All analytic solutions for geodesic motion in axially symmetric space-times

Recent observations of the orbits of star clusters around Sgr A⋆\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^\star $$\end{document}, imaging of black holes and gravitational waveforms of merging compact objects require a detailed understanding of the general relativistic geodesic motion. We came up with a method to provide all the possible geodesics in an axially symmetric space-time. The Kerr metric is explicitly worked out, recovering the Schwarzschild geodesics in the static limit. We also found the most general Killing tensor and its associated constant of motion for an axisymmetric space-time. The relevance of these results is crucial to understanding the different scenarios and the fundamental nature of the compact object at the galactic center.


Introduction
From the orbits of stars [1][2][3], and the imaging around supermassive black holes [4,5] through gravitational lensing [6], geodesic motions of particles and photons have a venerable history bringing results of General Relativity to observational grounds. Other important scenarios like frame dragging [7], radiation transport effects from accretion flows in the vicinity of relativistic stars [8], and gravitational waveform of merging compact objects [9], also require a detailed understanding of the geodesic motion in a general relativistic background.
Rotation is a crucial feature for celestial bodies from the an astrophysical viewpoint, and the Kerr solution of Einstein equations describes the gravitational field outside spinning compact objects and black holes [10]. Thus, geodesics across a Kerr gravitational background are very important and have a long history, since Carter's study of the existence of a new a e-mail: j.ospino@usal.es (corresponding author) b e-mail: jlhp@usal.es c e-mail: lnunez@uis.edu.co conserved quantity associated with each geodesic [11,12]. The astrophysical relevance of tracking particles & photons in Kerr space-times motivates a significant effort to obtain analytical and numerical trajectories in this gravitation background (see [13,14] and references therein).
We have recently implemented a tetrad formalism by an orthogonal splitting of the Riemann tensor, introducing a complete set of equations equivalent to the Einstein system and applying it to the spherical case [15][16][17]. This formalism provides coordinate-free results expressed in terms of structure scalars related to the kinematical and physical properties of the fluid.
We devise an exhaustive classification for all geodesic motion for any axisymmetric source establishing the equations which describe each alternative. Based on the tetrad scheme, we provide a method to integrate all possible orbits for any stationary axially symmetric solutions, illustrating each case with the Kerr metric.
We present the most general Killing tensor corresponding to any axisymmetric space-time and its associated constant of motion, which has not previously been obtained. This Killing tensor and its conserved quantity allow us to obtain the orbits by solving a system of algebraic equations. We recovered the famous Carter constant along the geodesic as a particular case.

The tetrad and kinematical variables
We shall consider stationary and axially symmetric sources with the line element written as with A = A(r, θ), B = B(r, θ), R = R(r, θ), and ω 3 = ω 3 (r, θ). In this case the tetrad is:

The scalars and the tetrad covariant derivative
The covariant derivative of V α in the 1+3 formalism can be written as V α;β = −a α V β + αβ , where the kinematical variables (a α the acceleration and αβ the vorticity) can be written, in terms of the tetrad, as Follows the covariant derivative of K, i.e.
Now, the covariant derivative of L, can be written as Finally the covariant derivative of S is

Scalars for a general axisymetric metric
Assuming ω 3 = A 2 ψ, the scalars for the axisymmetric metric (1) are:
In what follows we shall make use of the geodesic equations Z α;β Z β = 0, written in the tetrad formalism as and Because the norm of the tangent vector Z α is constant: where we have the derivatives z † ≡ 1 B z ,r and z θ ≡ 1 C z ,θ .

First order geodesic equations
Substracting (10) from (8) and (11) from (9) we have This system describes all possible geodesic equations for any axially symmetric space-time, having Kerr and Schwarzschild metrics as particular cases.
This is an exhaustive classification containing all possible cases for geodesic motions. Following subsections present the equations for each alternative and an application to the Kerr space-time.

Circular orbit on a plane
In the first case the system (12)-(13) is solved by z 2 = 0 and z 1 = 0, i.e the radial and the angular coordinate, θ , are constant. Thus, we obtain bounded orbits confined to a plane and the set of equations becomes,

General orbital motion on a constant plane
The second solution for Eqs. (12)- (13), emerges from the conditions z 2 = 0 and z θ 1 = j 1 z 1 . Then, the geodesic equations reduce to 3.5 General orbits on the two-sphere The third set of solutions we shall consider are general orbits circumscribed on a 2-sphere (θ − φ), recently reported in reference [13]. This occurs having z 1 = 0 and z † 2 = − j 2 z 2 , and the corresponding geodesic equations are 3.6 The general case The last case emerges from z † 2 − z θ 1 + j 1 z 1 + j 2 z 2 = 0. This is the most general case, and the system of geodesic equations becomes and with The solution of the equation is with f = f (r, θ) an arbitrary function of its arguments. Consequently, z θ 1 = j 1 z 1 and z † 2 = − j 2 z 2 , which in turn allows us to transform (8) and (9) into and

Symmetry and geodesic equations
In this section we shall discuss the consequences on imposing symmetries, i.e. Killing vectors and tensors, on the source generating the geodesic equations.

Killing vectors
From the Killing equation we can identify temporal and axial Killing vectors as These Killing vectors provide the energy E = τ α Z α and angular momentum l = ξ α Z α .
. Now the set of equations for the parallel transport of the vector Z α , can be written as The above equations lead to the following characteristic expressions:

Killing tensor
Killing tensors are useful because they also provide conserved quantities for geodesic motion. The most famous is obtained for the Kerr space-time where the Killing tensor leads to the Carter constant [12]. For the stationary axially symmetric space-time, the Killing tensor ξ αβ satisfies ξ αβ;μ + ξ μα;β + ξ βμ;α = 0, which can be written as Integrating the Killing tensor equation we obtain We have = (r 2 − 2mr + a 2 ) sin θ and ξ(r ) an r -function. Thus, the general conserved quantity Q for ξ αβ is Since Z has a constant modulus we found obtaining that the scalars z 1 and z 2 are with and The new conserved quantity Q recovers the Carter constant Q c for the Kerr metric [11,12] The following sections implement all the previous cases to the particular example of the Kerr space-times.

The Kerr metric
To illustrate the different cases mentioned above, we consider = r 2 + a 2 cos 2 θ in the Kerr metric, i.e.

Kerr circular orbit on a plane
Considering the set of Eqs. (14)- (16) for the metric (44) with θ = π 2 , Q c = l 2 we get: and Equations (46) and (47) determine the radius of the circumference and related the physical constants. For instance, when we have the specific case = a = 0 we obtain that r = 3m and l = √ 27Em.

Kerr general orbital motion on a constant plane
Equations (17)- (19) with θ = π 2 , and Q = l 2 lead to For a null geodesics and a = − l E we integrate (34) as where φ 0 is a constant of integration. Now for a time-like geodesic we have with and x 2 (r ) = β 4 + β 5 r where β k (k = 1, 2, 3, 4, 5, 6) are functions of the physical parameters.

The Kerr general case
The module of Z for the Kerr metric can be written as Substituting (26) into (56) we find which is an equation for the function f with a solution f = Next, substituting (25) into (34) and considering z θ 1 = j 1 z 1 and z † 2 = − j 2 z 2 , we obtain dθ f 2,θ = dr f 1 (r 2 − 2mr + a 2 ) and dφ = (F 2 − a 2 sin 2 θ)l − 2mar E sin 2 θ where F 2 = r 2 + a 2 − 2mr. Now, combining Eq. (59) we get ldθ sin 2 θ f 2,θ − (a 2 l + 2ma Er)dr and by introducing we get Next, substituting r = x + b into (61) we get and Consequently first integrals in r and θ of the Eq. (60) can be obtained as where (x) = b 2 +x b 3 +x , where γ i and α j , with i = 1, 2, 3 and j = 0, 1, 2, 3. Now, implementing the procedure described above for the polynomial (63) we get Next, integrating the Eq. (60) we get and becomes its general solution.

Conclusions
This work presents a method to classify and solve all geodesic motion analytically around any stationary axially symmetric source. The method summarises all these possible geodesic trajectories into two simple Eqs. (12) and (13), with ease in obtaining solutions. These distinct solutions allow us to classify the different trajectories for particles and photons. All the possible geodesics have been implemented for the Kerr metric, representing the gravitational field produced by a rotating compact object. In particular, those orbiting on a two-sphere surface could be especially relevant for describing observational data ( see [3,18,19] and references therein). Now, it will be possible to build templates from exact General Relativistic analytical solutions, i.e. without any approximations for the orbits of stars [1][2][3], and the imaging of black holes [4,5,20]. The method shown here allows us to find solutions of the form f 1 (r, θ) = C 1 and f 2 (r, θ) = C 2 (equations (70) and (71)) which complement the standard elliptic integrals procedure handling in writing down the geodesics trajectories ( see for example references [21][22][23] ). It is clear that in the equations mentioned above (i.e.(70) and (71)), the elliptic integrals can be avoided only for very particular choices of α i .
We found the most general for this Killing tensor corresponding to any axisymmetric space-time and its linked constant of motion. The existence this general constant of motion -along the geodesic-is clear from a simple system of algebraic equations (39) and (40). Again, the general expression for the constant along the geodesic could help to obtain solutions for the geodesic in a more general context where the Kerr metric may not adequately describe the gravitational field (see [24] and references therein). This new conserved quantity recovers the Carter constant for the Kerr metric ( [11,12]).
Although we have considered Kerr space-time a helpful example, the equations for each case in our classification are general and valid for any axisymmetric metric. Analytic solutions for geodesic with more complex Kerr-like sources describing richer rotational compact objects could fit better the trajectories of the stars or represent more accurate black hole imaging or open the possibility of new information from gravitational wave astronomy.