Gravitational Rutherford scattering of electrically charged particles from a charged Weyl black hole
GGravitational Rutherford scattering of electrically charged particles from a chargedWeyl black hole
Mohsen Fathi, ∗ Marco Olivares, † and J.R. Villanueva ‡ Instituto de F´ısica y Astronom´ıa, Universidad de Valpara´ıso, Avenida Gran Breta˜na 1111, Valpara´ıso, Chile Facultad de Ingenier´ıa y Ciencias, Universidad Diego Portales,Avenida Ej´ercito Libertador 441, Casilla 298-V, Santiago, Chile
Considering electrically charged test particles, we continue our study of the exterior dynamics ofa charged Weyl black hole which has been previously investigated regarding the motion of mass-lessand (neutral) massive particles. In this paper, the deflecting trajectories of charged particles aredesignated as being gravitationally Rutherford-scattered and detailed discussions of angular andradial particle motions are presented.
I. INTRODUCTION
The scattering of charged particles in electric fields is indeed one of the most re-known phenomena in physics and hashad numerous applications in small and large scale observations. Regarding the former, and without loss of generality,the famous Rutherford scattering experiment that led to the discovery of the atomic nucleus, is described in terms ofelastic deflecting trajectories of charged particles from a heavy charged central mass. Such particle trajectories, besidebeing well-known in small atomic scales, have been also investigated widely in black hole spacetimes. In fact, the studyof motion of test particles in the gravitational field of black holes, dates back to the early days of general relativityand ever since, it has found its way in classic textbooks (see for example Refs. [1–3] and the reviews in Refs. [4, 5]).The interest in performing such studies, beside their applicability in testing general relativity and modified theoriesof gravity, stems mostly in the opportunity that they provide to correctly analyze the dynamics of extremely warpedregions around black holes. In these regions, based on the effective gravitational potential that affects the particles,they can lie on different types of orbits, among which, and in particular, the deflecting trajectories relate tightlyto the scattering phenomena. It is well-known that the charge parameter of charged black hole spacetimes (likeReissner-Nordstr¨om and Kerr-Newman), contributes in the gravitational potential of the black hole and therefore,can affect the motion of neutral particles. In the case of charged test particles moving around such black holes, theadditional electromagnetic potential changes the nature of deflecting trajectories to a special form of the Rutherfordscattering. The importance of this kind of motion is such that it has received a large number of performed studies inanalyzing, numerically and analytically, the respected equations of motion and the scattering cross-sections. Thesestudies have been done in the contexts of general relativity and alternative gravity (see for example Refs. [6–23] andRefs. [24–29]). Although black holes with net electric charge are still remained as purely theoretical objects, however,studying them can pave the way in understanding physical phenomena like radiation reaction of particles [30, 31] andblack hole evaporation [32]. Hence, the interest in investigating particle motion around charged black holes becomesmore justified and, as well as in general relativity, it has found its way into alternative theories of gravity.Along the same effort, we investigate the motion and the scattering of charged test particles, as they travel in aparticular charged black hole spacetime, which has been obtained as a non-vacuum solution to the Weyl conformaltheory of gravity. In this discussion, we focus on studying the possible radial and angular motions of charged testparticles that approach the aforementioned charged Weyl black hole (CWBH). Our method of study is based on thestandard Hamilton-Jacobi formulation of particle motion. In particular, we concern with the deflecting trajectorieswhich are strongly related to the gravitational version of Rutherford scattering of charged particles. The equationsof motion are those corresponding to hyperbolic orbits and it is found that, finding analytical solutions to theseequations, demands specific mathematical tricks. Aside form this, we also calculate the Rutherford scattering forparticles on radial trajectories and the evolution of temporal parameters are derived. Additionally, assuming thecongruence deviation of a bundle of infalling world-lines, we discuss the internal interactions between the particlesand point out their effects on the kinematical congruence expansion.The paper is organized as follows: In Sec. II, we give a review on Weyl conformal gravity and bring in the black holesolution that we are intended to investigate. In Sec. III, the basic equations governing the motion of charged particlesin the spacetime generated by the CWBH, are given by means of the Hamilton-Jacobi formalism. In particular, the ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ g r- q c ] N ov angular motion is analyzed in detail in Sec. IV, which is followed by that for the radial motion in Sec. V. In thesetwo sections, the gravitational Rutherford scattering is formulated analytically and the trajectories are demonstratedrespectively. In Sec. VI, a bundle of particle world-lines is considered and we use its deviation as a tool to discuss thecongruence’s internal acceleration and expansion. We close our discussion in Sec. VII. II. THE WEYL CONFORMAL THEORY OF GRAVITY AND ITS STATIC BLACK HOLE SOLUTIONS
To obtain the Weyl conformal theory of gravity, the common Einstein-Hilbert action of general relativity I EH = K EH (cid:90) d x √− g R, (1)is replaced by I W = −K W (cid:90) d x √− g C αβµν C αβµν . (2)In the above actions, K EH and K W are appropriate coupling constants, g = det( g µν ), R = g νµ R αµαν = R νν is theRicci curvature scalar, and the Weyl tensor C αβµν = R αβµν −
12 ( g αµ R βν − g αν R βµ − g βµ R αν + g βν R αµ ) + R g αµ g βν − g αν g βµ ) , (3)is invariant under the conformal transformation g µν ( x ) = e (cid:96) ( x ) g µν ( x ), where 2 (cid:96) ( x ) is the local spacetime stretching.Accordingly, the action (2) can be recast as I W = −K W (cid:90) d x √− g (cid:18) R αβµν R αβµν − R µν R µν + 13 R (cid:19) , (4)in which, the Gauss-Bonnet term √− g ( R αβµν R αβµν − R µν R µν + R ) is a total divergence and hence, the action issimplified as [33, 34] I W = − K W (cid:90) d x √− g (cid:18) R µν R µν − R (cid:19) . (5)Applying δI W δg µν = 0, the equations of motion are derived as W µν = P αβ C µανβ + ∇ α ∇ α P µν − ∇ α ∇ µ P να = 14 K W T µν , (6)in which P µν = 12 (cid:18) R µν − g µν R (cid:19) , (7)is the Schouten tensor. The field equation (6) is known as the Bach equation. Based on the fact that Weyl gravityis a theory of fourth order in the metric, the Bach equation therefore contains derivatives up to the fourth order.Such field equations may lead to more convergent and renormalizable theories of gravity, although, the consistencyof their quantization is still under debate, mostly because the field equations are conceived as fluctuations on a fixedbackground [35]. Additionally, the fourth-order theories of gravity usually give rise to the presence of ghost fields,although some solutions to this issue have been reported in Ref. [36]. It is of worth mentioning that the standardmodel of particle physics can be endowed with conformal symmetry, if an appropriate gravitational term is added toits action. This gravitational coupling helps fixing a conformal gauge in accordance with a reference mass. This way,the action can generate particle mass, without encountering any symmetry breaking (see Ref. [37] for more details).In 1989, Mannheim and Kazanas obtained a static spherically symmetric solution to the vacuum Bach equation( T µν = 0), which was given by the metric d s = − B ( r ) d t + d r B ( r ) + r (d θ + sin θ d φ ) , (8) In this paper, we work in the geometric unit system, by considering G = c = 1. Accordingly, the dimensions of mass, electric chargeand length are given in meters. with the lapse function [33] B ( r ) = 1 − ζ (2 − ζρ ) r − ζρ + ρr − σr . (9)The first two terms are in common with general relativistic vacuum solutions, in the sense that by letting ζ = M and ρ = σ = 0, the Schwarzschild solution is obtained for a spherically symmetric source of mass M . The last twoterms, however, are peculiar to the fourth order Bach equation. Essentially, this solution was intended to explain theflat galactic rotation curves [38], by introducing an extra gravitational potential in the spacetime, and therefore, triesto recover one of the cosmological evidences that support the dark matter scenario. For a typical galaxy of about10 kpc in length, this potential grows with distance for r >
10 kpc, is approximately constant for r ∼
10 kpc, andbecomes Newtonian for r <
10 kpc (for this latter condition, see Refs. [39, 40] for more discussions). Accordingly,the estimated value of ρ , for which, the r and 1 /r related terms are comparable, is ρ ≈ − m − . This way, thesolution encounters the dimensionless quantity ζρ ≈ − and predicts the flat rotational velocity of 10 kms − [33].Furthermore, the last term of the solution (9), accounts for supporting the accelerated expansion of the universe, andtherefore is related to the dark energy scenario [41]. So, as it is observed, the theory generates a substantially differentgravitational potential to general relativity. Moreover, in the case that the energy-momentum tensor is associatedwith the electrostatic vector potential A α = (cid:16) q r , , , (cid:17) , (10)corresponding to a spherically symmetric massive source of electric charge q , the same line element as in Eq. (8) hasbeen applied by Mannheim and Kazanas, to obtain the Reissner-Nordstr¨om form of solution to Weyl gravity, withthe lapse function [42] B ( r ) = w + u r + v r − k r , (11)for which, the parameters can be rewritten as w = 1 − ζρ, (12a) v = ρ, (12b) u = − ζ (2 − ζρ ) − q ρ K W , (12c)in terms of the coefficients presented in the vacuum solution (9). In Ref. [43], the vector potential (10) has beenapplied, once again, to the Bach equation (6), however, for the reference lapse function B ( r ) = 1 + 13 (cid:0) c r + c r (cid:1) , (13)in which, the two coefficients c and c replace, respectively, the dark matter and dark energy terms, as in the lapsefunction (9). These coefficients can be determined by describing the solutions to the Bach equation, as perturbationson the Minkowski spacetime (appearing in the last two terms of Eq. (13)). Accordingly, we can decompose thespacetime metric as g µν = h µν + η µν , where h µν and η µν are respectively, the perturbation and the Minkowskimetrics. Therefore, for a spherically symmetric source of mass ˜ m , electric charge ˜ q and radius ˜ r , the Poisson equationfor the perturbation field, i.e. ∇ h µν = 8 πT µν , yields ∇ h = 8 πT = 8 π (cid:18) ˜ m π ˜ r + 18 π ˜ qr (cid:19) , (14)as its 00 component. Here, T corresponds to the scalar potentials produced by the mass and charge of the source.Applying Eq. (13) in Eq. (14), we get [43] c = − (cid:18) m r ˜ r + 32 ˜ q r + 3 c r (cid:19) , (15)that provides B ( r ) = 1 − r λ − Q r , (16)in which 1 λ = 3 ˜ m ˜ r + 2˜ ε , (17) Q = √ q. (18)Because we have excluded c by defining it in terms of c , the lapse function (16) does not contain the dark matterrelated term, and ˜ ε ≡ c in Eq. (17), adds a cosmological term to the spacetime. Regarding this, the lapse function(16) describes the exterior geometry of a CWBH with a cosmological component. This spacetime has been recentlyinvestigated regarding the motion of mass-less and neutral massive particles [44–46]. When λ > Q , this spacetimeadmits an event and a cosmological horizon, which are obtained by solving B ( r ) = 0, and are given respectively by[46] r + = λ sin (cid:18)
12 arcsin (cid:18) Qλ (cid:19)(cid:19) , (19) r ++ = λ cos (cid:18)
12 arcsin (cid:18) Qλ (cid:19)(cid:19) . (20)These horizons unify at the distance r ex = r + = r ++ = λ/ √
2, in the case of an extremal CWBH, when λ = Q .Accordingly, a naked singularity is encountered for λ < Q . Letting ˜ r to be a variable radial distance, r , 3 ˜ m → M ,2˜ ε → ± Λ, and Q → iQ , we get to the Reissner–Nordstr¨om–(Anti-)de Sitter black hole of mass M , charge Q and cosmological constant Λ. However, as it is noticed, this transition requires an imaginary transformation. Webegin studying the motion of charged test particles around the CWBH, from the next section, by employing theHamilton-Jacobi formalism. III. MOTION OF CHARGED PARTICLES AROUND THE CWBH
The Hamilton-Jacobi method of describing the motion of particles of mass m and charge q in an electromagneticfield, is based on the superhamiltonian [1] H = 12 g µν p µ p ν , (21)in which the 4-momentum p satisfies p µ p µ = − m and is defined as p µ = g µν d x ν d τ = ( π µ + qA µ ) , (22)in terms of the affine parameter τ , the vector potential A and the generalized momentum π , which is given accordingto the canonical Hamilton equation d π µ d τ = − ∂ H ∂x µ . (23)Recasting H in terms of the characteristic Hamilton function (i.e. the Jacobi action) H = − ∂S∂τ , (24)we have π µ = ∂S/∂x µ and the Hamilton-Jacobi equation of the wave crests can be written as12 g µν (cid:18) ∂S∂x µ + qA µ (cid:19) (cid:18) ∂S∂x ν + qA ν (cid:19) + ∂S∂τ = 0 . (25)The generalized momentum π is indeed responsible for the possible constants of motion. For stationary sphericallysymmetric spacetimes, such as that in Eq. (8), these constants are π t . = − E = g tt d t d τ − qA t , (26a) π φ . = L = g φφ d φ d τ − qA φ . (26b)The constant L corresponds to the particles’ angular momentum and for motion in asymptotically flat spacetimes, E is their energy. Now, confining the motion to the equatorial plane ( θ = π/
2) and taking into account the onlynon-zero term of the vector potential of a CWBH (i.e. A t = ˜ qr = Q √ r ), we can specify Eq. (25) as − B ( r ) (cid:18) ∂S∂t + qQ √ r (cid:19) + B ( r ) (cid:18) ∂S∂r (cid:19) + 1 r (cid:18) ∂S∂φ (cid:19) + 2 ∂S∂τ = 0 . (27)Based on the method of separation of variables of the Jacobi action, Eq. (27) can be solved by defining [47] S = − Et + S ( r ) + Lφ + 12 m τ, (28)for which, interpolation in Eq. (27) results in S ( r ) = ± (cid:90) d rB ( r ) (cid:112) ( E − V − )( E − V + ) , (29)where the radial potentials are given by V ± ( r ) = V q ( r ) ± (cid:115) B ( r ) (cid:18) m + L r (cid:19) , (30a) V q ( r ) . = qQ √ r . (30b)Note that, both of the ± branches of V ± ( r ) converge to the value E + = qQ √ r + at r = r + , which can be either positiveor negative, depending on the sign of the electric charges, q and Q . Here we adopt the condition qQ >
0, so thatthe V − branch is always negative (in the causal region r + < r < r ++ ), and we can consider the positive branch asthe effective potential, i.e. V eff . = V + ≡ V . Furthermore, applying Eqs. (26) and (29), it is possible to obtain thefollowing three velocities: u ( r ) ≡ d r d τ = ± (cid:112) ( E − V − )( E − V ) , (31) v t ( r ) ≡ d r d t = ± B ( r ) u ( r ) E − V q ( r ) , (32) v φ ( r ) = d r d φ = ± r u ( r ) L . (33)The zeros of the above velocities do correspond to the so-called turning points, r t , which are specified by the condition V ( r t ) = E t . Additionally, these equations lead to the quadratures that determine the evolution of the trajectories.This is dealt with in the forthcoming sections and the corresponding analytical solutions are obtained. In the nextsection, we begin with the angular motion of the test particles. IV. ANGULAR MOTION
In this section we focus on analyzing the trajectories followed by charged particles with non-zero angular momentum( L (cid:54) = 0). The effective potential in Eq. (30) has been plotted in Fig. 1, in which the turning points r t correspondto the values of E = E t that satisfy E t = V ( r t ). The significance of these points is that they do reveal the possibleorbits of the test particles. In fact, according to Fig. 1, three turning points are highlighted; r t = r U (the radius ofunstable circular orbits), r t = r S (the distance from the point of scattering) and r t = r F (the point of no return, orthe capturing distance). r U r S r F E U E t E + r + r ++ rV ( r ) FIG. 1. The effective potential for test particles with angular momentum, plotted for m = 1, L = 1, Q = 1, q = 0 . λ = 10.The turning points are determined by the intersection of E and the effective potential (i.e. E t = V ( r t )). These include theradius of unstable circular orbits r U , and two other points, r S and r F . A. Unstable circular orbits
The radius of the unstable circular (critical) orbits, r U , is given by the condition V (cid:48) ( r ) ≡ ∂V ( r ) ∂r (cid:12)(cid:12)(cid:12) r U = 0. Hence,from Eq. (30) we get (cid:32)(cid:115) G ( r ; L ) B ( r ) B (cid:48) ( r )2 − (cid:115) B ( r ) G ( r ; L ) L r − qQ √ r (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r U = 0 , (34)in which the function G ( r ; L ) is defined as G ( r ; L ) = m + L r . (35)In fact, the left hand side of Eq. (34) leads to an incomplete polynomial of twelfth degree in r , and hence, it can besolved only numerically. It is however still possible to calculate the proper ( T τ ) and the coordinate ( T t ) periods ofthese orbits. Combining Eqs. (31), (32) and (33), and the fact that for a complete orbit ∆ φ U = 2 π , we have T τ ≡ ∆ τ = 2 πr U L U , (36) T t ≡ ∆ t = 2 πr U L U E U − V q ( r U ) B ( r U ) = T τ (cid:114) G U B U , (37)where G U ≡ G ( r U ; L U ) and B U ≡ B ( r U ). Solving Eq. (34), we then obtain an expression for L U as (appendix A) L U = (cid:115) b − √ b − ac a , (38)as the angular momentum for the circular orbits, where a = ( Q − r U ) r U , (39a) b = 2 Q (1 + q ) r U − Q (2 + q )2 r U − r U − Q (1 − q ) λ , (39b) c = Q (1 + 2 q )4 r U − q Q + 4 r U λ − Q r U (1 − q ) λ . (39c)Accordingly, one can obtain the proper frequency ω τ = 2 πT τ = (cid:115) b − √ b − ac a r U , (40)straightly from Eqs. (36) and (38). The coordinate frequency can then be given by the ratio ω τ ω t = (cid:114) G U B U . (41)These values correspond to the velocity of particles on a surface, where they can maintain a circular orbit beforefalling into the event horizon or escape from it. In the study of particle trajectories, the critical orbits can locate theinnermost possible stable orbits around black holes and therefore are of great importance. The test particles, however,can also be scattered at the turning point r S , pursue a hyperbolic motion and escape the black hole. For electricallycharged particles, this corresponds to the so-called Rutherford scattering. We continue our discussion by analyzingthis kind of orbit. B. Orbits of the first kind and the gravitational Rutherford scattering
The particle deflection by the CWBH happens when the condition E + < E < E U is satisfied. This indeed resultsin two points of approach, r t = r S and r t = r F , at which, d r d φ | r t = 0 or E t = V ( r t ) (see Fig. 1). The relevant equationof motion can be derived from Eqs. (31) and (33), giving (cid:18) d r d φ (cid:19) = P ( r ) υ , (42)where P ( r ) ≡ r + A r + B r + C r + D , (43a) υ = Lλm , (43b)with A = υ (cid:18) E − m L + 1 λ (cid:19) , (44a) B = − Eυ (cid:18) qQ √ L (cid:19) , (44b) C = D L − B E − υ , (44c) D = υ (cid:18) mQ (cid:19) . (44d)To determine the turning points r S and r F , one therefore needs to solve P ( r t ) = 0, which is an incomplete equationof sixth degree in r , and values of r ( φ ) can therefore be obtained through numerical methods. To deal with thisproblem, we pursue the inverse process and find an analytical expression for φ ( r ). The behavior of r ( φ ) can then bedemonstrated by means of numerical interpolations.To proceed with this method, let us consider that P ( r ) has two distinct real roots, corresponding to the turningpoints r = r S and r = r F , two equal and negative real roots, say r = r < r and r = r ∗ . Accordingly, we can recast P ( r ) as P ( r ) = (cid:89) j =1 ( r − r j )= ( r − r S )( r − r F )( r − r )( r − r ) ( r − r ∗ ) . (45)The equation of motion (42) can then be written as φ ( r ) = υ (cid:90) rr S d r (cid:112) P ( r ) . (46)Particles reaching r S , experience a hyperbolic motion around the black hole and then escape to infinity. This kindof motion, known as orbit of the first kind (OFK) [3, 48], has the significance of gravitational Rutherford scatteringwhen the test particles are electrically charged. Considering the change of variable u j . = 1( r j /r S ) − , j = { , , , } , (47)the above integral results in (appendix B) φ ( r ) = κ (cid:104) ß( U ) − u F ( U ) (cid:105) , (48)where ß( x ) ≡ ß( x, g , g ) is the inverse Weierstraß ℘ function, and F ( U ) = 1 ℘ (cid:48) (Ω S ) (cid:20) ζ (Ω S )ß( U ) + ln (cid:12)(cid:12)(cid:12)(cid:12) σ (ß( U ) − Ω S ) σ (ß( U ) + Ω S ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , (49)also contains the Weierstraß Zeta and Sigma functions ( ζ ( y ) and σ ( y ), respectively) . Here, we have defined U ≡ U ( r ) = 14 (cid:16) rr S − (cid:17) + a , (50a)Ω S = ß a − (cid:16) r r S − (cid:17) , (50b) g = a − b , (50c) g = 116 (cid:18) ab − a − c (cid:19) , (50d)with a = u + u + u , (51a) b = u ( u + u ) + u u , (51b) c = u u u . (51c)The scattering angle in Eq. (48) gives the change in the particles’ orientation as they approach and recede the blackhole at the scattering point r S . To illustrate their corresponding trajectories, we make a list of points ( r t , φ ( r t ))and then find the numerical interpolating function of r ( φ ). The resultant OFK trajectories have been illustrated inFig. 2 for particles of different values for E . As it is observed, the scattering can be formed convexly (approaching)or concavely (receding). For particles coming from infinity, the scattering angle can be written as [24, 46] ϑ = 2 φ ∞ − π, (52) By definition, we have [49] ß( x ) ≡ y = (cid:90) ∞ x d t (cid:112) t − g t − g . Then the inverse function x = ℘ ( y, g , g ) ≡ ℘ ( y ) defines the elliptic Weierstraß ℘ function with the coefficients g and g , for which ℘ (cid:48) ( y ) ≡ dd y ℘ ( y ) = − (cid:113) ℘ ( y ) − g ℘ ( y ) − g . The two other related functions, namely the Weierstraß Zeta and Sigma functions, are defined as ζ ( y ) = − (cid:90) ℘ ( y )d y,σ ( y ) = e (cid:82) ζ ( y )d y . - - - E E E E E FIG. 2. The Rutherford scattering plotted for m = 1, Q = 1, q = 0 . λ = 10 and L = 1. For these values, r + = 0 . r U = 0 . E U = 1 .
72. The trajectories have been plotted for E = 0 . E = 1 . E = 1 . E = 1 . E = 1 .
7, while theircorresponding scattering distance ( r S ) have been indicated by dashed circles. As it is observed, the condition E ≈ E U hasmade the corresponding shape of the scattering to be of a convex form, showing an appeal to the critical orbits. in which φ ∞ ≡ φ ( ∞ ). Accordingly, from Eq. (48) we have ϑ = − π +2 κ ß (cid:16) a (cid:17) + u (cid:113) (4 a − ab + 27 u ( b − a u + u )) (cid:34) ζ (Ω S )ß (cid:16) a (cid:17) + ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ (cid:0) ß (cid:0) a (cid:1) − Ω S (cid:1) σ (cid:0) ß (cid:0) a (cid:1) + Ω S (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:35) . (53)The value of ϑ is specified directly by the initial E and the corresponding particular solutions r j , which are determinedby the equation E = V ( r ). These values therefore, cannot be considered to evolve in terms of a single variable.However, one can calculate the scattering angle for each particular trajectory, by applying Eq. (53). Additionally, thedifferential angular range of the scattered particles at the angle ϑ , is given by the solid angle element dΩ = sin ϑ d ϑ d φ .In this regard, and defining the impact parameter b = LE , the cross-sectional area of the scattered particles has thedifferential form dΣ = b d φ d b [46]. Therefore, the differential cross section of the scattering is given byΣ( ϑ ) . = dΣdΩ = b sin ϑ (cid:12)(cid:12)(cid:12)(cid:12) ∂b∂ϑ (cid:12)(cid:12)(cid:12)(cid:12) . (54)From Eqs. (48) and (52) we have 12 κ ( ϑ + π ) = ϕ + ϕ , (55)in which ϕ ≡ ß (cid:16) a (cid:17) , (56a) ϕ ≡ − u F (cid:16) a (cid:17) . (56b)0We define Ψ( L ) . = ℘ (cid:18) ϑ + π κ (cid:19) = ℘ ( ϕ + ϕ ) , (57)or [49] Ψ( L ) = 14 (cid:20) ℘ (cid:48) ( ϕ ) − ℘ (cid:48) ( ϕ ) ℘ ( ϕ ) − ℘ ( ϕ ) (cid:21) − ℘ ( ϕ ) − ℘ ( ϕ ) . (58)Now, applying the definition in Eq. (58), we can recast Eq. (54) asΣ( ϑ ) = b csc ϑ (cid:12)(cid:12)(cid:12)(cid:12) ∂ Ψ ∂ϑ (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ∂b∂ Ψ (cid:12)(cid:12)(cid:12)(cid:12) = 14 κ csc ϑ (cid:12)(cid:12)(cid:12)(cid:12) ℘ (cid:48) (cid:18) ϑ + π κ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ∂b ∂ Ψ (cid:12)(cid:12)(cid:12)(cid:12) , (59)for which, the identity ∂b ∂ Ψ = ∂b /∂L∂ Ψ /∂L yieldsΣ( ϑ ) = L κ E csc ϑ (cid:12)(cid:12)(cid:12)(cid:12) ℘ (cid:48) (cid:18) ϑ + π κ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ∂ Ψ ∂L (cid:12)(cid:12)(cid:12)(cid:12) − . (60)The expression of Ψ is analytically complicated. However, as before, the value of Eq. (60) can be numerically calcu-lated regarding definite initial values for distinct scattered trajectories.In this section, we analyzed the angular trajectories of infalling particles and demonstrated the correspondingRutherford scattering which is the most significant feature of such trajectories. However, scattering can happen, aswell, in the absence of the particles’ angular momentum, which is what we deal with in the next section by studyingthe radial trajectories of the test particles. V. RADIAL TRAJECTORIES
To discuss the horizon crossing process, as viewed by the comoving and the distant observers, it is common to usea framework, in which, the particles fall radially onto the black hole, without possessing any angular momentum. Aswell as in the case of angular orbits, purely radial motion can itself be ramified into several kinds. Further in thissection, these kinds of motion are discussed regarding the horizon crossing of the comoving observers and the frozen particles as viewed by distant observers (see Refs. [50, 51] for the notion of a frozen infalling object onto a black hole).The vanishing angular momentum of the radially moving particles, reduces the effective potential in Eq. (30) to V r ( r ) = V q ( r ) + m (cid:112) B ( r ) , (61)whose behavior has been plotted in Fig. 3. Accordingly, the motion becomes unstable where V (cid:48) r ( r ) = 0, solving which,leads to the maximum distance of the unstable motion, reading as r u = (cid:20) ˜ α − (cid:113) ˜ α − ˜ β (cid:21) / , (62)where (see appendix C) ˜ α = (cid:114) ˜ U − ˜ a , (63a)˜ β = 2˜ α + ˜ a b α , (63b)given that ˜ U = 2 (cid:114) ˜ η (cid:32)
13 cosh − (cid:32)
32 ˜ η (cid:115) η (cid:33)(cid:33) , (64)1 r s r f E E + r + r ++ r u R s E u E ++ rV ( r ) FIG. 3. The effective potential for radially moving particles plotted for m = 1, Q = 1, q = 0 . λ = 10. The maximumdistance of unstable motion, r u , and the two turning points r s and r f have been indicated in accordance with their correspondingvalues of E . In particular, the point R s is related to the distance at which the particles of the constant of motion E + , experiencetheir Rutherford scattering. with ˜ a = − Q λ (cid:18) − q m (cid:19) , (65a)˜ b = − q Q λ m , (65b)˜ c = Q λ (cid:18) q m (cid:19) , (65c)˜ η = ˜ a
48 + ˜ c , (65d)˜ η = ˜ a
864 + ˜ b − ˜ a ˜ c . (65e)Taking into account E u ≡ V r ( r u ), as in the angular case, possible motions are categorized based on the value of E compared with its critical value, E u : • Frontal Rutherford scattering of the first and the second kinds (RSFK and RSSK) : For E ++ < E < E u , thepotential allows for a turning point r s ( r u < r s < r ++ ) which corresponds to the scattering distance (RSFK). Inthe case that E + < E < E u , there is also another turning point r f ( r + < r f < r u ), from which, the trajectoriesare captured into the event horizon (RSSK). • Critical radial motion : For E = E u , the particles can stay on an unstable radial distance of radius r = r u .Therefore, those coming from the initial distances r i or d i ( r u < r i < r ++ and r + < d i < r u respectively), willultimately fall on r u , or cross the horizons.Now, let us rewrite the radial velocity relations, given in Eqs. (31) and (32), as (cid:18) d r d τ (cid:19) = m p ( r ) λ r , (66) (cid:18) d r d t (cid:19) = m ( r − r ) ( r − r ) p ( r ) E λ r ( r − √ qQE ) , (67)with p ( r ) ≡ r + ¯ ar + ¯ br + ¯ c, (68)2where ¯ a = ( E − m ) λ m , (69a)¯ b = − √ qQEλ m , (69b)¯ c = Q ( m + 2 q ) λ m . (69c)Applying the above equations, in this section, the radial behavior of the time parameter for the moving particles iscalculated, together with that for the distant observers. A. Frontal scattering
As it is inferred from the effective potential in Fig. 3, particles can encounter two turning points r s and r f whichare located at either sides of the critical distance ( r f < r u < r s ). These turning points do lead the trajectories todifferent fates. Particles with E ++ < E < E + , however, can only escape the black hole by being scattered at theonly possible turning point r s . Same as discussed in Sec. IV, the turning points are where the particles’ coordinatevelocity vanishes, which for the radial trajectories requires p ( r ) = 0 in Eq. (68), giving r s = ¯ α + (cid:113) ¯ α − ¯ β, (70) r f = ¯ α − (cid:113) ¯ α − ¯ β. (71)These radii are basically based on the same components as in Eqs. (63)–(65), and we only need to replace ˜ a → ¯ a ,˜ b → ¯ b and ˜ c → ¯ c , according to the values given in Eqs. (69). Having determined the turning points, the polynomial p ( r ) can be decomposed accordingly. As described above, the first kind scattering (RSFK) happens when the particlesapproach at r s , which is now considered as their initial position. Therefore Eq. (66) can be solved as (appendix D) τ ( r ) = − λm √ γ (cid:20) ß(U) + 14 F(U) (cid:21) , (72)where F(U) = 1 ℘ (cid:48) (Ω s ) (cid:20) ζ (Ω s )ß(U) + ln (cid:12)(cid:12)(cid:12)(cid:12) σ (ß(U) − Ω s ) σ (ß(U) + Ω s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , (73)and the function U( r ) and the Weierstraß coefficients are given asU( r ) = r s r − r s ) + γ γ , (74a)Ω s = ß (cid:18) γ γ (cid:19) , (74b)¯ g = γ γ − γ , (74c)¯ g = 116 (cid:18) γ γ − γ γ − γ (cid:19) , (74d)with γ = 6 + ¯ ar s , (75a) γ = 4 + 2¯ ar s + ¯ br s . (75b)The relation in Eq. (72) measures the radial change of the time parameter for observers comoving with the particles.For distant observers, such measurement is done on the coordinate time, whose evolution can be obtained by exploiting3 r ti m ea x i s E= r s (cid:8776) E + = . , R s (cid:8776) E= . , r s (cid:8776) . r + + r + r u rsrs R s FIG. 4. The radial behavior of the proper and coordinate times in the RSFK, for three scattering points and their correspondingvalues of E . After the scattering, the comoving observers (thick line) see a horizon crossing. This is while a distant observer(thin line) never observes this (frozen falling particles). The plots have been done for m = 1, Q = 1, q = 0 . λ = 10. the velocity in Eq. (67). Applying the same method as before, we obtain t ( r ) = − δ (cid:34) ß(U) + 14 (cid:88) k =1 δ k F k (U) (cid:35) , (76)where F k (U) = 1 ℘ (cid:48) (Ω k ) (cid:20) ζ (Ω k )ß(U) + ln (cid:12)(cid:12)(cid:12)(cid:12) σ (ß(U) − Ω k ) σ (ß(U) + Ω k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , (77)and Ω k = ß (cid:18) γ γ + z k (cid:19) , (78)in which z k . = r k /r s ) − , with r ≡ r + , r ≡ − r + , r ≡ r ++ and r ≡ − r ++ , and the coefficients are expressed as δ = λ Em √ γ z z z z z r s , (79a) δ = ( z + 1) z ( z − z )( z − z )( z − z )( z − z ) , (79b) δ = ( z + 1) z ( z − z )( z − z )( z − z )( z − z ) , (79c) δ = ( z + 1) z ( z − z )( z − z )( z − z )( z − z ) , (79d) δ = ( z + 1) z ( z − z )( z − z )( z − z )( z − z ) . (79e)Since these trajectories escape the black hole, they will eventually confront the cosmological horizon. In Fig. 4, thetemporal relations in Eqs. (72) and (76) have been used to demonstrate the RSFK, as observed by comoving anddistant observers, for three different scattering distances. As it is seen, for the comoving observers, a horizon crossingoccurs within a finite time after the particles are scattered at r s . This is while for the distant observers it takes aninfinite amount time for the particles to pass the horizon; in other words, they appear frozen. B. Frontal scattering of the second kind
By switching the scattering distance to r f , the particles experience the RSSK and they confront the event horizon.The corresponding equations of motion are the same as those in the case of RSFK and are given by exchanging4 r ti m ea x i s E= . , r f (cid:8776) . E= . , r f (cid:8776) . r u r + r f r f FIG. 5. The RSSK for two different scattering points and their corresponding values of E , plotted for m = 1, Q = 1, q = 0 . λ = 10. r s → r f in the relations. The corresponding temporal parameters have been demonstrated in Fig. 5. Same as before,the comoving and distance observers see different fates for the infalling particles, but here, regarding the captureprocess by the event horizon. C. Critical radial motion
In the case that E = E u , the unstable (critical) motion of particles depends on whether they approach from r i > r u or from d i < r u . According to the discontinuity of d τ d r and d t d r at r i and d i , we can expect two different behaviors forthe approaching particles, in the sense that they either fall on r = r u (fate I ) or be pulled towards the horizons (fate II ). These are revealed by integrating the equations of motion for the time parameters. For particles coming from r i , we obtain τ I ( r ) = ± λm [ τ A ( r ) − τ B ( r ) − τ A ( r i ) + τ B ( r i )] , (80) τ II ( r ) = ∓ λm [ τ A ( r ) − τ B ( r ) − τ A ( d i ) + τ B ( d i )] , (81)for the comoving observers, where τ A ( r ) = arcsinh (cid:32) r + r u (cid:112) ¯ a + 2 r u (cid:33) , (82a) τ B ( r ) = r u (cid:112) r u + ¯ a arcsinh (cid:32) r u + ¯ a + 2 r u ( r − r u ) | r − r u | (cid:112) ¯ a + 2 r u (cid:33) . (82b)For the distant observers, we get t I ( r ) = ± λ Emr u (cid:88) n =0 (cid:36) n [ t n ( r ) − t n ( r i )] , (83) t II ( r ) = ∓ λ Emr u (cid:88) n =0 (cid:36) n [ t n ( r ) − t n ( d i )] , (84)where t n ( r ) = r u (cid:112) R n arcsinh (cid:32) R n + ( r u + r n )( r − r n ) | r − r u | (cid:112) ¯ a + 2 r u (cid:33) , (85a) R n = 3 r u + ¯ a + 2 r u r n + r n , (85b)5 t i m e a x i s r + r u r i r ++ r (a) t i m e a x i s r + r u d i r (b)FIG. 6. The critical radial motion for fates I and II , plotted for comoving (thick line) and distant (thin line) observers, byletting m = 1, Q = 1, q = 0 . λ = 10. The trajectories have been specified for particles approaching from ( a ) r = r i = 5and ( b ) r = d i = 0 . and the coefficients are given as (cid:36) = r u ( r u − r )( r − r u )( r − r u )( r − r u )( r − r u ) , (86a) (cid:36) = r u r ( r − r )( r − r u )( r − r )( r − r )( r − r ) , (86b) (cid:36) = r u r ( r − r )( r − r u )( r − r )( r − r )( r − r ) , (86c) (cid:36) = r u r ( r − r )( r − r u )( r − r )( r − r )( r − r ) , (86d) (cid:36) = r u r ( r − r )( r − r u )( r − r )( r − r )( r − r ) , (86e)in which r ≡ r u , r ≡ r + , r ≡ − r + , r ≡ r ++ and r ≡ − r ++ . The critical radial behavior of temporal parameters,as measured by the comoving and the distant observers, have been plotted in Fig. 6, separately for the initial points r i and d i . In each of the diagrams, the cases I and II have been demonstrated and the horizon crossing is shownaccordingly. Once again, particles appear frozen to distant observers as they approach the horizons.So far, both the angular and the radial trajectories of charged particles were studied and possible analytical solutionsto the equations of motion were given. It is however worth mentioning that the study of the physical properties ofmoving particles is not summarized to the evolution of a single particle’s trajectory. In the case that a bundle oftrajectories is taken into account, definite kinematical parameters will play important roles in the characterization ofa flow of particle trajectories. Accordingly, and in the next section, we consider such a flow of particles and studyhow it reacts to the internal and external forces acting on the world-lines. VI. A CONGRUENCE OF INFALLING CHARGED PARTICLES
In this section, we consider a bundle of particle trajectories, which together, constitute a congruence of world-linesthat fall onto the CWBH. Essentially, the congruence kinematics is a tool to inspect the Penrose–Hawking singularitytheorems [52–55] and is accurately formulated by the well-known Raychaudhuri equation [56]. This equation formu-lates the way the congruences would evolve their cross-sectional (transverse) area (for a good review see Ref. [57]).Here, we switch our discussion to the possibility of applying some geometrical methods in order to demonstrate thedeviation of a congruence of time-like trajectories while they pass the black hole. For particles passing a RN blackhole, this deviation has been studied in detail in Refs. [58, 59].In the geometric sense, the congruence deviation gives the relative acceleration between the curves that are generatedby the tangential vector u , in terms of the Jacobi (deviation) vector field ξ . This vector field resides on the curves6that connect points of equal τ on smooth planes of world-lines. These vectors satisfy [60, 61] L u ξ = L ξ u , (87)where L X indicates the Lie differentiation with respect to a vector field X . The above equation therefore can berecast as ξ µ ; ν u ν = u µ ; ν ξ ν . (88)In above, the semicolons correspond to covariant differentiation. Note that, the quantity ξ · u varies along thecongruence as [61] D d τ ( ξ · u ) ≡ ( ξ · u ) ; ν u ν = 12 ( u · u ) ; ν ξ ν + a µ ; ν ξ µ u ν , (89)where a µ = u µ ; ν u ν , (90)is the four-acceleration of the non-inertial frames, according to non-gravitational effects. In this regard, a non-zero a corresponds to a vector field which is not parallel-transported along the world-lines. Accordingly, the congruencedeviation equation can then be written as A µ . = D ξ µ d τ ≡ (cid:0) ξ µ ; ν u ν (cid:1) ; γ u γ = a µ ; ν ξ ν − R µναβ u ν ξ α u β . (91)This vector, measures the relative acceleration between two world-lines, as measured by the change in ξ , and connectsit to the spacetime curvature [62, 63].According to the Eqs. (31), (32) and (33), we know that a congruence of charged particles with angular motion,that fall onto the charged black hole, is generated by the following four-velocity: u µ = (cid:18) E − V q ( r ) B ( r ) , (cid:112) ( E − V − )( E − V ) , , Lr (cid:19) , (92)which satisfies u · u = − m (we let m = 1). The congruence deviation (Jacobi) field, related to the vector field (92),can then take the generic form ξ µ = (cid:0) ξ ( r ) , ξ ( r ) , , ξ ( r ) (cid:1) , (93)for which, the consideration of the Lie transportation condition (i.e. L u ξ = ), provides ξ ( r ) = 2 / λ E − rV q ( r )4 − λ + Q λ − E − V q ( r ) r (cid:16) − λ + Q λ r (cid:17) , (94) ξ ( r ) = 2 / (cid:112) ( E − V )( E − V − ) , (95) ξ ( r ) = − / L (cid:18) − r (cid:19) . (96)The above vector field results in a non-zero rate of change of ξ · u , indicating that the Jacobi field ξ is nowhereorthogonal to the congruence. For two vectors x and y , we notate x · y = g µν x µ y ν . rE plot of || || li n e o f c on ti nuou s v a l u e s rE plot of Θ FIG. 7. The behaviors of || A || and Θ for 0 . < E < .
9, considering Q = 1, q = 0 . λ = 10 and L = 1. The corresponding eventand cosmological horizons are located respectively at 0 . .
98. The contours indicate discrete values for the parametersfor specific ranges of r and E . In particular, the parameter Θ, beside discrete ones, can have very close values that reside on aline tangent to the contours. The four-acceleration of the infalling charged particles in electromagnetic fields, obey the following relation [1]: a µ = − qm g µν F να u α , (97)which is given in terms of the field strength tensor F µν = A ν ; µ − A µ ; ν . (98)Accordingly, the congruence deviation equation (91) can be recast as (see also Refs. [58, 59]) A µ = − R µναβ u ν ξ α u β − qm g µα (cid:0) F αβ ; ν u β ξ ν + F αβ u β ; ν ξ ν (cid:1) . (99)Since, this acceleration is related to the internal interaction of the world-lines, it naturally affects the expansion of thecongruence. This expansion is defined as the fractional rate of change of the transverse subspace of the congruence,and in our case is defined as [61] Θ = u µ ; µ . (100)Accordingly, we can compare the behavior of A with that of the congruence expansion as the particle world-linesapproach the black hole. For this, we consider the norm of the aforementioned vector field, i.e. || A || , and plot it for adefinite range of E , inside the causal region. Same is done for the congruence expansion (see Fig. 7). As it is seen in thefigures, the approaching congruence is of positive expansion, so that its transverse cross-section increases in area andthe world-lines recede from each other. This is in agreement with the positive acceleration between the world-lines, asit is shown in the diagram of || A || . As the particles approach the event horizon, the congruence’s internal accelerationmerges to a single value at a specific E , which indicates that only distinct particles can reach that region and therethey will maintain a constant mutual force. In other regions, distant from the event horizon, the particle deflection(and scattering) can happen under positive congruence expansion and positive internal acceleration. According to the The norm of a vector X is defined as || X || . = √ X · X . E , the internal interactions between the world-lines remain repulsive at all distances,however, this repulsion is smaller at regions near the event horizon. This is while the congruence expansion reachesits maximum values for the same initial conditions. This is therefore a signature of scattering, where the expansionof the scattered congruence is a result of interactions with the source. On the other hand, for higher E , the relativeacceleration and the congruence expansion take their maximum values near the black hole. The expansion in thiscase is naturally a result of internal interactions between the world-lines. We can therefore infer that the dynamicalcharacteristics of a bundle of infalling world-lines on the CWBH, can indicate the effect of such interactions on theway the particles approach and recede the source, through their specific type of orbit. VII. FINAL REMARKS
The gravitational effects of the electrical charge constituents of charged black holes are apparent in the motion ofneutral particles in their vicinity. Charged test particles on the other hand, are also affected by an additional coulombpotential and this makes it more complicated to analyze their motion. Moving in the exterior geometry of suchblack holes, test particles feel this electric charge through a scalar potential, which is an external classical parameter.Despite this, the way this parameter is distributed inside the source, can change the total charge in the effectivepotential and therefore, can affect the particle trajectories. In general relativity, this can be inferred by applying theEinstein-Maxwell equations to obtain interior solutions of a relativistic star. In 1917, one year before proposing histheory of gravity, Weyl discovered a relationship between the metric and electrical potentials and found a class ofinterior axially symmetric solutions of a static source which had a quadratic dependence on the coulomb potential [64].This was later proved to be the case even in the absence of axial symmetry and could stabilize the source by balancingthe gravitational pull and the coulomb repulsion [65, 66]. These Weyl-type interior solutions, were then classifiedregarding the total integral of the interior charge density [67, 68], with extensions to higher dimensions [69, 70]. Thetotal charge parameter of the black hole, is therefore proved to be a consequence of the type of the interior solution itobeys and this can be distinguishable regarding the particle trajectories. For example, in Ref. [11], it has been shownthat switching between the Weyl-type interior solutions for a RN black hole, can change the intensity and the shapeof possible orbits of approaching charged test particles.The exterior geometry of the static charged black hole proposed by Mannheim and Kazanas in Ref. [42] and thatin the current study, are derived from Weyl-Maxwell equations. However, besides a few Schwarzschild-like interiorsolutions [71, 72], there is not yet a study specified to the interior structure of charged relativistic stars in Weylconformal gravity. Regarding the general interest of this paper, which was the Rutherford scattering of chargedparticles, such possible interior solutions could provide the chance of classifying the bending angle and the range ofinitial conditions to obtain a particular shape of scattering. As shown earlier in this paper, despite the similaritybetween the electric charge of the CWBH and that of the test particles, the scattering at some distances can beconvex and attractive, which indicates the unbalance between the gravitational and coulomb potentials felt by thetest particles. If the source is endowed with a particular charge density function and definite interior solutions, thenthe scattering can be categorized in accordance with the total charge integral for each of the solutions. So, an outlookfor future studies can be looking for obtaining charged interior solutions to Weyl conformal gravity and matchingthem with the exterior geometry of the respected charged black holes. This way, beside categorizing the types ofmotion of test particles, we will also be able to investigate them around a star under gravitational collapse.
ACKNOWLEDGMENTS
M. Fathi has been supported by the Agencia Nacional de Investigaci´on y Desarrollo (ANID) through DOCTORADOGrant No. 2019-21190382. J.R.V. is partially supported by Centro de Astrof´ısica de Valpara´ıso (CAV).
Appendix A: The method of finding L U in Eq. (38) Equation. (34) allows for obtaining an expression for L U , by solving a L U − b L U + c = 0 , (A1)9in which a = ( Q − r U ) r U , (A2a) b = 2 Q (1 + q ) r U − Q (2 + q )2 r U − r U − Q (1 − q ) λ , (A2b) c = Q (1 + 2 q )4 r U − q Q + 4 r U λ − Q r U (1 − q ) λ . (A2c)Solving Eq. (A1) for L U , then yields the value in Eq. (38). Appendix B: Finding the angular equation of motion
Since the closest approach happens at r S , to deal with the integral in Eq. (46), we define the following non-linearchange of variable: u . = 1 rr S − , (B1)which reduces Eq. (46) to φ ( r ) = κ (cid:34)(cid:90) ∞ u d u (cid:112) P ( u ) − u (cid:90) ∞ u d u ( u + u ) (cid:112) P ( u ) (cid:35) , (B2)where u j . = r j /r S ) − , with j = { , , , } , and P ( u ) ≡ u + a u + b u + c , (B3)with a = u + u + u , (B4a) b = u ( u + u ) + u u , (B4b) c = u u u . (B4c)Defining κ = υr S u √ u u u , (B5)and applying another change of variable U . = 14 (cid:16) u + a (cid:17) , (B6)we can rewrite Eq. (B2) as φ ( r ) = κ (cid:34)(cid:90) ∞ U d U (cid:112) P ( U ) − u (cid:90) ∞ U d U ( U + U ) (cid:112) P ( U ) (cid:35) , (B7)given that U = (cid:0) u + a (cid:1) , and P ( u ) ≡ U − g U − g . (B8)Direct integration of Eq. (B7), results in the expression in Eq. (48).0 Appendix C: Solving depressed quartic equations
The condition V (cid:48) r ( r ) = 0, provides the following equation of eighth degree: r + ˜ ar + ˜ br + ˜ c = 0 . (C1)To solve this equation, we firstly make the change of variable r . = x . Afterwards, we combine the methods of Ferrariand Cardano to solve a depressed quartic equation of the form (originally studied by Cardano in Ref. [73]) x + ˜ ax + ˜ bx + ˜ c = 0 , (˜ a, ˜ b, ˜ c ) ∈ R . (C2)This equation can be rewritten as the product of two quadratic equations, as follows: x + ˜ ax + ˜ bx + ˜ c = ( x − αx + ˜ β )( x + 2˜ αx + ˜ γ ) = 0 . (C3)Accordingly, we obtain ˜ a = ˜ β + ˜ γ − α , (C4a)˜ b = 2˜ α ( ˜ β − ˜ γ ) , (C4b)˜ c = ˜ β ˜ γ. (C4c)Solving the first two equations for ˜ β and ˜ γ , yields˜ β = 2˜ α + ˜ a b α , (C5a)˜ γ = 2˜ α + ˜ a − ˜ b α , (C5b)which together with Eq. (C4c), results in an equation of sixth degree in ˜ α :˜ α + ˜ a α + (cid:18) ˜ a − ˜ c (cid:19) ˜ α − ˜ β
64 = 0 . (C6)Applying the change of variable ˜ α = ˜ U − ˜ a , (C7)we obtain the depressed cubic equation ˜ U − ˜ η ˜ U − ˜ η = 0 , (C8)where ˜ η = ˜ a
48 + ˜ c , (C9a)˜ η = ˜ a
864 + ˜ b − ˜ a ˜ c . (C9b)The real solution to this cubic equation is obtained as [74, 75]˜ U = 2 (cid:114) ˜ η (cid:32)
13 cosh − (cid:32)
32 ˜ η (cid:115) η (cid:33)(cid:33) . (C10)Therefore, the roots of Eq. (C2) are x = ˜ α + (cid:113) ˜ α − ˜ β, (C11a) x = ˜ α − (cid:113) ˜ α − ˜ β, (C11b) x = − ˜ α + (cid:112) ˜ α − ˜ γ, (C11c) x = − ˜ α − (cid:112) ˜ α − ˜ γ. (C11d)1 Appendix D: Solving the equation of motion for frontal scattering
Equation (66) can be recast as (cid:18) d r d τ (cid:19) = m ( r − r s ) p ( r ) λ r , (D1)in which p ( r ) ≡ r + r s r + ( r s + ¯ a ) r + r s + r s ¯ a + ¯ b. (D2)Considering r s as the initial position, we can rewrite Eq. (D1) as τ ( r ) = λm (cid:90) rr s r d r (cid:112) ( r − r s ) p ( r ) , (D3)which by the linear change of variable z . = rr s − , (D4)reduces to τ ( z ) = λm (cid:90) z ( z + 1)d z (cid:112) z ˜ p ( z ) , (D5)where ˜ p ( z ) ≡ z + 4 z + γ z + γ , (D6)and γ = 6 + ¯ ar s , (D7a) γ = 4 + 2¯ ar s + ¯ br s . (D7b)Now, letting u . = 1 z , (D8)yields the following reduced integral form of Eq. (D5): τ ( u ) = − λm √ γ (cid:32)(cid:90) u ∞ d u (cid:112) ¯ p ( u ) + (cid:90) u ∞ d uu (cid:112) ¯ p ( u ) (cid:33) , (D9)in which ¯ p ( u ) ≡ u + γ γ u + 4 γ u + 1 γ . (D10)Applying the last change of variable u . = 4U − γ γ , (D11)we get τ (U) = − λm √ γ (cid:32)(cid:90) U ∞ dU (cid:112) ¯ P (U) + 14 (cid:90) U ∞ dU(U − γ γ ) (cid:112) ¯ P (U) (cid:33) , (D12)2where we have defined ¯ P (U) ≡ − ¯ g U − ¯ g . (D13)The direct integration of the elliptic integral in Eq. (D12), now results in the expression in Eq. (72). [1] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Princeton University Press, 2017).[2] J. A. H. Futterman, F. A. Handler, and R. A. Matzner,
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