reprintAPS/123-QED
The complete set of solutions of the geodesic equations in the spacetime of GMGHS black holes Saheb Soroushfar, Reza Saffari,∗ and Ehsan Sahami
arXiv:1601.03143v1 [gr-qc] 13 Jan 2016
Department of Physics, University of Guilan, 41335-1914, Rasht, Iran. (Dated: January 14, 2016)
Abstract In this paper we consider the timelike and null geodesics around a spherically symmetric charged dilaton black hole, described by the solution of Einstein-Maxwell equations found by Gibbons and Maeda and independently by Garfinkle, Horowitz and Strominger (GMGHS). Similarly, we discuss about the Magnetically charged and Electrically charged GMGHS black holes. The geodesic equations are solved in terms of Weierstrass elliptic functions. In order to classify the trajectories around the black holes, we use the effective potential and characterized the different types of the resulting orbits in terms of the conserved energy and angular momentum.
∗
Electronic address:
[email protected]
1
1.
INTRODUCTION
The well-known exact solution of the vacuum Einstein equations described by Schwarzschild in 1916 [1] as a spherically symmetric black hole in a four dimensional spacetime. Addition of an electric charge, change the Schwarzschild solution to a charged black hole. This solution was discovered by Reissner (1916), Weyl (1917) and Nordstr¨om (1918), independently and now it is known as Reissner-Nordstr¨om metric [2]. Also, another solution of charged black hole in four dimensions was obtained by Gibbons and Maeda [3], independently, by Garfinkle, Horowitz and Strominger [4] using a scalar field in range of low-energy of heterotic string theory, which is called GMGHS solution [5]. The GMGHS black hole can be explained in string or Einstein frame, which are connected to each other by conformal transformation despite of differences of the physical properties in each frames [6–8]. Study of motion of massive and massless particles give a set of comprehensive information about the gravitational field around a black hole. Analysis of geodesic equation of motion predict some observational phenomena such as perihelion shift, gravitational time-delay and light deflection. The first analytic solution for Schwarzschild spacetime using Weierstrassian elliptic functions and their derivatives presented by Hagihara in 1931 [9]. The theoretical and mathematical properties of Weierstrassian elliptic functions demonstrated by Jacobi [10], Abel [11], Riemann [12, 13], Weierstrass [14] and Baker [15]. Analytical solutions of geodesic equations were investigated for different spacetimes such as Reissner-Nordstr¨om, Schwarzschild-(anti)de Sitter and Reissner-Nordstr¨om–(anti)–de Sitter spactime in four dimensions and in higherdimensions [16–18]. Also the motion of test particles around rotating black holes [19, 20] and in the spacetime of a black hole which is combined by cosmic string was studied extensively [21, 22]. Recently, geodesic equations were solved analytically in the spacetime of black hole in f(R) gravity [23, 24]. Analysis of geodesics, include null, timelike [25, 26], circular null and timelike geodesics [27, 28], were studied in the spacetime of GMGHS black hole in the special cases. The aim of this paper is to determine the complete set of analytic solutions of the geodesic equations in the spacetime of GMGHS, Magnetically charged GMGHS and Electrically charged GMGHS black holes. We discussed the motion of test particles and light rays in the spacetime of these black holes and present the analytic solutions of the geodesic equations in terms of the elliptic Weierstrass functions. Then we determined the type of the 2
orbits for test particles and light rays in the vicinity of GMGHS black holes. Our paper is organized as follows: in Section (2) we introduce the metrics and their histories. In Section (3) we derive the geodesic equations from the Lagrangian corresponding to the metric Eqs.(2),(4) and (5) and discuss the effective potential. In Section (5) we classify the solutions of the timelike and null geodesic equations using the effective potential and plot the possible orbits for test particles and light rays around each black hole in acceptable regions. Our conclusions are drawn in section (6).
2.
METRICS
In this section, we review all the spacetimes which are used in this paper. In the Einstein frame, the GMGHS action is [4] Z √ S = d4 x −g(R − 2(▽φ)2 − e−2φ Fµν F µν ),
(1)
where φ is a dilaton, R is the scalar curvature, and Fµν is the Maxwell field. The spherically symmetric static charged solutions to equations of motion of the action (1) is ds2GM = −(1 −
2M −1 2 Q2 2M 2 )dt + (1 − ) dr + r(r − )(dθ2 + sin2 θdϕ2 ), r r M
(2)
where M and Q are mass and charge, respectively. This solution was obtained by Gibbons and Maeda, and independently Garfinkle, Horowitz and Strominger with use a transformation to the Schwarzschild solution [2, 29]. The GMGHS action in the string frame, is Z √ S = d4 x −ge−2φ (R + 4(▽φ)2 − Fµν F µν ),
(3)
where φ is a dilaton, R is the scalar curvature, and Fµν is the Maxwell’s field strength. String frame is related to the Einstein frame action by the conformal transformation of s E gµν = e2φ gµν , [3, 4]. By going from an electrically to a magnetically charged black hole, the
string metric does change with the change in sign of dilaton φ, but the Einstein metric does not change. Thus, the magnetically charged GMGHS black hole metric in the string frame is given by [8, 30]: ds2M ag
2M ) dr 2 r dt2 + + r 2 (dθ2 + sin2 θdϕ2 ), =− Q2 2M Q2 (1 − (1 − ) )(1 − ) Mr r Mr (1 −
3
(4)
And the electrically charged GMGHS solution in the string frame is given by: ds2El = −
3.
Q2 − 2M 2 ) dr 2 2 2 2 2 2 Mr dt + 2 2 2 + r (dθ + sin θdϕ ). Q 2 Q − 2M (1 + (1 + ) ) Mr Mr
(1 +
(5)
THE GEODESIC EQUATIONS
We consider the geodesic equation σ ρ d2 xµ µ dx dx + Γ = 0, ρσ ds2 ds ds
(6)
where ds2 = gµν dxµ dxν corresponds to proper time and 1 Γµρσ = g µν (∂ρ gσν + ∂σ gρν − ∂ν gρσ ), 2
(7)
is the christoffel coefficient, in a spacetimes given by metric Eqs.(2),(4) and (5). The geodesic equations can be derived by compute the Lagrangian for each metric as [16] dxµ dxν 1 1 = ǫ, L = gµν 2 ds ds 2
(8)
where ǫ = 0, 1 for null and timelike geodesics respectively. Thus the Lagrangian for the metric (2) is: 2LGM = −(1 −
2M −1 2 Q2 ˙ 2 Q2 2M ˙2 )t + (1 − ) r˙ + r(r − )θ + r(r − )sin2 θϕ˙ 2 , r r M M
(9)
for the metric (4) is: 2LM ag
2M ) 1 r t˙2 + =− r˙ 2 + r 2 (θ˙ 2 + sin2 θϕ˙ 2 ), 2 Q 2M Q2 (1 − (1 − ) )(1 − ) Mr r Mr (1 −
(10)
and for the metric (5) is: Q2 − 2M 2 ) 1 Mr t˙2 + 2LEl = − r˙ + r 2 (θ˙2 + sin2 θϕ˙ 2 ). (11) 2 2 Q 2 Q − 2M 2 (1 + (1 + ) ) Mr Mr The Killing vectors respect to the spacetime from the Euler-Lagrange for time and latitude ∂ ∂ and . The energy E and the angular momentum L are the constants of motion are ∂t ∂ϕ which are given by the generalized momenta Pt and Pϕ (1 +
Pt =
∂L = −E, ∂ t˙ 4
Pϕ =
∂L = L. ∂ ϕ˙
(12)
From the Euler-Lagrange equation for t we get to the energy conservations EGM = (gtt )GM
dt 2M dt = −(1 − ) , ds r ds
2M (1 − ) dt r dt , =− EM ag = (gtt )M ag Q2 ds ds ) (1 − Mr Q2 − 2M 2 ) dt (1 + dt Mr EEl = (gtt )El = − , Q2 2 ds ds (1 + ) Mr and for ϕ we obtained the angular momentum conservations LGM = (gϕϕ )GM
Q2 dϕ dϕ = r(r − )sin2 θ , ds M ds
dϕ dϕ = r 2 sin2 θ , ds ds dϕ dϕ = (gϕϕ )El = r 2 sin2 θ . ds ds
LM ag = (gϕϕ )M ag LEl
(13)
(14)
(15)
(16) (17) (18)
We consider the motion is took place in a equatorial plane because of the existence of spherically symmetry and choose θ = π and θ˙ = 0 as the initial conditions. Therefore with 2
substitute t˙ and ϕ˙ from Eqs.(13)-(18), in Eqs.(9)-(11), we get 2M L2GM dr 2 2 + ǫ), )( ( )GM = EGM − (1 − 2 ds r r(r − QM ) dr 2 Q2 2 2 2M Q2 L2M g ( )M ag = (1 − ) EM ag − (1 − )(1 − )( + ǫ), ds Mr r Mr r 2 dr Q2 2 2 Q2 − 2M 2 L2El ( )2El = (1 + ) EEl − (1 + )( 2 + ǫ). ds Mr Mr r
(19)
(20) (21)
We obtain the corresponding equation for r as a function of ϕ and as a function of t, with energy and angular momentum conservation (
dr 2 E2 ǫ −2E 2 Q2 2ǫQ2 2Mǫ 3 E 2 Q4 )GM = ( 2 − 2 )r 4 + ( + + )r + ( − dϕ L L ML2 ML2 L2 M 2 L2 4ǫQ2 2ǫQ4 Q2 ǫQ4 + + 2M)r − 2Q2 = RGM (r), − 2 2 − 2 − 1)r 2 + ( M L L ML2 M E2 ǫ −2E 2 Q2 ǫQ2 2Mǫ dr 2 )M ag = ( 2 − 2 )r 4 + ( + + 2 )r 3 + 2 2 dϕ L L ML ML L 2 4 2 2 E Q 2ǫQ Q +( 2 2 − 2 − 1)r 2 + ( + 2M)r − 2Q2 = RM ag (r), M L L M
(22)
(
5
(23)
(
E2 ǫ 2E 2 Q2 ǫQ2 2Mǫ dr 2 )El = ( 2 − 2 )r 4 + ( − + 2 )r 3 + 2 2 dϕ L L ML ML L 2 4 2 E Q Q +( 2 2 − 1)r 2 + (− + 2M)r = REl (r), M L M
(24)
and (
(
(
1 2M 2 2 2M L2 dr 2 + ǫ)], )GM = 2 (1 − ) [E − (1 − )( 2 dt E r r r(r − QM )
)2 dr 2 1 (1 − 2M 2M Q2 L2 Q2 2 2 r )M ag = 2 ) E − (1 − )(1 − )( + ǫ)], [(1 − Q2 2 dt E Mr r Mr r 2 ) (1 − Mr
dr 2 ) dt El
Q2 − 2M 2 2 ) (1 + 1 Q2 2 2 Q2 − 2M 2 L2 Mr = 2 [(1 + ) E − (1 + )( 2 + ǫ)]. 2 E Mr Mr r (1 + Q )2
(25)
(26)
(27)
Mr
Eqs.(19)-(27) gives a complete definition of the dynamics. In these set of equations, the values of L and E 2 in the right hand side refer to indices of the left hand side of them, that we ignore indices for L and E 2 for simplicity. We get the effective potential by comparing Eqs.(19)-(21) with r˙ 2 + Vef f = E 2 , (Vef f )GM = (1 −
L2 2M + ǫ), )( 2 r r(r − Q )
(28)
M
2M L2 )( 2 + ǫ) r r (Vef f )M ag = , (29) Q2 ) (1 − Mr Q2 − 2M 2 L2 (1 + )( 2 ) Mr r + ǫ (Vef f )El = − , (30) Q2 2 ) (1 + Mr which depends on radial coordinate r, charge Q and mass M of the black hole, the type of (1 −
the geodesics ǫ and the angular momentum L of the particles. In these set of equations, again the value of L in the right hand side refer to indices of the left hand side of them, that we ignore indices for L and E 2 for simplicity. Using dimensionless quantities is convenient way for analyzing the dependence of the possible types of orbits on the parameters of the spacetime and the test particle or light ray. Thus, we introduce r˜ =
r , M
˜ = Q, Q M 6
2 ˜=M . L L2
(31)
and rewrite Eqs.(22)-(24) as d˜ r 2 ˜ r 4 + (2Q ˜ 2 ǫ + 2ǫ − 2E 2 Q ˜ 2 )L˜ ˜ r 3 + (E 2 Q ˜4L ˜ ) = (E 2 − ǫ)L˜ dϕ GM ˜4L ˜ − 4Q ˜ 2 ǫL ˜ − 1)˜ ˜ 4L ˜+Q ˜ 2 + 2)˜ ˜ 2 = RGM (˜ −ǫQ r 2 + (2ǫQ r − 2Q r), (
d˜ r 2 ˜ r 4 + (Q ˜ 2 ǫ − 2Q ˜ 2 E 2 + 2ǫ)L˜ ˜ r3+ ) = (E 2 − ǫ)L˜ dϕ M ag ˜ 4E 2L ˜ − 2Q ˜ 2 ǫL ˜ − 1)˜ ˜ 2 + 2)˜ ˜ 2 = RM ag (˜ (Q r 2 + (Q r − 2Q r),
(32)
(
(
(33)
d˜ r 2 ˜ r 4 + (2Q ˜ 2E 2 − Q ˜ 2 ǫ + 2ǫ)L˜ ˜ r3 )El = (E 2 − ǫ)L˜ dϕ ˜ 4E 2L ˜ − 1)˜ ˜ 2 + 2)˜ +(Q r 2 + (−Q r = REl (˜ r ).
(34)
˜ and E 2 in the right hand side refer to indices Also, in these set of equations, the values of L ˜ and E 2 for simplicity. of the left hand side of them, that we ignore indices for L
4.
ANALYTICAL SOLUTION OF GEODESIC EQUATIONS
In this section we present the solution of the equations of motion analytically. In the Eqs.(32)-(34) for the test particle(ǫ = 1) and light ray(ǫ = 0) we have polynomials of degree P four in the form R(˜ r) = 4i=0 ai r˜i with only simple zeros, which for solving them in this
way, we can apply up to two substitutions. The first substitution is r˜ =
1 z
+ r˜R , where r˜R
is a zero of R, transforms the problem to 3
X dz bj z j , ( )2 = R3 (z) = dϕ j=1
z(ϕ0 ) = z0 ,
(35)
With a polynomial R3 of degree 3. Where bj =
1 d(4−j) R (˜ rR ), (4 − j)! d˜ r (4−j)
(36)
where bj , (j = 1, 2, 3) is an arbitrary constant for each metric which is related to the parameter of the relevant metric. A second substitution z =
1 (4y b3
−
b2 ) 3
cause that R3 (z)
changes into the Weierstrass form so that Eq.(35) turns into: (
dy 2 ) = 4y 3 − g2 y − g3 , dϕ 7
(37)
where g2 =
1 4 2 ( b − 4b1 b3 ), 16 3 2
g3 =
1 1 2 ( b1 b2 b3 − b32 − b0 b23 ), 16 3 27
(38)
are the Weierstrass invariants. The differential equation(37) is of elliptic type and we used the Weierstrass ℘ function to solve it [17, 23] y(ϕ) = ℘(ϕ − ϕin ; g2 , g3),
(39)
b2 b3 dy p with ϕ0 = 14 ( + ) depends only on the 3 r˜0 − r˜R 3 4y − g2 y − g3 initial values ϕ0 and r˜0 . Therefore, the solution of Eqs.(32)-(34) takes the form where ϕin = ϕ0 +
R∞ y0
r˜(ϕ) =
b3 b2 4℘(ϕ − ϕin ; g2 , g3) − 3
+ r˜R .
(40)
This is the analytic solution of the equation of motion of a test particle and light ray in a GMGHS, Magnetically charged GMGHS and Electrically charged GMGHS spacetimes. This solution is valid in all regions of this spacetimes.
5.
ORBITS
In a special spacetime with electric charge, the shape of an orbit depends on three paremeters, in which the angular momentum, L and the energy, E are the specifications of test particle or light ray and the electric charge, Q comes from the related spacetime (the mass can be absorbed through a rescaling of the radial coordinate). The polynomial R(r) defined in Eqs.(22)-(24) are included all these quantities. Since r should be real and positive the physically admissible regions are given by those r for which E 2 ≥ Vef f which is presented on the left hand side of Eqs.(19)-(21). Therefore, the form of the resulting orbits are characterized uniquely by the number of positive real zeros of R. In the following, we introduce four different types of orbit of the test particle or light ray can be recognized in spacetimes described by the metric Eqs.(2),(4) and (5) 1. Bound orbits (BOs): r fluctuates between two boundary values rp ≤ r ≤ ra with 0 < rp < ra < ∞. 2. Terminating escape orbits (TEOs): r comes from ∞ and falls into the singularity at r = 0. 8
3. Terminating bound orbits (TBOs): r starts in (0, ra ] for 0 < ra < ∞ and falls into the singularity at r = 0. 4. Flyby orbits (FOs): r starts from ∞, then approaches a periapsis r = rp and goes back to ∞. Other types of orbits are exceptional and treated separately. They are connected with the appearance of multiple zeros in R(r) or with parameter values which reduce the degree of R(r). In both cases the differential Eqs.(22)-(24) have much simplified structure. These orbits are radial geodesics with L = 0, circular orbits with constant r, orbits asymptotically approaching circular orbits. Defining the borders of R(r) ≥ 0 or, equivalently, E 2 ≥ Vef f is done by the four regular types of geodesic motion correspond to various arrangements of the real and positive zeros of R(r). If R(r) has no real and positive zeros at all a terminating escape orbit is possible if R(r) > 0 for all r > 0, but else no geodesic motion is allowed. If R(r) has at least one real and positive zero then a flyby orbit is possible if limr→∞ R(r) = ∞, and a terminating bound orbit if R(r) > 0 for 0 < r < r1 where r1 is the smallest positive zero. If R(r) has at least two real zeros r1 < r2 with R(r) > 0 for r1 < r < r2 a bound orbit is permitted. If R(r) is such that multiple types of orbits are possible the actual orbit depends on the initial position of the test particle or light ray. In the following we will analyse possible types of orbits dependent on the parameters ˜ and L. ˜ The major point in this analysis is that of the test particle or light ray ǫ, E 2 , Q Eqs.(32)-(34) implies R(˜ r ) ≥ 0 as a necessary condition for the existence of a geodesic. Thus, the zeros of R(˜ r ) are extremal values of r˜(ϕ) and determine (together with the sign of R(˜ r ) between two zeros) the type of geodesic. The polynomial R(˜ r ) is in our metrics of degree 4 and, therefore, has 4 (complex) zeros of which the positive real zeros are of interest for the type of orbit. ˜ and L ˜ the polynomial R(˜ For a given set of parameters ǫ, E 2 , Q, r ) has a certain number ˜ are varied this number can change only if two zeros of of positive real zeros. If E 2 and L dR(˜ r) ˜ for ǫ = 1 yields = 0 for E 2 and L, R(˜ r ) merge to one. Solving R(˜ r) = 0 , d˜ r 2 EGM =
˜ 2 r˜2 − 4Q ˜ 2 r˜ − 2˜ ˜ 2 + 8˜ Q r 3 + 4Q r 2 − 8˜ r , ˜ 2 r˜ − 4Q ˜ 2 − 2˜ (Q r 2 + 6˜ r )˜ r 9
˜2 ˜2 r 2 + 6˜ r ˜ GM = − 1 Q r˜ − 4Q − 2˜ L , (41) ˜ 4 − 2Q ˜ 2 r˜ + r˜2 ) 2 r˜(Q
2 EM ag = −
2 EEl =
2(˜ r 2 − 4˜ r + 4) , ˜ 2 r˜ − 4Q ˜ 2 − 2˜ Q r 2 + 6˜ r
˜ 4 + 2Q ˜ 2 r˜ − 4Q ˜ 2 + r˜2 − 4˜ 2(Q r + 4)˜ r , ˜ 2 + r˜)(Q ˜ 4 + 3Q ˜ 2 r˜ − 2Q ˜ 2 + 2˜ (Q r 2 − 6˜ r)
˜2 ˜2 r 2 + 6˜ r ˜ M ag = Q r˜ − 4Q − 2˜ L , ˜ 2 − 2) r˜3 (Q
(42)
˜4 ˜2 ˜2 r 2 − 6˜ r ˜ El = Q + 3Q r˜ − 2Q + 2˜ L . ˜4 + Q ˜ 2 r˜ − 2Q ˜ 2 + 2˜ r˜2 (Q r)
(43)
In Fig.1, the result of this analysis is shown for test particles (ǫ = 1).
FIG. 1: Regions of different types of geodesic motion for test particles (ǫ = 1) in GMGHS, magnetically charged and Electrically charged GMGHS spacetime.
For light rays (ǫ = 0), the analysis is the same as in the (ǫ = 1) case and the result of this analysis is shown in Fig.2
10
FIG. 2: Regions of different types of geodesic motion for light (ǫ = 0) in GMGHS, magnetically charged and Electrically charged GMGHS spacetime.
11
Four different regions can be identified in Fig.1 for timelike geodesic but for null geodesic, in Fig.2, we have only two regions. It should be noted that for Electrically charged GMGHS metric, the number of zeros is one more less than zeros of GMGHS and Magnetically charged GMGHS metrics in each region, in both timelike and null geodesics. Summary of possible orbit types in Tabels (I) and (II) can be found. region pos.zeros
range of r˜
orbit
I
4
2 BO
II
3
BO, FO
III
1
FO
IV
2
BO
TABLE I: Types of orbits of GMGHS and Magnetically GMGHS black holes. The range of the orbits is represented by thick lines. The dots show the turning points of the orbits. The single vertical line indicates r˜ = 0.
region pos.zeros
range of r˜
orbit
I
3
TBO, BO
II
2
TBO, FO
III
0
TEO
IV
1
TBO
TABLE II: Types of orbits of Electrically GMGHS black hole. The range of the orbits is represented by thick lines. The dots show the turning points of the orbits. The single vertical line indicates r˜ = 0.
In following, for each region, we present effective potential diagram, The numbers of positive real zeros and orbit types in that region for metric Eqs.(2),(4) and (5) for timelike geodesic (ǫ = 1). Ia: In region I for GMGHS blackhole, R(˜ r ) has 4 positive real zeros r1 < r2 < r3 < r4 with R(˜ r ) ≥ 0 for r1 ≤ r˜ ≤ r2 and r3 ≤ r˜ ≤ r4 . Therefore the possible orbit types are two different bound orbits, (see Figs. 3–5). 12
Veff 0.904
0.902
0.900
0.898
0.896
2
4
6
8
10
12
r
FIG. 3: Effective potential of region I for GMGHS blackhole.
40
20
-40
20
-20
40
-20
-40
(a)
FIG. 4: Bound orbit of region I for GMGHS blackhole for E 2 = 0.9,
13
M2 = 0.08. L2
30
20
10
-30
-20
10
-10
20
30
-10
-20
(b) -30
FIG. 5: Bound orbit of region I for GMGHS blackhole for E 2 = 0.9,
14
M2 = 0.08. L2
Ib: In region I for Magnetically charged GMGHS blackhole, R(˜ r ) has 4 positive real zeros r1 < r2 < r3 < r4 with R(˜ r ) ≥ 0 for r1 ≤ r˜ ≤ r2 and r3 ≤ r˜ ≤ r4 . Therefore the possible orbit types are two different bound orbits (see Figs. 6–8). Veff 0.96
0.95
0.94
0.93
0.92
0.91
r 5
10
15
20
25
30
FIG. 6: Effective potential of region I for Magnetically charged GMGHS blackhole.
20
10
-20
10
-10
20
-10
-20
(a)
FIG. 7: Bound orbit of region I for Magnetically charged GMGHS blackhole for E 2 = 0.98, 0.06.
15
M2 = L2
200
100
-200
100
-100
200
-100
-200
(b)
FIG. 8: Bound orbit of region I for Magnetically charged GMGHS blackhole for E 2 = 0.98, 0.06.
16
M2 = L2
Ic: In region I for Electrically charged GMGHS blackhole, R(˜ r ) has 3 positive real zeros r1 < r2 < r3 with R(˜ r ) ≥ 0 for 0 ≤ r˜ ≤ r1 and r2 ≤ r˜ ≤ r3 . Therefore the possible orbit types are terminating bound and bound orbits respectively (see Figs. 9–10). Veff 1.02
1.00
0.98
0.96
0.94
10
20
30
40
50
60
70
r
FIG. 9: Effective potential of region I for Electrically charged GMGHS blackhole.
5
5
-5
-5
(b)
FIG. 10: Terminating bound orbit of region I for Electrically charged GMGHS blackhole for M2 E 2 = 0.98, 2 = 0.06. L
17
40
20
-40
20
-20
40
-20
-40
(a)
FIG. 11: Bound orbit of region I for Electrically charged GMGHS blackhole for E 2 = 0.98, 0.06.
18
M2 = L2
IIa: In region II for GMGHS blackhole, R(˜ r) has 3 positive real zeros r1 < r2 < r3 with R(˜ r ) ≥ 0 for r1 ≤ r˜ ≤ r2 and r3 ≤ r˜. Therefore the possible orbit types are bound and flyby orbits respectively (see Figs. 12–14). Veff 3.0
2.5
2.0
1.5
1.0
0.5
0.0
0
2
4
6
8
10
12
r
FIG. 12: Effective potential of region II for GMGHS blackhole.
20
10
-20
10
-10
20
-10
-20
(a)
FIG. 13: Bound orbit of region II for GMGHS blackhole for E 2 = 1.5,
19
M2 = 0.02. L2
100
50
-100
50
-50
100
-50
-100
(b)
FIG. 14: Flyby Orbit of region II for GMGHS blackhole for E 2 = 1.5,
20
M2 = 0.02. L2
IIb: In region II for Magnetically charged GMGHS blackhole, R(˜ r ) has 3 positive real zeros r1 < r2 < r3 with R(˜ r ) ≥ 0 for r1 ≤ r˜ ≤ r2 and r3 ≤ r˜. Therefore the possible orbit types are bound and flyby orbits respectively (see Figs. 15–17). Veff
2.0
1.5
1.0
0.5
0.0
0
2
4
6
8
r
FIG. 15: Effective potential of region II for Magnetically charged GMGHS blackhole.
20
10
-20
10
-10
20
-10
-20
(a)
FIG. 16: Bound orbit of region II for Magnetically charged GMGHS blackhole for E 2 = 1.5, 0.02.
21
M2 = L2
60
40
20
-60
-40
20
-20
40
60
-20
-40
-60
(b)
FIG. 17: Flyby Orbit of region II for Magnetically charged GMGHS blackhole for E 2 = 1.5, 0.02.
22
M2 = L2
IIc: In region II for Electrically charged GMGHS blackhole, R(˜ r) has 2 positive real zeros r1 < r2 with R(˜ r ) > 0 for 0 ≤ r˜ ≤ r1 and r2 ≤ r˜. Therefore the possible orbit types are terminating bound and flyby orbits respectively (see Figs. 18–20). Veff
2.0
1.5
1.0 r 2
4
6
8
10
12
FIG. 18: Effective potential of region II for Electrically charged GMGHS blackhole.
2
1
-2
1
-1
2
-1
-2
(a)
FIG. 19: Terminating bound orbit of region II for Electrically charged GMGHS blackhole for M2 E 2 = 1.5, 2 = 0.02. L
23
300
200
100
-300
-200
100
-100
200
300
-100
-200
(b)
-300
FIG. 20: Flyby Orbit of region II for Electrically charged GMGHS blackhole for E 2 = 1.5, 0.02.
24
M2 = L2
IIIa: In region III for GMGHS blackhole, R(˜ r ) has 1 positive real zeros r1 with R(˜ r) ≥ 0 for r1 ≤ r˜. Therefore the possible orbit type is flyby orbit (see Figs. 21 and 22). Veff 3.0
2.5
2.0
1.5
1.0
0.5
0.0
0
2
4
6
8
10
12
r
FIG. 21: Effective potential of region III for GMGHS blackhole.
20
10
-20
10
-10
20
-10
-20
FIG. 22: Flyby Orbit of region III for GMGHS blackhole for E 2 = 1.6,
25
M2 = 0.09. L2
IIIb: In region III for Magnetically charged GMGHS blackhole, R(˜ r ) has 1 positive real zero r1 with R(˜ r ) ≥ 0 for r1 ≤ r˜. Therefore the possible orbit type is flyby orbit (see Figs. 23 and 24). Veff
1.5
1.0
0.5
0.0
0
1
2
3
4
5
6
7
r
FIG. 23: Effective potential of region III for Magnetically charged GMGHS blackhole.
600
400
200
-600
-400
200
-200
400
600
-200
-400
-600
FIG. 24: Flyby Orbit of region III for Magnetically charged GMGHS blackhole for E 2 = 1.6, 0.09.
26
M2 = L2
IIIc: In region III for Electrically charged GMGHS blackhole, R(˜ r) has 0 positive real zeros and R(˜ r) ≥ 0 for 0 ≤ r˜. Therefore the possible orbit type is terminating escape orbit (see Figs. 25 and 26). Veff
2.0
1.5
1.0
0.5
0.0
r 0
2
4
6
8
10
12
FIG. 25: Effective potential of region III for Electrically charged GMGHS blackhole.
3
2
1
-3
-2
1
-1
2
3
-1
-2
-3
FIG. 26: Terminating escape orbit of region III for Electrically charged GMGHS blackhole for M2 E 2 = 1.6, 2 = 0.09. L
27
IVa: In region IV for GMGHS blackhole, R(˜ r ) has 2 positive real zeros r1 ,r2 with R(˜ r) ≥ 0 for r1 ≤ r˜ ≤ r2 . Therefore the possible orbit type is bound orbit (see Figs. 27 and 28). Veff 3.0
2.5
2.0
1.5
1.0
0.5
0.0
0
2
4
6
8
10
12
r
FIG. 27: Effective potential of region IV for GMGHS blackhole.
40
20
-40
20
-20
40
-20
-40
FIG. 28: Bound orbit of region IV for GMGHS blackhole for E 2 = 0.5,
28
M2 = 0.04. L2
IVb: In region IV for Magnetically charged GMGHS blackhole, R(˜ r) has 2 positive real zeros r1 ,r2 with R(˜ r ) ≥ 0 for r1 ≤ r˜ ≤ r2 . Therefore the possible orbit type is bound orbit (see Figs. 29 and 30). Veff 2.0
1.5
1.0
0.5
0.0
0
2
4
6
8
r
FIG. 29: Effective potential of region IV for Magnetically charged GMGHS blackhole.
20
10
-20
10
-10
20
-10
-20
FIG. 30: Bound orbit of region IV for Magnetically charged GMGHS blackhole for E 2 = 0.5, 0.04.
29
M2 = L2
IVc: In region IV for Electrically charged GMGHS blackhole, R(˜ r ) has 1 positive real zero r1 with R(˜ r ) ≥ 0 for positive r. Therefore the possible orbit type is terminating bound orbit (see Figs. 31 and 32). Veff 2.0
1.5
1.0
0.5
0.0
r 0
2
4
6
8
10
12
14
FIG. 31: Effective potential of region IV for Electrically charged GMGHS blackhole.
2
1
-2
1
-1
2
-1
-2
FIG. 32: Terminating bound orbit of region IV for Electrically charged GMGHS blackhole for M2 E 2 = 0.5, 2 = 0.04. L
30
6.
CONCLUSIONS
In this paper we considered the motion of test particle in the spacetime of three metrics of static spherically symmetric charged black hole known as GMGHS, Magnetically charged GMGHS and Electrically charged GMGHS blackhole. We have derived geodesic equations of motion by Euler-Lagrange equations and classified them according to their energy E and angular momentum L. The geodesic equations of motion can be solved in terms of the elliptic Weierstrass ℘ functions. The complete set of analytical solutions of the geodesic equations of test particles and light ray were presented. Possible types of orbits were derived by using effective potential techniques and parametric diagrams. For Electrically charged GMGHS black hole, FO, TBO, BO and TEO are possible while for GMGHS and Magnetically charged GMGHS black holes, FO and BO are possible and any type of terminating orbit are not possible for these metrics. Some observational phenomena such as the periastron shift of bound orbits as shown in Fig.5, and the deflection angle of flyby orbits as shown in Fig.22, are some results of these solutions which can be calculated by metheds given in Refs.[17, 31, 32].
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