Geodesic equation in Schwarzschild-(anti-)de Sitter space-times: Analytical solutions and applications

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1 PHYSICAL REVIEW D 78, (2008) Geodesic equation in Schwarzschild-(anti-)de Sitter sace-times: Analytical solutions and alications Eva Hackmann* and Claus Lämmerzahl + ZARM, University of Bremen, Am Fallturm, Bremen, Germany (Received 5 May 2008; ublished 22 July 2008) The comlete set of analytic solutions of the geodesic equation in a Schwarzschild-(anti-)de Sitter sace-time is resented. The solutions are derived from the Jacobi inversion roblem restricted to the set of zeros of the theta function, called the theta divisor. In its final form the solutions can be exressed in terms of derivatives of Kleinian sigma functions. The different tyes of the resulting orbits are characterized in terms of the conserved energy and angular momentum as well as the cosmological constant. Using the analytical solution, the question whether the cosmological constant could be a cause of the Pioneer anomaly is addressed. The eriastron shift and its ost-schwarzschild limit is derived. The develoed method can also be alied to the geodesic equation in higher dimensional Schwarzschild sace-times. DOI: 0.03/PhysRevD PACS numbers: Jb, Hq I. INTRODUCTION AND MOTIVATION All solar system observations and almost all other observations related to gravity are erfectly described within Einstein s general relativity. This includes light deflection, the erihelion shift of lanets, the gravitational time-delay (Shairo effect) the Lense-Thirring, and the Schiff effect related to the gravitomagnetic field, as well as strong field effects governing the dynamics of binary systems and, in articular, binary ulsars [ 3]. However, there are two henomena which do not fit into this scheme and still reresent a mystery; that is dark matter and dark energy. Dark matter has been introduced to exlain the galactic rotation curves, gravitational lensing, structure formation, or articular features in the cosmic microwave background. Dark energy is needed to describe the accelerated exansion of the universe. All related observations like the fluctuations in the cosmic microwave background, structure formation, and SN Ia are consistently described by an additional energy-momentum comonent which aears in the Einstein field equation as an additional cosmological term R 2 Rg þ g ¼ T () where is the cosmological constant which, using the indeendent observations mentioned above, has a value of jj 0 52 m 2. As a consequence, it is necessary in rincile to describe all observations related to gravity within a framework including the cosmological constant. However, due to the smallness of the cosmological constant it seems unlikely that this quantity will have a large effect on smaller, that is, on solar system scales. In fact, it has been shown within an *hackmann@zarm.uni-bremen.de + laemmerzahl@zarm.uni-bremen.de aroximation scheme based on the frame given by the Schwarzschild-de Sitter sace-time that the cosmological constant lays no role in all the solar system observations and also not in strong field effects [4,5]. Also within a rotating version of this solution, the Kerr-de Sitter solution, no observable effects arise [6]. Nevertheless, there has been some discussion on whether the Pioneer anomaly, the unexlained acceleration of the Pioneer 0 and sacecraft toward the inner solar system of a Pioneer ¼ ð8:47 :33Þ0 0 m=s 2 [7] which is of the order of ch where H is the Hubble constant, may be related to the cosmological exansion and, thus, to the cosmological constant. The same order of acceleration is resent also in the galactic rotation curves which astonishingly successfully can be modeled using a modified Newtonian dynamics involving an acceleration arameter a MOND which again is of the order of 0 9 m=s 2. Because of this mysterious coincidence of characteristic accelerations aearing at different scales and due to the fact that all these henomena aear in a weak gravity or weak acceleration regime, it might be not clear whether current aroximation schemes hold. This is one motivation to try to solve the equations of motion of test articles in sace-times with cosmological constant analytically. Furthermore, by looking at the effective otential of a oint article moving in the Schwarzschild-de Sitter sacetime it can be seen that for a certain range of orbital arameters a switching on of the cosmological constant may result in a dramatic change of the orbital shae: for a ositive cosmological constant bound orbits may become escae orbits and for a negative cosmological constant escae orbits will become bound orbits, see Fig. 2. The characteristic distance where this haens is given by =2 which is of the order of 5 Gc which is roughly the radius of the visible universe and, thus, far outside the solar system and our galaxy [8]. However, an orbit which is =2008=78(2)=024035(22) Ó 2008 The American Physical Society

2 EVA HACKMANN AND CLAUS LÄMMERZAHL PHYSICAL REVIEW D 78, (2008) near to the searatrix of Schwarzschild geodesics may have a larger sensitivity to a cosmological constant which erhas may not be accounted for to the required accuracy in a erturbative aroach. In other words, it might be that a comaratively large acceleration ch a Pioneer a MOND at solar system or galactic distances may be the result of a very small cosmological constant. Therefore, a definite answer to this question can be given with the hel of an analytical solution only. In addition, the orbits of the Pioneer sacecrafts had been reconstructed using orbit determination rograms relying on the first order ost- Newtonian aroximation. The difference between this aroximation and the exact orbits with cosmological constant may be even more ronounced. There is further interest to understand exlicitly the structure of geodesics in the background of black holes in anti-de Sitter sace in the context of string theory and the AdS/CFT corresondence. In addition, recently there also has been a lot of work dealing with geodesics and integrability in black hole backgrounds in higher dimensions in the resence of a cosmological constant [9 3]. Besides these hysically motivated reasons, it is also of mathematical imortance to derive an exlicit analytical solution of the geodesic equation in a Schwarzschild-de Sitter sace-time. Orbits of articles and light rays have long been used to discuss the roerties of solutions of Einsteins field equations. In fact, the observation of light and articles is the only way to exlore the gravitational field. All solutions of the geodesic equation in a Schwarzschild gravitational field have been resented in a seminal aer of Hagihara [4]. The solution is given in terms of the Weierstrass }-function. With the same mathematical tools one can solve the geodesic equation in a Reissner-Nordström sace-time [5]. The analytic solutions of the geodesic equation in a Kerr and Kerr- Newman sace-time have also been given analytically (see [5] for a survey). Here we exose for the first time the comlete elaboration of the analytic solution of a oint article moving in a Schwarzschild-(anti) de Sitter sacetime resented in [6]. Also the entire set of ossible solutions is described and characterized. For a secialized case, orbits in a Schwarzschild-(anti) de Sitter sace-time have been resented [7,8]. Here we consider the general case of geodesics in the gravitational field of a sherically symmetric mass in a universe with cosmological constant (of any value), described by the Schwarzschild-(anti) de Sitter sacetime. Because of the static metric and the sherical symmetry of the roblem, the geodesic equation reduces to one ordinary differential equation which can be integrated formally by means of a hyerellitic integral. Here we exlicitly solve this integral. Our calculations are based on the mathematically very interesting inversion roblem of hyerellitic Abelian integrals studied first by Jacobi, Abel, Riemann, Weierstrass, and Baker in the 9th century [9 22]. The general ansatz was stated by Kraniotis and Whitehouse [23] and Drociuk [24] (see also [25]), but in addition to these considerations we exlicitly solve the equations of motion by restricting the roblem to the set of zeros of the theta function, the so-called theta divisor. This rocedure makes it ossible to obtain a one-arameter solution of the, in our case, two-arameter inversion roblem. This rocedure was suggested by Enolskii, Pronine, and Richter [26] who alied this method to the roblem of the double endulum. The resulting orbits are classified in terms of the energy and the angular momentum of the test article as well as of the value of the cosmological constant. A detailed discussion of the resulting orbits is given. The found analytical solution then is alied to the question whether the cosmological constant might be the origin of the anomalous acceleration of the Pioneer sacecraft. Over the whole mission, the influence of the cosmological constant leads to a modification in the orbit of the Pioneers of the order of 0 4 m only. The found solution is also used to derive the exact ost-schwarzschild aroximation of the eriastron shift. We also give one examle for the alication of this method to analytically solve the geodesic equation in higher dimensional Schwarzschild, Schwarzschild-(anti-)de Sitter or Reissner-Nordström- (anti-)de Sitter sace-times. II. THE GEODESIC EQUATION We consider the geodesic equation 0 ¼ d2 x dx ds 2 þ dx (2) ds ds where ds 2 ¼ g dx dx is the roer time along the geodesics and ¼ 2 g ð@ g g Þ (3) is the Christoffel symbol, in a sace-time given by the metric ds 2 ¼ r S r 3 r2 dt 2 r S r 3 r2 dr 2 r 2 ðd 2 þ sin 2 d Þ; (4) which describes the sherically symmetric vacuum solution of (). This Schwarzschild-de Sitter metric is characterized by the Schwarzschild-radius r S ¼ 2M related to the mass M of the gravitating body, and the cosmological constant (unless stated otherwise we use units where c ¼ G ¼ ). The main features of this metric deending on the value of the cosmological constant are shown in Fig.. For a general discussion of this metric, see e.g. [27,28]. The geodesic equation has to be sulemented by the dx normalization condition g dx ds ds ¼ where for massive articles ¼ and for light ¼

3 GEODESIC EQUATION IN SCHWARZSCHILD (ANTI-)DE... PHYSICAL REVIEW D 78, (2008) FIG.. The tt-comonent of the Schwarzschild-de Sitter metric for various values for. The dotted line corresonds to the Schwarzschild metric. For 0 < < =ð9m 2 Þ there are two horizons. The dashed line corresonds to the extremal Schwarzschild-de Sitter sace-time where the two horizons coincide. For r<r and r>r þ the radial coordinate becomes timelike. Because of the sherical symmetry we can restrict our consideration to the equatorial lane. Furthermore, due to the conserved energy and angular momentum dt E ¼ g tt ds ¼ r S r dt 3 r2 ds ; (5) L ¼ r 2 d ds ; (6) the geodesic equation reduces to one ordinary differential equation dr 2 r 4 ¼ d L 2 E 2 r S r 3 r2 þ L2 r 2 : (7) Together with energy and angular momentum conservation we obtain the corresonding equations for r as functions of s and t dr 2 ¼ E 2 r S ds r 3 r2 þ L2 r 2 ; (8) dr 2 ¼ dt E 2 r S r 2 3 r2 E 2 r S r 3 r2 þ L2 r 2 : (9) Equations (7) (9) give a comlete descrition of the dynamics. Equation (8) suggests the introduction of an effective otential V eff ¼ 2 3 L2 r S r þ L2 r 2 r SL 2 r 3 3 r2 (0) shown in Fig. 2. It is worthwhile to note that for light, i.e. ¼ 0, the cosmological constant just gives a constant contribution to the effective otential and, thus, does not influence (7) and (8). However, it still influences the motion of light through the timing formula (9). FIG. 2. The effective otential of a oint article with some given L in a Schwarzschild-de Sitter sace-time for different cosmological constants. As usual, we introduce a new variable u ¼ r S =r and obtain du 2 ¼ u d 3 u 2 þ u þðð ÞþÞþ u 2 () with the dimensionless arameters :¼ r2 S L 2 ; :¼ E 2 and :¼ 3 r2 S : (2) We rewrite () as u du 2 ¼ P5 ðuþ d (3) with P 5 ðuþ :¼ u 5 u 4 þ u 3 þðð ÞþÞu 2 þ : (4) If not stated otherwise, we take ¼ in the following. Note that 0 and 0. A searation of variables in (3) yields 0 ¼ Z u u 0 du 0 ffiffiffiffiffiffiffiffiffiffiffiffiffi ; (5) u 0 P 5 ðu 0 Þ where u 0 ¼ uð 0 Þ. In solving integral (5) there are two major issues which have to be addressed. First, the integrand is not well defined in the comlex lane because of the two branches of the square root. Second, the solution uð Þ should not deend on the integration ath. If denotes some closed integration ath and I udu ffiffiffiffiffiffiffiffiffiffiffiffi ¼! (6) P 5 ðuþ this means that 0! ¼ Z u u 0 du 0 ffiffiffiffiffiffiffiffiffiffiffiffiffi (7) u 0 P 5 ðu 0 Þ should be valid, too. Hence, the solution uð Þ of our roblem has to fulfill uð Þ ¼uð!Þ (8)

4 EVA HACKMANN AND CLAUS LÄMMERZAHL PHYSICAL REVIEW D 78, (2008) for every! Þ 0 obtained from an integration (6). A function u with the roerty (8) is called a eriodic function with eriod!. These two issues can be solved if we consider Eq. (5) to be defined on the Riemann surface X of the algebraic function x ffiffiffiffiffiffiffiffiffiffiffi P 5 ðxþ. III. THE INVERSION PROBLEM Let X be the comact Riemannian surface of the algebraic function x ffiffiffiffiffiffiffiffiffiffiffi P 5 ðxþ. It can be reresented as the algebraic curve X :¼ fz ¼ðx; yþ 2C 2 jy 2 ¼ P 5 ðxþg (9) ffiffiffiffiffi [29] or as the analytic continuation of P 5. The last one can be realized as a two-sheeted covering of the Riemann shere which can be constructed in the following way: let e i, i ¼ ;...; 5, be the zeros of P 5 and e 6 ¼ (for a olynomial of 6th order the zero e 6 is finite). These are the so-called branch oints. Now take two coies of the Riemann ffiffiffiffiffi shere, one for each of the two ossible values of P 5, and cut them between every two of the branch oints e i in such a way that the cuts do not touch each other. These are the so-called branch cuts, see Fig. 3. Of course, the two coies have to be ffiffiffiffiffi identified at the branch oints where the two values of P 5 are identical. They are then glued ffiffiffiffiffi together along the branch cuts in such a way that P 5 together with all its analytic continuations is uniquely defined on the whole surface. On this surface x ffiffiffiffiffiffiffiffiffiffiffi P 5 ðxþ is now a single-valued function. This construction can be visualized as a retzel, see Fig. 3. For a strict mathematical descrition of the construction of a comact Riemannian surface, see [30], for examle. Every Riemannian surface can be equied with a homology basis fa i ;b i ji ¼ ;...;gg2h ðx; ZÞ of closed aths as shown in Fig. 3, where g is the genus of the Riemannian surface, see the next section. From the construction of the Riemannian surface it is already clear that integrals over these closed aths indeed do not evaluate to zero and, hence, have to be eriods of the solution of (5). The task now is to analyze the details of eriodic functions on such Riemannian surfaces. A. Preliminaries Comact Riemannian surfaces are characterized by their genus g. This can be defined as the dimension of the sace of holomorhic differentials on the Riemannian surface or, toologically seen, as the number of holes of the Riemannian surface. Let P d ¼ P d s¼0 s x s be a olynomial of degree d with ffiffiffiffiffiffi only simle zeros and X be the Riemann surface of P d. Then the genus of X is equal to g ¼½ d 2 Š, where ½xŠ denotes the largest integer less or equal than x [3]. Hence, in our case of P 5 the genus of the Riemann surface is g ¼ 2. In order to construct eriodic functions on a Riemann surface we first have to define a canonical basis of the sace of holomorhic differentials fdz i ji ¼ ;...;gg and of associated meromorhic differentials fdr i ji ¼ ;...;gg on the Riemann surface by dz i :¼ xi dx ffiffiffiffiffiffiffiffiffiffiffi ; (20) P d ðxþ dr i :¼ 2gþ i X k¼i x k dx ðk þ iþ kþþi 4 ffiffiffiffiffiffiffiffiffiffiffi ; (2) P d ðxþ with j being the coefficients of the olynomial P d [3]. In our case these differentials are given by dz :¼ ffiffiffiffiffiffiffiffiffiffiffi dx ; dz 2 :¼ ffiffiffiffiffiffiffiffiffiffiffi xdx ; (22) P 5 ðxþ P 5 ðxþ dr :¼ 3x3 2x 2 þ x 4 ffiffiffiffiffiffiffiffiffiffiffi dx; dr 2 :¼ x2 dx P 5 ðxþ 4 ffiffiffiffiffiffiffiffiffiffiffi ; (23) P 5 ðxþ where is defined in (2). We also introduce the eriod matrices ð2!; 2! 0 Þ and ð2; 2 0 Þ related to the homology basis FIG. 3. Riemannian surface of genus g ¼ 2, with real branch oints e ;...;e 6. Uer figure: Two coies of the comlex lane with closed aths giving a homology basis fa i ;b i ji ¼ ;...;gg. The branch cuts (thick solid lines) are chosen from e 2i to e 2i, i ¼ ;...;gþ. Lower figure: The retzel with the toologically equivalent homology basis. 2! ij :¼ I a j dz i ; 2! 0 ij : ¼ I b j dz i ; 2 ij :¼ I a j dr i ; 2 0 ij : ¼ I b j dr i : (24) The differentials in (20) and (2) have been chosen such that the comonents of their eriod matrices fulfill the Legendre relation

5 GEODESIC EQUATION IN SCHWARZSCHILD (ANTI-)DE... PHYSICAL REVIEW D 78, (2008)!! 0 0 g!! 0 t ¼ 0 g i 0 g ; g 0 C. Theta functions (25) where g is the g g unit matrix, [3]. Finally we introduce the normalized holomorhic differentials 0 dz dz 2 d ~v :¼ ð2!þ d~z; d~z ¼ C A : (26) dz g The eriod matrix of these differentials is given by ð g ;Þ, where is defined by :¼!! 0 : (27) It can be shown [32] that this normalized matrix always is a Riemannian matrix, that is, is symmetric and its imaginary art Im is ositive definite. B. Jacobi s inversion roblem Let us consider now the Abel ma A x0 :¼ X! JacðXÞ; x Z x d~z (28) x 0 from the Riemannian surface X to the Jacobian JacðXÞ ¼ C g = of X, where ¼f!v þ! 0 v 0 j v; v 0 2 Z g g is the lattice of eriods of the differential d~z. The image A x0 ðxþ of X by this continuous function is of comlex dimension one and, thus, an inverse ma A x 0 is not defined for all oints of JacðXÞ. However, the g-dimensional Abel ma A x0 : S g X! JacðXÞ; ðx ;...;x g Þ t Xg i¼ Z xi x 0 d~z (29) from the gth symmetric roduct S g X of X (the set of unordered vectors ðx ;...;x g Þ t where x i 2 X) to the Jacobian is one-to-one almost everywhere. Jacobi s inversion roblem is now to determine ~x for given ~ from the equation ~ ¼ A x0 ð~xþ: (30) In our case g ¼ 2 this reads ¼ Z x dz ffiffiffiffiffiffiffiffiffiffiffi þ Z x 2 dz ffiffiffiffiffiffiffiffiffiffiffi ; x 0 P 5 ðzþ x 0 P 5 ðzþ 2 ¼ Z x zdz ffiffiffiffiffiffiffiffiffiffiffi þ Z x 2 zdz ffiffiffiffiffiffiffiffiffiffiffi : x 0 P 5 ðzþ x 0 P 5 ðzþ (3) We will see later, that we can solve our roblem (5) asa limiting case of this Jacobi inversion roblem. The Riemannian surface of genus g has 2g indeendent closed aths, each corresonding to a eriod of the functions defined on these surfaces and, hence, to a eriod of the solution u of (5). In order to construct 2g-eriodic functions, we need the theta function #: C g! C, #ð~z; Þ :¼ X ~m2z g e i ~mtð ~mþ2~zþ : (32) The series on the right-hand side converges absolutely and uniformly on comact sets in C g and, thus, defines a holomorhic function in C g. This is obvious from the estimate Re ð ~m t ðiþ ~m t Þ c ~m t ~m for some constant c> 0, what follows from the fact that Re ðiþ is negative definite. The theta function is already eriodic with resect to the columns of g and quasieriodic with resect to the columns of, i.e., for any n 2 Z g the relations #ð~z þ g ~n; Þ ¼#ð~z; Þ; (33) #ð~z þ ~n; Þ ¼e i ~nt ð ~nþ2~zþ #ð~z; Þ (34) hold. We will also need the theta function with characteristics [33] ~g, ~h 2 2 Zg defined by #½ ~g; ~hšð~z; Þ :¼ X ~m2z g e ið ~mþ ~gþtðð ~mþ ~gþþ2~zþ2 ~hþ ¼ e i ~gt ð ~gþ2~zþ2 ~hþ #ð~z þ ~g þ ~h; Þ: (35) Later it will be imortant that for every ~g, ~h the set ~gþ ~h : ¼f~z 2 C g j #½ ~g; ~hšð~z; Þ ¼0g, called a theta divisor,isa(g )-dimensional subset of JacðXÞ, see [32]or (49). The solution of Jacobi s inversion roblem (3) can be exlicitly formulated in terms of functions closely related to the theta function. First, consider the Riemann theta function Z x # e ðx; Þ :¼ # d ~v ~e; ; (36) x 0 with some arbitrary but fixed ~e 2 C g. The Riemann vanishing theorem, see e.g. [32], states that the Riemann theta function is either identically to zero or has exactly g zeros x ;...;x g for which X g i¼ Z xi x 0 d ~v ¼ ~e þ ~K x0 (37) holds (modulo eriods). Here ~K x0 2 C g is the vector of Riemann constants with resect to the base oint x 0 given by ( jj is the jth diagonal element) K x0 ;j ¼ þ jj X I Z x dv 2 j dv l ðxþ: (38) lþj a l x 0 If the base oint x 0 is equal to, this vector can be

6 EVA HACKMANN AND CLAUS LÄMMERZAHL PHYSICAL REVIEW D 78, (2008) determined by ~K ¼ Xg i¼ Z e2i d ~v; (39) where e 2i is the starting oint of one of the branch cuts not containing for each i, see [3]. Hence, ~K can be exressed as a linear combination of half-eriods in this case. For roblems of hyerellitic nature it is usually assumed that the Riemann theta function # e does not vanish identically. However, here we are interested in the oosite case: we want to restrict Jacobi s inversion roblem (3) to the set of zeros of #ð þ ~K x0 ; Þ, which is called the theta divisor K ~ x0. The solution of (3) and, thus, of (3) can be formulated in terms of the derivatives of the Kleinian sigma function : C g! C, ð~zþ ¼Ce ð=2þ~zt! ~z #ðð2!þ ~z þ ~K x0 ; Þ; (40) where the constant C can be given exlicitly, see [3], but does not matter here. Jacobi s inversion roblem can be solved in terms of the second logarithmic derivative of the sigma function called the generalized Weierstrass functions } ij logð~zþ ¼ ið~zþ j ð~zþ ð~zþ ij j 2 ; ð~zþ (4) where i denotes the derivative of the sigma function with resect to the ith comonent. D. The solution of the Jacobi inversion roblem The solution of Jacobi s inversion roblem (30) can be given in terms of generalized Weierstrass ffiffiffi functions. Let X be the Riemannian surface of P where P is without restriction defined by PðxÞ :¼ P 2gþ i¼0 i x i (this form can always be achieved by a rational transformation). Then the comonents of the solution vector ~x ¼ðx ;...;x g Þ t are given by the g solutions of 2gþ x g Xg } gi ð ~ Þx i ¼ 0; (42) 4 i¼ where ~ is the left-hand side of (30). Since ~x 2 S 2 X there is no way to define an order of the comonents of ~x. In our case of g ¼ 2, we can rewrite this result with the hel of the theorems by Vieta in the form x þ x 2 ¼ 4 5 } 22 ð ~ Þ; x x 2 ¼ 4 5 } 2 ð ~ Þ: (43) IV. SOLUTION OF THE EQUATION OF MOTION IN SCHWARZSCHILD-(ANTI-)DE SITTER SPACE-TIME Now we aly the results of the receding section to the roblem of the equation of motion in Schwarzschild-(anti-) de Sitter sace-time, Eqs. (3) and (5). As already mentioned before, the solution of the equation of motion can be found as a limiting case of the solution of Jacobi s inversion roblem in the case of genus g ¼ 2. A. The analytic exression To start with, we rewrite Jacobi s inversion roblem (3) in the form ¼ Z u dx ffiffiffiffiffiffiffiffiffiffiffi þ Z u 2 dx ffiffiffiffiffiffiffiffiffiffiffi ; P 5 ðxþ P 5 ðxþ 2 ¼ Z u xdx ffiffiffiffiffiffiffiffiffiffiffi þ Z (44) u 2 xdx ffiffiffiffiffiffiffiffiffiffiffi ; P 5 ðxþ P 5 ðxþ where ~ ¼ ~ 2 Z d~z: (45) u 0 Note that the right-hand side of (44) is exactly ~A ð ~uþ, the image of the Abel ma defined in (29), i.e. ~ ¼ ~A ð ~uþ.we use the obvious identity (comare [26]) u u ¼ lim u 2 (46) u 2! u þ u 2 and insert the solution of Jacobi s inversion roblem (43). Then } u ¼ lim 2 ð Þ ~ u 2! } 22 ð Þ ~ ð Þ ¼ lim ~ 2 ð Þ ~ ð Þ ~ 2 ð Þ ~ u 2! 2 2 ð Þ ~ 22 ð Þ ~ ¼ ð ~ Þ 2 ð ~ Þ ð ~ Þ 2 ð ~ Þ 2 2 ð ~ Þ ð ~ Þ 22 ð ~ ; (47) Þ where ~ ¼ lim ~ ¼ ~A ð ~u Þ (48) u 2! with ~u ¼ð u Þ. Note that the definition of the sigma function (40) and, hence, of the generalized Weierstrass functions (4) includes the vector of Riemann constant ~K x0 with x 0 ¼in our case, which is given by ~K ¼ =2 þ 0 =2 =2 (see (39) or[32]). The above limiting rocess also transfers Jacobi s inversion roblem to the theta divisor K ~. With ð2!þ ~ ¼ ð2!þ ~A ð ~u Þ¼ R u d ~v and the theorem

7 GEODESIC EQUATION IN SCHWARZSCHILD (ANTI-)DE... PHYSICAL REVIEW D 78, (2008) =2 0 # ; ð~z; Þ ¼0,9x: ~z ¼ Z x d ~v =2 =2 (49) rð Þ ¼ r S uð Þ ¼ r 2 ð ~ Þ S ð ~ Þ ¼ r 2 ð ~ Þ S ð ~ Þ : (57) roven by Mumford [32] it follows that =2 0 0 ¼ # ; ðð2!þ =2 =2 ~ ; Þ: (50) Via Eq. (35), this is equivalent to 0 ¼ # ð2!þ ~ þ =2 =2 and with (40) this means þ 0 ; ; (5) =2 ð ~ Þ¼0: (52) We first use this result in (47) and obtain u ¼ ð ~ Þ 2 ð ~ Þ : (53) Theorem (49) also tells us that ð2!þ ~ is an element of the theta divisor K ~, i.e. the set of zeros of =2 0 # ; ; =2 =2 and that, in the case g ¼ 2, K ~ is a manifold of comlex dimension one. Note that the restriction to the theta divisor is only ossible because is a branch oint what is essential for the validity of theorem (49). Since K ~ is a one-dimensional subset of C 2, there is a one-to-one functional relation between the first and the second comonent of ð2!þ ~. By the definition of ~ in (47) and Eq. (45) we have ~ ¼ lim ~ ¼ lim ~ 2 Z d~z ¼ Z u d~z Z d~z: u 2! u 2! u 0 u 0 u 0 (54) The hysical coordinate is given by (5), ¼ Z u zdz ffiffiffiffiffiffiffiffiffiffiffi þ 0 ¼ Z u dz 2 þ 0 : (55) u 0 P 5 ðzþ u 0 We insert this in (54) and obtain R u u ~ ¼ 0 dz R! R u0 dz 0 R u u ¼ 0 dz R! u0 dz u0 dz 2 0 ; 0 (56) where 0 0 ¼ 0 þ R u0 dz 2 deends only on the initial values u 0 and 0. We choose for each a such that ~ :¼ 0 0 is equal to ~. Then ð2!þ ~ ¼ð2!Þ ~ is an element of the theta divisor K ~ and we finally obtain This is the analytic solution of the equation of motion of a oint article in a Schwarzschild-(anti-)de Sitter sacetime. This solution is valid in all regions of the Schwarzschild-(anti-)de Sitter sace-time and for both signs of the cosmological constant and can be comuted with arbitrary accuracy. The exlicit comutation of the solution is described in Aendix A. B. Light trajectories In the case of light trajectories, the situation simlifies considerably. The equation of motion is then given by du 2 ¼ u 3 u 2 þ þ d 3 ¼ : P 3 ðuþ: (58) Light rays are uniquely given and, thus, uniquely characterized by the extremal distance to the gravitating body, that is, the smallest or largest distance (in the case of a Schwarzschild sace-time, it is also ossible due to its asymtotic flatness to take the imact arameter as characteristic of a light ray). This extremal distance u 0 is characterized by du ¼ 0; (59) d u¼u0 which gives u 3 0 u2 0 þ þ 3 ¼ 0. Then our equation of motion is du 2 ¼ u 3 u 2 u 3 0 d þ u2 0 (60) which is the same tye of equation as in Schwarzschild geometry. With a substitution u ¼ 4x þ 3 this reads dx 2 ¼ 4x 3 g d 2 x g 3 (6) where g 2 :¼ 2 ; g 3 : ¼ 8 27 þ 2 ðu3 0 u2 0 Þ (62) are the Weierstrass invariants. This differential equation can be solved directly in terms of ellitic functions, i.e. rð Þ ¼ r S uð Þ ¼ r S r 4xð Þþ ¼ S 3 4}ð 0 0 ; g 2;g 3 Þþ ; 3 (63) where } is the Weierstrass function [34,35] and 0 0 is given by the initial values 0 and x 0, 0 0 ¼ 0 þ R dx x0 ffiffiffiffiffiffiffiffi. The P 3 ðxþ corresonding light trajectories have been exhaustively discussed in [4]. In a recent aer [36], Rindler and Ishak discussed light deflection in a Schwarzschild-de Sitter sace-time. Though

8 EVA HACKMANN AND CLAUS LÄMMERZAHL PHYSICAL REVIEW D 78, (2008) the equation of motion is the same as in Schwarzschild sace-time, the measuring rocess for angles reintroduces the cosmological constant in the observables. According to their scheme, the exact angle between the radial direction and the satial direction of the light ray is now given by v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r S u rð Þ 3 r2 ð Þ tan ¼ t ð r S r 0 Þ ð r S rð Þ Þj ; (64) j r2 ð Þ r 2 0 where in the exression dr=d from (7) the E2 L 2 þ 3 has been relaced by the r 0 related to u 0. This now is valid for all light rays, not only for those rays showing a small deflection as discussed in [36]. V. THE CLASSIFICATION OF THE SOLUTIONS A. General classification Having an analytical solution at hand we can exlore the set of all ossible solutions in a systematic manner. The shae of an orbit deends on the energy E and the angular momentum L of the article under consideration as well as the cosmological constant (the Schwarzschild radius has been absorbed through a rescaling of the radial coordinate). These quantities are all contained in the olynomial P 5 ðuþ through the arameters, and (2). Since r (and u) should be real and ositive it is clear that the hysically accetable regions are given by those u for which E>V eff. The zeros of P 5 are related to the oints of intersection of E and V eff, and a real and ositive P 5 is equivalent to E>V eff as can also be seen from (3). Hence, the number of ositive real zeros of P 5 uniquely characterizes the form of the resulting orbit. Since P 5 goes to if x! and to if x!, P 5 ð0þ is ositive if the number of ositive real zeros of P 5 is even and negative if it is odd. If we denote by e ;...;e n the ositive real zeros, then it follows that the hysically accetable regions are given by ½0;e Š; ½e 2 ;e 3 Š;...; ½e n ; Š if n is even and by ½e ;e 2 Š;...; ½e n ; Š if n is odd. With resect to r we have the following classes of orbits (see Fig. 4): (i) the region ½0;e Š corresonds to escae orbits, (ii) the region ½e n ; Š corresonds to orbits falling into the singularity, i.e. to terminating orbits, and (iii) the regions ½e i ;e iþ Š corresond to bound orbits. This means that for any arrangement of zeros of P 5 there exist terminating orbits. Furthermore, for an even number of ositive real zeros we have escae orbits and for more than three real ositive zeros we have bound orbits. The case that there is no ositive real zero corresonds to a article coming from infinity and falling to the singularity, see Fig. 4. Quasieriodic bound orbits exists only if there are three or more ositive zeros. It can be shown that there are no more than four real ositive zeros for our olynomial (4): We decomose the olynomial P 5 into its (in general comlex) zeros P 5 ðuþ ¼ ðu u Þðu u 2 Þðu u 3 Þðu u 4 Þðu u 5 Þ. Multilication and comarison of the coefficients of the terms linear in u yields u u 2 u 3 u 4 þ u u 2 u 3 u 5 þ u u 2 u 4 u 5 þ u u 3 u 4 u 5 þ u 2 u 3 u 4 u 5 ¼ 0: (65) The assumtion that all zeros are real and ositive contradicts Eq. (65). Therefore, in any case there are at most four real ositive zeros. Figure 5 shows the arrangement of zeros of P 5 ðuþ for some chosen values of as a ð; Þ diagram. The code of gray scales is as follows: black corresonds to four, dark gray to three, gray to two, light gray to one, and white to no ositive real zero. B. Discussion with resect to Based on Fig. 5 we are now discussing the orbits related to different values of. Each Plot in Fig. 5 comrises all effective otentials (of the form shown in Fig. 2) for all ossible values of L and all article energies E and, thus, contains the comlete information about all orbits in Schwarzschild-(anti-)de Sitter sace-times for a given value of. (a) Let us first consider ¼ 0. In this case the constant term in P 5 ðuþ vanishes and P 5 ðuþ ¼u 2 ~P 3 ðuþ so that u ¼ 0 is a zero of P 5 ðuþ with multilicity 2. The olynomial ~P 3 corresonds to the Schwarzschild sace-time and has been extensively discussed in [4]. Nevertheless, let us examine some regions of FIG. 4. The five ossibilities of having real ositive zeros of P 5 ðuþ. The allowed regions of article motion are shaded in gray. The zeros corresond to the zeros of V eff ¼ E in Fig. 2 (note that u ¼ 0 corresonds to r ¼ 0 and u ¼ to r ¼ 0). Bound nonterminating, quasieriodic orbits exist only if there are three or more ositive zeros

9 GEODESIC EQUATION IN SCHWARZSCHILD (ANTI-)DE... PHYSICAL REVIEW D 78, (2008) FIG. 5. The zeros of P 5 ðuþ in a ð; Þ-diagram for different values for the cosmological constant ( is along the x-axis, along the y-axis). The gray scales encode the numbers of ositive real zeros of the olynomial P 5 : black ¼ 4, dark gray ¼ 3, gray ¼ 2, light gray ¼, white ¼ 0. In the lot for ¼ 0 characteristic lines are shown (the left uer oint of the dark gray region has the coordinates ( ¼ 8 9, ¼ 3 ); the uer intersection oint of the ¼ line with the dark gray region is at ¼ 4 ). ð; Þ and ossible orbits so that they can be directly comared with orbits for Þ 0. As seen in Fig. 5 (d), the straight line ¼ divides the lot in two arts. For < there is an odd number of ositive real zeros, i.e. we may not have any escae orbits in these regions. The light gray area corresonds to one real ositive zero and, therefore, it is only a terminating orbit ossible whereas in the dark gray region there may be in addition a bound orbit. For there is an even number of ositive real zeros and, thus, there is always an orbit which reaches infinity. The gray region corresonds to two real ositive zeros and, hence, to a terminating and an escae orbit. The white region reresents the case of no ositive real zeros, i.e. an orbit which comes from infinity and falls into the singularity. Also, beside u ¼ 0 there is a further real zero with multilicity 2 on the straight line ¼. (b) Let us comare now this with the case > 0, see Fig. 5(e) 5(g). We immediately recognize that left to the ¼ line the lot significantly changed. In addition, we notice that for growing this straight line shifts a bit to the left. Left of the straight -line there is now one more ositive real zero in each region. This means that a article which for ¼ 0 is in a light gray region now is in a gray region and, thus, may reach infinity. The same haens in the region which was dark gray for ¼ 0 and is now black. A article with < in the small band now right of our straight line switched to a gray or white region deending on its value. For a large ositive cosmological constant the black area will disaear, that is, there will no longer be any bound orbit. This is clear from the following (cf. Fig. 2): First we introduce r :¼ r=r S.For ¼ 0 the effective otential V eff ossesses two different ex

10 EVA HACKMANN AND CLAUS LÄMMERZAHL PHYSICAL REVIEW D 78, (2008) trema r if < 3. The smaller extrema r is a maximum whereas the larger r þ is a minimum. The extrema are bounded by r < 3 and r þ > 2 3 > 3. If we add the term containing > 0, which is of the form of a arabola, r shifts to the left and r þ to the right. Thus, r < 3 and r þ > 3 remain valid. In general, a second maximum r > r þ will aear and V eff! for r!. It follows that there will be no bound orbit if the minimum r þ and, thus, the maximum r disaears. This is fulfilled if we choose such large that the gradient is negative for all r>3, for examle r 2 S > Since > 3, the choice r2 S > 9 ensures that for any choice of < 3 there will be no bound orbits. This is of course only a rough estimate which can be imroved. (c) If < 0, the situation changed the other way around. We again immediately see that the right side of the lot significantly changed. The straight -line is no longer so easy to identify, but if we take into account the number of all real zeros, we can say that it shifts to the right when the absolute value of growths. An excetion to this is the art for small. There the line bends to the right and allows a switch from the gray art for ¼ 0 to the light gray art of < 0. Nevertheless, we can say that on the right side of the lot we have now an additional real ositive zero and, thus, also an odd number of ositive real zeros. This means that a article is no longer able to reach infinity for any ð; Þ (cf. Fig. 2). In the region being white for ¼ 0 and which now is light gray we have now a bound terminating orbit. In the for ¼ 0 gray region which now is dark gray the escae orbit becomes bound. C. Plots of orbits In Figs. 6 8 some of the ossible orbits are lotted. The figures are organized in order to highlight the influence of the cosmological constant. For all orbits in each figure the arameters and, that is, E and L, are the same. The absolute value of the cosmological constant is chosen as jj ¼0 5 in all lots. All lots are created from the analytical solution derived in Sec. IVA. In Fig. 6 the arameters are ¼ 0:92 and ¼ 0:28 which belong to the dark gray region in Fig. 5(d). Fora vanishing cosmological constant this corresonds to a bound eriodic orbit and to a bound terminating orbit ending in the singularity. The corresonding orbits are shown in Figs. 6(a) and 6(b). For a ositive cosmological constant the overall structure changes considerably since there will be a third tye of orbits not resent in the Schwarzschild case. Beside the terminating and bound orbits in Fig. 6(c) which both look quite similar to the corresonding orbits in the Schwarzschild case there is an escae orbit which is reelled from the otential barrier related to the ositive cosmological constant, Fig. 6(e). FIG. 6. Orbits for ¼ 0:92 and ¼ 0:28. The uer row is for vanishing, the lower row for ositive. (a) and (c): bound orbits with erihelion shift. (b) and (d) terminating orbit ending in the singularity. (e): Reflection at the -barrier. There is no analogue of (e) for ¼ 0. Black circles always indicate the Schwarzschild radius

11 GEODESIC EQUATION IN SCHWARZSCHILD (ANTI-)DE... PHYSICAL REVIEW D 78, (2008) FIG. 7. Orbits for ¼ : and ¼ 0:2. The uer row is for vanishing, the lower row for negative. The next arameter choice is ¼ : and ¼ 0:2. For vanishing these arameters lay in the gray area of Fig. 5 (d) denoting two zeros and, thus, corresond to a quasihyerbolic escae orbit, Fig. 7(a), and a terminating orbit ending in the singularity, Fig. 7(b). For the chosen ¼ 0 5 km 2 the situation changes dramatically as can be read off from Fig. 5(a) in comarison to Fig. 5(d): Now we have three zeros and, thus, one bound orbit and one terminating orbit ending in the singularity, see Figs. 7(c) and 7 (d). Switching on the negative cosmological constant makes an escae orbit a bound orbit. This of course has to be exected as one can see from Fig. 2 that there are no escae orbits for negative cosmological constant. Our third choice of arameters is ¼ 0:8 and ¼ 0:2. For ¼ 0, Fig. 5(d) this lays in the light gray region of one zero where is a terminating orbit only. This orbit is shown in Fig. 8(a). For a ositive cosmological constant > 0 these arameters are in a gray region with two zeros indicating a terminating and an escae orbit, see Figs. 8 (b) and 8(c). The orbit in Fig. 8(c) again is a reflection at the -barrier. VI. ON THE PIONEER ANOMALY We aly the obtained analytical solution in order to decide whether a nonvanishing cosmological constant may have an observable influence on the Pioneer satellites. From [37] we may deduce the energy and angular momentum of the Pioneers after their last flybys at Juiter and Saturn, resectively, with resect to the barycenter of the inner solar system, i.e. the Sun, Mercury, Venus, and Earth- Moon. This means that we used the value r S ¼ 2GM c 2 ¼ 2: km; (66) for the Schwarzschild radius, derived from GM ¼ : k 2 ðau 3 =day 2 Þ with Gauss constant k ¼ 0: defining the astronomical unit AU. Here all numbers are taken with 2 digits what corresonds to the today s accuracy of solar system ehemerides. In the case of Pioneer 0, the velocity at infinity v ¼ :322 km s taken from [37] gives us the energy er unit mass E M ¼ c 2 þ 2 v2 and therefore the arameter, ¼ E2 M ¼ : : (67) c4 The angular momentum er unit mass is given by L M ¼ q qv, where v ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2GMð q þ 2a Þ is the velocity at eriasis distance q; a is the semimajor axis. From this we derive the arameter, ¼ r2 S c2 L 2 ¼ 2: : (68) M In the case of Pioneer we obtain for the arameters and ¼ : ; ¼ : : (69)

12 EVA HACKMANN AND CLAUS LÄMMERZAHL PHYSICAL REVIEW D 78, (2008) FIG. 8. Orbits for ¼ 0:8 and ¼ 0:2. The uer grah is for vanishing, the lower grahs for ositive. There is no analogue of (c) for ¼ 0. With these coefficients we now can determine the exact orbits of Pioneer 0 and in the cases ¼ 0 and ¼ 0 45 km 2. From these exact orbits we calculated the differences in osition (in m) for a given angle (in rad) and the difference in the angle (in rad) for a given distance r (in m) of a test article moving in a sace-time with and without cosmological constant. The Pioneer anomaly aeared in a heliocentric distance from about 20 to 70 AU. For r in this range, we comute now ¼0 ðrþ Þ0 ðrþ with and without cosmological constant for both craft. Regarding Pioneer 0, the difference is in the scale of 0 9 rad, which corresonds to an azimuthal difference in osition of about 0 6 m. For Pioneer, the difference is in the scale 0 8 rad, which corresonds to an azimuthal difference in osition of about 0 5 m. The range of 20 to 70 AU corresonds to an angle between 0:4 and 0:6 if 0 ¼ 0 corresonds to the eriasis. In this range, we comute the radial difference r ¼0 ð Þ r Þ0 ð Þ also for both craft. For Pioneer 0 we obtain a difference in the scale of 0 5 m, for Pioneer in the scale of 0 4 m. Therefore we can say, that for the resent value of the cosmological constant the form of the Pioneer 0 orbit ractically does not change. For a definite estimate of the differences of the Pioneer orbits in Schwarzschild and Schwarzschild-de Sitter sace-time one of course has to analyze the time course of these orbits. However, the time variable is influenced by the cosmological constant in the same way as the radial coordinate so that no change in our statement will occur. Therefore, the influence of the cosmological constant on the orbits cannot be held resonsible for the observed anomalous acceleration of the Pioneer sacecraft. VII. PERIASTRON ADVANCE OF BOUND ORBITS In the case that P 5 has at least three real and ositive zeros, we may have a bound orbit for some initial values. The eriastron advance eri for such a bound orbit is given by the difference of the 2-eriodicity of the angle and the eriodicity of the solution rð Þ (which is the same as the eriodicity of uð Þ). Let us assume that the bound orbit corresonds to the interval ½e k ;e kþ Š, where e k and e kþ are real and ositive zeros of P 5, and that the ath a i surrounds this real interval. Then the eriastron advance is given by eri ¼ 2 2! 2i ¼ 2 2 Z e kþ xdx ffiffiffiffiffiffiffiffiffiffiffi ; (70) e k P 5 ðxþ where 2! 2i is an element of the (canonically chosen) 2 4 matrix of eriods ð2!; 2! 0 ffiffiffiffiffi Þ of P 5, see Eq. (24). We now calculate the ost-schwarzschild limit of this eriastron advance in the case that the considered bound orbit is also bound in Schwarzschild sace-time. For doing so we first exand x= ffiffiffiffiffiffiffiffiffiffiffi P 5 ðxþ to first order in x ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi x 2 þ P 5 ðxþ P 3 ðxþ 6 r2 S x 2 P 3 ðxþ ffiffiffiffiffiffiffiffiffiffiffi ; (7) P 3 ðxþ where P 3 ðxþ ¼x 3 x 2 þ x þ ð Þ is the olyno

13 GEODESIC EQUATION IN SCHWARZSCHILD (ANTI-)DE... PHYSICAL REVIEW D 78, (2008) mial for the corresonding Schwarzschild case given by ¼ 0. In the next ste we have to integrate both terms involving P 3 within the Weierstrass formalism, see for examle [38]. Emloying the substitution x ¼ 4z þ =3 we rewrite P 3 in a Weierstrass form where P 3 ðxþ ¼4 2 ð4z 3 g 2 z g 3 Þ¼4 2 P W ðzþ; (72) g 2 ¼ 2 4 (73) g 3 ¼ 6 ð 2 27 þ 2 3 Þ (74) are the Weierstrass invariants. We assume that the orbit under consideration is bound not only in the Schwarzschild-de Sitter but also in the corresonding Schwarzschild sace-time. This means that the three largest real zeros of P 5 are ositive and, thus, the zeros z > z 2 >z 3 > 2 of P ffiffiffiffiffiffiffi W are all real. The square root P W is branched over ffiffiffiffiffiffiffi z, z 2 and z 3 and, thus, the ellitic function } based on P W has a urely real eriod! and a urely imaginary eriod! 2. They are given by! ¼ I dz ffiffiffiffiffiffiffiffiffiffiffiffiffi A P W ðzþ! 2 ¼ I dz ffiffiffiffiffiffiffiffiffiffiffiffiffi (75) B P W ðzþ where the ath A runs around the branch cut from z 3 to z 2 and the ath B around z 2 and z, both clockwise. The branch ffiffiffiffiffiffiffi of the square root in (75) ffiffiffiffiffiffiffi is chosen such that P W > 0 on ½z 3 ;z 2 Š and, thus, P W negatively imaginary on ½z 2 ;z Š. The branch oints of } can be exressed in terms of the eriods: z ¼ }ð Þ, z 2 ¼ }ð 2 Þ and z 3 ¼ }ð 3 Þ with ¼! =2, 2 ¼ð! þ! 2 Þ=2 and 3 ¼! 2 =2. The fundamental rectangle in the comlex lane sanned by the eriods!,! 2 of } is denoted by R ¼ fx! þ y! 2 j0 x; y < g, see Fig. 9. Let the three biggest real and ositive zeros of P 5 be given by x >x 2 >x 3 > 0. Then, ffiffiffiffiffi for the canonical choice of the matrix of eriods! of P 5, the integration ath ai runs from x 3 to x 2 and back with conversed sign of the square root. Let the ath be the reimage of a i by u }ðuþ ¼z in the fundamental rectangle R. For a ositive cosmological constant, wehavex 3 <z 3 <z 2 <x 2 and, thus, starts at some urely imaginary ð0þ ¼u 2 R with 0 < Im ðu ÞIm ð 3 Þ and goes straight to ðþ ¼ u þ!. Then, for any rational function F, we obtain I dz FðzÞ ffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Z Fð}ðuÞÞdu: (76) a i P W ðzþ This is derived from the differential equation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi } 0 ðuþ ¼ 4}ðuÞ 3 g 2 }ðuþ g 3 ¼ P W ð}ðuþþ; (77) where the branch of the square root was chosen to be consistent with the sign of } 0. FIG. 9. The fundamental rectangle R. The integration of the first art on the right-hand side of (7) is straightforward and yields the Schwarzschild eriod dx ffiffiffiffiffiffiffiffiffiffiffi ¼ Iai I dz ffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Z u þ! du ¼! : (78) P 3 ðxþ a i P W ðzþ u The integration of the second art on the right-hand side of (7) is more involved and is erformed in Aendix B. As a result we obtain the first order aroximation of the eriastron shift with resect to : eri ¼ 2 I xdx a i ¼ 2 ffiffiffiffiffiffiffiffiffiffiffi P 5 ðxþ! þ r2 S X 3 þ z j! 96 j¼ } 00 ð j Þ 2 þ ð4z j þ 3 Þ2 2 6 þ! 6} 0 ðu 0 Þ 4 þ 6 } 00 ðu 0 Þ 6 } 0 ðu 0 Þ 5 ð u 0 þ ðu 0 ÞÞ þ Oð 2 Þ; (79) where u 0 is such that }ðu 0 Þ¼ 2. The terms in this exression involving j and u 0 can artly be relaced by terms containing the Weierstrass invariants g 2 and g 3. From the differential equation (77) we derive qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi } 0 ðu 0 Þ¼ 4}ðu 0 Þ 3 g 2 }ðu 0 Þ g 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 432 þ g 2 2 g 3: (80) The first derivative of (77) yields 2} 00 ¼ 2} 2 g 2 and, thus, gives } 00 ð j Þ¼6z 2 j 2 g 2 and } 00 ðu 0 Þ¼ 24 2 g 2; (8) where the g 2, g 3, as well as the zeros of P W can be exressed by and. The result (79) gives the ost-schwarzschild eriastron shift in a closed algebraic form. The advantage of this result is that no further integration is needed. Another advantage lies in the fact that only ellitic functions and

14 EVA HACKMANN AND CLAUS LÄMMERZAHL PHYSICAL REVIEW D 78, (2008) related quantities are used which are well described and tabulated in mathematical books and which are also well imlemented in common commercial math rograms. What is still left to do is to exress the result (79) in terms of, e.g., r min and r max or, equivalently, in terms of the semimajor axis and the eccentricity. These quantities are directly observable and also have the advantage that an exansion in terms of m=r min and m=r max can be erformed giving in addition a ost-newtonian exansion. This will be described elsewhere. Let us aly these formulas to the erihelion advance of Mercury and comare with the results of Kraniotis and Whitehouse, [23]. We take the values r S ¼ r S L 2 M ¼ 2953:25008 m for the Schwarzschild-radius, : m 2 =s 2 for the angular ffiffiffiffiffiffiffi momentum er unit mass L M and E M ¼ 0: m=s for the energy er unit mass E M given in [23]. These values lead to the zeros of P W and to the eriods z ¼ 0: ; z 2 ¼ 0: ; z 3 ¼ 0: ! ¼ 3: ;! 2 ¼ 20: i; ¼ 6: i; which all comare well to the results in [23]. Also the hysical data, i.e. the ahel r A, the erihel r P and the erihelion advance in Schwarzschild-sace-time S, comare well to [23] and also to observations [39]: r A ¼ 6: m; r P ¼ 4: m; S ¼ 42: arcsec cy : (82) Here we used the rotation eriod days of Mercury and 00 SI-years er century to determine the unit arcsec cy. The first order ost-schwarzschild correction SdS corr to the erihelion advance can now be calculated from formula (79). For a cosmological constant of ¼ 0 5 m 2 we obtain for the arameters which aear in the exansion (79) ¼ 0: ; } 02 ðu 0 Þ¼ : ; } 00 ðu 0 Þ¼: ; ðu 0 Þ¼: i: This leads to a correction of SdS corr ¼ 5: arcsec cy : (83) This result also comares well to [23] where the erihel advance of Merkur does not change within the given accuracy when considered in Schwarzschild-de Sitter sace-time. The value of the correction is also far beyond the measurement accuracy of 0:002 arcsec cy for the erihelion advance of Mercury. However, for an extreme case the influence of the cosmological constant on the eriastron advance may become more ronounced. The orbital data of quasar QJ287 reorted in [40,4] indicates that the correction to the eriastron advance SdS corr will be some orders of magnitude larger than the correction in the case of Merkur. Indeed, when we calculate from this data the energy arameter and the angular momentum arameter, ¼ 0:982 66; ¼ 0:092 37; (84) we obtain SdS corr 0 3 arcsec cy : (85) VIII. GEODESICS IN HIGHER DIMENSIONAL SCHWARZSCHILD SPACE-TIMES We want to show here that our method for solving the equation of motion in Schwarzschild-(anti-)de Sitter sace-times can also be alied to solve the geodesic equation in, e.g., higher dimensional Schwarzschild sace-times. The metric of a Schwarzschild sace-time in d dimensions is given by [42] ds 2 rs d 3 ¼ dt 2 r rs d 3 dr 2 r r 2 d 2 d 2 ; (86) where d 2 ¼ d 2 and d 2 iþ ¼ d i þ sin 2 i d 2 i for i. Because of sherical symmetry, we again restrict the considerations to the equatorial lane by setting i ¼ 2 for all i. With the conserved energy E and angular momentum L as well as the substitution u ¼ r S r the geodesic equation reduces to du 2 ¼ u d þ u d 3 u 2 þ ð Þ ¼P d d ðuþ; (87) where the arameters ¼ r2 S and ¼ E 2 have the same L 2 meaning as in the Schwarzschild-(anti-)de Sitter case (2). For d ¼ 4 this equation reduces of course to the Schwarzschild case [4]. For d ¼ 5 a substitution u ¼ x þ l where l is a zero of P 4 reduces the differential equation (87) to dx 2 ¼ b3 x 3 þ...þ b d 0 x 0 : (88) With an additional substitution x ¼ b 3 ð4y b 2 3 Þ this equa

15 GEODESIC EQUATION IN SCHWARZSCHILD (ANTI-)DE... PHYSICAL REVIEW D 78, (2008) FIG. 0. Arrangement of zeros of the olynomial P d for d ¼ 6. The gray scale code is the same as in Fig. 5. tion acquires the form (6) which can be solved by Weierstrass ellitic functions. In the case of a d ¼ 6-dimensional Schwarzschild sace-time, however, the differential equation (87) comrises a olynomial of degree five on the right-hand side. This now can be solved by means of our method. The only difference is that the hysical angle is now given by 0 ¼ Z u du 0 ffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Z u dz u 0 P 5 ðu 0 (89) Þ u 0 what corresonds to dz rather than to dz 2 as it was in the Schwarzschild-(anti-)de Sitter case. This means that the solution of the geodesic equation in six-dimensional Schwarzschild sace-time is given by where rð Þ ¼ r S uð Þ ¼ r 2 ð ~;6 Þ S ð ~;6 Þ ; (90) ~ ;6 ¼ 0 0 Here is chosen in such a way that ð2!þ ~ ;6 is an element of the theta divisor K ~ and 0 0 ¼ 0 þ R u0 dz deends only on the initial values u 0 and 0. The case d ¼ 7 corresonds to a olynomial P 6 of degree six. If we aly a substitution u ¼ x þ l where l is a zero of P 6, we obtain the differential equation x dx 2 ¼ b5 x 5 þ...b d 0 x 0 (9) with some constants b i. This can be solved in exactly the same way as the differential equation (3). The solution is rð Þ ¼ r S uð Þ ¼ r 2 ð ~;7 Þ S ð ~;7 Þ ; (92) where ~;7 ¼ 0 0 and, again, is selected such that ð2!þ ~;7 is an element of the theta divisor ~ K and 0 0 ¼ 0 þ R u0 dz 2. The only difference to the solution of the geodesic equation : FIG.. Orbits for chosen values of and in six-dimensional Schwarzschild sace-time. The black circle indicates the Schwarzschild radius