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PoS( 2015)073 1 F lementary http://pos.sissa.it/ 80803. pell’s hypergeometric function t for spherical polar and non-polar omplete theory for the propagation of its fundamental parameters enter of a rotating electrically Newman (KN) . Various s in terms of Appell and Lauricella KE problem of treating a KN black hole ing the contribution from the cosmo- ctly null geodesics on the sphere and Λ ic source of light and a static observer ric functions of Appell-Lauricella and n space is solved analytically. For this all le for a non-spinning charged Reisser– E black hole spacetime. Our solutions are m analytic solution for the deflection an- lectroweak Scale oretical Physics r kind invitation and for organising such an exciting . , MIS 375734, code THALIS“Beyond the Standard Model:Theoretical Physics of E ive Commons onal License (CC BY-NC-ND 4.0). † ∗ [email protected] The null geodesics that describe orbits in the curved logical constant. Thus the theory producedof light in signals this in work the is field a of rotating c charged black holes: (Kerr-Newman) are solved exactly includ Speaker. Contributed talk, the author thanks the organisers for thei Weierstraßelliptic function. We then derive the closed for the analytic solutions on an equal footing. We first solve exa light orbits in the Kerr-Newman-(anti)deSitter (KN(a)dS) expressed elegantly in terms of multivariable hypergeomet gle that an equatorial unbound light ray undergoes by a Kerr- produce closed form solutions for the frame dragging of ligh limiting cases are studied. In particularNordström the black deflection hole ang is computed analytically in terms of Ap hypergeometric functions and the Weierstraß modular form. both located far away but otherwisemodel, at arbitrary we positions derive i the analytic solutions of the equation or in terms of inverse Jacobian functions.as The a more gravitationallens, involved i.e. a KN black hole along with a stat ∗ † Copyright owned by the author(s) under the terms of the Creat c Work partially supported by research grant: G. V. Kraniotis Gravitational lensing and frame-dragging of lightthe in Kerr-Newman and the Kerr-Newman-(anti) de Sitter black hole Ioannina, Greece GR 451 10 E-mail: University of Ioannina, Physics Department, Section of The 25-29 May 2015 Ioannina, Greece 18th International Conference From the Planck Scale to the E Particles and under the light of LHC” Attribution-NonCommercial-NoDerivatives 4.0 Internati

PoS(PLANCK 2015)073 = G (1.9) (1.1) (1.5) (1.7) (1.2) (1.4) (1.8) (1.6) (1.3) 2 ) φ d G. V. Kraniotis ) 2 a ,  + , 2 ) θ r aL L ( 2 f the black hole respec- − − − t cos E θ , with vector potential ( ) d 2 2 ) 2 2 , a 4 2 ′ θ . A ′ a ac c are published in [2], in which ) d a sin ∆ ( Θ Ge + φ + 2 θ √ − d = 2 + r 2 2 aE time is completely integrable and r 1 tionary black hole solution of the θ ± r ( ⇔ ( ρ ( F 2 2 =  2 2 . the singularity surrounded by the sin 2 = 2 Ξ odesics in the spacetime surrounding 2 2 Ξ c / θ a sin istic Hamilton-Jacobi equation by sep- GM 1 ρ KN r Ξ θ λ ∆ stem of geodesics in the KN(a)dS black a GM 2 # a d d ∆ , − 2 , namely the Kerr-Newman-(anti) de Sitter 2 − ] − 4 − ≤ Λ denote Newton’s and + 2 t ρ c 3 2  must be restricted by the relation: 2 Ge c d  aL 2 θ a e , , ( J L ′ a d ) + − G 2 + R θ ⇒ − 2 θ + E ρ 1 ∆ 2 2 √ 2 2 )  θ / r 2 2 1 ± = − a dynamical system. : J  cos 2 Mc  = r 2 2 sin Ξ + 2 4 er d a  2 r c r KN r , λ r Ge " d 2 aE + d 3 ∆ Λ θ KN r 2 ρ 2 2 )[( + ≥ ∆ r 2 − ρ θ 2 ( a 1 2 a − 2 Ξ cos , 2  c ′ 2  + GM ) − Λ Ξ sin 2 θ Θ 3 2 ≥ φ = r θ d : a = ( √ d ∆ 2 2 θ A + c Ξ Z − KN r 2 GM 1 ∆ = = = sin = ′ completely integrable : a t φ λ λ r R d θ d d d d − √ 2 2 ∆ t ρ ρ d c Z c ( is the cosmological constant. Also, 2 denote the Kerr parameter (spin), and electric charge o ρ Λ KN r 2 e , ∆ Ξ M , = a 2 s In this talk I am going to review recent results of mine, which The metric in Boyer-Lindquist coordinates is [1]: This is accompanied by a non-zero electromagnetic field 1): d = Gravitational lensing by charged, rotating black holes 1. Motivation I have obtained closed forman analytic electrically solutions charged, of rotating cosmological the black null hole black ge hole. The KN(a)dS)metricEinstein-Maxwell system is of the differential most equations. general Thehole sy (BH) exact solution sta is a c For the surrounding spacetimehorizon, to the represent electric charge a and black angular momentum hole, i.e The dynamical system of geodesicthe equations geodesic in equations can KN(a)dS be space derivedaration by of solving variables the [2]: relativ where respectively. tively, while PoS(PLANCK 2015)073 (2.4) (2.7) (2.8) (2.2) (2.5) (2.1) (2.6) (2.3) (1.10) (1.11) G. V. Kraniotis . 2 .  , L )  2 αβ 48 z − ) θ Q θ − 2 Φ aE 2 1 −  sin − 2 a sin + L α a ( − are first integrals of motion. We 48 ] 2 aE Q ′ − θ L Ξ 0) is: [2] 2 2 , KN 2 L − aP + Q ∆ . Ξ = sin KN 3 , 2 aP ) . Q ∆  θ − E 2 Φ + Φ β 2 L  θ 2 + E × .  + + / 2 sin ∆ E 3 × r ε β 2  g Q α 2 4 z Q + a + 12 ( θ / + 0 and a constant radius. It is convenient to µ α − 2 = A − Q ( = : [ α φ : 1 aE ξ = + θ 2 216 − − KN r cos 3 Q β 2 g − )+ ξ ξ , Λ 2 ∆  d = β a − , E − KN 2 − 3 cos aP 2 / The quantities 3 ℘ ∆ + g 2 a µ . L = ξ and the Weierstraß invariants take the form , − z  : = 0 α 4 − d ) − ( z α = Q 2 : ξ aL 2 4 2 = p = z a Q : Ξ by the Weierstraß modular Jacobi form q − Φ 2 µ 1 2 − + ) α − E 3 , assuming 2 = ) Z 2 z r 2 ) aE θ φ α a 1, unless it is stipulated otherwise. = Mr β d d − 2 = ( p + φ = ′ + inverted θ L d 2 − 2 1 P c 2 r α 2 , ( ( Z − + ( e  =  cos 1 2 Q a 12 = + G  Ξ = 2 − : φ = a = = d ′ z : : 2 KN ′ ′ + g aP ∆ R 2 Θ 0 and using  r 2 1 = = : − Φ = : KN ∆ A The relevant differential equation is: introduce the parameters: Assuming where Now the closed form solution for the spherical polar orbit ( Gravitational lensing by charged, rotating black holes where Using the variable: we obtain the integral equation: where Null geodesics are derived by setting and this orbit integral can be where use geometrized units, 2. Null geodesics on the sphere and frame dragging of light PoS(PLANCK 2015)073 0) 6= (2.9) (2.10) (2.11) (2.14) (2.15) (2.12) (2.16) (2.13) Φ denotes a ] and param- denotes the 2 ε is [2]: . The Weier- y ) )) ˜ G. V. Kraniotis , z ξ + r Λ , x ) and γ 2 + , ) 2 / ′ 1 non-polar ( β 1 M ( A , Q ( 2 , 3 Γ a 2 α 4 ′ − ( , Γ ( , ring effect) of light that . A = F r 2  2 ) Q ! 2 ξ a 2 ′ A 2 3 ) + Q ) 4 / A 1 Ξ 2 r 2 1 ( / = 2 e ( + ′ : a , and + Γ a 2 2 − 2 . ˜ 2 ξ Q ] ℘ Γ a ) + 3 M , Q ( 3 = Λ ! 1 Q 4 ) + ( ity. After a complete oscillation in Q is the initial value of ewman-(anti)de Sitter black hole is ′ 2 , 2 )= 3 ′′ 2 a 0 A ′′ + a 1 2 . Ξ ˜ 4 ξ Mr em 2 in the presense of 2 , 2 Q nt of dragging for the spherical photon β + KN black hole, increases by: a Q 2 , a a 1 2 ns from the vanishing of the polynomial 2 Mr a − 2 + ′ 2 − − 2  + A Q ′′ , ( Q , 2 Q e 2 2 − 3 a Λ ( F α a Λ Mr a ω 1 2 ( 2 1 3 2 12 2 3 4 π , where a a = a − 1 2 4 + 12 [ ) + 2 = = 2 )) Q ′ r 2 − 2 2 2 a 4 ) 0 e a e , r ′′ A 2 r ˜ − Q ξ − 1 / 4 + ( , α − , ′ θ + GTR 2 pKN 1 4 a 1 ′′ 1 2 2 2 2 β 2 , − φ 3 − ) r a = , r Q 2 1 ∆ 1 2 +( ℘ = − , cos 2 , : − Q = e 6 − 2 1 2 : + 1 2 ′′ , ′′ 1 e a + 2 4 ′ ( 3 2 ) α 2 φ

β 2

r a A ′′ , a 1 − )( − 2 F ′ [ β + ( F 1 Q A 1 2 6 2 )= ′ + + ℘ ) Q ε is given by the expression: 2 − A 1 2 ′′ a ′ 6 = Mr ( a √ + 1 α a A = Q 3 ˜ ( a ξ + φ ) 2 q 2 ′ √ 2 r − 2 1 − A is Appell’s first hypergeometric function of two variables 432 12 Ξ α r r 2 ( = ( − 2 a KN r ) 2 − e y = = ℘ ∆ ω , 2 M Ξ = ′′ 3 ′′ 2 x = ( a g , g + 2 a ′ Λ γ r , A 2 ′ / [4]. The generalisation of Theorem 3 to the case of spherical a ′ GTR β pKN , γ φ α + The closed form solution for the frame dragging (Lense-Thir , β ′ ∆ , = M β : 2 α , is the real half-period of the Weierstraß elliptic function ( a ′′ β 1 , α ω F = α and its first derivative: Φ ) r ( where Explicitely eqn (2.9) reads: eters a null spherical polar undergoes[2]: in the spacetime of a Kerr-N orbits in KN(a)dS spacetime is [2]: Gravitational lensing by charged, rotating black holes Theorem 1. constant of integration. Also straßinvariants are: where Gauß hypergeometric function. The generalisation of Theor Theorem 3. The assumption of spherical orbits, resultsR in two conditio where which are also the conditionslatitude, for the the angle photon of longitude, to which escapepolar determines orbit to the in infin amou the general (GTR)Theorem for 2. the PoS(PLANCK 2015)073 ± > 2 BH (3.2) (3.3) (3.1) (3.5) (3.6) (3.4) − 2 c (2.17) µ GM α ≥ = . 2 1 ± − Ξ r G. V. Kraniotis 2 µ 2 a ows, α  ight orbit in the  0 1 3 ]+ Λ α z 2 z . , 2 r Ξ m − ) 3 2 2 z d . a ) , m R 1 a # z − Λ 2 , 4 √ − of an equatorial 1 ) β Φ  , , ( − 0 2 Φ 2 1 2 r 2 1 eqKN − , Λ α , e z ) A  , − + ( 1 2 a r − r 3 2 D d (  ) 3 and we have that , − Q F R ) [ , a 4 3 ) F a 2 Λ Φ 3 Z √ β Φ 3 , 2 ( 2 z π ) 2 , tions technique: a 2 − 1 1 } + e 3 2 1 ρ which appears in the analytic solution of − z r + − Φ Mr = Mr = α −  ( d µ m : D 2 2 has been introduced. The radii of the (di- i 1 2 1 − 2 z , α F D 2 R ( > i 1 Φ a − m F − Mr i 2 m − H z √ − 2 BH 2 2 z − + − m ) µ α r 0 and the relevant differential equation for the KN 2 e R z z 2 M 1 ∆ + α r + − ′ {z : p Φ a r η 5 > ] eqKN + √ = r + ω 1 µ 2 ( = ) p = A r 2 ν Φ α − − m i KN = a z a )+ Q α z r Φ ± 1 Mr α a ∆ horizons are given (In the usual units: ( r ω 2 1 + ) − − > 3 ) | 2 − ) )+ z 1 Z µ | r = = 2 ′ − and H a ( )+ r ′ Φ a | )( α m − + ′ ± 2 ( ( r µ z m r − 2 r + m m z Φ α e d z z 2 Φ through ( η r ( − R = ′ 2 m = 2 Φ 2 [( − r z ρ e √ a r 2 φ 1 ( α d ( p d , 2 KN r Ξ Z Z | | 2 r a ∆ a Φ + 2 H H 2 = = + " | | µ .) by: and Cauchy Ξ 4 4 Ξ 2 2 α  φ r ) 2 d = − + ′ a + = We organize all roots in ascending order of magnitude as foll = r Λ Z ν . ( + R α ∞ 2 α , 4 R c 1 GTR Ge nPKN 2 + φ  µ The exact amount of frame dragging for a spherical non polar l ∆ = α − 2 =  GTR eKN µ φ BH α 2 By applying the transformation c ∆ . For equatorial geodesics the parameter GM 3 unbound light ray in the Kerr-Newman spacetime −  µ We compute the following integral applying the partial frac Thus The parameters and variables of Lauricella’s function where the quartic radial polynomial has the form: KN(a)dS spacetime is: where a dimensionless variable exact computation of the deflection angle is: mensionless) event Theorem 4 are defined in Eqs.(52)-(54) in [2]. Also α Gravitational lensing by charged, rotating black holes Theorem 4. r 3. Closed form analytic computation for the deflection angle where PoS(PLANCK 2015)073 d (3.8) (3.7) (3.9) (3.12) (3.10) (3.14) (3.13) (3.15) (3.11) G. V. Kraniotis ! !#   . r + r − z z , ,  , 2 3 3 2 2  , , 2 Φ 9 9 ) 3 3 r A 6 a z β β − , , , 2 − 3 2 2 2 1 1 a , Φ    ,  ( ra A ) r + 2 2 D D r in the closed form solution of ) ic compact form in terms of the 1 β ) z rm in terms of Appell-Lauricella e 1 F F , , x a x ( , − 1 2 3 2 ′ ( , + + ) − , pell-Lauricella hypergeometric func- ) ) 1  ) ℘ ℘ 9 =   3 1 1 x 1 Φ 1 x x , 2 β r A r A ( x ( − − F ′ ( ( )  , , g z z ( ′ 2 ) ) 1 , , ) ′ ′ r − e 2 1 ) ℘ ℘ x ! 1 ℘ ℘ z 3 3 2 2 1 ( ) ω ω x ,  )+ − , , x − − ( − − 1 ′ ( α 2 3 ℘ 4 ) ) D + + ra ra ) ) ′ ′ x A A 6 ) , F ( , ℘ ℘ − 9 β β 3 ω ω ω ω ω ω ) 2 2 − Φ , , ) β 6 − − δ β 1 1 2 2 , + + + + + + ℘ a − ) ) = )( 3 2 1 2 2 2 2 2 2 2 2 − 1 a   2 − α ( / / / / / / / / − 1 1 with the result:  α Φ 1 1 1 1 1 1 1 1 , 1 4 F F Φ ( ) x x x x x x x x ) − D ( 1 3 − − − F 2 γ ω − − − − − − − − − x + g 2 ( ) e ( ( ( ( ( ( ( ( ( , 3

′ ′ ′ ′ ′ , of an equatorial unbound light ray in the gravitational fiel a √ H ) 2 β ) ) − ℘ ℘ ℘ ℘ p 2 g ℘ ℘ ℘ ℘ ℘ β β 2 ( − 4 Φ 2 2 2 2 1 1 1 1 eKN

) eqKN + 2 δ − − α ) are defined in [2]. = − ( A = = = = 1 Φ α α 2 2 2 2 2 ( ) ′ ′ ω ( ( − + γ ) a − δ β α 2 − Γ − ( κ κ eKN ω ω ) Φ / √ 2 H δ − + " 2 3 a √ √ 1 2 ( − / ( H H 216 1 Γ a eqKN − + ( 1 ( as follows [2]: 12 − A Γ 2 ′ eqKN − π eqKN + 2 A A = = Φ − ℘ 2 − 2 , 2 3 2 is determined from the equation g g = 1 ℘ : + + + + x The deflection angle eKN δ The roots of the quartic radial polynomial (3.2), that appea while the equations Theorem 5, have been computedelliptic functions in a very elegant closed analyt Gravitational lensing by charged, rotating black holes We computed analytically the radial integrals in terms of Ap tions [2] : Theorem 5. of a Kerr-Newman black holehypergeometric is functions [2]: computed in closed analytic fo determine the Weierstraß invariants The rest of the quantities in where the point PoS(PLANCK 2015)073 . π e , − , Φ C C e, the ! (3.16) (3.17) , 2 25 26 k , 10 10 δ β × × − − G. V. Kraniotis 65 94 α α . . 7 5 r on the black hole’s -2, we observe that

⇔ ⇔ 1 . Concerning tentative − eKN 1.5 1.0 esu esu 0.5 δ Φ for fixed value for the Kerr cn 35 36 e ) , γ 10 10 Φ 0.2 − π . For fixed values for a × × ) Reisser–Nordström black hole α − 0 29 77 )( eKN . . Φ  = δ δ 2 1 0.4 fixed values of the parameters 4 e on the electric charge, for smaller r A (Kerr and Kerr- alues of the impact factor parameter Λ z − = = e ) , 0 = 3 2 β ( , esu esu = ra A 0.6 p e 33 33 β 52 ( . , = 10 10 0 2 1 the smaller π π × × ,  = 7 − 1 − a 0.8 F , ! ! 2 2 ! 9884 9884 k 10 . . k ) eKN , 1 1 there is a strong dependence of , δ α γ β as a function of the parameters δ δ × × α − − 6 6 − − − δ eKN α α 10 10 β α δ )( 8 1 × × s α r Φ

− 06 06 1 . . 1 γ 4 4 − − ( 8 8 6 − − sn dn p ) ) 10 10 γ γ

decreases with increasing values for the parameter ) × × 52. − − . 1 ( ) . 0 α α 2 Γ β α eKN )( )( = ) / − δ − Φ Φ 6743 6743 2 3 δ δ γ γ a . . 4 4 ( / 6 6 The deflection angle for the non-spinning (a δ δ 1 Γ − − − ( From the 3-d plots of the deflection angle in Theorem 5, Figs.1 The deflection angle for the uncharged − p p β β β α Γ ( ( Plot of the deflection angle 11 85 Φ . . = p p : 2 0 0 2 = = = k = = e e -see Fig.1. For a fixed small distance eRN We also observe, the strong dependencevalues of of the the deflection spin angl ofΦ the black hole, particularly for small v is given by: values for the electric charge of the SgrA*: the smaller the Kerr parameter the larger the deflection, for Remark 8. electric charge: the larger the electric charge e parameter (spin) Remark 7. δ Gravitational lensing by charged, rotating black holes Corollary 6. deflection angle with (anti) de Sitter) has been investigated in [3]. Figure 1: PoS(PLANCK 2015)073 (4.2) (4.1) (4.4) (4.3) . 2  Φ G. V. Kraniotis θ − 2 θ 2 sin the number of turning sin for fixed Kerr parameter , a ime is given in terms of m . , e r θ 1.2 1.0 0.8 0 0 , 0.6 0.4 d d − Φ  Θ Newman spacetime )= )= 2 ) √ m m a , , θ S S 2 -axis and Φ e − y y z as the ones close to the extremals , , 0.05 Φ S S sin x x arity. , , ± i i tation that the electric charge e trapped + ( y y , , + i i Q x x  R ( ( 2 1 √ Φ = A A 2 a 0.10 Θ KN and: − − , ∆ 0.9939 8 − ) ) ) 1 i i = 2 in the form [2]: ) 12 ± y y ) ,a 7 , , ) . a i i 1 + x 1 x . eKN ( ( ( − ∆ ) 4 as a function of the parameters ( Φ KN 2 KN 1 R 10 Mr R R 2 √ eKN + ( and − δ + 1 ) KN ) Q Φ 2 n ( 7 ∆ e 8 , . S 1 ± − KN y ( ( , ∆ a S x 6 − ( = 2 φ  r φ ∆ d Φ d a denotes the number of windings around the − n ) 2 a The analytic solution of the lens equations for the KN spacet + 2 r . Plot of the deflection angle (  = 9939 : . The lens equations in this case are Eq. 0 R = In these equations we note that their likelihood is debatable:in There is the an galactic expec nucleouspredicted will in (1.6) not that likely allow the reach avoidance so of high a naked values singul 4. Exact analytic solution of the lens equations in the Kerr- points in the polar motion. Theorem 9. a We have written the lens Eqs. Gravitational lensing by charged, rotating black holes Figure 2: PoS(PLANCK 2015)073 (4.5) (4.7) (4.9) (4.6) (4.8) (4.10)  2 O  z 2 S ,  z 3 2 r A , z , G. V. Kraniotis 3 2 , 7 3 , 3 2 β 7 3 ) , , β  ε 1 ra , A 3   1 z β  , , 1 O ]+ D 1 S  m z , − 1 2 z F z  , ) D , m ···   3 F 2 m × z 2 3 2O A z , 1 , , + 2S A z  ) 4 × F 3 ) 4 , z 3 ) √ 3 m O 1 ) ) ) ) , β  ) z ) 1 z m z β 3 2 i ( S , 2 2 2 ( 3 , z , 2 3 2 , z m β z ) / Γ / / 1 2 − − − , / Γ 1 2 z ( − ) ra 1 3 3 A ) 3 m 3 m 1 − m ra 2 1 ( − ( (  A 2 1 (  z z , β z , 3 ( , Γ Γ Γ S ( β D m Γ , z D 1 )( , y sign Γ z ) F 2 1 Γ , 1 F , ( m )( , 2 1 , ) δ ) 2 1 S 1 z   1 m  2 1 m x , m 1  , z 1 z , − , z even odd − 1 i 1 1S A 1O A F 1 2 2 , etric functions of Appell-Lauricella [2] 2 1 y 2 − z z F O − ) ) , α − , , , 1 2 3 3 m m z 3 ) ) i S z z 1 2  1 Arccos ( 3 3 3 z )( 1 z x 3 3 2 2 , ( z z 2  z ( − − ( ( γ O , , F  − − m O S 1 3 2 z − 3 z z S z z m π mO ra − ra F z ( ( A A − z A F )= z m ( ) θ ( O m ) , p β β m z m π z z 9 S m − 3 p α − , , z z z m z 1 z ) ( 1 m − z √ m O )+ 2 2 1 1 mO 3 m m z S i q √ z z √ z z z π z − θ π q y p ) − | m   i − , | √ z m 1 − a √ i 1 1 − − 1 − y ( | 1 a z m x ) O ( ( | F F ) 1 S ⇔ ( z m 1 ( z z )] S O ) ) ) z = m ) )] z z ) 3 3 : 3 ) ( m O z S m z z ) z KN 2 ) S O z m z m z m mO m z i sign mS − − z 1 z z , z z | R ( y θ mS | − − θ r − S , a m m Φ a | θ − ◦ i − y | Φ − ◦ z z | m m m = r , a ] x ( ( ] z z z 1 | O S m | m ( S m S O S ( ( ( z 2 z θ a θ ( x φ 1 ( | ( m m σ σ ( ( s s , ) i KN 1 z z 2 p | | Arccos 1 − y p ) − − R | a a p , [ | | Φ Φ p p 1 i sgn sgn a − = 1 1 Φ S | 2 2 : x | | ( σ − − − a a m 1 1 1 | | 2 ( + [ 1 1 + + [ + ( 2 2 mO m A 2 ( θ ) = ℘ 2 + + [ + [ + = m , = − ) = S i S y y ξ , , i S x x ( , i y KN 1 , i R x ( 2 A is the possible position of the image and: i Gravitational lensing by charged, rotating black holes Weierstraßelliptic function and multivariable hypergeom y where PoS(PLANCK 2015)073 , non-zero Phys. Rev. G. V. Kraniotis and will therefore 11 for an observer at Λ . , 0 e Sitter spacetimes a , = J. Math.Pure Appl.8 rr-Newman and the e 3 π of Eq.(4.5) can be written as: 9939, aically special metrics, = . lactic centre black hole and its 0 O dependance of relativistic observ- iastron precession on the electric θ Fig.3 and [2])depends significantly = , a ; thus, one can separate from it the 11 ξ . e parameters M Precise analytic treatment of Kerr and , Rend.Circ.Mat. Palermo 7 (1893) . In establishing (4.7) we used the fact tect the electric charge of the galactic i 0 ) Jour. of Math. Physics,6,(1965)918; R P x 4 . = 3 ,Clas. Quantum Grav.28 (2011), 085021 2 e ( denotes a constant of integration. The Weier- , , Gen.Rel.Grav.46 (2014) 11, 1818 ε 2 and 9939 1 does contribute to the gravitational bending and 10 ) and . 8 0 . M the larger values for e the smaller the boundary Λ , mS 2 i y 1 2 2 1 = ( θ - - a ◦ 1 S - θ are extremal values of The motion of test particles in black-hole backgrounds with sign ) 0 = : ξ , . S ) are defined in m σ ξ ξ , ) ( ( 2 7 1 , Bull. of the Astronomical Inst. of Chech 34 (1983) 129 . θ − 4 Metric of a Rotating, Charged Mass, ( Sur les fonctions hypergéometriques de deux variables, , ℘ cos S 3. 1 / σ Frame dragging and bending of light in Kerr and Kerr-(anti) d Gravitational lensing and frame-dragging of light in the Ke θ et al π , where Sulle funzioni ipergeometriche a più variabili ∝ S The cosmological constant 0 ξ = S ξ ∞ cos ξ R O R ) θ = S S : σ σ 2 Gravitational field of a spinning mass as an example of algebr The boundary of the shadow of the KN black hole for θ − ◦ 1 1 θ 111-158; P. Appell (1882) 173-216 Clas.Quantum Grav.22 (2005),4391-4424; G. V. Kraniotis, Kerr-(anti) de Sitter black holes as gravitational Kerr-Newman-(anti) de Sitter black hole spacetimes Lett. 11(1963) 237; Z. Stuchlík, cosmological constant Kerr, +( m S ξ [4] G. Lauricella [3] G.V. Kraniotis, [2] G.V. Kraniotis, [1] E. T. Newman ξ polar position Gravitational lensing by charged, rotating black holes Also that theR sum of the second and third term on the right hand side straß invariants in Eq. expression Figure 3: determine the type of the galactic centre black hole. References ables such as the deflectioncharge of angle, the frame black hole dragging [2]. andon The the shadow the of per electric the black charge-for hole fixedof (see values the for shadow a ofrelativistic the observables black hole. may constrain Future significantly measurements or of de the ga frame dragging of light by a KN(A)dS BH. There is a significant centre rotating black hole as well as the rest of the black hol Conclusion 10.