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Revisiting timelike and null geodesics in the Schwarzschild spacetime: general expressions in terms of Weierstrass elliptic functions

The theory of Schwarzschild geodesics is revisited. Basing on a result by Weierstrass and Biermann, we derive a formula describing all non radial, timelike and null trajectories in terms of Weierstrass elliptic functions. Quite remarkably, a single formula works for an entire geodesic trajectory, even if it passes through turning points. Using this formula, we derive expressions for the proper and coordinate time along the geodesic.


Introduction
In this paper we revisit some elements of the theory of geodesics in the Schwarzschild spacetime. The motivation for repeating this classic calculation in a new form stems from the works on the kinetic description of the Vlasov gas on the Schwarzschild spacetime and the accretion of the Vlasov gas onto Schwarzschild black holes [54,53,44,45,43,23,25,13]. In particular, we were motivated by an ongoing project aiming at constructing Monte Carlo type simulations of the gas consisting of collisionless particles moving around the Schwarzschild black hole. Having those applications in mind, we put special emphasis on unbounded trajectories-they are neglected in many discussions, but they play a crucial role in the description of Bondi-type accretion processes.
Existing descriptions of Schwarzschild geodesics differ in the parametrization and also in types elliptic functions used to express the solutions. As usual, different prescriptions appear to be more or less convenient, depending on the actual problem at hand. Our goal was to specify the constants of motion (in particular the energy and the angular momentum) together with the initial location of the particle, and obtain the corresponding trajectory in an exact and reliable manner. We achieve this aim using Weierstrass elliptic functions, but our prescription turns out to be different than existing ones (or at least the ones we are aware of). The main result presented in this paper is a concise formula describing all types of timelike and null trajectories in the Schwarzschild metric (except for the purely radial ones), based on a theorem due to Biermann and Weierstrass.
The problem of an analytic description of the motion of test particles in the Schwarzschild spacetime is nearly as old as the Schwarzschild solution itself. The first attempt to solve geodesic equations in the Schwarzschild spacetime was published in 1917 by Droste, together with a derivation of the Schwarzschild metric [18] (this paper is now also available as a "Golden Oldie" reprint [19]). Droste expressed his solution in terms of the Weierstrass elliptic function. Thirteen years later Hagihara gave a full description of the motion of test particles around a Schwarzschild black hole [34]. His work contains a complete characterization of all types of allowed orbits and is now a classic position in the theory of Schwarzschild geodesics.
Simultaneously to the development of solutions based on Weierstrass functions, Forsyth, Greenhill, Morton, Darwin, Mielnik, and Plebański succeeded in expressing Schwarzschild geodesics in terms of Jacobi elliptic functions and Legendre integrals [21,30,49,16,17,47]. Publication years of these papers span across several decades. As a historical remark, let us note that the authors of Refs. [16,47], published in 1959 and 1962, already admitted that their calculations could had been made almost forty years earlier, as all required mathematical tools were already known at that time.
A slightly different approach to the geodesic motion in the Schwarzschild metric was recently proposed by Scharf [57]. Scharf's analysis is based on a simplified version of a result which we will refer to as the Biermann-Weierstrass formula.
According to Refs. [29,65,52], around 1860 Weierstrass obtained a general solution of an ordinary differential equation where α, β, γ, and δ are constant coefficients, which is roughly equivalent to an integral problem where f is a quartic polynomial, and x 0 = x(0). Weierstrass' solution was later published by his student Biermann, as a part of his inaugural dissertation [6]. The Weierstrass-Biermann formula is derived in Greenhill's textbook [29]; it appears also in the classic textbook by Whittaker and Watson [65]. On the other hand, it is absent in other textbooks dealing with elliptic functions from that time [ The Biermann-Weierstrass formula for x = x(t) simplifies, if x 0 is a zero of the polynomial f , and this version is used by Scharf. In the context of the geodesic motion this requirement restricts the choice of the starting (reference) point of the trajectory to turning points. In our work, we start with the general Biermann-Weierstrass expression, and hence this restriction is removed. A single formula [Eq. (36) of this article] describes all timelike and null geodesic trajectories, except purely radial ones. Moreover, it is valid along the entire trajectory, even if it passes through turning points. The latter fact is not immediately obvious from the derivation of Eq. (36), but it can be shown using addition theorems for elliptic functions.
The affine parameter and the coordinate time along a geodesic can be obtained as integrals involving the Biermann-Weierstrass expression. While, in principle, they can be evaluated assuming the general form of Eq. (36), the resulting formulas are lengthy and thus of little practical use. For completeness, we decided to provide appropriate expressions for the affine parameter and the coordinate time, assuming the simplified version of the Biermann-Weierstrass formula.
A proof of the Biermann-Weierstrass formula is given in Appendix A; we decided to provide this material, since existing, known to us proofs of the Biermann-Weierstrass formula are rather hard to follow in detail. Apart from a sketch of the proof given in Biermann's dissertation [6], there is a proof in Greenhill's textbook [29], and its more modern account in [52]. We fill some minor gaps missing in [52].
The order of this paper is as follows. The next section specifies horizon-penetrating coordinates used in this paper. Equations of motion are derived in Sec. 3. The main result of this paper-a single formula describing non-radial, timelike and null trajectories-is given in Sec. 4. In Section 5 we discuss the range of the true anomaly parameter. The proper time and the coordinate time along a geodesic are computed in Sec. 6 and 7, respectively. The proof of the Biermann-Weierstrass formula is given in Appendix A. Appendix B contains a brief classification of timelike and null geodesics. In Appendix C we evaluate certain elliptic integrals, required to control the range of the true anomaly parameter for unbounded orbits.
Throughout the paper we use geometric units with c = G = 1, where c is the speed of light, and G denotes the gravitational constant. The signature of the metric is assumed to be (−, +, +, +). Spacetime dimensions are labeled with Greek indices, µ = 0, 1, 2, 3.

Horizon-penetrating coordinates
We will work in spherical coordinates (t, r, θ, ϕ). In its simplest form (in the so-called Schwarzschild coordinates) the Schwarzschild metric can be written as where and M denotes the black hole mass. Since ultimately we envisage applications of geodesic solutions to accretion problems, we will also need coordinates in which the metric is explicitly regular at the horizon. Such coordinates can be easily obtained by a choice of the time foliation. The transformation where η = η(r) is a function of radiusr, yields the metric in the form The function η defines the time foliation. A popular choice η ≡ 1 leads to coordinates which are manifestly regular at the horizon r = 2M , and which are sometimes referred to as Eddington-Finkelstein coordinates. Note that since we only change the time foliation, the radial coordinate r retains its interpretation as the areal radius. Contravariant components of the metric (6) are given by Moreover, g tr 2 − g rr g tt = 1.

Geodesic motion
There are many well-known descriptions of the geodesic motion. In order to keep a connection with the works [54,53,44,45], we will work in the Hamiltonian framework. The Hamiltonian H describing the geodesic motion of a free particle can be chosen as Here (x µ , p µ ) are treated as canonical variables, and H depends on x α through g µν (x α ). It is easy to verify that the Hamilton equations lead to standard geodesic equations of the form For timelike geodesics we choose the affine parameters as a rescaled proper timeτ , i.e., s =τ /m, where m is the particle rest mass. The four velocity u µ = dx µ /dτ is normalized as g µν u µ u ν = −1. We require that p µ = dx µ /ds, and that H = 1 2 g µν p µ p ν = − 1 2 m 2 .
For null geodesics m = 0 and H = 1 2 g µν p µ p ν = 0. In this case the affine parameter s is unique up to a transformations with α > 0 and an arbitrary β. Such an affine reparametrization implies a rescaling of the tangent vector The explicit form of the Hamiltonian H, assuming metric (6), reads Since H depends neither on t nor on ϕ, the momentum components E ≡ −p t (the energy) and l z ≡ p ϕ are constants of motion. The Hamiltonian H is also independent of s, and hence it is also conserved. A simple calculation allows one to check that the total angular momentum is another constant of motion. The remaining momentum components p θ and p r can be expressed as and is the radial effective potential, and where we have introduced the signs θ = ±1, and r = ±1, corresponding to the directions of motion.
Here and in what follows, the symbol √ denotes the non-negative branch of the square root. Equation (16) follows directly from Eq. (15). Equation (17) can be obtained from the equality H = − 1 2 m 2 . Note also that, contrary to the formula for p r , the expression for p r , does not depend on η. On the other hand and this expression enters the equations of motion listed below. Also note that, while for r = +1 (outgoing motion) the expression for p r diverges at the horizon, the corresponding expression for r = −1 is perfectly regular (particles are allowed to fall into the black hole). An expression for p r with r = −1, manifestly regular at r = 2M , reads Equations dx µ /ds = ∂H/∂p µ can be written as Note that the dependence on η appears only in Eq. (22d). In standard Schwarzschild coordinates N η ≡ 1 and dt/ds = E/N . It is convenient to work in dimensionless rescaled variables. For timelike geodesics we define such variables as in [54], i.e., by t = M τ, r = M ξ, p r = mπ ξ , p θ = M mπ θ , E = mε, l = M mλ, l z = M mλ z .
In addition, a new affine parameter s is defined bỹ For null geodesics m = 0. We introduce an arbitrary mass parameterm > 0, and define ands = (M/m)s. In terms of these dimensionless variables, the equations of motion (22) can be written as where N (ξ) = 1 − 2/ξ. The dimensionless radial potential reads for timelike geodesics, and for null ones.
It is well known that geodesic motion in the Schwarzschild spacetime is confined to a plane. Choosing the coordinate system so that θ ≡ π/2, and dθ/ds ≡ 0, we get λ 2 = λ 2 z , and thus λ z = ±λ. We will adopt a convention with λ ≥ 0 and define the angle in the orbital plane (the so-called true anomaly) ψ = sgn(λ z )ϕ. The relevant equations of motion can be written as System (29) can also be obtained by introducing standard orbital elements such as the orbital inclination, the argument of periapsis, the argument of latitude, and the true anomaly (see, e.g., [39]). Another possibility to (partially) decouple the equations of motion (26) is to introduce the so-called Mino time [48].
A qualitative analysis of the effective radial potential allows for a general classification of different types of orbits. This is done, to some extent, in Appendix B, both for timelike and null orbits. In general, we divide trajectories into bound and unbound ones. Undbound trajectories can either start at infinity and plunge into the black hole (we refer to such trajectories as absorbed ones). The second large class of unbound trajectories consists of orbits characterized by sufficiently large angular momentum. In this case the particles arriving from infinity are scattered off the centrifugal barrier (these trajectories are referred to as scattered ones).

Timelike geodesics
We will start our analysis with timelike geodesics. Given the form of Eqs. (29), it is natural to treat ψ as a parameter and search for a solution of the form ξ = ξ(ψ). From (29a) and (29b) we get immediately Defining where one can write Eq. (30) as For a segment of the trajectory for which r is constant, we get where ξ 0 is an arbitrarily chosen radius corresponding to the angle ψ = 0. We emphasize that √ is assumed to be non-negative. Weierstrass invariants of the polynomial f read (see Appendix A) Therefore, thanks to the Biermann-Weierstrass theorem (see Appendix A for a statement of this theorem and the proof), we can write the formula for ξ = ξ(ψ) as Here ℘ is understood to be defined by the invariants g 2 , and g 3 given by Eq. (35), i.e., ℘(z) = ℘(z; g 2 , g 3 ), and f is defined in Eqs. (31) and (32). We emphasize that formula (36) works in a much more general setting than described above. It turns out to be valid also for trajectories along which the sign r changes. This can be checked numerically, but there is also a way to demonstrate this fact analytically. The argument can be sketched as follows.
Denote the functions defined by Eq. (36) and corresponding to two different signs r as and It follows from Eq.
Consider a particle moving initially inwards (i.e., with r = −1) from a starting position ξ 0 to the turning point ξ 1 , for which f (ξ 1 ) = 0, and then moving outwards (with r = +1) up to a point with the radius ξ. The angle ψ corresponding to this motion can be expressed as ψ = ψ 1 + ψ 2 , where The simplicity of the above formulas is, of course, due to the fact that f (ξ 1 ) = 0. Using expression (37) we get ξ 1 = ξ − (ψ 1 ; ξ 0 ). The fact that the formula (37) describes the continuation of the trajectory in the segment from ξ 1 to ξ means that While the above expression could, in principle, be checked directly, it is much easier to check the corresponding relations involving Weierstrass ℘ functions. According to the addition theorem for the Weierstrass elliptic function ℘, we have Inserting in the above formula the expressions for ℘(ψ 1 ), ℘(ψ 2 ), ℘ (ψ 1 ), and ℘ (ψ 2 ) given by Eqs. (40), we get, after some algebra, as predicted by the Biermann-Weierstrass formula (A. 19). Deriving Eq. (43), we have to remember that f is a fourth order polynomial given by Eq. (31), and f (ξ 1 ) = 0. The reason for considering the difference ψ 1 − ψ 2 , instead of the sum ψ 1 + ψ 2 , can be understood in the light of Eq. (41) and the fact that ξ + (ψ 2 ; ξ 1 ) = ξ − (−ψ 2 ; ξ 1 ). In summary, Eq. (36) can be used to describe any orbit with ψ = 0 for ξ = ξ 0 . The sign r in Eq. (36) can be understood as referring to the direction of motion at ψ = 0, and it need not be changed as the trajectory passes through a turning point. This stays in a clear contrast to the approaches based on Jacobi and Legendre elliptic functions, where one has to deal with different types of orbits separately. Apart from this universality, the main practical advantage of formula (36) is the fact that it does not require finding zeros of the polynomial f . Of course, there are applications in which the knowledge about zeros of the polynomial f is required-we require such knowledge indirectly in Secs. 5, 6, and 7, dealing with the allowed range of ψ, the proper and coordinate time s and τ , respectively. Note that and consequently zeros of the polynomial f are related to zeros of the expression ε 2 −U λ (ξ), corresponding to turning points and discussed in Appendix B. The Biermann-Weierstrass expression is based on a transformation of the integral appearing on the right-hand side of Eq. (34) to the Weierstrass form, i.e.,  Figure  1 depicts examples of bound inner orbits. Figure 2 shows a sample outer bound orbit. Unbound absorbed orbits are shown in Fig. 3. Finally, a family of unbound scattered orbits is plotted in Fig. 4. In all figures, the left panel depicts the radius ξ versus the angle ψ. Right panels show the orbits in the orbital plane with Cartesian coordinates x, y. For comparison, in all cases we draw the same orbits obtained by integrating geodesic equations numerically. These numerical results are depicted with dotted or dashed lines.

Null geodesics
The reasoning for null geodesics is analogous. The equation defining the trajectory reads Adhering to the same notation as for timelike orbits, we set i.e.,   and a 1 = a 4 = 0. The Weierstrass invariants can be written as With these modifications, remaining equations of the previous subsection hold for null geodesics as well. In particular, Eq. (36), with f (ξ) and the Weirestrass invariants given by Eqs. (47) and (49), is valid also for null geodesics.

The range of ψ
In practical applications, one may need to control the allowed range of the parameter ψ or to compute the values of ψ referring to certain points at the trajectory (pericenter, apocenter).
In particular, for unbound scattered orbits the angles ψ ∞± corresponding to the  asymptotics ξ → ∞ could be obtained from the equation i.e., from the requirement that the denominator in Eq.
meaning that ψ ∞ + is the relevant angle in this case. Note that in order to get ψ ∞± direcly form Eq. (50), one would have to invert (locally) the Weierstrass function ℘, which is troublesome in practical applications, as ℘ is not a one to one map.
In Appendix C, we express the function for an unbound scattered timelike or null trajectory in terms of the Legendre elliptic integrals. The result reads where y 1 < y 2 < y 3 are real zeros of the polynomial 4y 3 − g 2 y − g 3 , and k 2 = (y 2 − y 1 )/(y 3 − y 1 ). For a particle arriving from infinity at a scattered trajectory, the angles ψ ∞± can be obtained in a way illustrated in Fig. 8. Let ξ 0 be a location of an incoming particle, with r = −1, corresponding to ψ = 0. The angle ψ ∞+ < 0 can be expressed as ψ ∞+ = −X(ξ 0 ). Denote the location of the pericenter as ξ per ; suppose it corresponds to ψ =ψ. We haveψ + |ψ ∞+ | =ψ − ψ ∞+ = X(ξ per ).
Since the orbit is symmetric with respect to ξ per , one can express ψ ∞− as Figure 8. Trajectory of a scattered particle in the motion plane. The radius ξ per and the angleψ correspond to the pericenter.
For an unbound absorbed trajectory the parameter ψ ∞+ can be expressed as before by ψ ∞+ = −X(ξ 0 ) (we take r = −1), but an explicitly real expression for X(ξ 0 ) is different, and it is given by Eqs. (C. 15) and (C.17) of Appendix C.
For all types of trajectories-bounded and unbounded ones-one can define the integral which we also compute in Appendix C. Finally note that for scattered trajectories the values ψ ∞± yield the bending (or deflection) angle in the Schwarzschild spacetime, which has been studied both for timelike and null geodesics [22,41,63,64].

Affine parameter (proper time)
In this section we compute the affine parameter s associated with a given angle ψ. For timelike geodesics the value of s is related to the proper timeτ byτ = M s.
Given an expression for ξ = ξ(ψ), the corresponding affine parameter can be computed by integrating Eq. (29b), i.e., as Integrating the square of expression (36) is, in principle, possible, but it is tedious, and the result seems to be too complicated to be useful in practical applications. Much simpler formulas can be obtained using Eq. (29b) with the reference position taken at a zero of the polynomial f (ξ). Let ξ 1 denote a radius such that f (ξ 1 ) = 0 (usually a periapsis or an apoapsis); assume that it corresponds to ψ = 0. The radius ξ corresponding to an angle ψ reads (irrespectively of the radial direction of motion, i.e., the value of r ). The affine parameter elapsed during the motion from ψ = 0 to some ψ = ψ 2 can be written as The above integral can be computed with the help of the following two integral formulas ( [7], p. 312 and [27], p. 626): where ζ(x) and σ(x) denote the Weierstrass functions ζ(x; g 2 , g 3 ) and σ(x; g 2 , g 3 ), respectively. We have where ℘(y) = 1 24 f (ξ 1 ) or y = ℘ −1 1 24 f (ξ 1 ) . Usually, using the inverse of the Weierstrass function is troublesome, since ℘ is not a one-to-one function. Fortunately, in formula (62), one is permitted to choose any y satisfying the above condition.
We now invoke to the reasoning from the end of the previous section. Consider a motion of a particle starting from an arbitrary location ξ 0 and moving inwards, up to a periapsis with the radius ξ 1 (thus f (ξ 1 ) = 0). Next the particle moves outwards, up to a location with a radius ξ. Define the angles ψ 1 and ψ 2 by Eqs. (39). Both angles satisfy ψ 1 ≥ 0 and ψ 2 ≥ 0. Let ψ = ψ 1 + ψ 2 . Because of symmetry, the proper time of the entire motion can be written as Formula (63) can be understood as a replacement for integral (57) with ξ(ψ) given by Eq. (36). Note that, since s * (ψ 2 ; ξ 1 ) is an odd function of ψ 2 , we get s(ψ = 0) = 0, as expected. It can also be checked that the same formula holds for ξ 1 corresponding to an apoapsis, provided that definitions (39) are changed accordingly, so that ψ 1 ≥ 0 and ψ 2 ≥ 0.

Coordinate time
The coordinate time τ can be obtained in a way similar to the calculation of the affine parameter s. Consider a trajectory originating at ξ = ξ 0 , ψ = 0, τ = 0. The coordinate time corresponding to the lapse of the parameter ψ can be computed by integrating Eqs. (29). Combining Eqs. (29b) and (29c), one gets where and The second equality in Eq. (66) follows from Eq. (29a). The integral T 2 is clearly gauge-dependent. In the standard Schwarzschild coordinates η(ξ) = 1/N (ξ), and T 2 ≡ 0. Of course, With the help of the identity integral (65) can be written as In analogy to the discussion of the previous section, we will start the computation of T 1 considering at first the special case of a trajectory originating at a turning point ξ = ξ 1 , ψ = 0, such that f (ξ 1 ) = 0. In this case the radius ξ = ξ(ψ) is given by Eq. (58). The lapse of the function T 1 during the motion from ψ = 0 to ψ = ψ 2 can be expressed as where ℘(y) = 1 24 f (ξ 1 ) and the function I 1 is defined by Eq. (60). The last integral can be written in the form where . Equation (70) can now be written as The next step proceeds as in the previous section. Consider a particle on a trajectory originating at the radius ξ 0 and ψ = 0, moving inwards to the turning point ξ = ξ 1 , ψ = ψ 1 , and then continuing outwards, up to a location with an arbitrary radius ξ = ξ(ψ). Let ψ = ψ 1 +ψ 2 , ψ 1 ≥ 0, ψ 2 ≥ 0, where ψ 1 and ψ 2 are given by (39). Thanks to symmetry Again, the same formula holds for a particle moving initially outwards, provided that the signs in the definitions of ψ 1 and ψ 2 are suitably adjusted.

Summary
We have revisited the theory of timelike and null geodesics in the Schwarzschild spacetime. The novel aspect of our work is the application of the Biermann-Weierstrass theorem to the description of Schwarzschild geodesics. A single formula (36) describes all types of timelike or null geodesic orbits, except for purely radial ones. Working with a single formula gives an advantage in those applications, in which one is forced to deal with many different orbits at the same time. We should emphasize that, in contrast to standard numerical methods, Eq. (36) yields exact solutions for arbitrary evolution times, even in the case of dynamically unstable orbits. Our motivation comes from works on kinetic description of relativistic gases. Hence, we parametrize geodesics with conserved quantities (the energy and the angular momentum of the particle) and the particle initial location. Although such a parametrization is natural (and perhaps also the most popular), it might not be optimal in some applications, especially in the context of null geodesics, in which case specifying the locations of the emitter and the observer could by more convenient (cf. [28]).
The Biermann-Weierstrass method of this paper is fairly general, and it is deliberately presented as such in this paper. We choose as our example the Schwarzschild spacetime, but a generalization to a large class of spherically symmetric metrics is straightforward, the Reissner-Nordström spacetime being one of natural possibilities. This fact opens up a variety of applications, including astrophysical ones, related to testing the nature of astrophysical black holes, both in the context of light propagation and the motion of massive particles (see, e.g. [4]).

Acknowledgments
We would like to thank anonymous referees for useful comments and suggestions. A. C. acknowledges a support of the Faculty of Physics, Astronomy and Applied Computer Science of the Jagiellonian University, grant No. N17/MNS/000051. P. M. was partially supported by the Polish National Science Centre Grant No. 2017/26/A/ST2/00530.

Appendix A. Biermann-Weierstrass theorem
The proofs of Lemma Appendix A.1 and Theorem Appendix A.2 given below are adapted from Refs. [6,29,52].
In the following, we restrict ourselves to the range in which dw/dx [and hence also A(x, y)] has a definite sign, so that the map x → w(x) constitutes a valid change of the integration variable. One can show that A 2 (x, y) = W (w(x)), where W = 4w 3 − g 2 w − g 3 , and the invariants g 2 and g 3 are given by Eqs. (A.16). Consequently, A(x, y) = W (w(x)), where = ±1. This proves Eq. (A.20). More precisely, (A.24) Setting y = x 0 in Eq. (A.21), we get w → +∞ for x → x 0 . As a consequence, one obtains Note that = +1 for x > x 0 , and = −1 for x < x 0 . It follows that w(x) can be written as w(x) = ℘( z(x); g 2 , g 3 ) = ℘(z(x); g 2 , g 3 ) and W (w(x)) = − ℘ (z(x); g 2 , g 3 ) (the last relation can be obtained directly by differentiating Eq. (A.25) with respect to x).
It is easy to check that w defined by Eq. (A.21) can be also written as It is a positive root of the quadratic equation Using the relation one can transform Eq. (A.27) into the form which is a quadratic equation with respect to x − y. Solutions of this equation can be written as . (A.31) The value U λ (ξ min ) grows from 8/9 to 1, as λ 2 increases from 12 to infinity. Simultaneously, the value U λ (ξ max ) grows from 8/9 to infinity. For λ 2 ≥ 16, the value of the potential at the maximum is always greater than or equal to one; otherwise, it is smaller. Consider an equation i.e., a limiting case of inequality (B.1), where ξ max is given by Eq. (B.3). A solution of Eq. (B.5) with respect to λ 2 reads (B.6) In other words, λ c (ε) denotes the value of the angular momentum for which the radial potential at the local maximum is equal to ε 2 . It turns out to be particularly useful in classifying different types of timelike trajectories. There are three main types of orbits: radial, bound, and unbound. They can be characterized as follows.
(see, e.g., [23]). Note that circular orbits with either ξ = ξ max (stable) or ξ = ξ min (unstable) belong to this class. There are also limiting cases with λ = λ c (ε), similar to the limiting cases of Type IIa and Type IIIc.
• Type III (unbound orbits). In this case ε ≥ 1. Unbound trajectories are divided into the following two sub-types: a) Absorbed orbits. These trajectories originate at ξ = ∞ and end beneath the black hole horizon. The angular momentum associated with absorbed trajectories satisfies λ < λ c (ε). b) Scattered orbits. Both endpoints of scattered trajectories reach infinity. The particles never reach below ξ = 3, i.e., below the photon sphere. Their energy is bounded from below by (B.10) The total angular momentum of a scattered particle is limited from above, i.e., λ c (ε) < λ ≤ λ max (ε, ξ). c) A limiting case with λ = λ c (ε). The particle travels from infinity and spirals asymptotically to ξ = ξ max .
A comprehensive discussion of the classification of orbits can be found in [54,10,39]. Figure B1 shows the radial effective potential U λ (ξ) corresponding to different types of orbits listed above.
The classification of null trajectories is similar to that of timelike orbits, but it is in many respects much simpler. The dimensionless radial potential U λ (ξ), defined by Eq. (28), has a single maximum at ξ max = 3. The value of the potential at the maximum reads Consequently, the equivalent of the function λ c (ε), defined in Eq. (B.5), reads The orbits are divided into the following classes.
• Type I (radial orbits). As for timelike geodesics, this class consists of orbits with λ = 0.