqs. (101), (102) Orbit r/MQ/M2 EL/Me/Ma/M φGT R

(a) 7 2 0.9329746347090826 2.6203403438437856 0 0.999999 6.85204 (b) 7 2 0.9329917521582036 2.6193683198162256 0.11 0.9939 6.84936 (c) 7 2 0.9330249803496844 2.6231975525737035 0 0.9939 6.84944 (d) 7 2 0.9347100491303167 2.71403958181262 0.11 0.8 6.76005 (e) 7 2 0.9346178572238036 2.7048064351873267 0.2 0.8 6.75993 (f) 4 8 0.9187670265667813 0.9264577651658418 0 0.9939 7.77146 (g) 4 8 0.9185597216627694 0.9130252832603007 0.11 0.9939 7.77188 (h) 4 8 0.9263820446220102 1.1924993953748733 0 0.8 7.49676 (i) 4 8 0.9253520786762164 1.1444194140656208 0.2 0.8 7.49741

Table 2 Properties including frame-dragging (Lense–Thirring precession) of spherical timelike Kerr–Newman (retrograde) orbits, applying the exact analytic formula (95) and the second branch of solutions for the constants of motion in Eqs. (101), (102) Orbit r/MQ/M2 EL/Me/Ma/M φGT R

(a) 7 12 0.9500313121192591 −1.3504453996037784 0 0.999999 −5.58661 (b) 7 12 0.9499790519730756 −1.3443270259214553 0 0.9939 −5.59099 (c) 7 12 0.9497202606484783 −1.313994954227598 0.11 0.9939 −5.59138 (d) 7 12 0.9482593233936484 −1.1250483727157161 0.11 0.8 −5.72968 (e) 7 12 0.9477180186254757 −1.046386864788553 0.2 0.8 −5.73048 (f) 10 1 0.9626040652152477 −4.1133141793233525 0 0.9939 −5.8465 (g) 10 1 0.9625680840381455 −4.109699623451054 0.11 0.9939 −5.84647 (h) 10 1 0.9611085421961251 −4.008398488785058 0.11 0.8 −5.93674 (i) 10 1 0.961033868172724 −4.000352717478337 0.2 0.8 −5.93669

which is valid for Re(α) > 0, Re(γ − α) > 0. .Itcon- For producing the result in the last line of Eq. (91), we used verges absolutely inside the m-dimensional cuboid: the following transformation property of Appell’s hypergeometric function F1: |z j | < 1,(j = 1,...,m). (90) Lemma 2 For m = 2 FD in the notation of Appell becomes the two variable hypergeometric function F (α,β,β ,γ,x, y) with +β−γ γ −α− −β 1 y1 (1 − y) 1(x − y) integral representation given by Eq. (83). Setting z j = 0in y y (85) yields: × F − β,β, + α − γ, + β − γ, , 1 1 1 2 − − y x y 1 z− L 1 dz 1 1 +β−γ −β − √ √ = y1 x 0 1 − z 2 z |a| 1 − E2 (z − z+)(z − z−) y − × F 1 + β + β − γ,β,1 + α − γ,2 + β − γ, , y . L −1 −1/2 1 = √ (1 − z−) (z+ − z−) x 2|a| 1 − E2 (92) 1 ( ) − 2 1 1 1 1 3 z− z− × FD , 1, , , , , 1, 3 2 2 2 2 1 − z− z− − z+ On the other hand we compute analytically the second 2 − integral that contributes to frame-dragging and we obtain: L −1 −1/2 = √ (1 − z−) (z+ − z−) 2|a| 1 − E2 Proposition 3 ( / )2 3 − 1 2 2 1 1 z− z− × F1 , 1, , 1, , 3 2 2 1 − z− z− − z+ z− aP − aE 2 KN√ θ − d L −1/2 1 1 z− = √ z+ π F1 , , 1, 1, , z− . (91) 0 2|a| 1 − E2 2 2 z+ aP 1 −1 = − aE √ KN |a| 1 − E2 2

123 147 Page 18 of 37 Eur. Phys. J. C (2021) 81:147 1 1 ( , ; , , ) 2 2 1 1 z− Lα,β r Q a e M × √ F , , 1, − √ z+ − z− 2 2 z+ − z− 2 2 3 2 2 (r + a ) ∓ ϒ + aQ + 2aMr − ae r aP 1 π 1 =− , = − aE √ − √ √ (98) KN | | − 2 2 z+ r 2 2e2r 2 + r 3(r − 3M) − 2a(aQ ∓ ϒ) a 1 E × 1, 1, , z− . F 1 (93) where 2 2 z+ ϒ ≡ a2 Q2 + r 2 −2e2 Q + 3Qr M − (e2 + Q)r 2 + r 3 M . In Eq. (93) F(a, b, c, z) with a, b, c ∈ R and c ∈/ Z≤0 denotes the Gauß’ hypergeometric function which is deﬁned (99) by: The third and fourth classes of solutions are related to the ∞ ! (a) (b) ﬁrst two by: F(a, b, c, z) = n n zn. (94) (c)nn! (Eγ,δ, Lγ,δ) =−(Eα,β , Lα,β ). (100) n=0 We thus obtain the following fundamental result, in closed analytic form, for the amount of frame-dragging that a time- Theorem 5 Using dimensionless parameters or equivalently like spherical orbit in Kerr–Newman spacetime undergoes. setting M = 1, the constants of motion that solve the conditions for spherical Kerr–Newman orbits are given by the equations: Theorem 4 As θ goes through a quarter of a complete oscil- GTR Eα,β (r, Q; a, e) (101) lation we obtain the change in azimuth φ, φ : √ 2 2 + 3( − ) − ( ∓ ϒ) = e r r r 2 a aQ π √ GTR L −1/2 1 1 z− 2 2 2 + 3( − ) − ( ∓ ϒ) φ = √ z+ F1 , , 1, 1, , z− r 2e r r r 3 2a aQ |a| 1 − E2 2 2 2 z+ aP 1 π 1 + − √ Lα,β (r, Q; a, e) (102) aE √ KN |a| 1 − E2 2 z+ √ (r 2 + a2) ∓ ϒ + aQ + 2ar3 − ae2r 2 × 1, 1, , z− . =− √ , F 1 (95) 2 2 2 3 2 2 z+ r 2e r + r (r − 3) − 2a(aQ ∓ ϒ)

where Proof Using Propositions 1 and 3,weproveEq.(95) and Theorem 4. ϒ ≡ a2 Q2 + r 2 −2e2 Q + 3Qr − (e2 + Q)r 2 + r 3 . (103)

In Eq. (95) the quantities z± are given by: a2(1 − E2) + L2 + Q ∓ (a2(E2 − 1) − L2 − Q)2 − 4a2(1 − E2)Q z∓ = . (96) 2a2(1 − E2)

Equations (101), (102), of Theorem 5, for zero electric 5.1.2 Conditions for spherical Kerr–Newman orbits charge (e = 0) reduce correctly to the corresponding ones in Kerr spacetime [90,91]. In order for a non-equatorial spherical (NES) orbit in Kerr– In order that the smaller square root appearing in the solu- Newman spacetime to exist at radius r, the conditions R(r) = tions (101), (102) is real, we need to impose the condition dR = dr 0 must hold at this radius. As in Sect. 3 where we investigated equatorial circular geodesics, we solve these ϒ ≥ 0 (104) two equations simultaneously, and the solutions now take ⇔ a2 Q2 + Q(−2e2r 2 + 3r 3 M − r 4) − e2r 4 + r 5 M ≥ 0 an elegant compact form when parametrised in terms of r (105) and Carter’s constant Q. There are four classes of solutions (Ei , Li ) which we label by i = α, β, γ, δ. We obtain the Since ϒ is quadratic in Carter’s constant Q its two roots are: following novel relations for the first two: 2 √ r 2 Q , = r(r − 3M) + 2e ± S , (106) 1 2 2a2 Eα,β (r, Q; a, e, M) √ e2r 2 + r 3(r − 2M) − a(aQ ∓ ϒ) = √ (97) r 2 2e2r 2 + r 3(r − 3M) − 2a(aQ ∓ ϒ) 123 Eur. Phys. J. C (2021) 81:147 Page 19 of 37 147 where We also remark that the condition α,β > 0 will imply that the numerator of (101) is positive. Indeed, from (108) S ≡ r 4 − 6Mr3 + 9M2r 2 + 4e4 − 4e2r(3M − r) we deduce: +4a2(e2 − rM). (107) √ 2e2r 2 + r 3(r − 3M) − 2a(aQ ∓ ϒ) > 0 Now S is a quartic equation in r that is related to the study of √ 1 ⇔−a(aQ ∓ ϒ) > − r 3(r − 3M) − e2r 2 (110) circular photon orbits in the equatorial plane around a Kerr– 2 Newman black hole [92]. The two largest roots of S r1, r2 are the radii of the prograde abd retrograde photon orbits, respec- so that ≤ < < ≤ tively. They lie in the ranges M r1 3M r2 4M. √ 1 e2r 2 + r 3(r − 2M) − a(aQ ∓ ϒ) > r 3(r − M)>0. The locations of these two photon orbits will be significant 2 in what follows, as they will demarcate the allowed radii of (111) the timelike spherical orbits. It is useful to note that S is negative in the range r1 < r < r2. This means that real solu- The reality conditions from the square roots allow for the tions for Q1,2 only exist outside this range. We have checked Carter constant Q to be negative. In the uncharged Kerr black that Q2 ≤ Q1 < 0 when r ≤ r1, and 0 < Q2 ≤ Q1 when hole it was argued that negative Q for spherical orbits were r ≥ r2. not allowed [91]. In fact a necessary condition for negative Since ϒ is quadratic in Q with positive leading coefficient Q with E2 > 1 was presented in [91]: 2 (a > 0), inequality (105) is satisfied when Q ≤ Q2 or a2(1 − E2) + Q + L2 < 0. (112) Q ≥ Q1. Thus when r < r1 or r > r2, the allowed ranges for Carter’s constant are Q ≤ Q and Q ≥ Q . On the other 2 1 For the Kerr black hole it was argued that this condition hand, when r ≤ r ≤ r , there is no restriction on the range 1 2 cannot be satisfied thus ruling out the case of negative Q for of Q. spherical orbits [91]. Following a similar analysis with [91], For the solutions (101), (102) to be valid, the larger square we will discuss now the non-negativity of Q for the more root appearing in them also has to be real. Thus we must general Kerr–Newman black hole. Substituting (101), (102) impose the condition: into the left-hand side of (112) we obtain: √ 2 2 3 α,β ≡ 2e r + r (r − 3M) − 2a(aQ ∓ ϒ) ≥ 0. (108) KN 2 2 2 α,β a (1 − Eα,β ) + Q + Lα,β = , (113) r 2α,β We first find the values of r and Q for which α,β = 0. Initially we obtain: where

4 KN 3 3 2 2 αβ = r S, (109) α,β ≡ r (r − M)(Mr + a Q) − a ϒ √ 2 + Mr3 + a2 Q ∓ 2a ϒ thus we see that α may vanish only if r = r1 or r = r2, and √ similarly for β . For either value of r, we have checked that + e2r 2 −r 4 + r 2a2 ± 2a ϒ . (114) α = 0ifQ ≥ Q2 and α(r1,2)>0ifQ < Q2(r1,2).On the other hand, β = 0ifQ ≤ Q2, and β < 0ifQ > Q2. Since we want to impose α,β > 0, we deduce that the first The ranges of Carter’s constant Q we have found so far ensure α,β class of solutions (α) is allowed only if Q < Q2. The second that the quantities are positive. It remains to show that KN class of solutions (β) is not allowed at all. α,β are also positive for these ranges. KN KN When r < r1 or r > r2 recall that Q is allowed to√ take We have checked by explicit calculation that α β ≤ ≥ = 4S KN the ranges Q Q2 and Q Q√1. Note that α,β r is quadratic in the invariant Q. For fixed r, α may vanish 4 KN when Q = Q2, and α,β =− r S when Q = Q1. Thus only at the roots of this quadratic, and likewise for β . we deduce that α,β > 0ifQ ≤ Q2 and α,β < 0if We have checked that the discriminant of this quadratic is Q ≥ Q1. Consequently, the range Q ≥ Q1 is ruled out for negative, thus both roots are complex numbers in this case. KN these cases. We infer that α,β do not vanish for any real value of Q. < < KN When r1 r r2, recall that there is no√ restriction on Since α,β are continuous functions of Q for the ranges we the range of Q. We observe that α,β =± −r 4S when are interested in, it follows that their signs do not change as r2 Q = (2e2 + r(r − 3M)). Thus we conclude that α > 0 Q is varied. Thus we still have to check that KN and/or 2a2 α KN and β < 0 for any value of Q. As a result, the second class β are positive for a specific value of Q in each of the of solutions (β) is not allowed in this case. allowed ranges we have found. 123 147 Page 20 of 37 Eur. Phys. J. C (2021) 81:147

When r < r1 or r > r2, recall that Q is allowed to take 5.1.3 Frame-dragging for spherical polar Kerr–Newman the range Q ≤ Q2. We compute: geodesics

In this subsection we shall investigate the physically impor- KN α,β (Q2) tant class of polar orbits of massive particles, i.e. timelike 2 √ KN r 2 geodesics crossing the symmetry axis of the Kerr–Newman = α,β (r(r − 3M) + 2e − S) 2a2 spacetime. The ﬁrst integral in Eq. (10) for zero cosmological 1 √ constant reads: = r 4(r(r − M) + e2 − S)2 + Mr5KN 2 ρ4θ˙2 = Q + a2(E2 − 1) cos2 θ − L2 cot2 θ. (117) 2e2r 4(2rM − e2 − 2r 2) + , (115) It follows from (117) that in order for the orbit to reach the 4 polar axis, where cos2 θ = 1, it is necessary that [93]

KN L = 0. (118) which is positive. Thus α,β > 0ifQ ≤ Q2 When r = r1 or r = r2, recall that Carter’s constant Q The polar orbits considered in this subsection, mean that our is allowed to take the range Q < Q2, although only the first test particle is supposed not only to cross the symmetry axis class of solutions (α) is allowed. The above argument is still but also to sweep the whole range of the angular coordi- KN ˙ valid by continuity, and we deduce that α > 0ifQ < Q2. nate θ. Therefore, we demand that θ does not vanish for any When r1 < r < r2 we found that there is no restriction on θ ∈[0,π]. According to (117) and (118) , this condition is the range of Q, although only the first class of solutions (α) equivalent to the demand that 2 is allowed. We compute for Q = r (r(r − 3M) + 2e2): 2a2 Q > 0 when E2 ≥ 1 (119) and 2 KN r 2 α (r(r − 3M) + 2e ) > 2( − 2) 2 < . 2a2 Q a 1 E when E 1 (120) √ 1 Spherical polar orbits (i.e polar orbits with constant radius) = r 4(r(r − M) + e2 − −S)2 ( ) = , dR = 2 are given by the conditions R r 0 dr 0. Equivalently, 2e2r 4(2rM − e2 − 2r 2) polar spherical orbits correspond to local extrema of the fol- + Mr5KN + , (116) 4 lowing effective potential: KN(r 2 + K ) V 2 (r) := , KN ef f 2 2 2 (121) which is positive. Thus α > 0 for any value of Q. (r + a ) Thus we have shown that (113) is positive for all the ranges where K is the hidden constant of motion which for polar of Carter’s constant of motion Q. The condition (112) does = + 2 2 2 ( ) geodesics reads K Q a E . The local extrema of Vef f r not hold, in particular, when Q is negative. This rule out all occur at the roots of the equation: values of Q which lie in the negative range. 2 2 2 2 3 4 Having obtained exact analytic solutions of the geodesic e r(r + a ) − e (2r + 2rK) + Mr equations and solved the conditions for spherical orbits, we −(K − a2)r 3 + 3M(K − a2)r 2 present examples of particular timelike spherical orbits in −(K − a2)a2r − MKa2 = 0. (122) Kerr–Newman spacetime in Tables 1, 2. In Table 1 we display examples of prograde orbits: φGT R > 0 when L > 0 The general features of the r motion of a test particle in while in Table 2 retrograde orbits: φGT R < 0 when L < polar orbit about the source of the Kerr–Newman field can 2 ( ) 0, are exhibited. For a choice of values for the black hole be deduced from graphs of Vef f r , such as those in Figs. 26, parameters a, e and for Carter’s constant Q, the constants 27. As can be deduced from the graphs, a test particle follows of motion associated with the Killing vector symmetries of a bound orbit when its specific energy at infinity E is such Kerr–Newman spacetime are computed from formulae (101), that E2 < 1. It can also be seen that, when E2 < 1 and a (102). In the last column of Tables 1, 2 we present the change black hole is involved, the particle gets dragged, disappearing φGT R over one complete oscillation in latitude of the orbit, behind the event horizon, i.e., the surface r = r+, unless K 2 ( ) 2 which means we multiply Eq. (95) by four. In Sect. 5.1.3 we is such that Vef f r develops a local maximum Emax outside 2 < 2 will study spherical polar orbits ( a special case with zero the even horizon and E Emax. In this case, the particle angular momentum L = 0 ) and we will determine the value will not be swallowed by the black hole, provided it is initially of Carter’s constant Q separating prograde and retrograde found in the region r > r0, where r0 is the point on the r axis 2 ( ) = 2 orbits. where Vef f r Emax. Moreover, in the case of a rotating 123 Eur. Phys. J. C (2021) 81:147 Page 21 of 37 147

2 ( ) 2 ( ) Fig. 26 The effective square potential Vef f r for polar spherical Fig. 27 The effective square potential Vef f r for polar spherical orbits in the neighbourhood of a rotating charged black hole with orbits in the neighbourhood of a rotating charged black hole with a = 0.8, e = 0.2. The parameter K distinguishing the three curves a = 0.52, e = 0.85. For comparison we also plot the case of a Kerr is Carter’s hidden integral of motion in the Kerr–Newman black hole black hole with a = 52, e = 0. The parameter K distinguishing the corresponding curves is Carter’s hidden integral of motion in the Kerr– Newman black hole charged black hole with vanishing cosmological constant, i.e. 2 ≤ 2 − 2 2 ( ) when e M a , Vef f r necessarily vanishes√ at the radii The exact analytic result in Eq. (127) shows that during a of the event and Cauchy horizons r± = M ± M2 − a2 − e2 which are the roots of the equation KN = 0. Considering revolution of the particle along the polar orbit the lines of 2 2 nodes advance in a direction which coincides with that of the the case of spherical orbits, we note that E = V (r0), ef f rotation of the central Kerr–Newman black hole. Thus this is where r0 is a root of (122), is a necessary condition for a spherical orbit to obtain. Thus, with the aid (122) we derive a typical dragging effect, so that if the rotation vanishes, the φ the following relations for the speciﬁc energy and Carter’s change in azimuth vanishes too. constant: Using the convergent series expansion around the origin 2 ≡ a2(1−E2) < 2 < (k Q 1 for bound orbits with E 1) for the r(KN)2 Gauß’s hypergeometric function we obtain: E2 = , (123) (r 2 + a2)Z KN 4 2 3 2 2 4 2 3 2 2 Mr + a r − 3Ma r + a r − e r + e ra φGT R = Polar Q KN Z 4aE(2Mr − e2) π 1 − 2 2, = √ a E (124) KN 2 Q 2 1 a2(1 − E2) 9 a2(1 − E2) where × 1 + + +··· (128) 4 Q 64 Q Z KN := 2e2r + r 3 − 3Mr2 + a2r + Ma2. (125) In addition, using the same techniques as in the proof of Also using Eqs. (123)–(125) we express the variable of the Theorem 4, we derive the following Lense–Thirring preces- hypergeometric function k2 as follows: φGT R sion Polar per revolution for a polar spherical orbit in a 2 Kerr–Newman spacetime: k2 = r π 4 GT R aP 1 1 1 1 z− [ 4 + 2 2 + 4 − 3 − re + 2 2] φ = − √ √ , , , , r 2r a a 4Mr M 4r e Polar 4 KN aE F 1 × . | | − 2 2 z+ 2 2 z+ 2 2 4 2 2 4 2 3 a 1 E 4 + 4 + 2 2 − 2 − 2re a − a e − a e + 2 2 − e r r a 2a r 4Mra M rM rM 4a e M (126) (129)

Q where P = E(r 2 + a2) for L = 0, z+ = , z− = 1. a2(1−E2) Equation (126) simpliﬁes to: We present examples of spherical polar orbits in Table 3. The larger the Kerr parameter the larger the frame drag- 4aE(2Mr − e2) π 1 1 1 a2(1 − E2) φGT R = √ F , , 1, . ging precession. We note the small contribution of the elec- Polar KN 2 Q 2 2 Q tric charge on the Lense–Thirring precession for ﬁxed Kerr (127) parameter. 123 147 Page 22 of 37 Eur. Phys. J. C (2021) 81:147

Table 3 Properties including frame-dragging (Lense–Thirring precession) of spherical timelike polar Kerr–Newman orbits applying the exact analytic formula (127) and the relations (123), (124), for various values of the black hole’s spin and electric charge / / 2 / / φGT R Orbit r MQM EeMaM Polar (a) 10 14.198012556704303 0.955967687347955 0 0.80.316593 = 18.1394◦ (b) 10 14.175980906902574 0.9559475468735232 0.11 0.80.316593 = 18.1395◦ (c) 10 14.125237970658672 0.9559012400826655 0.2 0.80.316595 = 18.1396◦ (d) 10 14.149643079211955 0.9558495321451093 0 0.999999 0.394826 = 22.6219◦ (e) 10 14.149642728218016 0.9558495318270006 0.00044 0.999999 0.394826 = 22.6219◦ (f) 10 14.248389653442322 0.9560911507901617 0 0.52 0.206277 = 11.8188◦ (g) 10 14.175264277554161 0.9560237528884883 0.2 0.52 0.206278 = 11.8188◦

5.2 Periods We then write: z− a2zdz Squaring the geodesic differential equation for the polar vari- √ √ 2 2 z 2|a| 1 − E2 z (z − z+)(z − z−) able (10) (for = 0), multiplying by the term cos θ sin θ, j and making the change to the variable z, yields the following z− a2(z − z+ + z+)dz = √ √ (134) 2 differential equation for the proper polar period: z j 2|a| 1 − E z (z − z+)(z − z−)

2 2 (r + a z)dz = − + ( − −) dτθ = √ (130) Applying the change of variables:z z x z j z 2 z a2(1 − E2)z2 + (−a2(1 − E2) − L2 − Q)z + Q we compute the term: Finally, our closed form analytic computation for the z− a2(z − z+)dz √ √ proper polar period after integrating (130) yields: 2 z j 2|a| 1 − E z (z − z+)(z − z−) a2 (z+ − z−)(z − z−) Proposition 5 = √ j 2|a| 1 − E2 z−(z− − z+)(z j − z−) 2 1/2 r π 1 1 1 z− x(z−−z j ) τ = √ √ , , , 1 − θ 4 F 1 dx 1 z−−z+ |a| 1 − E2 2 z+ 2 2 z+ × x(z−−z ) √ 2 0 − j a 1 √ 1 1 z− 1 z− x − √ z+π F , − , , | | 1 2 a 1 − E2 2 2 z+ a2 (z+ − z−)(z − z−) = √ j 2 π z+a 1 1 1 1 z− 2|a| 1 − E2 z−(z− − z+)(z − z−) + √ √ F , , 1, . (131) j 2|a| − 2 2 z+ 2 2 z+ 1 E 1 1 1 3 z− − z j z− − z j ×F1 , − , , , , 2 2 2 2 z− − z+ z− Proof The integral in (130) can be split into two parts: 1 (1) × 2 (135) r 2dz 1 3 τθ = √ √ 2 2|a| 1 − E2 z (z − z−)(z − z+) Setting z j = 0 in the expression which involves Appell’s a2zdz + √ √ . (132) hypergeometric function F1 yields: | | − 2 ( − )( − ) 2 z a 1 E z z+ z z− z− a2(z − z+)dz We compute ﬁrst: √ √ 0 2|a| 1 − E2 z (z − z+)(z − z−)

1 3 3 1 1 −a2 1 (z+ − z−) − − z− r 2dz 1 = √ √ 2 2 2 2 2 √ √ 2|a| − 2 z+ − z− 3 3 − 1 3 − 1 1 E 2 2 2 2 2 0 2|a| 1 − E2 z (z − z−)(z − z+) 1 1 −z− 2 × F , − , 1, r 1 1 −z− 1 − = √ F , , , 2 2 z+ z− 1 − 2|a| 1 − E2 2 2 z+ z− 2 −a2 1 √ 1 1 1 1 z− = √ z+ F , − , 1, 1 1 2|a| − 2 2 2 2 2 z+ × √ 1 E 2 z+ − z− (136) r 2 π 1 1 1 z− = √ √ F , , 1, . (133) This yields the second term in Eq. (131). In our calculation in 2|a| 1 − E2 2 z+ 2 2 z+ (136), we made use of the following transformation property 123 Eur. Phys. J. C (2021) 81:147 Page 23 of 37 147

2 [( 2 + 2) − ] 2 1 of Gauß’s hypergeometric function F, discovered by Kum- GTR a r a E aL 2 φ = √ √ KN − − mer [94]: r z+ z− z− z −2 a4 4 − z1 c F(a − c + 1, b − c + 1, 2 − c, z) 1 1 1 z− z− × F1 , , , 1, , − − − z 2 2 2 z− − z+ z− − z = z1 c(1 − z)c b 1 F 1 − a, b − c + 1, 2 − c, . z − 1 2 + H aE2 1 (137) 4 2 2 a 3 We also used the special value for Appell’s hypergeometric 1 1 1 z− z− × FD , , , 1, 1, , , −η 2 2 2 z− − z+ z− − z function F1: − 2 2 1 HL 2 (α,β,β,γ, , ) + F1 x 1 a4 1 − z− 2 3 = F(a,β ,γ,1)F(α,β,γ − β , x) 1 1 1 z− z− −z− × F , , , , , , , , −η, , (γ)(γ − α − β ) D 1 1 1 − − − = F(α,β,γ − β, x), 2 2 2 z− z+ z− z 1 z− (γ − α)(γ − β) (141) (γ − α − β )>0, (138) where we deﬁne: where in the second equality in (138) we applied the Gauß 1 1 H ≡ √ √ summation theorem: 2 z+ − z− z− − z + a z− 1 3 ( ,β,γ, ) F a 1 2 −a z− (γ)(γ − α − β ) η ≡ . (142) = , (γ − α − β )>0. (139) + a2 (γ − α)(γ − β) 1 3 z− The variables z+, z−, z appearing in the hypergeometric functions in (141) are the roots of polynomial equation (205). 5.3 Spherical orbits in Kerr–Newman–(anti) de Sitter In our calculations we used the following property for the spacetime values of Lauricella’s multivariate function FD:

From (7) and (10) we derive the equation : (n)(α, β ,...,β ,γ, , ,..., ) FD 1 n 1 x2 xn (γ)(γ − α − β ) dφ a2 [(r 2 + a2)E − aL] = 1 = √ (γ − α)(γ − β ) dθ KN 1 r × (n−1)(α, β ,...,β ,γ − β , ,..., ), 2 2 θ − FD 2 n 1 x2 xn aE sin L − √ max{|x |,...,|x |} < 1, (γ − α − β )>0. (143) + a2 2 θ ( 2 θ) 2 n 1 1 3 cos sin a2 [(r 2 + a2)E − aL] 5.3.1 Conditions for spherical orbits in Kerr–Newman = √ KN spacetime with a cosmological constant r 2 aE(1 − z) − L − √ . (140) We now proceed to solve the conditions for timelike spherical + a2 ( − ) 1 3 z 1 z orbits in the Kerr–Newman spacetime in the presence of the cosmological constant. As before, for a spherical orbit to exist Using the variable z we obtain the following novel exact at radius r, the conditions R(r) = dR/dr = 0 must hold result in closed analytic form for the amount of frame- at this radius. The procedure of solving these conditions will dragging that a timelike spherical orbit in Kerr–Newman– lead us to a generalisation of Eqs. (101), (102) and (26), (27). (anti)de Sitter spacetime undergoes. As θ goes through a We solved these conditions simultaneously and the novel quarter of a complete oscillation we obtain the change in solutions take the following elegant and compact form when GTR azimuth φ, φ in terms of Lauricella’s FD and Appell’s parametrised in terms of r and Carter’s constant Q: F1 multivariable generalised hypergeometric functions:

123 147 Page 24 of 37 Eur. Phys. J. C (2021) 81:147

Our Theorem 7 also generalises in a non-trivial way our Theorem 7 results for the gravitational frequency shifts in Sect. 4.Using

our solutions for the invariant parameters in (144), (145)we Eα,β (r, Q; , a, e) (144) √ can investigate the redshift and blueshift of light emitted by r 2e2 + r 3(r − 2) − r 2(r 2 + a2) r 2 − a(aQ ∓ ϒ) = √ , timelike geodesic particles in spherical orbits that are not r 2 2e2r 2 + r 3(r − 3) − a2r 4 − 2a(aQ ∓ ϒ) necessarily conﬁned to the equatorial plane, around a Kerr– Newman–(anti) de Sitter black hole. We can work either with Lα,β (r, Q; , a, e) (145) √ Eq. (46) or the corresponding of Eq. (48). Indeed in Eq. (46) (r2 +a2)(∓ ϒ +aQ)+2ar3 +ar4 (r2 +a2)−ae2r2 r =− √ , with U = 0: r2 2e2r2 +r3(r −3)−a2r4 −2a(aQ ∓ ϒ) φ θ θ (kt E − k L + gθθk U )| where 1 + z = e ( t − φ + θ θ )| 2 4 Ek Lk gθθk U d ϒ ≡ a Q(Q + r ) (146) t φ θ θ (Eγ U − Lγ U + gθθ K U )|e + r 2(−2e2 Q − (e2 + Q)r 2 + 3Qr + r 3 − r 6 ). = t φ θ θ (150) (Eγ U − Lγ U + gθθ K U )|d

For vanishing cosmological constant Eqs. (144), (145) we substitute E, L through relations (144) and (145)to reduce to the ﬁrst integrals Eqs. (101) and (102) for spherical obtain: timelike orbits in Kerr–Newman spacetime. For vanishing φ θ θ (kt Eα,β (r, Q; , a, e) − k Lα,β (r, Q; , a, e) + gθθk U )| + = e Carter’s constant Q,Eqs.(144), (145) reduce to the con- 1 z t φ θ θ (Eα,β (r, Q; , a, e)k − Lα,β (r, Q; , a, e)k + gθθk U )|d stants of motion, Eqs. (26), (27), that describe prograde and (151) retrograde timelike circular orbits in the equatorial plane of the Kerr–Newman–(anti) de Sitter black hole. Also the velocity component U θ of the test massive particle Thus, in Theorem 7 we have derived the most general is given by: solutions for the constants of motion E, L that are associated with the two Killing vectors of the Kerr–Newman–(anti) de (U θ (r,θ,Q; , a, e)2 Sitter black hole spacetime, which satisfy the conditions for = ( + ( ( , ; , , ) − ( , ; , , ))2 − 2 2 θ) timelike spherical orbits. Q Lα,β r Q a e aEα,β r Q a e a cos θ To ensure that the smaller square root appearing in the (aEα,β (r, Q; , a, e) 2 θ − Lα,β (r, Q; , a, e))2 − sin ρ4 (152) solutions (144), (145) is real, we need to impose the condi- sin2 θ tion: The impact factor for such more general orbits (non- ϒ ≥ 0. (147) equatorial) is computed in Sect. 5.8 see Eq. (217). Note from (146) that ϒ is quadratic in Carter’s constant A thorough investigation will appear in a separate publi- Q, and its two roots are: cation. 2 r 2 2 2 Q , = (−a r + r(r − 3) + 2e ± G), (148) 1 2 2a2 5.3.2 Frame-dragging for spherical polar geodesics in Kerr–Newman–de Sitter spacetime where 4 4 2 2 2 2 2 4 2 3 G ≡ a r + 2a r (r + 3r − 2e ) + r + 9r − 6r The generalisation of the effective potential eqn (121)inthe +4e4 − 4e2r(3 − r) + 4a2e2 − 4a2r. (149) presence of the cosmological constant is: KN( 2 + ) We mention at this point, that G is a polynomial equation 2 r r K V . = . (153) in r that is familiar from the study of circular photon orbits in ef f (r 2 + a2)2 the equatorial plane around a Kerr–Newman–(anti) de Sitter From the local extrema of the effective potential (153)we black hole [39]. determine the following ﬁrst integrals of motion for spherical polar geodesics in Kerr–Newman–de Sitter spacetime:

−r 3(r 2 +a2)2 −r 3(2r 2 + a2)(r 2 + a2)+2r 5(r 2 +a2)+Mr4 +(a2 − e2)r 3 − 3Ma2r 2 + ra2(a2 + e2) K = , Z (154)

123 Eur. Phys. J. C (2021) 81:147 Page 25 of 37 147

In deriving the last line in Eq. (159)weusedthefollowing Lemma for Appell’s hypergeometric function F1: r(KN)2 E = r , (155) Lemma 6 (r 2 + a2)Z −β −β x y F (α, β, β ,γ,x, y) = (1 − x) (1 − y) F γ − α, β, β ,γ, , . where 1 1 x − 1 y − 1 2 3 2 2 2 Z ≡ 2e r + r − 3Mr + a r + Ma (160) + ( 2 + 2)( 2 + 2) − 3( 2 + 2). r 2r a r a 2 r r a (156) Likewise, using initially the transformation as before: z = 2 z− + ξ (z j − z−) and then setting z j = 0, integration of the second term gives: GT R Theorem 8 The Lense–Thirring precession φ per SpherPolar √ revolution for a polar spherical orbit in Kerr–Newman–de Sitter −aE2 θ H π √d = 1 √ spacetime is expressed in closed analytic form in terms of Appell’s 2 FD (1 + a z a4/3 π/2 F1 and Lauricella’s FD multivariable hypergeometric functions 3 as follows: 1 1 1 1 3 z− − z j z− − z j z− − z j × , , , , 1, , , , , −η 2 2 2 2 2 z− z− − z+ z− − z φGT R (157) SpherPolar z =0 1 1 1 1 j= √ √ 2 2 2 4a (r + a )E π 1 1 z+ − z− z− − z 1 + a2 z− a4/3 = √ √ √ 3 KN 4/ 2 z+ −z 2 r a 3 aE π 1 1 1 1 3 z− z− × √ FD , , , , 1, , 1, , , −η 1 1 1 z− z− 2 π/2 2 2 2 2 2 z− − z+ z− − z × F , , , 1, , 1 1 1 1 1 2 2 2 z+ z (143= ) √ √ − − 2 42aEπ 1 1 1 1 1 z+ z− z− z 1 + a z− a4/3 − √ √ √ 3 a2 − 2 π 2 1 + z− a4/3 z+ z 1 + η aE 1 1 1 z− z− 3 × FD , , , 1, 1, , , −η −η 2 2 2 2 z− − z+ z− − z × 1 , 1 , 1 , , , z− , z− , . FD 1 1 2aEπ 2 2 2 z+ z −η − 1 = 1 1 1 √ 1 √ 1 √ 1 + a2 z− a4/3 2 z+ −z 1 + η 3 1 1 1 z− z− −η × FD , , , 1, 1, , , , (161) 2 2 2 z+ z −η − 1 Proof The relevant differential equation is: φ 2 ( 2 + 2) 2 −(z −z−)(2aE/2) d a r a E aE = √ √ √j √ = √ − √ . (158) where H1 − − − ( + 2 ) . In deriv- θ KN 2 z− z j z− z− z+ z− z 1 a 3 z− d r + a 1 3 z ing the last line of (161) we used the following Lemma for The integration of Eq. (158) splits into two parts. Integrating Lauricella’s multivariate hypergeometric function FD: the ﬁrst part yields:

Lemma 7 −β −β −β x y z FD(α,β,β ,β ,γ,x, y, z) = (1 − x) (1 − y) (1 − z) FD γ − α, β, β ,β ,γ, , , . (162) x − 1 y − 1 z − 1

a2(r 2 + a2)E θ √d KN r −a2(r 2 + a2)E 1 1 1 π = √ √ The quantities z+, z−, z, of the hypergeometric func- KN 4/ z+ − z− z− − z 2 r a 3 tions in Eq. (157) are roots of the cubic equation: 1 1 1 z− z− × F1 , , , 1, , 4 z− − z+ z− − z a 2 2 2 z3 −a2(r 2 + a2)E 1 π 1 1 3 = √ √ KN 2 4/ 2 z+ −z 2 a 2 2 2 2 2 r a 3 − z (Q + a E ) − a (1 − E ) 3 × 1, 1, 1, , z− , z− . F1 1 (159) − (( + 2 22) + 2 + 2(− )) + = . 2 2 2 z+ z z Q a E a 2aE aE Q 0 (163)

123 147 Page 26 of 37 Eur. Phys. J. C (2021) 81:147

5.4 Frame-dragging effect for polar non-spherical bound The variables of the hypergeometric functions are given in orbits in Kerr–Newman spacetime terms of the roots of the quartic and the radii of the horizons by the expressions: In this section we will derive novel closed-form expressions α − β − γ α − β δ − γ t2 r± 2 for frame-dragging (Lense–Thirring precession) for polar κ± := ,κ := , (169) α − γ r± − β α − γ δ − β non-spherical timelike geodesics in Kerr–Newman spacetime. Thus we assume in this section that = L = 0. The while relevant differential equation for the calculation of frame- H± ≡ (1 − E2)(αμ+ − αμ− ) αμ − αμ+ dragging is: 1 1 1 αμ+1 − αμ−3 dφ (2ar − ae2)E = √ . (164) = ( − 2)(β − ) α − β β − δ. dr KN R 1 E r± (170) The quartic radial polynomial R is obtained from R in (11) 5.4.1 Exact calculation of the orbital period in for = L = 0. Using the partial fractions technique we non-spherical polar Kerr–Newman geodesics integrate from the periastron distance rP to the apoastron distance rA: In this section we will compute a novel exact formula for We apply the transformation: the orbital period for a test particle in a non-spherical polar Kerr–Newman geodesic. The relevant differential equation 1 r − αμ+ α − γ r − β z = 1 = (165) is: ω r − αμ+2 α − β r − γ cdt r 2 + a2 a2 E sin2 θ = √ E(r 2 + a2) − √ , (171) and denote the real roots of the radial polynomial R by dr KN R R α, β, γ, δ, α>β>γ>δ. We organise all the roots in the ascending order of magnitude: and we integrate from periapsis to apoapsis and back to periapsis. Indeed, our analytic computation yields: α >α >α >α, ρ σ ν i (166) Eβ2 GM 2 c2 ct ≡ cPKN = √ √ √ α = α = α, α = α = 2 1 − E2 2 α − γ 2 β − δ with the correspondence ρ μ σ μ+1 β,αν = αμ+ = γ,α = αμ− , i = 1, 2, 3,αμ− = 1 1 2 i i 1 × π F , 2, , 1,ω,κ2 = ,α = δ , 1 2 2 aμ−2 r± μ−3 , where r+ r− denote the radii of the event horizon and the inner or Cauchy horizon respectively. 2ωγ π 3 1 − F , 2, , 2,ω,κ2 We thus compute the following new exact analytic result β 2 1 2 2 for Lense–Thirring precession that a test particle in a non- γ 2ω2 3π 5 1 + F , 2, , 3,ω,κ2 spherical polar orbit undergoes, in terms of Appell’s hyper- β2 8 1 2 2 geometric function F1: GM ω 1 +2E(a2 − e2) √ c2 1 − E2 2 (α − β)(β − δ) GT R φ ×F(1/2, 1/2, 1,κ2)π tpKN ω3/2 π 4EGM ω β = − + 3, , 1, ,κt2,κ2 + √ 2 A F1 1 2 + c2 1 − E2 2 (α − β)(β − δ) H+ tpKN 2 2 2 √ 1 1 2 ω + 1 1 × π F1 , 1, , 1,ω,κ + A F , 1, , 1,κt2,κ 2 π 2 2 tpKN 1 + H+ 2 2 ωγ π / 3 1 2 ω3 2 π − F1 , 1, , 2,ω,κ − − 3, , 1, ,κt2,κ2 β 2 2 2 AtpKN F1 1 2 − H− 2 2 2 EG M ω √ + 4 2 √ 1 ω 2 − 1 1 t2 2 c2 1 − E2 (α − β)(β − δ) + A F1 , 1, , 1,κ− ,κ π (167) H− tpKN 2 2 4EG2M ×F( / , / , ,κ2)π − 1 2 1 2 1 2 c ω3/2 KN π where the partial fraction expansion parameters are given by: A+ 3 1 2 2 × − F1 , 1, , 2,κ+,μ H+ 2 2 2 − + 2 + r+2aE ae E ω1/2 AKN A = , + + 1 , , 1 , ,κ2 ,μ2 π tpKN − F1 1 1 + r− r+ H+ 2 2 2 / − +r−2aE − ae E ω3 2 AKN π = . − − 3 , , 1 , ,κ2 ,μ2 AtpKN (168) F1 1 2 − r− − r+ H− 2 2 2 123 Eur. Phys. J. C (2021) 81:147 Page 27 of 37 147 ω1/2 KN A− 1 1 2 2 of the Lense–Thirring precession Eq. (167) and its orbital + F1 , 1, , 1,κ−,μ π H− 2 2 period Eq. (172) as follows: GM +2E 2 2π P c := KN . ω3/2( 2 − 4) π LTP GT R (176) 4e r+ e 3 1 2 2 φ × − √ F1 , 1, , 2,κ+,μ tpKN (−2 1 − a2 − e2)H+ 2 2 2 ω1/2( 2 − 4) 4e r+ e 1 1 2 2 We now proceed to calculate using our exact analytic + √ F1 , 1, , 1,κ+,μ π (−2 1 − a2 − e2)H+ 2 2 solutions and assuming a central galactic Kerr–Newman ω3/2(e4 − 4e2r ) π black hole, the Lense–Thirring effect and the correspond- − √ − 3 , , 1 , ,κ2 ,μ2 F1 1 2 − ing Lense–Thirring period for the observed stars S2,S14 for (−2 1 − a2 − e2)H− 2 2 2 ω1/2( 4 − 2 ) various values of the Kerr parameter and the electric charge e 4e r− 1 1 2 2 + √ F1 , 1, , 1,κ−,μ π of the central black hole-see Tables 4 and 5. The choice (− − 2 − 2) 2 2 2 1 a e H− of values for the invariant parameters Q, E is restricted by −a2 E GM + √ c2 requiring that the predictions of the theory for the periap- 2 Q sis, apoapsis distances and the period PKN are in agree- 1 1 1 3 2 2 2 ment with the orbital data from observations in [15]. We note × sin(ϕ)F1 , , , , sin ϕ,κ sin ϕ 2 2 2 2 here the orbital data from observations that we use to con- = 1 1 1 3 2 2 2 strain our theory. For S2 the eccentricity measured is eS2 + − sin(ϕ)F1 , , , , sin ϕ,κ sin ϕ 2 2 2 2 0.8760 ± 0.0072, its orbital period PS2(yr) = 15.24 ± 0.36, ( ) = . ± . 1 1 1 3 1 and the semimajor axis aS2 arcsec 0 1226 0 0025 [15]. + sin(ϕ)F , , − , , sin2 ϕ,κ2 sin2 ϕ (172) 1 2 2 2 2 κ2 The corresponding orbital measurements for S14, reported in [15] eS14 = 0.9389 ± 0.0078, PS14(yr) = 38.0 ± 5.7, where aS14(arcsec) = 0.225±0.022. The theoretical prediction for the eccentricity of the S2 orbit for the choice of values for 4 1 1 π ϕ = am 2 Q √ √ √ the invariant parameters Q, E in the last two lines in Table 4 2 1 − E2 2 α − γ 2 β − δ 2 theory = . is eS2 0 883 . It is consistent with the upper allowed 1 1 a2(1 − E2) value by experiment. On the other hand the prediction of the × F , , 1,κ2 , (173) 2 2 Q exact theory, for the choice of values for the constants of 2 + 2 − motion Q, E in the first three lines in Table 4, for the orbital KN a e 2r+ theory A+ := − , eccentricity is e = 0.8755 in precise agreement with r− − r+ S2 experiment. We observe that the contribution of the electric − 2 − 2 + KN a e 2r− charge on the frame-dragging precession is small. A− := − , (174) r− − r+ and the moduli (variables) of the hypergeometric function of 5.5 Periapsis advance for non-spherical polar timelike Appell are given by: Kerr–Newman orbits α − β δ − γ μ2 = κ2 = In this section we shall investigate the pericentre advance for α − γ δ − β a non-spherical bound Kerr–Newman polar orbit, assuming α − β − γ 2( − 2) 2 r± 2 a 1 E a vanishing cosmological constant. We shall first obtain the κ± = ,κ = . (175) α − γ r± − β Q exact solution for the orbit and then derive an exact closed √ √ form formula for the periastron precession. The relevant dif- α−β 2 Also ω = α−γ , H± ≡ (1 − E )(β −r±) α − β β − δ ferential equation is: r± and r± = are the dimensionless horizon radii. In Eq. GM θ c2 r dr dθ (173) am(u, k2) denotes the Jacobi amplitude function [95]. √ =± √ (177) R The last three terms in (172) arise after integrating the second term on the right hand side of (171). Details of this par- Now applying the transformation (165) on the left hand side ticular angular integration are given in Appendix A.1, pages of (177) yields: 1801–1802 in [46]. For zero electric charge, e = 0, Eq. (172) reduces cor- r " √d = 1 √ dr rectly to Eq. (33) in [46] for the case of a Kerr black hole. GM ( − 2)(−)( −α)( −β)( −γ)( −δ) R 2 1 E r r r r c " √ The Lense–Thirring period for a non-spherical polar timelike dz ω = 1 √ √ √ , (178) geodesic in Kerr–Newman BH geometry, is defined in terms GM ( − 2) (α−β)(β−δ) ( − )( − 2 ) c2 1 E z 1 z 1 k z 123 147 Page 28 of 37 Eur. Phys. J. C (2021) 81:147

Table 4 Lense–Thirring precession for the star S2 in the central of the star S2 was performed using the exact result in Eq. (172). The arcsecond of the galactic centre, using the exact formula (167). We periapsis and apoapsis distances for the star S2 in the ﬁrst three lines 15 16 assume a central galactic Kerr–Newman black hole with mass MBH = are respectively 1.82 × 10 cm and 2.74 × 10 cm. In the last two 6 15 16 4.06 × 10 M and that the orbit of S2 star is a timelike non-spherical lines these distances are: rP = 1.68 × 10 cm, rA = 2.71 × 10 cm polar Kerr–Newman geodesic. The computation of the orbital period / 2 / / φGT R ( ) ( ) Star Q M EeMaM tpKN PKN yr LT P yr

. arcsec . × 6 S2 5693.30424 0.999979485 0.33 0.52 3 14284 revol. 15.15 6 25 10 . arcsec . × 6 S2 5693.30424 0.999979485 0.11 0.52 3 14295 revol. 15.15 6 25 10 . arcsec . × 6 S2 5693.30424 0.999979485 0 0.52 3 14297 revol. 15.15 6 25 10 . arcsec . × 6 S2 5273.53220 0.999979145 0.11 0.52 3 5261 revol. 14.78 5 43 10 . arcsec . × 6 S2 5273.53220 0.999979145 0.33 0.52 3 52596 revol. 14.78 5 43 10

Table 5 Lense–Thirring precession for the star S14 in the central of the star S14 was performed using the exact result in Eq. (172). The arcsecond of the galactic centre, using the exact formula (167). We periapsis and apoapsis distances for the star S14 in the ﬁrst two lines 15 16 assume a central galactic Kerr–Newman black hole with mass MBH = are respectively 1.64 × 10 cm and 5.22 × 10 cm, whereas in the 4.06 × 106 M and that the orbit of S14 star is a timelike non-spherical last two lines are predicted to be 1.29 × 1015 cm and 4.73 × 1016 cm polar Kerr–Newman geodesic. The computation of the orbital period respectively / 2 / / φGT R ( ) ( ) Star Q M EeMaM tpKN PKN yr LT P yr

. arcsec . × 6 S14 5321.06355 0.999988863 0.11 0.9939 6 64977 revol. 37.88 7 38 10 . arcsec . × 6 S14 5321.06355 0.999988863 0 0.9939 6 64981 revol. 37.88 7 38 10 . arcsec . × 6 S14 4204.76359 0.999987653 0.11 0.9939 9 47145 revol. 32.45 4 44 10 . arcsec . × 6 S14 4204.76359 0.999987653 0 0.9939 9 47151 revol. 32.45 4 44 10 where Equation (183) represents the ﬁrst exact solution that des- α − β δ − γ δ − γ cribes the motion of a test particle in a polar non-spherical k2 = ≡ ω . α − γ δ − β δ − β (179) bound orbit in the Kerr–Newman ﬁeld in terms of Jacobi’s 2 2 sinus amplitudinus elliptic function sn. The function sn (y, k ) The roots α, β, γ, δ of the quartic polynomial equation: ( 2) = π 1 , 1 , , 2 has period 2K k F 2 2 1 k which is also a period R = ((r 2 + a2)E)2 − (r 2 + a2 + e2 − 2r)(r 2 + Q + a2 E2) of r. This means that after one complete revolution the angular integration has to satisfy the equation = 0, (180) θ ∂ − √d = √1 are organised as αμ >αμ+1 >αμ+2 >αμ−3, and we have Q 1 − k2 sin2 the correspondence α = αμ,β = αμ+ ,γ = αμ+ ,δ = 1 2 π 4 1 1 1 1 2 αμ−3. = √ √ √ F , , 1, k , − 2 α − γ β − δ By setting, z = x2, we obtain the equation 1 E 2 2 2 (184) dx (1 − x2)(1 − k2x2) where the latitude variable := π/2 − θ has been intro- √ √ √ 1 − E2 α − β β − δ dθ duced. Equation (184) can be rewritten as = √ √ . (181) 2 ω dx Using the idea of inversion for the orbital elliptic integral on ( − 2)( − 2 2) 1 x 1 k x the left-hand side we obtain 4 1 1 π 1 1 √ √ √ = Q √ √ √ F , , 1, k2 1 − E2 α − γ β − δ dθ 1 − E2 α − γ β − δ 2 2 2 x = sn √ , k2 . (182) 2 (185)

In terms of the original variables we derive the equation and

√ √ 2 2 2 " a (1 − E ) β − γ α−β 2 1−E (α−γ )(β−δ) √dθ , 2 2 := . α−γ sn 2 k GM k (186) r = √ √ . (183) Q 2 " 2 − α−β 2 1−E (α−γ )(β−δ) √dθ , 2 c 1 α−γ sn 2 k 123 Eur. Phys. J. C (2021) 81:147 Page 29 of 37 147

Also x2 = sin2 = cos2 θ. Equation (185) determines By performing this calculation, we gain a more complete the exact amount that the angular integration satisfies after a appreciation of the effect of the electric charge of the rotat- complete radial oscillation for a bound non-spherical polar ing galactic black hole (we assume that the KN solution orbit in KN spacetime. Now using the latitude variable the describes the curved spacetime geometry around SgrA*) on change in latitude after a complete radial oscillation leads to this observable. We also assume that the angular momentum the following exact expression for the periastron advance axis of the orbit is co-aligned with the spin axis of the black for a test particle in a non-spherical polar Kerr–Newman hole and that the S-stars can be treated as neutral test parti- orbit, assuming a vanishing cosmological constant, in terms cles i.e. their orbits are timelike non-circular equatorial Kerr– of Jacobi’s amplitude function and Gauß’s hypergeometric Newman geodesics. Our results are displayed in Tables 8 and function 9. The choice of values for the constants of motion L, E is restricted by requiring that the predictions of the theory for GTR = − 2π the periapsis apoapsis distances and the orbital period are in KN 4 1 1 π agreement with published orbital data from observations in = am Q √ √ √ 1 − E2 α − γ β − δ 2 [15]. It is evident in this case that the value of electric charge plays a significant role in the value of the pericentre-shift 1 1 a2(1 − E2) × F , , 1,κ2 , − 2π. (187) as opposed to the results in Tables 6 and 7 especially for 2 2 Q moderate values of the spin of the black hole. We note at this point that a more precise analysis would The Abel–Jacobi’s amplitude am(m, k2) is the function that involve the calculation of relativistic periapsis advance for inverts the elliptic integral more general timelike non-spherical orbits, inclined non- ∂ equatorial and non-polar in the KN(a)dS spacetime, which = m (188) 0 1 − k2 sin2 is beyond the scope of the current publication. Such a generalisation is important because the observed high eccentricity We compute with the aid of (187) the periapsis advance orbits of S-stars are such that: neither of the S stars orbits is for the stars S2 and S14 assuming that they orbit in a timelike equatorial nor polar. Such a general analysis will be a subject non-spherical polar Kerr–Newman geodesic. Our results are of a future publication.12 displayed in Tables 6 and 7. We observe that the effect of e A few further comments are in order. The values of the is small. hypothetical electric charge of the central Kerr–Newman We also compute with the aid of the following exact black hole have been chosen so that the surrounding space- formula for the periapsis advance for an equatorial non- time represents a black hole, i.e. the singularity surrounded circular timelike geodesic in the Kerr–Newman spacetime, by the horizon, the electric charge and angular momentum J first derived in [28]: must be restricted by the relation: δteKN := φGT R − π, P teKN 2 (189) 1/2 where GM J 2 Ge2 ≥ + ⇔ (191) 2 4 ω3/2 π c Mc c φGT R = − + 3, , 1, ,κt2,κ2 teKN 2 A F1 1 2 + H+ teKN 2 2 2 √ / ω GM Ge2 1 2 + 1 1 t2 2 ≥ 2 + ⇒ + A F1 , 1, , 1,κ+ ,κ π a (192) H+ teKN 2 2 c2 c4 ω3/2 π 2 ≤ 2( − 2) − 3 1 t2 2 e GM 1 a (193) − A F1 , 1, , 2,κ− ,κ H− teKN 2 2 2 √ ω 1 1 a = a − t2 2 where in the last inequality GM/c2 denotes a dimen- + A F1 , 1, , 1,κ− ,κ π H− teKN 2 2 sionless Kerr parameter. √ ω The values of the electric charge used for instance in 2 L 1 1 2 + F , , 1,κ π Tables 7, 8, for the SgrA* galactic black hole correspond (1 − E2)(α − β)(β − δ) 2 2 (190) 12 For non-spherical non-polar orbits with orbital inclination 0◦ < ◦ the pericentre-shift for the stars S2 and S14 for various val- i < 90 one expects that the resulting periapsis advance has a value ues for the spin and charge of the central black hole. The between those calculated using formula (187)and(190) (assuming the three orbital configurations, polar,non-polar and equatorial have the derivation of the formula (190) as well as the definition of same eccentricity and semimajor axis and fixed values of the black hole ± , the coefficients AteKN H± can be found in [28, pp 24–26]. parameters). 123 147 Page 30 of 37 Eur. Phys. J. C (2021) 81:147

Table 6 Periastron precession for the star S2 in the central arcsecond of that the orbit of S2 star is a timelike non-spherical polar Kerr–Newman the galactic centre, using the exact formula (187). We assume a central geodesic 6 galactic Kerr–Newman black hole with mass MBH = 4.06×10 M and / 2 / / GTR Star Q M EeMaM KN Periapsis rP Apoapsis rA . . arcsec . × 15 . × 16 S2 5693.30424 0.999979485 0.11 0 9939 682 512 revol. 1 82 10 cm 2 74 10 cm . . arcsec . × 15 . × 16 S2 5693.30424 0.999979485 0.11 0 52 682 533 revol. 1 82 10 cm 2 74 10 cm

Table 7 Periastron precession for the star S14 in the central arcsecond and that the orbit of S14 star is a timelike non-spherical polar Kerr– of the galactic centre, using the exact formula (187) . We assume a cen- Newman geodesic 6 tral galactic Kerr–Newman black hole with mass MBH = 4.06×10 M / 2 / / GTR Star Q M EeMaM KN Periapsis rP Apoapsis rA . . arcsec . × 15 . × 16 S14 5321.06355 0.999988863 0.11 0 9939 730 351 revol. 1 64 10 cm 5 22 10 cm . . arcsec . × 15 . × 16 S14 5321.06355 0.999988863 0.11 0 52 730 376 revol. 1 64 10 cm 5 22 10 cm

Table 8 Periastron precession for the star S2 in the central arcsecond of and that the orbit of the star S2 is a timelike non-circular equatorial the galactic centre, using the exact formula Eq. (190). We assume a cen- Kerr–Newman geodesic 6 tral galactic Kerr–Newman black hole with mass MBH = 4.06×10 M / / / δteKN Star L ME eMaM p Periapsis rP Apoapsis rA

. . arcsec . × 15 . × 16 S2 75.4539876 0.999979485 0.11 0 9939 670 565 revol. 1 82 10 cm 2 74 10 cm . . arcsec . × 15 . × 16 S2 75.4539876 0.999979485 0.025 0 9939 671 876 revol. 1 82 10 cm 2 74 10 cm . . arcsec . × 15 . × 16 S2 75.4539876 0.999979485 0.025 0 52 677 571 revol. 1 82 10 cm 2 74 10 cm . . arcsec . × 15 . × 16 S2 75.4539876 0.999979485 0.1 0 52 676 5 revol. 1 82 10 cm 2 74 10 cm . . arcsec . × 15 . × 16 S2 75.4539876 0.999979485 0.85 0 52 595 1 revol. 1 82 10 cm 2 74 10 cm . . arcsec . × 15 . × 16 S2 72.6190898 0.999979145 0 0 9939 725 018 revol. 1 68 10 cm 2 71 10 cm . . arcsec . × 15 . × 16 S2 72.6190898 0.999979145 0.11 0 9939 723 525 revol. 1 68 10 cm 2 71 10 cm . . arcsec . × 15 . × 16 S2 72.6190898 0.999979145 0 0 52 731 407 revol. 1 68 10 cm 2 71 10 cm . . arcsec . × 15 . × 16 S2 72.6190898 0.999979145 0.11 0 52 728 176 revol. 1 68 10 cm 2 71 10 cm

6 Table 9 Periastron precession for the star S14 in the central arcsecond hole. We assume a central black hole mass MBH = 4.06 × 10 M of the galactic centre, using the exact analytic formula (189), (190)for and that the orbit of the star S14 is a timelike non-circular equatorial different values of the electric charge and the spin of the galactic black Kerr–Newman geodesic / / / δteKN Star L ME eMaM p Periapsis rP Apoapsis rA

. . arcsec . × 15 . × 16 S14 72.9456205 0.999988863 0.11 0 9939 717 128 revol. 1 64 10 cm 5 22 10 cm . . arcsec . × 15 . × 16 S14 72.9456205 0.999988863 0.025 0 9939 718 531 revol. 1 64 10 cm 5 22 10 cm . . arcsec . × 15 . × 16 S14 72.9456205 0.999988863 0.11 0 52 723 432 revol. 1 64 10 cm 5 22 10 cm . . arcsec . × 15 . × 16 S14 72.9456205 0.999988863 0.33 0 52 711 595 revol. 1 64 10 cm 5 22 10 cm . . arcsec . × 15 . × 16 S14 72.9456205 0.999988863 0.85 0 52 636 568 revol. 1 64 10 cm 5 22 10 cm . . arcsec . × 15 . × 16 S14 64.8441485 0.999987653 0 0 9939 907 686 revol. 1 29 10 cm 4 73 10 cm . . arcsec . × 15 . × 16 S14 64.8441485 0.999987653 0.11 0 9939 905 812 revol. 1 29 10 cm 4 73 10 cm . . arcsec . × 15 . × 16 S14 64.8441485 0.999987653 0 0 52 916 661 revol. 1 29 10 cm 4 73 10 cm . . arcsec . × 15 . × 16 S14 64.8441485 0.999987653 0.11 0 52 914 786 revol. 1 29 10 cm 4 73 10 cm to magnitudes: = 2.29 × 1035esu ⇔ 7.65 × 1025 C . (194) e = 0.85 6.6743 × 10−84.06 × 106 × 1.9884 × 1033esu Concerning these tentative values for the electric charge e we used in applying our exact solutions for the case of SgrA* = 1.77 × 1036esu ⇔ 5.94 × 1026 C, black hole we note that their likelihood is debatable: There is e = 0.11 6.6743 × 10−84.06 × 106 × 1.9884 × 1033esu an expectation that the electric charge trapped in the galactic 123 Eur. Phys. J. C (2021) 81:147 Page 31 of 37 147

nucleous will not likely reach so high values as the ones αμ−1 − αμ+1 r − αμ close to the extremal values predicted in (193) that allow z = (196) αμ − αμ+1 r − αμ−1 the avoidance of a naked singularity. However, more precise statements on the electric charge’s magnitude of the galactic The roots of the sextic radial polynomial are organised as black hole or its upper bound will only be reached once the follows: relativistic effects predicted in this work are measured and αν >αμ >αρ >α, (197) a comparison of the theory we developed with experimental i 13 data will take place. where αν = αμ−1,αρ = αμ+1 = rP ,αμ = rA,αi = αμ+i+1, i = 1, 3. Also we deﬁne: 5.6 Periapsis advance for non-spherical polar timelike Kerr–Newman–de Sitter orbits H ≡ (αμ − αμ+1)(αμ − αμ+2)(αμ − αμ+3)(αμ − αμ+4),

In this section we are going to derive a new closed form (198) expression for the pericentre-shift of a test particle in a time- αμ − αμ+ ω := 1 . (199) like non-spherical polar Kerr–Newman–de Sitter geodesic. αμ−1 − αμ+1 After one complete revolution the angular integration has to satisfy the equation: The integral on the left of Eq. (195) is an elliptic integral √ of the form: θ rP ω d dr θ z √ = 2 √ = 2 √d =−1 d , √ rA R H 2 3 z a4 (z − z)(z − z+)(z − z−) 3 1 1 1 1 2 2 2 (200) × −FD , , , , 1,κ ,λ ,μ π 2 2 2 2 Inverting the elliptic integral for z we obtain: π 3 1 1 1 2 2 2 + ωFD , , , , 2,κ ,λ ,μ (195) 2 2 2 2 2

−β1 z = . (201) √ √ √ √ ω π ω δ −β α −β 2 ω sn2 2 −F 1 , β, 1, x π + ωF 3 , β, 2, x 1 1 1 1 12a , 2 − 1 1 D 2 D 2 2 2ω1 3 3 H

Equivalently the change in latitude after a complete radial For computing the radial hyperelliptic integral in (195)14 in oscillation leads to the following exact novel expression for closed analytic form in terms of Lauricella’s multivariable the periastron advance for a test particle in an non-spherical hypergeometric function FD, we apply the transformation polar Kerr–Newman–de Sitter orbit: [46]:

√ θ = arccos ± z ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ − ⎜ −β1 ⎟ = cos 1 ⎜± ⎟ , (202) ⎜ √ √ √ ⎟ ω δ −β α −β 2 ⎝ 2 1 3 π 1 1√ 1 1a 2 ⎠ ω1sn 2 −FD , β, 1, x π + ωFD , β, 2, x , − 1 2 2 2 ω1 3 3 H

13 In this regard, we also mention that the author in [96], under the where: assumption that the curved geometry surrounding the massive object in 1 1 1 the Galactic Centre is a Reissner–Nordström (RN) spacetime, obtained β ≡ , , , x ≡ κ2,λ2,μ2 . (203) an upper bound of e 3.6 × 1027C. This upper bound does not distin- 2 2 2 guish yet between a RN black hole scenario and a RN naked singularity scenario. Also the Jacobi modulus of the Jacobi’s sinus amplitudi- 14 The sextic polynomial R is obtained by setting μ = 1andL = 0in nous elliptic function in formula (202), for the periapsis (11). advance that a non-spherical polar orbit undergoes in the 123 147 Page 32 of 37 Eur. Phys. J. C (2021) 81:147

Kerr–Newman–de Sitter spacetime, is given in terms of the r 2 − (3r 2 − 4rM + a2 + e2) roots of the angular elliptic integral by: E2 2 2 3 ω δ α − β δ re 2 2 2 2e r 2 1 1 1 1 1 + ( r − Mr + a + e ) − = = . (204) 2 2 3 δ − β α δ − β ME M 1 1 1 1 1 2 2 e r 2 The roots z, z+, z− appearing in (200) are roots of the poly- = r − ξ , (210) M nomial equation: e2r r 4 + η a2 − Mr + a4 a2 Q z3 − z2 [Q + (L − aE)22]+a2z2(1 − E2) M 3 3 2 r 2 − z{[Q + (L − aE)22] + a2 + aE2(L − aE)}+Q = , − r(r − M) − Mra 2 0 2 (205) E re2 2e2r 3 + (2r − 3Mr + a2 + e2) − after setting15 L = 0. Also the variables of the hypergeomet- ME2 M F e2r ric function D are: = Mr(ξ 2 − 2aξ)− ξ 2. (211) M αμ− − αμ+ α − β r 1 − γ κ2 = ω 1 2 = α − α 1 α − γ μ μ+2 r − β These equations can be solved for ξ and ηQ. Thus, elimi- 1 nating η between them, we obtain: αμ− − αμ+ α − β r − δ Q λ2 = ω 1 3 = (206) α − α 1 α − δ μ μ+3 r − β 2 1 2 2 2 2 2 e r αμ−1 − αμ+4 α − β r − r a (r − M)ξ − 2aM r − a − ξ μ2 = ω = M αμ − αμ+ r 1 − β α − r 2 4 a2e2 − (r 2 + a2)[r(r 2 + a2) − M(3r 2 − a2) + 2e2r − ] M 5.7 Computation of ﬁrst integrals for spherical timelike 2a2e2r e2 a2e4 (KN)2 geodesics in Kerr–Newman spacetime + r − + − = 0. (212) M M M2 E2 The equations determining the timelike orbits of constant radius in KN spacetime are: We ﬁnd that the solution of this quadratic equation is given by:

2 ( 2− ) 2( − ) 2 ( − )( 2+ 2) 2 2 M r 2 − a2 − e r ± rKN − − M 1 − e 2rM e r M a − r M r a a e M 1 1 r E2 rM2(KN)2 M(KN)2r2 ξ = . (213) a(r − M)

4 2 2 R = r + (2Mr − e )(ηQ + (ξ − a) ) η KN The parameter Q is then determined from equation: + r 2(a2 − ξ 2 − η ) − a2η − r 2 = , Q Q 2 0 (207) E ∂ 2 R 3 2 2 2 re = 4r + 2M ηQ + (ξ − a) + 2r(a − ξ − ηQ) − η r 2 − a2 − ∂r Q M KN 2 r 2 − 2r − (2r − 2M) = 0, (208) 4 2 2 re 2 2 =−3r − a r − E E M where we define: r 2 + (3r 2 − 4rM + a2 + e2) Q E2 ξ ≡ L/E,η≡ . 2 Q 2 (209) re E − ( r 2 − Mr + a2 + e2) 2 2 3 Equations (207)–(208) can be combined to give: ME 2e2r 3 e2r + + r 2 − ξ 2. (214) re2 re2 M M 3r 4 + a2 r 2 − − η r 2 − a2 − M Q M For zero electric charge e = 0, Eqs. (213) and (214) reduce 15 We have the correspondence α1 = z+,β1 = z−,δ1 = z. correctly to the corresponding equations for the first integrals 123 Eur. Phys. J. C (2021) 81:147 Page 33 of 37 147 of motion in Kerr spacetime [76]: dragging, pericentre-shift and orbital period for timelike non- spherical polar geodesics in Kerr–Newman spacetime and applied them for the computation of the corresponding rela- M(r 2 − a2) ± r 1 − 1 (1 − M ) E2 r tivistic effects for the orbits of stars S2 and S14 in the central ξ = , a(r − M) arcsecond of SgrA*, assuming the Galactic centre supermas- (215) sive black hole is a Kerr–Newman black hole for various values of the Kerr parameter and electric charge. Secondly, we r 3 η a2(r − M) = [4a2 M − r(r − 3M)2] computed in closed analytic the periapsis advance for time- Q M like non-spherical polar orbits in Kerr–Newman and Kerr– 2r 3 M 1 M Newman de Sitter spacetimes. In the Kerr–Newman case, − 1 ± 1 − 1 − r − M E2 r the pericentre-shift is expressed in terms of Jacobi’s amplitude function and Gauß hypergeometric function, while in r 2 + [r(r − 2M)2 − a2 M]. (216) the Kerr–Newman–de Sitter the periapsis-shift is expressed E2 in an elegant way in terms of Jacobi’s sinus amplitudinus elliptic function sn and Lauricella’s hypergeometric function 5.8 The apparent impact factor for more general orbits FD with three-variables. We also computed the first integrals of motion for The apparent impact parameter for the Kerr–Newman– non-equatorial Kerr–Newman and Kerr–Newman–de Sitter (anti) de Sitter black hole can also be computed in the case geodesics of constant radius. We achieved that by solving in which the considered orbits depart from the equatorial the conditions for timelike spherical orbits in KNdS and plane and therefore θ = π/2. Again, we compute this quan- μ KN spacetimes. We derived new elegant compact forms for tity from the k kμ = 0 relation just taking into account its the parameters (constants of motion) of these orbits. Our maximum character, i.e., that kr = 0. Our calculation yields: results are culminated in Theorems 5 and 7. These expres- −[ 2( 2 + 2) − 2KN]± 2KN[2 4 + Q ( 2 − KN)] a r a a r r r γ a r sions together with the analytic equation for the apparent γ = . − 22 + 2KN a r impact factor we derived in this work-Eq. (217), can be used (217) to derive closed form expressions for the redshift/blueshift of the emitted photons from test particles in such non-equatorial Our exact expression (217) for the apparent impact parameter constant radius orbits in Kerr–Newman and Kerr–Newman in the KN(a)dS spacetime, for zero cosmological constant (anti) de Sitter spacetimes. A thorough analysis will be a task ( = 0) and zero electric charge (e = 0), reduces to Eq. (59) for the future.16 Such a future endeavour will also involve the (the apparent impact parameter for the Kerr black hole) in computation of the redshift/blueshift of the emitted photons [55]. Also for zero value for Carter’s constant Qγ Eq. (217) for realistic values of the first integrals of motion associated reduces to Eq. (55). with the observed orbits of S-stars (the emitters) such as those of Sect. 5.5, especially when the first measurements of the pericentre-shift of S2 will take place. The ultimate aim of 6 Conclusions course is to determine in a consistent way the parameters of the supermassive black hole that resides at the Galactic In this work using the Killing-vector formalism and the asso- centre region SgrA*. ciated first integrals we computed the redshift and blueshift It will also be interesting to investigate the effect of a of photons that are emitted by geodesic massive particles and massive scalar field on the orbit of S2 star and in particular travel along null geodesics towards a distant observer-located on its redshift and periapsis advance by combining the results at a finite distance from the KN(a)dS black hole. As a concrete of this work and the exact solutions of the Klein–Gordon– example we calculated analytically the redshift and blueshift Fock (KGF) equation on the KN(a)dS and KN black hole experienced by photons emitted by massive objects orbiting backgrounds in terms of Heun functions produced in [99, the Kerr–Newman–(anti) de Sitter black hole in equatorial and circular orbits, and following null geodesics towards a 16 distant observer. Another interesting application of the theory of the frequency shift of the photons developed in this paper, is the possibility to focus on In addition and extending previous results in the litera- the measurements of the special class of the principal null congruences ture we calculated in closed analytic form firstly, the frame- (PNC) photons as was shown for PNC photons emitted from sources on dragging that experience test particles in non-equatorial equatorial circular orbits around a Kerr naked singularity in [97]. The spherical timelike orbits in KN and KNdS spacetimes in significance of the PNC photon trajectories lies in the fact that they mold themselves to the spacetime curvature in such a way that, if Cαβγ δ is the ∗ β γ terms of generalised hypergeometric functions of Appell and Weyl conformal tensor and Cαβγ δ is its dual, then Cαβγ [δk]k k = 0 ∗ β γ Lauricella. We also derived new exact results for the frame- and Cαβγ [δk]k k = 0[98]. 123 147 Page 34 of 37 Eur. Phys. J. C (2021) 81:147

100](seealso[74]) . This research will be the theme of a black hole is given by the expression: future publication.17 √ −a(2Mr − e2) ± r 2 r 2 + a2 + e2 − 2Mr The fruitful synergy of theory and experiment in this fas- γ = . (218) 2 + 2 − cinating research field will lead to the identification of the r e 2Mr resident of the Milky Way’s Galactic centre region and will Its preservation provides the following equation that must be provide an important test of General Relativity at the strong satisfied by the radius of a circular equatorial observer: field regime. (r 2 − 2Mr + e2) Acknowledgements I thank Dr. G. Kakarantzas for discussions. Dur- 4 2 2 2 ing the last stages of this work, the research was co-financed by Greece × (r + r (a − e) + 2Mr(e − a) and the European Union (European Social Fund - ESF) through the − 2( − )2) = Operational Programme “Human Resources Development, Education e e a 0 (219) and Lifelong Learning 2014–2020” in the context of the project “Scalar fields in Curved Spacetimes: Soliton Solutions, Observational Results The roots of the quartic equation in (219) can be obtained and Gravitational Waves” (MIS 5047648). I thank Prof. Z.Stuchlík and either by the Ferrari algorithm or in a more elegant way by Dr. P. Slaný for useful correspondence. I am grateful to the referees for their very constructive comments. the use of the Weierstraß functions making use of the addition theorem for points on the elliptic curve [28]: Data Availability Statement This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The datasets 1 ℘(−x /2 + ω) − ℘(x ) generated during and/or analysed during the current study are available α = 1 1 , (220) from the corresponding author on reasonable request.] 2 ℘(−x1/2 + ω) − ℘(x1) ℘(− / + ω + ω) − ℘( ) β = 1 x1 2 x1 , Open Access This article is licensed under a Creative Commons Attri- ℘(− / + ω + ω) − ℘( ) (221) bution 4.0 International License, which permits use, sharing, adaptation, 2 x1 2 x1 distribution and reproduction in any medium or format, as long as you 1 ℘ (−x /2 + ω ) − ℘ (x ) γ = 1 1 , (222) give appropriate credit to the original author(s) and the source, pro- 2 ℘(−x /2 + ω) − ℘(x ) vide a link to the Creative Commons licence, and indicate if changes 1 1 1 ℘(−x /2) − ℘(x ) were made. The images or other third party material in this article δ = 1 1 , (223) are included in the article’s Creative Commons licence, unless indi- 2 ℘(−x1/2) − ℘(x1) cated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended x use is not permitted by statutory regulation or exceeds the permit- where the point 1 is defined by the equation: ted use, you will need to obtain permission directly from the copy- 2 − 2 =− ℘( ), right holder. To view a copy of this licence, visit http://creativecomm a 6 x1 (224) ons.org/licenses/by/4.0/. ω,ω ℘ Funded by SCOAP3. and denotes the half-periods of the elliptic function . The equations

2 2 2 2 2(a − ) = 4℘ (x1), −3℘ (x1) + g2 =−e ( − a) Appendix A: Linking the equatorial circular radii for (225) emitter and detector determine the Weierstraß invariants (g2, g3) with the result: In this Appendix and for the case of a Kerr–Newman black hole we will show how the radii of the emitter and detec- 1 2 2 2 2 2 g2 = (a − ) − e ( − a) , (226) tor can be linked for circular equatorial orbits through the 12 constants of motion Lγ , Eγ . The procedure was followed 1 2 2 3 1 4 g3 =− (a − ) − (a − ) in [55] for the Kerr black hole. Indeed, from the fact that 216 4 2 2 the ﬁrst integrals of motion Lγ , Eγ and hence the apparent a − − e2( − a)2 . (227) impact parameter are preserved along the whole trajectory 6 followed by the photons, the latter quantity is the same when evaluated either at the emitter or detector positions, rendering The maximum root provides the circular radius of the detec- = the following relation e = d . For circular and equatorial tor. For zero electric charge, e 0 (i.e. Kerr black hole) the orbits the maximised impact parameter for a Kerr–Newman detector radius reduces to the result obtained in [55]: − 17 e a 2 An initial study of such hypothetical scalar effects has been per- rd = − 27M (e − a) formed by Gravity Collaboration for the Kerr background and in solv- 3 ing approximately the KGF equation for the case in which the Compton 1 3 wavelength of the scalar ﬁeld is much larger than the gravitational radius + 27M2( − a) + ( + a)3 of the black hole [68]. e e 123 Eur. Phys. J. C (2021) 81:147 Page 35 of 37 147

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