McGehee regularization of general SO(3)-invariant potentials and applications to stationary and spherically symmetric spacetimes
MMcGehee regularization of general SO (3)-invariant potentials andapplications to stationary and spherically symmetric spacetimes Pablo Galindo
Dept. de Geometr´ıa y Topolog´ıaUniversidad de Granada,Campus de Fuentenueva s/n,18071 Granada, [email protected]
Marc Mars
Inst. de F´ısica Fundamental y Matem´aticas(IUFFyM),Universidad de Salamanca,Plaza de la Merced s/n 37008 Salamanca, [email protected]
Abstract
The McGehee regularization is a method to study the singularity at the origin of thedynamical system describing a point particle in a plane moving under the action of a power-law potential. It was used by Belbruno and Pretorius [Belbruno and Pretorius, 2011] toperform a dynamical system regularization of the singularity at the center of the motion ofmassless test particles in the Schwarzschild spacetime. In this paper, we generalize the McGeheetransformation so that we can regularize the singularity at the origin of the dynamical systemdescribing the motion of causal geodesics (timelike or null) in any stationary and sphericallysymmetric spacetime of Kerr-Schild form. We first show that the geodesics for both massiveand massless particles can be described globally in the Kerr-Schild spacetime as the motion of aNewtonian point particle in a suitable radial potential and study the conditions under whichthe central singularity can be regularized using an extension of the McGehee method. As anexample, we apply these results to causal geodesics in the Schwarzschild and Reissner-Nordstr¨omspacetimes. Interestingly, the geodesic trajectories in the whole maximal extension of bothspacetimes can be described by a single two-dimensional phase space with non-trivial topology.This topology arises from the presence of excluded regions in the phase space determined by thecondition that the tangent vector of the geodesic be causal and future directed.
Kerr-Schild metrics [Kerr and Schild, 1965] are a well-known Ansatz to solve the Einstein fieldequations and leads to many physically important exact solutions of the four-dimensional case,such as the Schwarzschild black hole, the Reissner-Nordstr¨om, the Kerr black hole, the chargedKerr–Newman black hole, the Vaidya radiating star, Kinnersley photon rocket, pp-waves and alsosome of their higher dimensional analogues [M´alek, 2012]. Kerr-Schild metrics have played a crucialrole in the discovery of rotating black holes in higher dimensions [Myers and Perry, 1986, Gibbonset al., 2005] as well as in the recent work on so-called higher order gravities [Anabal´on et al.,2009, Anabal´on et al., 2011]. Also, most static and spherically symmetric spacetimes can bedisplayed in Kerr-Schild form and their analysis conforms a field that continues giving interestingresults nowadays [Parry, 2012, Hackmann et al., 2008]. Two of the best known static and spherically a r X i v : . [ g r- q c ] M a y ymmetric metrics which have been studied extensively and remain an area of current researchare the Schwarzschild metric and the Reissner-Nordstr¨om metric. Although the behavior of thegeodesics in both metrics is well-known, the geodesic equations have a large number of dynamicproperties that are still providing new results, such as the characterization of the circular motion inthe Reissner-Nordstr¨om spacetime for neutral and charged particles [Pugliese et al., 2011a, Puglieseet al., 2011b] or the dynamics of the chaotic motion in the Schwarzschild black hole surrounded byan external halo [de Moura and Letelier, 2000]. The dynamical system approach to the analysis ofthe geodesic flow in these spacetimes and their rotating Kerr generalizations is a novel approachwhich, besides providing many new and interesting results, also describes known results from adifferent perspective. Examples are the homoclinic orbits that asymptotically approach the unstablebranch of circular orbits [Levin and Perez-Giz, 2008, Levin and Perez-Giz, 2009, Perez-Giz andLevin, 2009, Misra and Levin, 2010] or the fact that perturbation of the geodesic flow possesses achaotic invariant set [Moeckel, 1992, Levin, 2000, Suzuki and Maeda, 1999]. One of the advantagesof this method is that a great amount of information can be obtained without integrating thegeodesic equations. Also, by the use of “blow-up” techniques in the dynamical system we candescribe the behavior of the geodesic equations near the singularity, of which little is known. Oneof the most recent works along this line [Belbruno and Pretorius, 2011] has analyzed the null caseof the geodesic flow by the use of the McGehee regularization [McGehee, 1981], which is a methoddesigned to deal with the singularities at the center in the motion of Newtonian particles subjectto a central power-law potential. In the context of geodesics in Schwarzschild, the limitation ofthe MacGehee method is the restriction to power-law potentials, which prevents its application totimelike geodesics. This is one of the reasons why null geodesics only where treated in [Belbrunoand Pretorius, 2011]. Also, the standard McGehee method involves a somewhat complicated phasespace which obscures the analysis. Other approaches to this problem [Stoica and Mioc, 1997]have studied timelike geodesic in Schwarzschild by the use of a variation of the McGehee method.However, the approach is such that one deals with a one-parameter family of energy-dependentdynamical systems in which only one curve in each phase space is relevant. This obviously obscuresand complicates unnecessarily the results (in fact, this drawback was not explicitly noticed in[Stoica and Mioc, 1997]).In this paper we generalize the McGehee regularization so that we can deal with centralpotentials of a very general form. With this method we can treat not only general causal geodesicsin Schwarzschild but also geodesics in the Reissner-Nordstr¨om spacetime. In fact, for a substantialfraction of the paper we work in full generality in stationary and spherically symmetric spacetimesof Kerr-Schild form, of which the previous are just particular cases. There are several possibleapproaches to derive the geodesics equations in such spacetimes. Explicit computation of theChristoffel symbols is tedious and not particularly enlightening. It is particularly cumbersometo incorporate the conserved quantities associated to Killing vectors into the equations. Morestraightforward and convenient is the use of Hamiltonian methods which, in particular, allows forthe incorporation of conserved quantities into the system in a straightforward way. Once we have thegeodesic equations for such spacetimes, we can apply the generalized McGehee regularization andsubsequently analyze the phase space defined by the geodesic equations, with particular emphasisat the vicinity of the singularity, where new and interesting dynamics appears. A remarkablefact is that the dynamics in the entire maximal extension of the spacetime can be described in a2ingle two-dimensional phase space, which has particular importance in the Reissner-Nordstromcase. The key for this lies in the presence of excluded regions in the phase space arising from thecondition that the trajectories correspond to future directed causal geodesics.The paper is organized as follows: In section 2 we follow a simple way to obtain the geodesicequations for a general stationary Kerr-Schild metric and obtain a simplified Hamiltonian withthe Killing conserved quantity already incorporated. In section 3 we particularize to the caseof stationary and spherically symmetric Kerr-Schild spacetimes. In particular, we find that thegeodesics can be described by a classical Hamiltonian of the form H = T + V with V a sphericallysymmetric potential. It is not at all clear a priori that this should be possible in the whole Kerr-Schild domain. The Hamilton equations already incorporate all the constants of motion associatedto the symmetries. We analyze under which conditions a Hamiltonian trajectory corresponds to acausal, future directed geodesic of the spacetime. These conditions will be translated into excludedregions in the corresponding phase spaces. In Section 4 we generalize the McGehee transformationto radial potentials of very general form and provide a method to choose the appropriate parameterto perform the regularization. This discussion also helps clarifying the original regularizationprocedure proposed by McGehee. The physical meaning of the generalized McGehee variablesis also discussed. In Section 5 we particularize the previous general results to the Schwarzschildspacetime paying particular attention to the collision manifold and to the excluded region forfuture-oriented geodesics. We recover the known results on null geodesics near the singularityobtained in [Belbruno and Pretorius, 2011] and extend them to timelike geodesics (in fact, allcausal geodesics are treated simultaneously). As already mentioned, the understanding of theexcluded regions is crucial to have a phase space of physical trajectories with a non-trivial topologycapable of dealing with all Kerr-Schild patches of the Kruskal spacetime. Finally, in Section 6 weperform a similar analysis for the maximal extension of the Reissner-Nordstr¨om spacetime. Throughout this paper, we will consider spacetimes {M = R × ( R \ C ) , g } where C ⊂ R is aclosed subset such that M is connected and g is a Lorentzian metric of Kerr-Schild form [Kerr andSchild, 1965]. More specifically, let { x α } = { T, x i } ( α, β, · · · = 0 , , , i, j, · · · = 1 , ,
3) beCartesian coordinates on R × R and endow M with the Minkowski metric η = − dT + δ ij dx i dx j .Let K be a smooth one-form on M which is null with respect to the metric η and h : M −→ R asmooth function. The metric g being of Kerr-Schild form means that it takes the form g αβ = η αβ + hK α K β . (1)It is well-known (and immediate to check) that the inverse metric g − is( g − ) αβ = η αβ − hK α K β , where all Greek indices are raised and lowered with the Minkowski metric η . This expressionshows, in particular, that the one-form K is also null in the metric g . We will assume from nowon that neither K nor h vanish on a non-empty open set on M .3ur aim in this section is to study the geodesic equations for a Kerr-Schild metric assumingthe spacetime to be stationary with Killing vector ξ = ∂ T . It is clear from (1) that ξ is a Killingvector of g if and only( £ ξ h ) K ⊗ K + h ( £ ξ K ) ⊗ K + + h K ⊗ ( £ ξ K ) = 0 . (2)where £ denotes Lie derivative. At any point p ∈ M where K | p (cid:54) = 0, let V p ∈ T (cid:63)p M be a vectorsubspace such that T (cid:63)p M = (cid:104) K | p (cid:105) ⊕ V p and use this direct sum to decompose £ ξ K | p = C | p K + U | p .Inserting this into (2) yields( £ ξ h + 2 Ch ) K ⊗ K + h ( U ⊗ K + K ⊗ U ) | p = 0which is equivalent to h U | p = 0 and ( £ ξ h + 2 Ch ) | p = 0. Using the fact that neither h nor K vanishon a non-trivial open set, it follows that ξ = ∂ T is a Killing vector of g if and only if there exists asmooth function C : M −→ R such that £ ξ K = C K and £ ξ h = − Ch . Let f : R \ C −→ R beany smooth positive function and let f : R × ( R \ C ) −→ R be the unique solution of ∂ T f + Cf = 0with initial data f | T =0 = f . It is immediate to check that f > h (cid:48) := hf and K (cid:48) := f K , they satisfy £ ξ h (cid:48) = 0 , £ ξ K (cid:48) = 0 , while the metric g takes the form g αβ = η αβ + h (cid:48) K (cid:48) α K (cid:48) β . (3)Dropping the primes, it follows that ξ is a Killing vector for g if and only h and K can be selectedto be Lie constant along ξ . We assume this from now on.In the Minkowskian coordinates { x α } let us write K α = ( (cid:98) K, (cid:126)K ) where (cid:98) K satisfies (cid:98) K = (cid:126)K := K i K i and Latin indices are raised and lowered with the Euclidean metric δ ij . The Killing vector ξ is timelike on the set { p ∈ M ; h (cid:126)K | p < } , null on the set { p ∈ M ; h (cid:126)K | p = 1 } and spacelike onthe set { p ∈ M ; h (cid:126)K | p > } . Note also that we are not assuming K to be future directed or pastdirected everywhere, so that a priori (cid:98) K may change sign.In any spacetime ( M , g ), affinely parametrized geodesics are the solutions of the Hamiltonequations of the Hamiltonian H = 12 ( g − ) αβ p α p β (4)defined on the cotangent bundle of M . The Hamilton equations fix p = g ( u, · ) where u is thetangent vector to the geodesic. Using the explicit expression (1) for the metric, this Hamiltoniantakes the form H = 12 (cid:16) η αβ p α p β − h ( K α p α ) (cid:17) . (5)Given that ξ is a Killing vector, the quantity E := − p ( ξ ) is conserved along geodesics. Note alsothat, with this definition, K α p α = − E (cid:98) K + (cid:126)K · (cid:126)p, (6)4here we have written p = { (cid:98) p, (cid:126)p } and dot means scalar product with δ ij .The Hamiltonian itself is a conserved quantity with the value of H = − µ where µ = 0 , ± µ = 1), spacelike ( µ = −
1) or null ( µ = 0). Inserting(6) and the conserved quantity E into (5) the following Hamiltonian arises naturally H (cid:48) := H + 12 E = 12 (cid:18) (cid:126)p − h (cid:16) (cid:126)K · (cid:126)p − E (cid:98) K (cid:17) (cid:19) , (7)which is now defined on the cotangent bundle of R \ C .The interest of this Hamiltonian lies in the fact (easy to check) that if a curve ( T ( s ) , (cid:126)x ( s ) } is a geodesic in ( M , g ) with tangent vector u satisfying g ( u, u ) = − µ and conserved quantity p ( u ) = − E , then { (cid:126)x ( s ) } is the projection to the base space R \ C of a solution of the Hamiltonequations of (7) satisfying H (cid:48) = (cid:15) := 12 (cid:0) E − µ (cid:1) (8)along the curve and T ( s ) satisfies the ODE (cid:16) − h (cid:126)K (cid:17) dTds + h (cid:98) K (cid:126)K · d(cid:126)xds = E, (9)which is simply the explicit form for g ( u, ξ ) = − E in the Cartesian coordinates { T, (cid:126)x } . Theconverse to this statement will be addressed in the following section in the spherically symmetriccase. In this section we want to particularize the problem to the spherically symmetric setting. So, weassume the group of rotations SO (3) acting on M as SO (3) × M −→ M , ( R, ( T, (cid:126)x )) −→ ( T, R ( (cid:126)x ))to be an isometry of g . Note that, for this definition to make sense, the set C must be invariant underthe SO(3) action, which we assume from now on. The isometry condition requires £ (cid:126)ζ ( h K ⊗ K ) = 0,for any generator ζ of the group SO (3). Pulling back this relation to the orbits of the isometrygroup and using the fact that the only symmetric 2-covariant tensor on the sphere which is invariantunder SO (3) is a constant times the standard metric on the sphere, it follows that h K ⊗ K pullsback to zero on the SO (3) orbits. Since h does not vanish on open sets, we conclude that K itselfpulls back to zero on these surfaces. Given the stationarity condition, this is equivalent to theexistence of a smooth function f : R \ C −→ R such that (cid:126)K = f (cid:126)x | (cid:126)x | where | (cid:126)x | := √ (cid:126)x · (cid:126)x . Hence, K α = (cid:16) ˆ K, (cid:126)K (cid:17) = (cid:18) ˆ K, f (cid:126)x | (cid:126)x | (cid:19) , with ˆ K = f . (10)The following lemma gives the most general form of g under a mild additional restriction.5 emma 1. Assume that h is non-zero on a dense set, that the null vector K α does not have anyflat zero (i.e. a point where K α and all its derivatives vanish) and that ( M , g ) is stationary andspherically symmetric. Then h and K α can be chosen so that h ( (cid:126)x ) is spherically symmetric and K α = (cid:18) σ, (cid:126)x | (cid:126)x | (cid:19) , where σ = ± is a constant on M . Proof.
The condition that f has no flat zeros implies that f (and hence K ) cannot vanish ona non-empty open set. So, as discussed in Section 2, we can assume £ ξ h = 0 and £ ξ K = 0where ξ = ∂ T , and that (10) holds. Let S I be the collection of arc-connected components of { f (cid:54) = 0 } ⊂ R \ C . On each one of these open sets we have ˆ K = σ I f , where σ I = ± S I . Let S + be the union of components S I with σ I = +1 and S − be the union of components S I with σ I = − S + ∪ S − is dense in R \ C and thelatter is connected it follows that there exists a point p ∈ R \ C that can can be approached by asequence { p + i ∈ S + } and by a sequence { p − i ∈ S − } . Since ˆ K is smooth everywhere, in particular at p , it follows that necessarily f and all its derivatives vanish at p , against assumption. Thus, either S − = ∅ (and we can write ˆ K = f everywhere) or S + = ∅ (and we can write ˆ K = − f everywhere).Consequently, the Kerr-Schild metric takes the form g = η + hf K (cid:48) ⊗ K (cid:48) with K (cid:48) α = ( σ, (cid:126)x | (cid:126)x | ).Defining h (cid:48) = hf , and given the spherically symmetric invariance of K (cid:48) α , it follows immediatelythat g is spherically symmetric if and only if h (cid:48) is spherically symmetric. Dropping the primes in K (cid:48) α and h (cid:48) the lemma follows. (cid:4) Remark 2 . As a consequence of this lemma, the Hamiltonian H (cid:48) in eq. (7) takes the form H (cid:48) = 12 (cid:126)p − h (cid:18) (cid:126)x · (cid:126)p | (cid:126)x | − σE (cid:19) . (11)An important question is to what extent the field equations of this Hamiltonian reproduce theinformation concerning the geodesics of g . Note first that the Hamilton equation ˙ (cid:126)x = ∂H (cid:48) ∂(cid:126)p readsexplicitly ˙ (cid:126)x = (cid:126)p − h ( (cid:126)x ) (cid:18) (cid:126)x · (cid:126)p | x | − σE (cid:19) (cid:126)x | (cid:126)x | (12)which can be written in matrix form as (we denote by (cid:126)x T the transpose of the vector column (cid:126)x and by Id the identity matrix) ˙ (cid:126)x = (cid:18) Id − h (cid:126)x(cid:126)x T | (cid:126)x | (cid:19) (cid:126)p + σEh (cid:126)x | (cid:126)x | . (13)Now, the relationship between the four-velocity u α of a geodesic and the corresponding four-momentum p α = g αβ u β is obviously invertible. For a geodesic { T ( s ) , (cid:126)x ( s ) } , the four-velocity is u = ˙ T ( s ) ∂ T + ˙ (cid:126)x ( s ) ∂ (cid:126)x . Lowering indices and using K α = − σdT + (cid:126)x | (cid:126)x | d(cid:126)x it follows p = (cid:32) ( h −
1) ˙ T − hσ (cid:126)x · ˙ (cid:126)x | (cid:126)x | (cid:33) dT + (cid:32) ˙ (cid:126)x + h (cid:32) (cid:126)x · ˙ (cid:126)x | (cid:126)x | − σ ˙ T (cid:33) (cid:126)x | (cid:126)x | (cid:33) d(cid:126)x. E = − p ( ∂ T ), we also have p = − EdT + (cid:126)p · d(cid:126)x or, equivalently, E = (1 − h ) ˙ T + hσ (cid:126)x · ˙ (cid:126)x | (cid:126)x | , (14) (cid:126)p = ˙ (cid:126)x + h (cid:32) (cid:126)x · ˙ (cid:126)x | (cid:126)x | − σ ˙ T (cid:33) (cid:126)x | (cid:126)x | . (15)The first equation is exactly equation (9) for the case under consideration and must be added to theHamiltonian system (11) in order to describe the geodesics. Concerning the second equation, itsrelationship to equation (12) is as follows. First of all, it is immediate to check that any trajectorysatisfying (14)-(15) also satisfies the pair of equations (14)-(12). To analyze the converse, observethat the matrix in parenthesis in (12) is invertible for all h (cid:54) = 1. So, given (cid:126)x ( s ), this equationcan be solved uniquely to obtain (cid:126)p ( s ) and hence, assuming that (14) holds, this solution mustbe necessarily (15). This shows the equivalence between (14)-(15) and (14)-(12) at points where h (cid:54) = 1. However, at points where h = 1 (corresponding to the set where the Killing vector ∂ T isnull) the matrix in parenthesis is the projector orthogonal to (cid:126)x and hence not invertible. Thus, thecomponent of (cid:126)p parallel to (cid:126)x is not determined by (13). It follows that, at points where h = 1, theset of Hamilton equations of H (cid:48) and the ODE (14) must be supplemented by the component of (cid:126)p in (15) parallel to (cid:126)x which is, for any value of h , (cid:126)x · (cid:126)p = (1 + h )( (cid:126)x · ˙ (cid:126)x ) − σh | (cid:126)x | ˙ T . (16)Note finally that, at points where h = 1 the dependence of ˙ T drops completely from (14). Givena solution { (cid:126)x ( s ) , (cid:126)p ( s ) } of the Hamiltonian equations of H (cid:48) , it is precisely (16) that allows one tosolve for ˙ T ( s ) at points satisfying h = 1, and hence must be added to the system.The next lemma shows that the trajectories of the Hamiltonian (11) can be also obtained bysolving a much simpler Hamiltonian. Lemma 3.
Let Ω ⊂ R be a domain and { (cid:126)x } Cartesian coordinates on Ω . Consider the phasespace F := Ω × R with global canonical coordinates { (cid:126)x, (cid:126)p } and let π : Ω × R −→ Ω be theprojection. Define on F the two Hamiltonians H (cid:48) = 12 (cid:126)p − h ( (cid:126)x )2 (cid:18) (cid:126)x · (cid:126)p | (cid:126)x | − σE (cid:19) , E ∈ R ˆ H = 12 ˆ p − h ( (cid:126)x )2 (cid:18) L | (cid:126)x | + µ (cid:19) , L, µ ∈ R (17) where h : Ω → R is rotationally symmetric and σ = ± . Denote by γ ( (cid:126)x , (cid:126)p )( s ) (resp. ˆ γ (ˆ x , ˆ p )( s ) )the H -trajectory (resp. H (cid:48) -trajectory) passing at s = 0 through the point ( (cid:126)x , (cid:126)p ) (resp. (ˆ x , ˆ p ) ).Assume that in some neighborhood of (cid:126)x , h ( (cid:126)x ) is not of the form h ( (cid:126)x ) = α | (cid:126)x | (cid:0) β + γ | (cid:126)x | ) (cid:1) − with α, β, γ ∈ R . Then, the two projection curves π ( γ ( (cid:126)x , (cid:126)p )( s )) and π (ˆ γ (ˆ x , ˆ p )( s )) are the same ifand only if ˆ x = (cid:126)x , ˆ p = (cid:126)p − h ( (cid:126)x ) (cid:18) (cid:126)x · (cid:126)p | (cid:126)x | − σE (cid:19) (cid:126)x | (cid:126)x | , | (cid:126)x × (cid:126)p | = | L | , H (cid:48) ( (cid:126)x , (cid:126)p ) = 12 (cid:0) E − µ (cid:1) . roof. First of all, we note that a curve (cid:126)x ( s ) in R satisfying (cid:126)x ( s ) × ˙ (cid:126)x ( s ) = (cid:126)J (18)˙ (cid:126)x ( s ) V ( | (cid:126)x ( s ) | ) = (cid:15), (19)where (cid:126)J and (cid:15) are constants, is uniquely determined by the initial data (cid:126)x (0) and ˙ (cid:126)x (0). This is aknown result of central forces in R .Let { (cid:126)x ( s ) , (cid:126)p ( s ) } = γ ( (cid:126)x , (cid:126)p )( s ) and { ˆ x ( s ) , ˆ p ( s ) } = ˆ γ (ˆ x , ˆ p )( s ). Both Hamiltonians H (cid:48) and ˆ H are spherically symmetric and time independent, so there exist constants (cid:126)J , ˆ J , H (cid:48) and ˆ H suchthat (cid:126)x ( s ) × (cid:126)p ( s ) = (cid:126)J , H (cid:48) ( (cid:126)x ( s ) , (cid:126)p ( s )) = H (cid:48) , ˆ x ( s ) × ˆ p ( s ) = ˆ J , ˆ H (ˆ x ( s ) , ˆ p ( s )) = ˆ H . The respective Hamilton equations imply˙ˆ x ( s ) = ˆ p ( s ) , (20)˙ (cid:126)x ( s ) = (cid:126)p ( s ) − h ( (cid:126)x ) (cid:18) (cid:126)x · (cid:126)p | x | − σE (cid:19) (cid:126)x | (cid:126)x | (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)x = (cid:126)x ( s ) ,(cid:126)p = (cid:126)p ( s ) (21)and hence ˆ x ( s ) × ˙ˆ x ( s ) = ˆ J , (cid:126)x ( s ) × ˙ (cid:126)x ( s ) = (cid:126)J . We next write down explicitly H (cid:48) ( (cid:126)x ( s ) , (cid:126)p ( s )) − H (cid:48) = 0. For any vector (cid:126)a , we can compute itssquare norm as (cid:126)a = ( (cid:126)x × (cid:126)a ) + ( (cid:126)x · (cid:126)a ) | (cid:126)x | . (22)From (21) we have (cid:126)x ( s ) · ˙ (cid:126)x ( s ) = (cid:16) ( (cid:126)x · (cid:126)p )(1 − h ) + σh | (cid:126)x | E (cid:17)(cid:12)(cid:12)(cid:12) (cid:126)x = (cid:126)x ( s ) ,(cid:126)p = (cid:126)p ( s ) . (23)Decomposing (cid:126)p ( s ) and ˙ (cid:126)x ( s ) according to (22) and inserting (23), a straightforward calculationtransforms (1 − h ) ( H (cid:48) − H (cid:48) ( (cid:126)x ( s ) , (cid:126)p ( s )) ) = 0 into H (cid:48) = 12 ˙ (cid:126)x ( s ) − h ( (cid:126)x ( s ))2 (cid:32) (cid:126)J | (cid:126)x ( s ) | + E − H (cid:48) (cid:33) := ˙ (cid:126)x ( s ) V ( | (cid:126)x ( s ) | ) . (24)where the second equality defines V ( | (cid:126)x | ). For the trajectory ˆ x ( s ), the form of the Hamiltonian ˆ H immediately impliesˆ H = 12 ˙ˆ x ( s ) − h (ˆ x ( s ))2 (cid:32) (cid:126)L | ˆ x ( s ) | + µ (cid:33) := ˙ˆ x ( s ) V ( | ˆ x ( s ) | ) , (25)8here the second equality defines ˆ V ( | ˆ x | ). Comparing (24) and (25) we conclude that the twotrajectories (cid:126)x ( s ) and ˆ x ( s ) agree if and only if they have initial position, initial velocity and therespective potential functions V ( | (cid:126)x | ) and ˆ V ( | (cid:126)x | ) agree up to an additive constant c . The condition V ( | (cid:126)x | ) − ˆ V ( | (cid:126)x | ) − c = 0 reads explicitly h ( (cid:126)x ) (cid:16) (cid:126)J − L + (cid:0) E − H (cid:48) − µ (cid:1) | (cid:126)x | (cid:17) = − c | (cid:126)x | . Since by hypothesis h ( (cid:126)x ) is not of the form h ( (cid:126)x ) = α | (cid:126)x | (cid:0) β + γ | (cid:126)x | (cid:1) − in any neighborhood of (cid:126)x , this equation has as only solution c = 0, (cid:126)J = L and H (cid:48) = ( E − µ ). We conclude that thetrajectories (cid:126)x ( s ) and ˆ x ( s ) agree if and only if (cid:126)x = ˆ x , ˙ (cid:126)x | s =0 = ˙ˆ x | s =0 , | (cid:126)J | = | L | and H (cid:48) = ( E − µ ).Given the relation (21) between velocity and momentum, the lemma follows. (cid:4) Remark 4 . It is interesting that the Hamiltonian ˆ H is independent of σ , so that we will be ableto describe the geodesics in ( M , g ) both for the case when K is future directed (plus sign) or pastdirected (negative sign). Moreover, the Hamiltonian ˆ H is a standard Hamiltonian in Newtonianmechanics for a point particle in a central potential. This a substantial simplification over theoriginal problem of solving the geodesic equations in a stationary and spherically symmetricspacetime of Kerr-Schild form, because we can exploit all the information known for trajectories ofpoint particles in Newtonian mechanics under the influence of a radial potential of the form V ( | (cid:126)x | ) = − h ( (cid:126)x )2 (cid:18) L | (cid:126)x | + µ (cid:19) . (26)The main consequence of Lemma 3 is, thus, that the spatial part of all geodesics in a stationaryand spherically symmetric spacetimes of Kerr-Schild form turns out to be equivalent to the (muchsimpler) problem of solving the trajectory of a Newtonian point particle in the potential (26).Once the spatial part of the geodesics is solved, the temporal part is dealt with by solving equation(14) (at points where h (cid:54) = 1) and equation (16) (at points where h = 1). Since we are interestedin causal and future directed geodesics we need to find the restrictions on the initial data whichguarantee this. The following Proposition summarizes the results above and addresses the issue offuture directed initial data for both choices of σ . Proposition 5.
Let M = R × ( R \ C ) be connected with C ⊂ R closed. The most generalstationary and spherically symmetric metric of Kerr-Schild form g = η + h K ⊗ K such that h and K are smooth and with no flat zeros can be written in the form g = − dT + d(cid:126)x · d(cid:126)x + h ( r )( dr − σdT ) ⊗ ( dr − σdT ) (27) where σ = ± and r = √ (cid:126)x · (cid:126)x . Assume that h ( (cid:126)x ) is not of the form h ( (cid:126)x ) = α | (cid:126)x | (cid:0) β + γ | (cid:126)x | ) (cid:1) − with α, β, γ ∈ R , in any domain. Then, the g -geodesic trajectories ( T ( s ) , (cid:126)x ( s )) with normalizedtangent vector correspond exactly to the solutions of ¨ (cid:126)x = − ∂V ( r ) ∂(cid:126)x = ∂∂(cid:126)x (cid:20) h ( | (cid:126)x | )2 (cid:18) L | (cid:126)x | + µ (cid:19)(cid:21) (28) (cid:126)L = (cid:126)x × ˙ (cid:126)x (29)9 here (cid:126)L is an arbitrary constant vector, µ takes the values µ = +1 for timelike geodesics, µ = 0 for null geodesics and µ = − for spacelike geodesics and ˙ r − h ( r )) L r − h ( r ) µ (cid:0) E − µ (cid:1) := (cid:15) (30) where E is a constant. Moreover, the tangent vector u ( s ) := ( ˙ T ( s ) , ˙ (cid:126)x ( s )) satisfies E = (1 − h ) ˙ T + hσ (cid:126)x · ˙ (cid:126)x | (cid:126)x | . (31) In addition, if the time orientation of ( M , g ) is chosen so that the null vector ∂ T + σ (cid:126)x | (cid:126)x | ∂ (cid:126)x isfuture directed, then a geodesic with µ = 0 , starting at a point ( T , (cid:126)x (cid:54) = 0) is future causal if andonly if ˙ (cid:126)x satisfies (with r := | (cid:126)x | , ˙ r := (cid:126)x · ˙ (cid:126)x | (cid:126)x | and h := h ( | (cid:126)x | ) )if h > , (cid:40) σ ˙ r ∈ [ a , ∞ ) E = ± (cid:113) ˙ r − a if h < , (cid:40) σE ∈ [ a , ∞ )˙ r = ± (cid:113) E − a if h = 1 , (cid:26) σ ˙ r ∈ [0 , ∞ ) with ˙ r = 0 = ⇒ µ = L = 0 E = σ ˙ r where a ( r , L, µ ) := (cid:114) | − h ( r ) | (cid:16) L r + µ (cid:17) ≥ . Proof.
The first part of the Proposition is a consequence of Lemma 3 in combination withRemark 3. Note, in particular, that (16) (at points where h ( (cid:126)x ) = 1) must be used to reconstructthe spacetime trajectory ( T ( s ) , (cid:126)x ( s )) from the solutions of equations (28)-(29).For the statements on the initial data, let ( T , (cid:126)x (cid:54) = 0) be the initial point of the geodesic and u = ( ˙ T , ˙ (cid:126)x ) the initial velocity, normalized to satisfy g ( u , u ) = − µ ( µ = 0 ,
1) and assumed tobe future directed. The initial data ( ˙ T , ˙ (cid:126)x ) is equivalent to ( ˙ T , (cid:126)L, ˙ r ). Recall that the Kerr-Schildvector is K α = ( σ, (cid:126)xr ). The choice of time orientation means that σK α is future directed. Thus, u being future directed is equivalent to g ( u , σK | s =0 ) < u = bσK | s =0 , with b ≥
0. To compute g ( u , σK | s =0 ) observe that g ( σK, · ) = − dT + σr (cid:126)x · d(cid:126)x which implies g ( u , σK | s =0 ) = σ ˙ r − ˙ T . (32)On the other hand, the condition u = bσK | s =0 ( b ≥
0) is ( ˙ T = b, ˙ (cid:126)x = bσ | (cid:126)x | (cid:126)x ) or equivalently( ˙ T = σ ˙ r ≥ , (cid:126)L = 0). Equations (30) and (31) evaluated at s = 0 read E = ˙ r + sign(1 − h ) a , (33) E = (1 − h ) ˙ T + h σ ˙ r , (34)where sign(1 − h ) takes the values 1 , , − h < h = 1 or h > a is as defined in the statement of the Theorem. At points h (cid:54) = 1, equations1033)-(34) imply g ( u , u ) = − µ . However, when h = 1, (33) is a trivial consequence of (34) and g ( u , u ) = − µ must be imposed additionally. We compute (with h = 1) − µ = g ( u , u ) = η ( u , u ) + h ( K | s =0 ( u )) = − ˙ T + ˙ (cid:126)x + g ( σK | s =0 , u ) == 2 σ ˙ r (cid:16) σ ˙ r − ˙ T (cid:17) + L r , (35)where (32) has been used in the last equality.We can now find the most general u satisfying all these restrictions. The analysis depends onwhether h > h < h = 1. We start with h (cid:54) = 1. Because of (34), the initial data ˙ T canbe substituted by the value of E . Moreover,(1 − h ) g ( u , σK | s =0 ) = (1 − h ) (cid:16) − (1 − h ) ˙ T + (1 − h ) σ ˙ r (cid:17) = ( h −
1) ( E − σ ˙ r ) , where (34) has been again inserted in the last equality. Thus, the statement g ( u , σK | s =0 ) < u = bσK | s =0 with b ≥ h − E − σ ˙ r ) < E = σ ˙ r ≥ , the second inequality being a consequence of ˙ T = σ ˙ r ≥ h >
1. Theconditions to be imposed are { E < σ ˙ r or E = σ ˙ r ≥ } , together with E = ˙ r − a (fromequation (33)). The locus of this quadratic equation is a hyperbola with two branches (degeneratingto two straight lines when a = 0) and with asymptotes E = ± ˙ r . The condition { E < σ ˙ r or E = σ ˙ r ≥ } selects precisely the branch satisfying σ ˙ r ≥ a , as claimed in the Proposition.The case h < { E > σ ˙ r or E = σ ˙ r ≥ } together with E = ˙ r + a . The solution to these inequalities is the branch of the hyperbola satisfying σE ≥ a .For the case h = 1, rewrite equation (35) as2 σ ˙ r (cid:16) σ ˙ r − ˙ T (cid:17) = − (cid:18) µ + L r (cid:19) ≤ . (36)Thus, the condition { σ ˙ r − ˙ T < T = σ ˙ r ≥ } is equivalent to σ ˙ r ≥ µ = L = 0. This is because, when σ ˙ r >
0, equation (36) can be solved uniquely for ˙ T with thesolution satisfying σ ˙ r − ˙ T ≤
0, that is, either σ ˙ r − ˙ T < T = σ ˙ r >
0. When σ ˙ r = 0 then µ = L = 0 and ˙ T ≥ { σ ˙ r − ˙ T < T = σ ˙ r ≥ } . Finally,the statement E = σ ˙ r when h = 1 follows directly from (34). (cid:4) Remark 6 . Note that when L = µ = 0 we have a ≡ r = 0 , E = 0 irrespectively of the value of h . When h (cid:54) = 1, this boundary case correspondsto the situation when the initial tangent four-vector vanishes, and hence the geodesic is a trivialcurve. This is consistent with the fact that the zero vector is null and future directed. Admittingtrivial curves as null future directed geodesics has the advantage that allows one to treat at oncethe cases µ = 0 and µ = 1. 11 orollary 7. The variation ranges for (cid:15) are (cid:40) (cid:15) ∈ [ − µ , ∞ ) if h ≥ (cid:15) ∈ [ a − µ , ∞ ) if h < independently of the sign of σ and of the function h ( (cid:126)x ) in the Kerr-Schild metric. Proof.
Immediate from the ranges of variation of E in Proposition 5 and the relation (cid:15) = ( E − µ ). (cid:4) In his original paper [McGehee, 1981], McGehee proposes a method of blowing-up the singularityby introducing a coordinate system that regularizes the origin for power-law radial potentials V ( | (cid:126)x | ) = | (cid:126)x | − σ in R , σ >
0. The field equations are¨ (cid:126)x = − grad (cid:0) | (cid:126)x | − σ (cid:1) = σ | (cid:126)x | − σ − (cid:126)x (38)where dot is derivative with respect to τ and grad = ∂∂x i is the gradient operator. Since thetrajectories lie in a plane, this system can be restricted to R without loss of generality. Introducing,as usual, an auxiliary vector variable (cid:126)y this system can be rewritten as a first order system on R as ˙ (cid:126)x = (cid:126)y, ˙ (cid:126)y = σ | (cid:126)x | − σ − (cid:126)x. At this point McGehee proposes identifying R with the complex plane C . Writing { x, y } for { (cid:126)x, (cid:126)y } after this identification, the change of coordinates x = r χ e iθ y = r − σ χ ( u + iv ) e iθ , (39)where χ = σ has the two properties of (i) regularizing the system at r = 0 and (ii) decouplingthe system in the pair of variables { u, v } . Thus the original system transforms into an autonomousdynamical system on the plane { u, v } (with no singularities) together with a pair of first orderODEs in { r, θ } (also free of singularities) which can be integrated afterwards. The new variablestake values u, v ∈ R , r ∈ [0 , ∞ ) and θ ∈ [0 , π ).It is natural to ask whether such a regularization and decoupling procedure also occursfor more general potentials V ( | (cid:126)x | ). We prove in appendix A that only the power-law and thelogarithmic potentials V ( | (cid:126)x | ) ∝ ln | (cid:126)x | decouple in the variables { u, v } , even after introducingarbitrary functions of r in the transformation (39). Despite this impossibility, the system can stillbe simplified substantially by a suitable choice of generalized McGehee transformation.12 heorem 8. Let N be an open annulus in C and V : N → R be a radially symmetric function V ( x ) = V ( | x | ) . Assume that V ( | x | ) is C as a function of | x | and define ∇ = ∂ x + i∂ x where x = x + ix , x , x ∈ R . Then the dynamical system ˙ x = y, ˙ y = −∇ V ( | x | ) := Λ( | x | ) x, (40) on N × C is equivalent to the system r (cid:48) = ruθ (cid:48) = vv (cid:48) = − ( β + 1) u v (41) u (cid:48) = r − β Λ( r ) − βu + v , where β is an arbitrary constant. The coordinates { r, θ, u, v } take values in r ∈ ( a, b ) ⊂ R + , θ ∈ S and u, v ∈ R . The coordinate change is defined by x = re iθ y = r β ( u + iv ) e iθ (42) dτ = r − β ds, where τ is the flow parameter in (40) and s is the flow parameter in (41). Remark 9 . In the case when the potential V ( | x | ) is a power-law V ( | x | ) = | x | − σ the transformation(42) does not agree with the original McGehee transformation (39) even after making the choice β = − σ . The reason lies in the specific choice χ = 2 / (2 + σ ) made by McGehee. In fact, anychoice of non-zero constant χ in the transformation (39) preserves the same properties for thetransformed system. We prefer to make the choice χ = 1 because then r measures directly thedistance of the particle to the origin. In order to recover the specific form used by McGehee, itis necessary to apply an additional coordinate change r → r χ to the system (41). However, thishas no benefits for the dynamical system and has the drawback of obscuring the clear geometricinterpretation of r . Proof.
The coordinate change (42) is a particular case of the coordinate change (67) introducedin appendix A with ξ ( r ) = r , ξ ( r ) = r β and ξ ( r ) = r − β . In particular, equations (70) and (71)hold with α = 1 and c = 1. Thus, the dynamical system in the new coordinates takes the form(72) which is exactly (41) after setting α = 1, c = 1. N being an open annulus, it must be of theform N = { x ∈ C ; x = re iθ with r ∈ ( a, b ) , θ ∈ S } for some 0 < a < b . This proves the claim onthe domain of the new coordinates. (cid:4) Concerning the properties of the new dynamical system we have13 heorem 10.
With the same conditions as in Theorem 8, the transformed dynamical systemadmits the following two constants of motion L = vr β +1 (43) (cid:15) = 12 r β (cid:0) u + v (cid:1) + V ( r ) . (44) Moreover, if ∈ N and we assume that there is γ > such that | x | γ V ( | x | ) admits a C extension(as a function of | x | ) to | x | = 0 , then for any β ≤ − γ the system (41) admits a C extension to [0 , b ) × S × R × R . Proof.
The dynamical system (40) describes the motion of a point particle under the influence ofa radial potential V . Thus, the angular momentum L = (cid:126)x × ˙ (cid:126)x and the energy (cid:15) = ˙ (cid:126)x + V ( (cid:126)x ) areconserved. In terms of the complex variables { x, y } they take the form [McGehee, 1981]: L = (cid:61) (¯ xy ) (cid:15) = 12 | y | + V ( | x | ) , where (cid:61) is the imaginary part and ¯ x is the complex conjugate of x . Applying the coordinatechange (42) one finds L = vr β (cid:15) = 12 r β (cid:0) u + v (cid:1) + V ( r ) , as claimed. Assume now that ∃ γ > f ( | x | ) := | x | γ V ( | x | ) can be C extended to | x | = 0.The function Λ( | x | ) (see expression (40)) is defined to beΛ( | x | ) = − | x | dV ( | x | ) d | x | = γ f ( | x | ) | x | γ − | x | γ df ( | x | ) d | x | . (45)Inserting this into (41) we see that a sufficient condition for the dynamical system to admit a C extension to r = 0 is that 2 β + γ ≤ (cid:4) The existence of the first integral L can be used to remove v from the equations and reducethe dimensionality of the system, as well as to decouple a two-dimensional subsystem. Lemma 11.
The subset of trajectories of Theorem 8 with constant of motion L are equivalent tothe dynamical system r (cid:48) = ru (46) u (cid:48) = r − β +1) (cid:0) L + r Λ( r ) (cid:1) − βu (47) θ (cid:48) = Lr − ( β +1) (48) defined on ( a, b ) × S × R . This system is decoupled in the { r, u } variables and admits the firstintegral (cid:15) = L r + 12 u r β + V ( r ) . (49)14 roof. Solve for v in the constant of motion (43) and substitute in the dynamical system (41)and in the expression for (cid:15) (44). (cid:4) The physical meaning of the coordinates { r, θ, u, v } follows easily from their definition (42):1. The coordinates r, θ are the standard polar coordinates on the plane.2. The coordinates u, v are proportional to the radial and the angular components of the velocity.This follows from the first equation in (40) because r β ( u + iv ) e iθ = ˙ x = ˙ re iθ + ire iθ ˙ θ ⇐⇒ u = r − β ˙ rv = r − β ˙ θ (cid:27) . (50)Thus, u carries all the radial information of the velocity whereas v encodes the angular partof the velocity,The decoupling of the system in the { r, u } variables is adequate since it corresponds to the usualdecoupling of the radial motion of a point particle under the influence of a radial potential. Oncethis motion is solved, the angular motion θ ( s ) follows by simple integration of θ (cid:48) ( s ) = Lr ( s ) − ( β +1) . We have already discussed in Theorem 10 the range of values for β which regularize the dynamicalsystem (41) at r = 0. The reduced dynamical system (46)-(47) incorporates extra powers of r which are potentially divergent at r = 0. The following lemma determines the range of β whichregularizes the reduced system. Corollary 12.
Under the same assumptions as in Theorem 10, the reduced system (46)-(47) for L (cid:54) = 0 admits a C extension to [0 , b ) × S × R for β ≤ min {− , − γ } . Proof.
We already know that β ≤ − γ regularizes the term in Λ of equation (47) at r = 0. For L (cid:54) = 0, equation (48) admits a C extension at r = 0 if and only if 1 + β ≤
0. This condition alsoregularizes the first term in equation (47). Thus, β ≤ min {− , − γ } is a sufficient condition for theexistence of a continuous extension to r = 0. (cid:4) Remark 13 . The optimal choice of β for a detailed study of the dynamical system (46)-(47) at r = 0 is β = min {− , − γ } with γ selected in such a way that | x | γ V ( | x | ) admits a C extension to | x | = 0 and lim | x |→ | x | γ V ( | x | ) (cid:54) = 0. Indeed, a larger value of β is not capable of regularizing thesystem at r = 0. On the other hand, a smaller value of β overkills the singularity. This has theeffect that the invariant submanifold { r = 0 } (which is called the collision manifold ) has u = 0as the unique fixed point, and this is always non-hyperbolic. Thus, all details of the phase spacestructure of the dynamical system at { r = 0 } are lost by this choice of β . We will see belowan example of this behavior when considering the Schwarzschild limit of the dynamical systemdescribing causal geodesics in the Reissner-Nordstr¨om spacetime.15 The Schwarzschild dynamical system
Figure 1: The Schwarzschild Penrose-Carter diagram
As is well-known, the Kruskal spacetime of mass
M > R × R + × S with respective metrics ds = − (cid:18) − Mr (cid:19) dV − σdV dr + r d Ω (51)where σ = − σ = 1 for the retarded one. The coordinates take values in V ∈ R , r ∈ R + and d Ω is the round unit metric on the sphere. Furthermore the time orientation ischosen so that V increases along any timelike curve. The coordinate change V = T − σr transformsthe metric (51) into ds = − dT + dr + r d Ω + 2 Mr ( dT − σdr ) with range of variation T ∈ R , r ∈ R + . Transforming the flat metric dr + r d Ω to Cartesiancoordinates brings the metric to Kerr-Schild form g = − dT + d(cid:126)x · d(cid:126)x + 2 Mr ( dr − σdT ) ⊗ ( dr − σdT ) (52)where r = √ (cid:126)x · (cid:126)x and the manifold is R × ( R \ { } ). The choice of time orientation in (51) impliesthat the null vector field ∂ T + σ (cid:126)x | (cid:126)x | ∂ (cid:126)x is future directed.This form of the metric was obtained in [Kerr and Schild, 1965]. The case σ = − r = 0. The case σ = +1 covers the lower andright quadrants of the diagram and approaches the white hole singularity at r = 0. Similarly, thespacetime (52) with σ = − σ = 1 covers the left-lower quadrants. The only set of points the Kruskalspacetime not covered by these patches is the bifurcation surface at r = 2 M .16s noticed in Remark 3, the spatial part of the geodesic equations do not depend of the choiceof sign in σ and therefore the dynamical system will also be independent of this choice. Thishas the following interesting consequence. Consider for instance a future directed causal geodesicstating in the region r < M in the lower part of the diagram (for simplicity we call this the white hole region and by black hole region we refer to the domain r < M in the upper part ofthe diagram). This geodesic can be described in the Kerr-Schild metric (52) with σ = 1. Aftera finite value of its affine parameter, the geodesic will approach r = 2 M . This can happen witheither v → + ∞ or v → v finite. In either case, since the dynamical system only involves thespatial coordinates, this portion of geodesic will have a limit point p in the phase space satisfying r ( p ) = 2 M . Irrespectively of which spacetime point q is approached (even it the point lies onthe bifurcation surface), the geodesic can be continued further as a portion of a geodesic in thespacetime (52) with σ = − q . The fact that the dynamical system forthe spatial coordinates is independent of σ implies that the trajectory will be described in a singlephase space, i.e. the change of spacetime chart will pass fully unnoticed in the phase space of thedynamical system. Thus, we will be able to describe the full geodesic as a single trajectory in thephase space, without having to determine in which spacetime coordinate chart we are working ateach portion. As we will see below, this is only possible due to the presence of a excluded regionin the phase space. In turn, this excluded region arises as a consequence of the restrictions inthe initial data imposed by the condition that the trajectory describes a future directed causalgeodesic. Lemma 14.
The McGehee regularization for the dynamical system that describes the spatial partof the set of geodesic trajectories with angular momentum L in the Kruskal spacetime is r (cid:48) = ru, (53) u (cid:48) = r (cid:0) L − µM r (cid:1) − L M + 32 u , (54) θ (cid:48) = L √ r. (55) where µ = 1 , , − for timelike, null and spacelike geodesics, respectively. The system admits theenergy first integral (cid:15) = u r + L r − M (cid:18) L r + µr (cid:19) . (56) Proof.
We can apply Proposition 5 with h = Mr , which corresponds to the spacetime (52).Equations (28), (29) and (30) become¨ (cid:126)x = − (cid:18) M L r + µMr (cid:19) (cid:126)x, (57) (cid:126)L = (cid:126)x × ˙ (cid:126)x,(cid:15) : = ˙ r L r − M (cid:18) L r + µr (cid:19) . µ and the energy is E = ˙ T (cid:0) Mr − (cid:1) + Mr ˙ r , from (31).Given a geodesic, we can rotate the Cartesian coordinates so that the geodesic lies in the plane { x , x } and define x as the complex coordinate x = x + ix ,. The equations of motion (57)become ˙ x = y, ˙ y = − (cid:18) M L | x | + µM | x | (cid:19) x = −∇ (cid:18) − M L | x | − µM | x | (cid:19) := Λ( | x | ) x. This flow is singular at r = 0 and we can apply the McGehee regularization described above. FromCorollary 12 and the fact that | x | V ( | x | ) admits a C extension to x = 0 with non-zero value atthis point, we find that the optimal value for regularization is β = min {− , − γ } = − . ApplyingLemma 11 the following regularized system is obtained r (cid:48) = ruu (cid:48) = r (cid:0) L − µM r (cid:1) − L M + 32 u θ (cid:48) = L √ r(cid:15) = u r + L r − M (cid:18) L r + µr (cid:19) . (cid:4) The dynamical system (53)-(55) describes all future directed causal geodesics in the Kruskalspacetime. However, not all trajectories in this dynamical system correspond to future directedcausal geodesics in this spacetime. The reason is that the set of initial data { r , ˙ r } for futuredirected causal geodesics is constrained by Proposition 5. Given the relation between u and ˙ r ,this implies that not all points { r, u, θ } in the phase space describe future causal geodesics in thespacetime. We will call the allowed region the set of points in the phase space corresponding tofuture directed causal geodesics and the excluded region its complement. Let us determine thesesets.As discussed in Proposition 5, at points where h <
1, i.e. r > M , there is no restriction onthe possible values of ˙ r , and consequently no restrictions on u arise. On the other hand, when h >
1, i.e. r < M , then σ ˙ r ∈ [ a , ∞ )where a = (cid:115)(cid:18) Mr − (cid:19) (cid:18) L r + µ (cid:19) . For h = 1 ( r = 2 M ), ˙ r is restricted to satisfy σ ˙ r ∈ [0 , ∞ )18ith ˙ r = 0 only if L = 0 and the geodesic is null ( µ = 0). Given that u = r − β ˙ r = r ˙ r the allowedregion for geodesics in the advanced Eddington-Finkelstein spacetime (i.e. σ = −
1) is u ∈ (cid:32) −∞ , − r (cid:115)(cid:18) Mr − (cid:19) (cid:18) L r + µ (cid:19) (cid:35) , r ≤ M with ( u = 0 , r = 2 M ) allowed only if L = 0 , µ = 0 ,u ∈ ( −∞ , ∞ ) r > M So, the excluded region is u ∈ (cid:32) − r (cid:115)(cid:18) Mr − (cid:19) (cid:18) L r + µ (cid:19) , ∞ (cid:33) , r ≤ M with ( u = 0 , r = 2 M ) excluded if L (cid:54) = 0 or µ (cid:54) = 0Similarly, for geodesics in the retarded Eddington-Finkelstein spacetime ( σ = 1) the excludedregion is u ∈ (cid:32) −∞ , r (cid:115)(cid:18) Mr − (cid:19) (cid:18) L r + µ (cid:19)(cid:33) , r ≤ M with ( u = 0 , r = 2 M ) excluded if L (cid:54) = 0 or µ (cid:54) = 0 . As discussed above, the bifurcation surface at r = 2 M is not included in either the advanced orretarded Eddington-Finkelstein spacetime. This is the reason why the the point ( u = 0 , r = 2 M )when either L (cid:54) = 0 or µ (cid:54) = 0 is excluded. Since future causal geodesics with L (cid:54) = 0 in the Kruskalspacetime do cross the bifurcation surface, we must incorporate this set of points of the phasespace into the allowed region in order to describe all causal geodesics of the Kruskal spacetime.We conclude that the set of phase space points not describing future directed causal geodesics inthe Kruskal spacetime is the intersection of both excluded regions after adding ( u = 0 , r = 2 M ) tothe allowed regions, namely u ∈ (cid:32) − r (cid:115)(cid:18) Mr − (cid:19) (cid:18) L r + µ (cid:19) , r (cid:115)(cid:18) Mr − (cid:19) (cid:18) L r + µ (cid:19)(cid:33) , r ≤ M. The existence of these three types of excluded regions is crucial for describing all future directedcausal geodesics in the Kruskal spacetime in a single phase space diagram. Indeed, any trajectorypassing through an allowed point in the region r < M with u > r < M regionof a retarded Eddington-Finkelstein chart of the Kruskal spacetime, i.e. to the white hole region ofthe spacetime. A trajectory passing through an allowed point in the region r < M with u > r > M belongs to both charts.The boundary of the allowed region is given by the set of points satisfying u = r (cid:18) Mr − (cid:19) (cid:18) L r + µ (cid:19) , r ≤ M igure 2: The left image correspond to the excluded regions in the phase space { r, u } for null geodesics( µ = 0). Different values of the angular momentum L are displayed (the darker the zone, the lower thevalue of L ). The right image corresponds to the analogous case for timelike geodesics ( µ = 1). In both cases M = 1. which can be rewritten as the set of points with (cid:15) = − µ . In fact, the excluded region can beequivalently defined as the set of points for which (cid:15) < − µ , r ≤ M , cf. Corollary 7.The graphical representation of the excluded regions for null and spacelike geodesics is displayedin Fig. 2. Notice that, in the null case, there is no excluded region in the limit L = 0. However,in this case the set of points u = 0 correspond to trivial null geodesics consisting of single pointswith vanishing tangent vector. For non- null geodesics, the excluded region is always non-emptyirrespectively of the value of the angular momentum L . The submanifold r = 0 is clearly invariant under the flow. Since r = 0 corresponds to the spacetimesingularity, this submanifold is called collision manifold . It can be described globally by thecoordinates { ( θ, u ) } so its topology is R × S . The dynamical system (54)-(55) restricted to thecollision manifold reads u (cid:48) = 32 u − L M,θ (cid:48) = 0 . This system has two lines of critical points: one line of stable points at ( θ, u ) = ( θ , −√ M L )and one line of unstable nodes at ( θ, u ) = ( θ , √ M L ), where θ ∈ S is an arbitrary value. Thephase space portrait in the collision manifold is shown in Fig. 3. For each value of θ , there isa trajectory extending from u = −∞ and approaching u = −√ M L as its future limit point, atrajectory from u = −√ M L to u = √ M L and a trajectory having u = √ M L as its pastlimit point and extending to u = + ∞ , all of them with θ = θ . One may wonder how thesetrajectories relate to causal geodesics in the Kruskal spacetime. To see this, note that any suchgeodesics must have a finite value of (cid:15) . On the other hand (cid:15) diverges at r = 0 (see (56)). In fact, it20 igure 3: The flow in the collision manifold (left) with the critical lines in u = ±√ M L (we have chosen L = 5 , M = 1) and the collision manifold embedded cylinder (right) with the flow and the critical lines overit. does so in the following waylim r → (cid:15) = (cid:40) + ∞ if | u | > √ M L , −∞ if | u | < √ M L . The limit when | u | approaches the critical points on the collision manifold depends on the details ofhow this limit is taken. Since, the trajectories joining the stable critical points to the unstable oneswithin the collision manifold are interior to the excluded region of the phase phase, it follows thatsuch trajectories are completely unrelated to causal geodesics in the spacetime. This is consistentwith the fact that (cid:15) → −∞ on these trajectories, while future directed causal geodesics in theregion r < M must have (cid:15) ≥ − µ . On the other hand, the trajectories on the collision manifoldleaving u = + √ M L and those approaching u = −√ M L correspond to the limit of trajectoriesof causal geodesics leaving the white hole singularity at r = 0 and approaching the black holesingularity at r = 0 when their energy (cid:15) diverges to + ∞ . Thus, this set of trajectories on thecollision manifold carries interesting information on the causal geodesics in the spacetime.To analyze the behavior of the particles near the collision manifold, we can linearize the systemat its first order as r ( s ) = δr ( s ) ,u ( s ) = u ( s ) + δu ( s ) , where u ( s ) its a solution that corresponds to an orbit in the collision manifold and thereforesatisfies u (cid:48) = − L M + 3 u . | u | > √ M L is u ( s ) = −√ M L coth (cid:32) (cid:114) M L s (cid:33) . The branch s ∈ ( −∞ ,
0) corresponds to the solution leaving the unstable fixed point at u =+ √ M L and approaching u −→ + ∞ , while the branch s ∈ (0 , + ∞ ) corresponds to the solutionextending from u = −∞ and approaching the fixed point u = −√ M L . The first-order linearizedsystem in the variables δr and δu is δr (cid:48) = ( δr ) u ,δu (cid:48) = L ( δr ) + 3( δu ) u . We can easily solve for δr ( s ): δr ( s ) = δr (cid:12)(cid:12)(cid:12)(cid:12) sinh (cid:18) (cid:113) ML s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , where δr > s . To analyze the behavior in the τ variable we recall the relation(42), namely τ (cid:48) ( s ) = ( δr ( s )) . This integration can be performed explicitly, but it is not particularly enlightening. Instead, wewill determine a series expansion of δr ( τ ) near τ = 0 where τ is chosen so that the particle reachesthe singularity r = 0 at τ = 0 (note that τ < τ > x ( s ) as x ( s ) = 1 (cid:12)(cid:12)(cid:12)(cid:12) sinh (cid:18) (cid:113) ML s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) so that ( δr )( s ) = ( δr ) x ( s ) and hence dxds = u x . In terms of x , u = ∓√ M L (cid:112) x , where the sign depends on the branch we are considering (upper sign for the approach to the blackhole and lower sign for the white hole). Then dτdx = dτds dsdx = ( δr x ) xu = ∓ ( δr ) √ M L x √ x . Expanding the right-hand side as a series en x , integrating and inverting we find x ( τ ) = (cid:18) (cid:19) ( ∓ aτ ) + 522 (cid:18) (cid:19) ( ∓ aτ ) + O ( τ ) + . . . where a = √ ML ( δr ) . A plot of the function δr ( τ ) in each of the two branches is given in Fig. 4.These plots describe the approach to the singularity of very energetic particles in the Kruskalspacetime for different values of the angular momentum. Note that the result is the same formassive and massless particles, as one could expect given the very high energy involved.22 igure 4: Plot of the function δr ( τ ). The image on the left corresponds to the branch in witch u ≤ −√ M L (black hole solution) in the collision manifold and the image on the right corresponds to the branch in witch u ≥ √ M L (white hole solution). In both plots δr = 1 and M = 1. The reduced dynamical system with µ = 0 takes the form r (cid:48) = r u, (58) u (cid:48) = rL − L M + 3 u . (59)Its phase space is displayed for different values of L in Fig. 5. The fixed points are (assuming L (cid:54) = 0) ( r = 0 , u = ±√ M L ) ( r = 3 M, u = 0) . The linearization of the system around these points has the following eigenvalues (cid:40) λ = √ M L , λ = −√ M L for ( r = 3 M, u = 0) ,λ = ± √ M L λ = ±√ M L for ( r = 0 , u = ±√ M L ) . Figure 5: Phase space for massless particles with L = 0 (left picture) and L = 1 . M (right picture). Thedark zone correspond to the forbidden region given by (cid:15) < µ . igure 6: Phase spaces for massive particles with L = 0, L = 1 . M , L = 2 √ M and L = 3 . M . Thedark zone correspond to the forbidden region given by (cid:15) < µ . The larger the value of L/M the larger theexcluded region in the phase space.
Thus, all critical points are hyperbolic. The point ( r = 0 , u = + √ M L ) is an unstable node(repulsor), ( r = 0 , u = −√ M L ) is a stable node (attractor) and ( r = 3 M, u = 0) is saddle point.This saddle point obviously corresponds to the unstable circular orbit for massless particles.
When µ = 1 the reduced dynamical system is r (cid:48) = r u,u (cid:48) = r (cid:0) L − M r (cid:1) − L M + 3 u , with phase spaces displayed for different values of L in Fig. 6. The fixed points are( r = 0 , u = ±√ M L ) , ( r = r ± ( M, | L | ) := L ± | L |√ L − M M , u = 0) , the second pair under the additional condition L ≥ M . For L > M all fixed pointsare hyperbolic, with ( r = r + ( M, | L | ) , u = 0) being a center (purely imaginary eigenvalues) and( r = r − ( M, | L | ) , u = 0) being a saddle. When L = 12 M , there is a bifurcation in the phasespace, which can be visualized in the transition between the third and fourth plots in Fig. 6.Given that r + ( M, | L | ) is an increasing function of | L | with values ranging from (6 M, ∞ ) we recoverthe well-known fact that the innermost stable circular orbit (ISCO) in Schwarzschild is r = 6 M .The fixed point r − ( M, | L | ) is a decreasing function of | L | with values ranging from 6 M (when | L | → √ M ) to 3 M when | L | → + ∞ ). We thus recover easily all well-known results for geodesicsin Schwarzschild outside the horizon. The approach here, however, is perfectly regular both acrossthe horizon at r = 2 M and even at the singularity r = 0. Moreover, it allows us to treat all pointsin the Kruskal spacetime with a single dynamical system. The Reissner–Nordstr¨om spacetime [Reissner, 1916, Nordstr¨om, 1918] corresponds to the mostgeneral solution for Einstein equations with electromagnetic field when spherical symmetry is24ssumed. The maximal extension of this spacetime is covered by a infinite amount of copiesof four basic patches. As in the Kruskal spacetime two of the patches are isometric to theReissner–Nordstr¨om advanced Eddington-Finkelstein spacetime and the remaining two to theReissner–Nordstr¨om retarded Eddignton-Finkelstein. Each one of these spacetimes consist of themanifold R × R + × S with respective metrics ds = − (cid:18) − Mr + Q r (cid:19) dV − σdV dr + r d Ω , where σ = − σ = 1 for the retarded one. The coordinates takes valuesin V ∈ R , r ∈ R + and d Ω is the round unit metric on the sphere. Performing the same coordinatechange as in the Kruskal case, the metric is transformed into its Kerr-Schild form [Kerr and Schild,1965]: g = − dT + d(cid:126)x · d(cid:126)x + 2 M r − Q r ( dr − σdT ) ⊗ ( dr − σdT ) (60)where r = √ (cid:126)x · (cid:126)x and the manifold is R × ( R \ { } ). As before we choose the orientation so thatthe nowhere-zero, null vector ∂ T + σ (cid:126)x | (cid:126)x | ∂ (cid:126)x is future directed. Figure 7: The Reissner–Nordstr¨om Penrose-Carter diagram.
As is well-known, the global structure of the maximal extension of the Reissner-Nordstr¨omspacetime depends strongly on whether | Q | > M , | Q | = M or | Q | < M . For the discussion25elow we concentrate on the latter case because it corresponds to a non-extremal black hole. Weemphasize, however, that all the dynamical systems in this Section are valid for any value of Q and M , so they can be used to study geodesics in the extremal black hole or naked singularitycases as well.When | Q | < M , the basic structure of the maximal extension of the Reissner–Nordstr¨omspacetime has now two bifurcation surfaces located, respectively, at the intersection of the twosmooth null hypersurfaces with r = r + and the two smooth null hypersurfaces with r = r − , where r ± = M ± (cid:112) M − Q . We call each one of these four hypersurfaces a horizon . The patch with σ = − r = ∞ → r = r + → r = r − → r = 0 (starting from either the right or the left) while the patch with σ = 1 covers thestructure unit that goes in an ascending way from r = 0 → r = r − → r = r + → r = ∞ (startingfrom the right or the left). Notice that there is no causal geodesic starting from a left-mostquadrant which, after crossing the null hypersurface r = r + then goes across the portion of thenull hypersurface at r = r − lying at the left of the diagram (and similarly for geodesics starting ata right-most quadrant). The reason is that the Killing 1-form ξ := g ( ξ, · ) where ξ = ∂ T (definedon a single Eddington-Finkelstein patch) is integrable with orthogonal hypersurfaces foliating theregion r − < r < r + with timelike leaves. Let us label these leaves by t . As a consequence we have ξ = Gdt on r − < r < r + where G is a smooth function. Consider the conserved energy for thegeodesic, i.e. (cid:104) ∂ T , u (cid:105) = − E where u is the tangent vector. In the region between r − and r + , inorder for the geodesic to enter from the left portion of the null hypersurface r = r + and leaveacross the left portion of the hypersurface r = r − , the geodesic must become somewhere tangentto a hypersurface t = const. At this point we have − E = ξ ( u ) = Gdt ( u ) = 0. But E = 0 isimpossible for a geodesic starting on the left-most region where ξ is timelike. A similar argumentapplies to geodesics starting at the right-most quadrant.As discussed in Section 5, the fact that the dynamical system for the spatial coordinatesis independent of σ implies that the trajectory will be described in a single phase space. Tounderstand the geodesic flow along the maximal extension of the Reissner–Nordstr¨om we need toanalyze the excluded regions. We first write down the regularized dynamical system. Lemma 15.
The McGehee regularization for the dynamical system that describes the spatial partof the set of geodesic trajectories with angular momentum L in the Kruskal spacetime is r (cid:48) = ru, (61) θ (cid:48) = L r, (62) u (cid:48) = r (cid:0) r (cid:0) L − µM r + µQ (cid:1) − L M (cid:1) + 2 (cid:0) L Q + u (cid:1) . (63) where µ = 1 , , − for timelike, null and spacelike geodesics, respectively. The system admits theenergy first integral L (cid:0) r ( r − M ) + Q (cid:1) + µr (cid:0) Q − M r (cid:1) + u = 2 r (cid:15). (64)26 roof. We can apply Proposition 5 with h = Mr − Q r , from (60). Equations (28), (29) and (30)become ¨ (cid:126)x = (cid:18)(cid:18) Q L r + µQ r (cid:19) − (cid:18) M L r + µMr (cid:19)(cid:19) (cid:126)x,(cid:126)L = (cid:126)x × ˙ (cid:126)x,(cid:15) : = ˙ r L r − M (cid:18) L r + µr (cid:19) + Q (cid:18) L r + µ r (cid:19) . Adapting coordinates so that the geodesic lies in the { x , x } plane and defining the complexvariable x = x + ix , the equations are˙ x = y, ˙ y = (cid:18)(cid:18) Q L | x | + µQ | x | (cid:19) − (cid:18) M L | x | + µM | x | (cid:19)(cid:19) x = −∇ (cid:18) − M (cid:18) L | x | + µ | x | (cid:19) + Q (cid:18) L | x | + µ | x | (cid:19)(cid:19) := Λ( | x | ) x. We now apply the McGehee regularization. Since | x | V ( | x | ) admits a C extension to x = 0,Corollary 12 fixes the optimal value for regularization as β = min {− , − γ } = − (cid:15) . (cid:4) As in the Schwarzschild case, not all trajectories of the dynamical system (61)-(63) correspond tofuture directed causal geodesics in the Reissner-Nordstr¨om spacetime. Proceeding as in Schwarz-schild and exploiting Proposition 5, it is straightforward to find that the excluded region of thephase space diagram is u ∈ (cid:32) − r (cid:115)(cid:18) Mr − Q r − (cid:19) (cid:18) L r + µ (cid:19) , r (cid:115)(cid:18) Mr − Q r − (cid:19) (cid:18) L r + µ (cid:19)(cid:33) , r − ≤ r ≤ r + . In addition, for geodesics in the σ = − u must be lie below theforbidden region (i.e. u ≤ r − ≤ r ≤ r + ), while geodesics in the σ = +1 Eddington-Finkelstein patch must lie above the forbidden region (i.e. u ≥ r − ≤ r ≤ r + ). Notethat, as in Schwarzschild, the boundary of the allowed region is defined by u = r (cid:18) Mr − Q r − (cid:19) (cid:18) L r + µ (cid:19) , r − ≤ r ≤ r + , which correspond to the set of points with (cid:15) = − µ and the excluded region is precisely the set (cid:15) < − µ , r − ≤ r ≤ r + , cf. Corollary 7.The excluded regions for timelike geodesics are displayed in Fig. 8 with representative valuesof Q and L . The excluded regions for timelike/null geodesics show the same behavior on L as in27 igure 8: The three pictures displayed correspond to the excluded regions in the phase space { r, u } fortimelike geodesics ( µ = 1) with Q = 0, Q = 0 . Q = 0 .
975 respectively. Different values of the angularmomentum L are displayed in each one (the darker the zone, the lower the value of L ). Units are chosen sothat M = 1. The vertical lines correspond to r ± . the Schwarzschild case, i.e. in the null case there is no excluded region in the limit L = 0 but thenthe line u = 0 consists of a family of trivial geodesics and in the timelike case the excluded regionis always non-empty for all values of L .We can now discuss how the behavior of geodesics across the horizons can be described in thephase space diagram. The crucial information is the restriction for u (which, recall, is proportionalto ˙ r ) in the domain r − < r < r + . Let us see how this implies that any geodesic traveling from r = r > r + → r + → r − → r and back to r − → r + must have changed the Kerr-Schild patchalong the way (by changing the value of σ ). Assume for definiteness that the particle starts ina left-most quadrant. After the crossing of r + the particle must necessarily cross r = r − . Thiswell-known fact can be directly deduced also from the dynamical system because the allowedregion in the domain r − < r < r + when σ = − u < r = r − happens in the right part of the Kerr-Schild patch, as discussed before. Thus the trajectory isstill contained in the original Kerr-Schild patch and, at the same time, it has also entered a newpatch with σ = −
1. Since the collision manifold cannot be attained (see below) the geodesicmust necessarily reach a point where u = 0 and cross to positive values of u . From then on, thecurve crosses r = r − towards larger values of r . At this crossing, the trajectory necessarily leavesthe original Kerr-Schild patch. It does so with u > σ = − r = r + and enters a different asymptotic region from which it started. Thisbehavior is of course well-known. What is important here is that a single phase space allows for acomplete description of the complicated spacetime trajectory by simply noticing that each timethat the forbidden region is encircled, we have moved one step in the ladder in Kerr-Schild patchesin the maximal extension of Reissner-Nordstr¨om. This is possible only because (i) the phase spacehas an excluded region and (ii) the crossing at r = r − is not ambiguous, forcing the particle tostay in the same Kerr-Schild patch it started. 28 .2 The collision manifold The Reissner-Nordstr¨om collision manifold has again topology R × S and global coordinates { ( θ, u ) } . The dynamical system (62)-(63) restricted to the collision manifold reads u (cid:48) = 2 (cid:0) L Q + u (cid:1) ,θ (cid:48) = 0 , (65)which has no fixed points. This is a manifestation of the fact that, in Reissner-Nordstr¨om, thesingularity has a repulsive behavior, unlike the Schwarzschild case. This is already indicated bythe fact that, for | LQ | (cid:54) = 0, lim r → (cid:15) = + ∞ irrespective of the value of u . So, no physical particleswith finite energy can access the collision manifold and, the larger the value of (cid:15) they have, thecloser to the collision manifold r = 0 they can get.To analyze the repulsive nature of the collision manifold in more detail let us linearize thedynamical system to its first order around the collision manifold. Thus, we write r ( s ) = δr ( s ) ,u ( s ) = u ( s ) + δu ( s ) , where u ( s ) is a trajectory on the collision manifold, so that it satisfies u (cid:48) = 2 L Q + 2 u . The general solution for this differential equation satisfying u (0) = −∞ is u ( s ) = −| LQ | cot(2 | LQ | s ) . Where s ∈ (0 , π | LQ | ). The first-order linearized system in the variables δr and δu is δr (cid:48) = ( δr ) u ,δu (cid:48) = − L M δr + 4( δu ) u . We can easily solve for δr ( s ): δr ( s ) = δr (cid:112) sin(2 | LQ | s ) , where δr ( π | LQ | ) = δr . If follows that, no matter how close to the collision manifold we get (at s = π | LQ | where δr ( s ) is minimum), the trajectory never touches the collision manifold and in fact,it separates from it very quickly. To understand how fast this happens we need to change to the τ variable. Given the relation (42), namely τ (cid:48) ( s ) = ( δr ( s )) . As in the Schwarzschild case we will determine a series expansion of δr ( τ ) near τ = 0, where τ ischosen to fulfill τ = 0 when s = π | LQ | . Define x ( s ) as x ( s ) = 1 (cid:112) sin(2 | LQ | s )29o that ( δr )( s ) = ( δr ) x ( s ) and hence dxds = u x . In terms of xu = −| LQ | (cid:112) x − , then dτdx = dτds dsdx = ( δr x ) u x = − ( δr ) | LQ | x √ x − . Expanding the right-hand side as a series en x , integrating and inverting we find x ( τ ) = 1 + ( aτ ) −
56 ( aτ ) + 2318 ( aτ ) + O ( τ ) where a = | LQ | ( δr ) .It is interesting to note that inserting Q = 0 in (65) we do not recover the Schwarzschild case.This is because the value of the parameter β adapted to Reissner-Nordstr¨om is different to thatof Schwarzschild. Thus, in the Schwarzschild subcase of Reissner-Nordstr¨om we have overkilledthe singularity and the fixed points that previously existed at r = 0, u = u ± have both collapsedto u = 0. This collapse can be detected directly on the Reissner-Nordstr¨om phase space becausethe fixed point { u = 0 , θ = θ } is no longer hyperbolic when Q = 0. Another way of seeing this isby comparing the shape of the excluded regions of phase space { r, u } for Q = 0 with the shapeof the excluded regions in the Schwarzschild phase space. While in the latter case the excludedregions covered a non-empty interval of { r = 0 } , the Reissner-Nordstr¨om regularization is suchthat the bubble displayed in Fig. 8 , which stays separated from the collision manifold when Q (cid:54) = 0,just touches the line { r = 0 } in the limit Q = 0. Thus the whole line of excluded points in theSchwarzschild regularization has collapsed to a point in the Reissner-Nordstr¨om regularization ofthe Schwarzschild spacetime. Figure 9: Phase space for massless particles for the values ( L = 0, Q = 0) , ( L = 1 . M , Q = 0 . M ),( L = 1 . M , Q = 0 . M ) and ( L = 4 M , Q = 0 . M ) from left to right. The dark zone corresponds to theforbidden region. µ = 0 is r (cid:48) = ruu (cid:48) = r (cid:0) L r − L M (cid:1) + 2 (cid:0) L Q + u (cid:1) . The phase portrait can be viewed for different values of L and Q in Fig. 9. The fixed points ofthis system (assuming LQ (cid:54) = 0) are( r = R ± ( M, Q ) := 12 (cid:16) M ± (cid:112) M − Q (cid:17) , u = 0)provided 0 < | Q | ≤ (cid:113) M . In the strict inequality case, the two fixed points are hyperbolic,with the point ( u, r ) = (0 , R + ( M, Q )) being a saddle point (with real eigenvalues of oppositesign) and the point ( u, r ) = (0 , R − ( M, Q )) being a center (purely imaginary eigenvalues). It isstraightforward to check that the point ( u = 0 , r = R + ( M, Q )) always lies in the allowed region.The point ( u = 0 , r = R − ( M, Q )) lies in the excluded region as soon as this region is non-empty,i.e. for | Q | < M .As discussed in Section 6.1.1, curves encircling the excluded region when Q (cid:54) = 0 have periodicproperties in the phase space but they are really moving upwards in the Penrose-Carter diagramchanging form the white hole patch to the black hole patch as many times as needed. Note thatwith Q = 0 we recover the Schwarzschild fixed points but, as previously noticed in corollary 12,the phase portrait is nevertheless different because of the different choice of β . This is also thecase when µ = 1. 31 .3.2 Massive particles Figure 10: Phase space for mass particles with L = 0, L = 3 M , L = 3 . M (from left to right) and with Q = 0 (upper row) y Q = 0 . M (lower row). The dark zone correspond to the forbidden region given by (cid:15) < µ . When µ = 1 the dynamical system takes the form r (cid:48) = ru,u (cid:48) = r (cid:0) r (cid:0) L − M r + Q (cid:1) − L M (cid:1) + 2 (cid:0) L Q + u (cid:1) , with phase spaces displayed for different values of L and Q in Fig. 10. The portraits show veryclearly the repulsive nature of the singularity discussed above. The fixed points lie on the line u = 0 and are given by the roots of r M − r (cid:0) L + Q (cid:1) + 3 rL M − L Q = 0 . The root structure of this polynomial is not uniform in the parameters { M, Q, L } . Let us concentratefor definiteness in the most interesting case L (cid:54) = 0 and 0 < | Q | < M . It turns out that this equationalways has one real solution which lies inside the excluded region and corresponds to a hyperboliccritical point that happens to be a center (purely imaginary eigenvalues). Moreover, there exists L ( M, Q ) > < | L | ≤ L this is the only root. For | L | = L , there is second rootwhich is double (and hence a non-hyperbolic fixed point for the dynamical system). For | L | > L there are two additional hyperbolic points, both lying outside the excluded region to its right. Theone closer to the excluded region is a saddle and the one with largest value of r is a center. Thefunction L ( M, Q ) is defined as the only positive and real solution of L (cid:0) Q − M (cid:1) + 6 L (cid:0) M − M Q + 4 Q (cid:1) + 3 L (cid:0) Q − M Q (cid:1) + 8 Q = 0 . < | Q | < M . Thefunction L ( Q, M ) is displayed in Fig. 11, note that L ( Q = 0) = 2 √ M . These points areanalogous to the well-known critical points in the Schwarzschild spacetime and of course theycoincide in the limit Q = 0. Figure 11: The image shows the variation of the function L with respect to the value of Q . The plot isgiven in units where M = 1. The main result of this work is a dynamical systems method of analyzing causal geodesics instationary and spherically symmetric Kerr-Schild spacetimes. A remarkable result is that thegeodesics can be globally described as the motion of a Newtonian particle in the presence of a radialpotential. For spacetimes with singularities at the center we have developed a generalization of theMacGehee transformation that allows for a regularization of the origin and hence for a descriptionof the approach to the singularity in terms of regular variables. In particular, the dynamics at thecollision manifold can be analyzed, which gives us useful information for the physical trajectories.We have applied this method to the Schwarzschild and Reissner-Nordstr¨om spacetimes. Besidesthe regularized analysis of the singularity in these spacetimes, we have emphasized the importanceof the presence of excluded regions, which in effect makes the phase space diagram acquire anon-trivial topology. This topology and the property that the phase space portrait is independentof whether we deal with an advanced or a retarded Kerr-Schild patch allows one to study in thegeodesic motion is spacetimes with complicated global behavior (e.g. the Reissner-Nordstr¨omspacetime) in terms of a single two-dimensional phase space portrait.
Acknowledgements
M.M. acknowledges financial support under the projects FIS2012-30926 (MINECO) and P09-FQM-4496 (Junta de Andaluc´ıa and FEDER funds).P.G. acknowledges the useful comments and help provided by Ester Ramos.33
Absence of Mcgehee transformation decoupling the system in ( u, v ) In this appendix we prove a no-go Theorem for McGehee-type transformations capable of decouplingthe dynamic equations of a point particle on a plane under the influence of a general radial potential V ( | x | ). Specifically, we intend to analyze the dynamical system˙ x = y ˙ y = −∇ V ( | x | ) = Λ( | x | ) x. (66)where { x, y } are coordinates on an open subset of N of C and ∇ = ∂ x + i∂ x . In view of thetransformation proposed by McGehee for the power-law potential (39), we define the generalizedMacGehee transformation x = e iθ ξ ( r ) ,y = e iθ ξ ( r )( u + iv ) , (67) dτ = ξ ( r ( s )) ds, where ξ , ξ , ξ : R + → R are smooth and non-zero functions to be determined. Note that ξ ( r )must be invertible for this transformation to be well-defined. The new coordinates { u, v, r, θ } takevalues on R (for u, v ), in R + (for r ) and on S (for θ ). We note that making ξ complex does notdefine a more general transformation since it can be reduced to the above one by redefining thevariable θ . Given that we are replacing the power-law potential V ( | x | ) = | x | − σ by a general radialpotential, it is reasonable to keep the general structure of the original McGehee transformationand introduce general functions of r in the transformation. In this sense, we can consider (67) asthe most general Mcgehee transformation in this context. We prove the following result Lemma 16.
Except for potentials which are either power-law ( V ( r ) = Cr − σ ) or logarithmic( V ( r ) = C ln r ) there exists no generalized McGehee transformation capable of decoupling the system(66) in the coordinates ( u, v ) . Proof.
In the new parameter s , the dynamical system is x (cid:48) = ξ ( r ) yy (cid:48) = ξ ( r )Λ( | x | ) x, where prime is derivative with respect to s . Inserting the transformation (67) yields iθ (cid:48) ξ + dξ dr r (cid:48) = ξ ξ ( u + iv ) , (cid:18) iθ (cid:48) ξ + dξ dr r (cid:48) (cid:19) ( u + iv ) + ξ ( u (cid:48) + iv (cid:48) ) = Λ( ξ ) ξ ξ . Taking real and imaginary parts in the first equation determines r (cid:48) and θ (cid:48) as r (cid:48) = uξ ξ dξ /dr ,θ (cid:48) = vξ ξ dξ /dr . (68)34ubstituting into the second equation and separating real and imaginary parts gives v (cid:48) = − uv ξ ( r ) (cid:18) dξ /drdξ /dr + ξ ξ (cid:19) ,u (cid:48) = ξ ( r ) (cid:18) − u dξ /drdξ /dr + v ξ ( r ) ξ ( r ) (cid:19) + ξ ( r ) ξ ( r )Λ( ξ ( r )) ξ ( r ) . (69)To uncouple the system in ( u, v ) it is necessary that no function of r appears in the equations for( u (cid:48) , v (cid:48) ). From the equation for u (cid:48) , we need to impose ξ ( r ) (cid:18) dξ /drdξ /dr (cid:19) = βαξ ( r ) ξ ( r ) ξ ( r ) = α, (70)where α, β are constants with α (cid:54) = 0. The second one fixes ξ as ξ ( r ) = αξ ( r ) ξ ( r ) , which inserted into the first one gives ξ ( r ) dξ /drξ ( r ) dξ /dr = β This equation can be integrated to obtain: ξ ( r ) = c ξ ( r ) β . (71)where c (cid:54) = 0 is a constant. Inserting these expressions into (68)-(69) the dynamical system becomes r (cid:48) = αu ξ dξ /drθ (cid:48) = αvv (cid:48) = − α (1 + β ) uv (72) u (cid:48) = αc ξ ( r ) − β ) Λ( ξ ( r )) + α (cid:0) v − βu (cid:1) . From the expression for u (cid:48) , the system is uncoupled if and only if ξ ( r ) − β ) Λ( ξ ( r )) = α for some constant α , i.e. Λ( r ) is a power-law. Since the potential V is related to Λ by dVdr = − Λ( r ) r it follows that the only case for which the generalized McGehee transformation decouples thesystem is when the potential itself is a power-law or logarithmic (recall that an additive constantis completely irrelevant in the potential V ). (cid:4) eferences [Anabal´on et al., 2009] Anabal´on, A., Deruelle, N., Morisawa, Y., Oliva, J., Sasaki, M., Tempo,D., and Troncoso, R. (2009). Kerr–Schild ansatz in Einstein–Gauss–Bonnet gravity: an exactvacuum solution in five dimensions. Classical and Quantum Gravity , 26(6):065002.[Anabal´on et al., 2011] Anabal´on, A., Deruelle, N., Tempo, D., and Troncoso, R. (2011). Remarkson the Myers-Perry and Einstein–Gauss–Bonnet rotating solutions.
International Journal ofModern Physics D , 20(05):639–647.[Belbruno and Pretorius, 2011] Belbruno, E. and Pretorius, F. (2011). A dynamical system’sapproach to Schwarzschild null geodesics.
Classical and Quantum Gravity , 28(19):195007.[de Moura and Letelier, 2000] de Moura, A. P. and Letelier, P. S. (2000). Chaos and fractals ingeodesic motions around a nonrotating black hole with halos.
Physical Review E , 61(6):6506.[Eddington, 1924] Eddington, A. S. (1924). A comparison of Whitehead’s and Einstein’s formulae.
Nature , 113:192.[Finkelstein, 1958] Finkelstein, D. (1958). Past-future asymmetry of the gravitational field of apoint particle.
Physical Review , 110(4):965.[Gibbons et al., 2005] Gibbons, G. W., L¨u, H., Page, D. N., and Pope, C. (2005). The generalKerr–de Sitter metrics in all dimensions.
Journal of Geometry and Physics , 53(1):49–73.[Hackmann et al., 2008] Hackmann, E., Kagramanova, V., Kunz, J., and L¨ammerzahl, C. (2008).Analytic solutions of the geodesic equation in higher dimensional static spherically symmetricspacetimes.
Physical Review D , 78(12):124018.[Kerr and Schild, 1965] Kerr, R. P. and Schild, A. (1965). A new class of vacuum solutions of theEinstein field equations.
Atti del Congregno Sulla Relativita Generale: Galileo Centenario .[Levin, 2000] Levin, J. (2000). Gravity waves, chaos, and spinning compact binaries.
Physicalreview letters , 84(16):3515.[Levin and Perez-Giz, 2008] Levin, J. and Perez-Giz, G. (2008). A periodic table for black holeorbits.
Physical Review D , 77(10):103005.[Levin and Perez-Giz, 2009] Levin, J. and Perez-Giz, G. (2009). Homoclinic orbits around spinningblack holes. I. Exact solution for the Kerr separatrix.
Physical Review D , 79(12):124013.[M´alek, 2012] M´alek, T. (2012). Exact solutions of general relativity and quadratic gravity inarbitrary dimension. arXiv preprint arXiv:1204.0291 .[McGehee, 1981] McGehee, R. (1981). Double collisions for a classical particle system withnongravitational interactions.
Commentarii Mathematici Helvetici , 56(1):524–557.[Misra and Levin, 2010] Misra, V. and Levin, J. (2010). Rational orbits around charged blackholes.
Physical Review D , 82(8):083001. 36Moeckel, 1992] Moeckel, R. (1992). A nonintegrable model in general relativity.
Communicationsin Mathematical Physics , 150(2):415–430.[Myers and Perry, 1986] Myers, R. C. and Perry, M. (1986). Black holes in higher dimensionalspace-times.
Annals of Physics , 172(2):304–347.[Nordstr¨om, 1918] Nordstr¨om, G. (1918). On the energy of the Gravitation field in Einstein’sTheory.
Koninklijke Nederlandse Akademie van Weteschappen Proceedings Series B PhysicalSciences , 20:1238–1245.[Parry, 2012] Parry, A. R. (2012). A survey of spherically symmetric spacetimes. arXiv preprintarXiv:1210.5269 .[Perez-Giz and Levin, 2009] Perez-Giz, G. and Levin, J. (2009). Homoclinic orbits around spinningblack holes. II. The phase space portrait.
Physical Review D , 79(12):124014.[Pugliese et al., 2011a] Pugliese, D., Quevedo, H., and Ruffini, R. (2011a). Circular motion ofneutral test particles in Reissner-Nordstr¨om spacetime.
Physical Review D , 83(2):024021.[Pugliese et al., 2011b] Pugliese, D., Quevedo, H., and Ruffini, R. (2011b). Motion of charged testparticles in Reissner-Nordstr¨om spacetime.
Physical Review D , 83(10):104052.[Reissner, 1916] Reissner, H. (1916). ¨Uber die eigengravitation des elektrischen Feldes nach derEinsteinschen Theorie.
Annalen der Physik , 355(9):106–120.[Stoica and Mioc, 1997] Stoica, C. and Mioc, V. (1997). The Schwarzschild problem in astrophysics.
Astrophysics and Space Science , 249(1):161–173.[Suzuki and Maeda, 1999] Suzuki, S. and Maeda, K.-i. (1999). Signature of chaos in gravitationalwaves from a spinning particle.