Analytic solutions of the geodesic equation in static spherically symmetric spacetimes in higher dimensions

Analytic solutions of the geodesic equation in static spherically symmetric spacetimes in higher dimensions Eva Hackmann 2, Valeria Kagramanova, Jutta Kunz, Claus Lämmerzahl 2 Oldenburg University, Germany 2 ZARM, Bremen University, Germany MG 2, Paris 2.07. 8.07.2009 Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 / 54

Table of contents Preliminaries to geodesics 2 Topology of orbits 3 Analytical solution of geodesic equation 4 Geodesics in higher dimensional spacetimes Schwarzschild Schwarzschild in 9D Schwarzschild in D Schwarzschild de Sitter Schwarzschild de Sitter in 9D Schwarzschild de Sitter in D Reissner Nordström Reissner Nordström in 7D Reissner Nordström de Sitter Reissner Nordström de Sitter in 4D Reissner Nordström de Sitter in 7D 5 Conclusion and Outlook Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 2 / 54

Outline Preliminaries to geodesics Preliminaries to geodesics 2 Topology of orbits 3 Analytical solution of geodesic equation 4 Geodesics in higher dimensional spacetimes Schwarzschild Schwarzschild in 9D Schwarzschild in D Schwarzschild de Sitter Schwarzschild de Sitter in 9D Schwarzschild de Sitter in D Reissner Nordström Reissner Nordström in 7D Reissner Nordström de Sitter Reissner Nordström de Sitter in 4D Reissner Nordström de Sitter in 7D 5 Conclusion and Outlook Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 3 / 54

Preliminaries to geodesics Table: which one is integrable? AIM is: Integrate analytically geodesics in higher dimensional space times Space time 4 5 6 7 8 9 0 2 Dimension Schwarzschild - - - Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 4 / 54

Preliminaries to geodesics Table: which one is integrable? AIM is: Integrate analytically geodesics in Schwarzschild 9,) Space time 4 5 6 7 8 9 0 2 Dimension Schwarzschild - - - Schwarzschild de Sitter - - - - Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 4 / 54

Preliminaries to geodesics Table: which one is integrable? AIM is: Integrate analytically geodesics in Schwarzschild anti-)de Sitter 9,) Space time 4 5 6 7 8 9 0 2 Dimension Schwarzschild - - - Schwarzschild de Sitter - - - - Reissner Nordström - - - - - - Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 4 / 54

Preliminaries to geodesics Table: which one is integrable? AIM is: Integrate analytically geodesics in Reissner Nordström 7) Space time 4 5 6 7 8 9 0 2 Dimension Schwarzschild - - - Schwarzschild de Sitter - - - - Reissner Nordström - - - - - - Reissner Nordström de Sitter - - - - - - integration by elliptic functions integration by hyperelliptic functions Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 4 / 54

Preliminaries to geodesics Table: which one is integrable? AIM is: Integrate analytically geodesics in Reissner Norström anti-)de Sitter 4,7) Space time 4 5 6 7 8 9 0 2 Dimension Schwarzschild - - - Schwarzschild de Sitter - - - - Reissner Nordström - - - - - - Reissner Nordström de Sitter - - - - - - integration by elliptic functions integration by hyperelliptic functions Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 4 / 54

Preliminaries to geodesics Table: which one is integrable? AIM is: Integrate analytically geodesics in higher dimensional space times: Space time 4 5 6 7 8 9 0 2 Dimension Schwarzschild - - - Schwarzschild de Sitter - - - - Reissner Nordström - - - - - - Reissner Nordström de Sitter - - - - - - integration by elliptic functions integration by hyperelliptic functions Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 4 / 54

Preliminaries to geodesics The METRIC in static spherically symmetric spacetime in d dimensions: ds 2 = g tt dt 2 g rr dr 2 r 2 dω 2 d2 with dω 2 = dϕ2 and dω 2 i = dθ2 i sin2 θ i dω 2 i, i ) The motion of a test particle with 4 velocity u is given by the geodesic equation D u u = 0 with gu,u) = ǫ In Reissner Nordström de Sitter spacetime in d dimensions: g tt = grr rs ) d3 2Λr 2 q ) 2d3) = r d )d 2) with r s = 2M r The equation of motion ) 2 [ d r = λ r 4 µ ) dϕ r 2 Λ r 2 d3 d )d 2) ηd3 ǫ )] r 2d3) λ r 2 λ = r2 s L, µ = 2 E2, η = q2, rs 2 Λ = Λr 2 s, r = r r s The effective potential V eff = ) r 2 Λ r 2 d3 d )d 2) ηd3 ǫ ) r 2d3) λ r 2 Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 5 / 54

Preliminaries to geodesics Geodesic equation For a space time of a dimension d a degree of a polynomial P n in the RHS of the equation of motion is: Λ = 0 Λ 0 d r dϕ d r dϕ η = 0 η 0 ) 2 ) 2 = P n d r r with n = d n4 dϕ = P n r with n = 2d 2) ) n4 2 ) 2 = P n d r r with n = d n6 dϕ = P n r with n = 2d ) n6 For odd dimensions a transformation u = r 2 reduces the EOM to 4 ) 2 du = η d3 u d λη d3 u d2 u 2 d) λu 2 d) u 2 dϕ u λµ ) ) 2 Λ d )d 2) 2 Λλ d )d 2) Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 6 / 54

Outline Topology of orbits Preliminaries to geodesics 2 Topology of orbits 3 Analytical solution of geodesic equation 4 Geodesics in higher dimensional spacetimes Schwarzschild Schwarzschild in 9D Schwarzschild in D Schwarzschild de Sitter Schwarzschild de Sitter in 9D Schwarzschild de Sitter in D Reissner Nordström Reissner Nordström in 7D Reissner Nordström de Sitter Reissner Nordström de Sitter in 4D Reissner Nordström de Sitter in 7D 5 Conclusion and Outlook Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 7 / 54

Topology of orbits Depending on the ST EOM through a transformation r = fx) takes one of two forms 2 2 x dϕ) dx = dx P5 x) dϕ) = P5 x) with P 5 x) = a 5 x 5 a 4 x 4 a 3 x 3 a 2 x 2 a x a 0. Separation of variables yields the hyperelliptic integral x in = xϕ in )) ϕ ϕ in = x x dx x in P5x ) ϕ ϕ in = x dx x in P5x ) P 5 P 5 P 5 x ) No real positive zero P 5 x 2) real positive zero P 5 x 3) 2 real positive zeros P 5 Up to 5 real positive zeros of P 5 x) Possible orbits: infinite) terminating, escape, bound x x x 4) 3 real positive zeros 5) 4 real positive zeros 6) 5 real positive zeros Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 8 / 54

Outline Analytical solution of geodesic equation Preliminaries to geodesics 2 Topology of orbits 3 Analytical solution of geodesic equation 4 Geodesics in higher dimensional spacetimes Schwarzschild Schwarzschild in 9D Schwarzschild in D Schwarzschild de Sitter Schwarzschild de Sitter in 9D Schwarzschild de Sitter in D Reissner Nordström Reissner Nordström in 7D Reissner Nordström de Sitter Reissner Nordström de Sitter in 4D Reissner Nordström de Sitter in 7D 5 Conclusion and Outlook Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 9 / 54

Analytical solution of geodesic equation A homology basis of hyperelliptic curve of genus 2 All periods Copy of C b b 2 Copy 2 of C e e 2 e 3 e 4 e 5 e 6 a a 2 e e 2 e 3 e 4 e 5 e 6 b b 2 a a 2 Riemann surface of genus 2: b b 2 A basis of holomorphic differentials d z and matrices of a and b periods 2ω ij and 2ω ij i =,..., g): dx xdx dz =, dz2 = P5x) P5x) 2ω ij = dz i, 2ω ij = dz i. a j b j Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 0 / 54

Analytical solution of geodesic equation soln The analytical solution of the geodesic equation is: rϕ) = fxϕ)) = f σ ) ϕ ) = f σ 2 ϕ ) σ x x in d z σ 2 x x in d z ) x d z) in x in d z) The corresponding form of EOM determined by the ST) defines the physical angle ϕ: ϕ = x x in dz 2 ϕ in = x x in xdx P5 x) ϕ in, ϕ = Thereafter depending on the problem ϕ is: x x in dz ϕ in = x x in dx P5 x) ϕ in ϕ,z2 = x x in dz ) x in dz ϕ ϕ ϕ ϕ ) ϕ,z = x x in dz 2 x in dz 2 where ϕ = ϕ in x in dz 2 and ϕ = ϕ in x in dz E. Hackmann and C. Lämmerzahl, Phys. Rev. Lett. 00, 70 2008) E. Hackmann and C. Lämmerzahl, Phys. Rev. D 78, 024035 2008) Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 / 54

Outline Geodesics in higher dimensional spacetimes Preliminaries to geodesics 2 Topology of orbits 3 Analytical solution of geodesic equation 4 Geodesics in higher dimensional spacetimes Schwarzschild Schwarzschild in 9D Schwarzschild in D Schwarzschild de Sitter Schwarzschild de Sitter in 9D Schwarzschild de Sitter in D Reissner Nordström Reissner Nordström in 7D Reissner Nordström de Sitter Reissner Nordström de Sitter in 4D Reissner Nordström de Sitter in 7D 5 Conclusion and Outlook Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 2 / 54

Outline Geodesics in higher dimensional spacetimes Schwarzschild Preliminaries to geodesics 2 Topology of orbits 3 Analytical solution of geodesic equation 4 Geodesics in higher dimensional spacetimes Schwarzschild Schwarzschild in 9D Schwarzschild in D Schwarzschild de Sitter Schwarzschild de Sitter in 9D Schwarzschild de Sitter in D Reissner Nordström Reissner Nordström in 7D Reissner Nordström de Sitter Reissner Nordström de Sitter in 4D Reissner Nordström de Sitter in 7D 5 Conclusion and Outlook Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 3 / 54

TABLE Geodesics in higher dimensional spacetimes Schwarzschild Table: Types of geodesics in Schwarzschild space times; lim r 0 r V eff = ) 2 d r [ dϕ = r λµ ) r 3 λ r 2 r ] d = 4 µ > terminating bound, escape, terminating escape orbit µ < terminating bound, periodic bound orbit 2 d r dϕ) = λµ ) r 4 λ ) r 2 d = 5 µ > λ > λ < λ > terminating escape terminating bound, escape, terminating escape orbit terminating bound orbit µ < λ < terminating bound orbit ) 2 d r [ dϕ = r λµ ) r d r d3 λ r 2 ] d5 d 6 µ > terminating bound, escape, terminating escape orbit µ < terminating bound orbit Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 4 / 54

Geodesics in higher dimensional spacetimes Schwarzschild space time in 9D Schwarzschild ) 2 du = 4u u 4 λu 3 u λµ ) ) = 4P 5 u) dϕ The solution rϕ) = σ2 ϕ,z ) σ ϕ,z ) V eff 4 µ = 3.9 λ 3 µ = 2.6 2 µ 0 0 0 20 green 2 positive roots, grey and white no real roots; V eff for λ = 0.5 r Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 5 / 54

Geodesics in higher dimensional spacetimes Schwarzschild Schwarzschild space time in 9D: Orbits for λ = 0.5 0.5 20 0 0.5 0 0.5 0 0.5 20 0 0 0 20 0.5 0 0.5 20 0 0 0.5 0 0.5 20 0 7) µ = 2.6: terminating bound and escape orbit 8) µ = 3.9: terminating bound and escape orbit 0 5 0.5 0.5 0.5 0 0.5 20 0 0 0.5 0 0.5 0 5 0 0.5 0.5 0 5 9) µ = 3.9203: terminating bound and escape orbit 0) µ = 3.92046627: terminating bound and escape orbit Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 6 / 54

Geodesics in higher dimensional spacetimes Schwarzschild 5 0 5 0 5 ) µ = 4.0: terminating escape orbit A deflection angle for an escape orbit of a test particle in Schwarzschild space times in 9 dimensions is ϕ = 2 0 u e { du 4P5 u) π = 4ω π if Re i > u e, 4ω 2 π if Re i < u e where u e is a zero of P 5 u) corresponding to an escape orbit and Re i is the real part of a corresponding complex root e i. Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 7 / 54

Geodesics in higher dimensional spacetimes Schwarzschild Schwarzschild space time in D ) 2 du = 4u u 5 λu 4 u λµ ) ) = 4P 6 u) x dx ) 2 = 4P 5 x) dϕ dϕ with u = x u 6; The solution rϕ) = ) σ2 ϕ,z 2 ) σ ϕ,z2 ) u 2 6 V eff λ 2 µ = 2. µ =.6 µ =.00 µ 0 0 5 30 45 r green 2 positive roots, grey and white no real roots; V eff for λ = 0.4 Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 8 / 54

Geodesics in higher dimensional spacetimes Schwarzschild Schwarzschild space time in D: Orbits for λ = 0.4 0 5 0.5 0.5 0.5 0 0.5 20 0 0 0.5 0 0.5 0 5 0 0.5 0.5 0 5 2) µ = 2.: terminating bound and escape orbit 3) µ = 2.76008452: terminating bound and escape orbit The deflection angle for an escape orbit of a test particle in the Schwarzschild space time in dimensions is ϕ = 2 x e xdx 4P5 x) π = 4ω 2 4ω 22 π, where x e is a zero of P 5 x) corresponding to an escape orbit. Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 9 / 54

Outline Geodesics in higher dimensional spacetimes Schwarzschild de Sitter Preliminaries to geodesics 2 Topology of orbits 3 Analytical solution of geodesic equation 4 Geodesics in higher dimensional spacetimes Schwarzschild Schwarzschild in 9D Schwarzschild in D Schwarzschild de Sitter Schwarzschild de Sitter in 9D Schwarzschild de Sitter in D Reissner Nordström Reissner Nordström in 7D Reissner Nordström de Sitter Reissner Nordström de Sitter in 4D Reissner Nordström de Sitter in 7D 5 Conclusion and Outlook Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 20 / 54

Geodesics in higher dimensional spacetimes Schwarzschild de Sitter Schwarzschild de Sitter space time in 9D Table ) d r 2 = [ ) ] 2λ Λ 2 Λ dϕ r d5 d )d 2) rd λµ ) r d r d3 λ r 2 d )d 2) ) du 2 = 4 u 5 λu 4 u 2 λµ ) Λ ) u Λ ) dϕ 28 28 λ = 4P 5 u) V eff Λ < 0 A µ B Λ > 0 r Figure: Potential for negative and positive cosmological parameter Λ. For the same value of a test particle energy µ it can be on a bound orbit with perihelion and aphelion points A and B or on an escape one with a point of nearest approach A. Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 2 / 54

Geodesics in higher dimensional spacetimes Schwarzschild de Sitter Schwarzschild de Sitter space time in 9D λ λ Figure: Λ > 0; green 2 positive roots, white no real roots µ µ Figure: Λ < 0; blue 3 positive roots, grey Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 22 / 54

Geodesics in higher dimensional spacetimes Schwarzschild de Sitter Schwarzschild de Sitter space time in 9D: Orbits Λ > 0 20 40 0.5 20 0.5 0 0.5 0 0.5 0 20 40 60 80 00 0.5 0 0.5 20 0 0 0 20 0.5 20 0.5 0 40 20 ) µ =.008: terminating bound and escape orbit 2) µ =.8: terminating bound and escape orbit 0 5 0.5 0.5 0.5 0 0.5 20 0 0 0.5 0 0.5 0 5 0 0.5 0.5 0 5 3) µ = 3.3: terminating bound and escape orbit 4) µ = 3.392073469: terminating bound and escape orbit Figure: Examples of a test particle s motion in 9 dimensional SdS space time with positive cosmological constant Λ = 8.7 0 5 and λ = 0.2 Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 23 / 54

Geodesics in higher dimensional spacetimes Schwarzschild de Sitter Schwarzschild de Sitter space time in 9D 2: Orbits Λ > 0 5 0 5 5 0 5 0 0 5 ) µ = 3.392: terminating escape orbit 2) µ = 3.5: terminating escape orbit Figure: Examples of a test particle s motion in 9 dimensional SdS space time with positive cosmological constant Λ = 8.7 0 5 and λ = 0.2 Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 24 / 54

Geodesics in higher dimensional spacetimes Schwarzschild de Sitter Schwarzschild de Sitter space time in 9D: Orbits Λ < 0 40 40 0.5 20 0.5 20 0.5 0 0.5 40 20 0 20 40 0.5 0 0.5 400 200 0 200 400 0.5 20 0.5 20 40 40 ) µ =.008: terminating bound and periodic bound orbit 2) µ =.5: terminating bound and periodic bound orbit Figure: Orbits of a test particle in 9 dimensional SadS space time for Λ = 8.7 0 5 and λ = 0.2 For a periodic bound orbit restricted to the interval [r min, r max] there is a perihelion shift perihel = 2π 2 ei e i dx 4P5 x) = 2π 4ω k, where e i and e i are that zeros of P 5 x) related to r max and r min. The ω k depends on the position of the complex zeros of P 5 x). Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 25 / 54

Geodesics in higher dimensional spacetimes Schwarzschild de Sitter Schwarzschild de Sitter space time in D ) 2 du = 4 u 6 λu 5 u 2 λµ ) Λ ) u Λ ) dϕ 45 45 λ = 4P 6 u) with u = x u 6 x dx dϕ) 2 = 4P5 x) with polynom P 5 x) V eff µ = 4.38 λ 4 3 2 µ = 3.0 µ =.0 0 µ 0 20 40 green 2 positive roots, white no real roots; V eff for λ = 0.5 r Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 26 / 54

Geodesics in higher dimensional spacetimes Schwarzschild de Sitter Schwarzschild de Sitter space time in D: Orbits 40 20 0.5 20 0.5 0 0.5 0 0.5 0 20 40 60 80 0.5 0 0.5 20 0 0 0 20 0.5 20 0.5 0 40 20 ) µ =.0: terminating bound and escape orbit 2) µ = 3.0: terminating bound and escape orbit 0 0.5 0.5 0 0.5 0.5 20 0 0 0 3) µ = 4.38: terminating bound and escape orbit Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 27 / 54

Outline Geodesics in higher dimensional spacetimes Reissner Nordström Preliminaries to geodesics 2 Topology of orbits 3 Analytical solution of geodesic equation 4 Geodesics in higher dimensional spacetimes Schwarzschild Schwarzschild in 9D Schwarzschild in D Schwarzschild de Sitter Schwarzschild de Sitter in 9D Schwarzschild de Sitter in D Reissner Nordström Reissner Nordström in 7D Reissner Nordström de Sitter Reissner Nordström de Sitter in 4D Reissner Nordström de Sitter in 7D 5 Conclusion and Outlook Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 28 / 54

TABLE Geodesics in higher dimensional spacetimes Reissner Nordström Table: Types of geodesics in Reissner Nordström space times; lim r 0 r V eff = ) 2 d r dϕ = λµ ) r 4 λ r 3 λη) r 2 r η d = 4 µ > periodic bound, escape µ < 2 periodic bound ) 2 [ d r dϕ = r λµ ) r 6 2 λ ) r 4 λη 2 ) r 2 η 2] λη 2 > 0 escape λ > λη 2 < 0 escape d = 5 µ > λη 2 > 0 periodic bound, escape λ < λη 2 < 0 escape λη 2 > 0 periodic bound λ > λη 2 < 0 periodic bound µ < λη 2 > 0 periodic bound λ < λη 2 < 0 not allowed ) 2 [ d r dϕ = r 2d4) λµ ) r 2d2) r 2d3) λ r d r d3 λη d3 r 2 η d3] d 6 µ > periodic bound, escape µ < periodic bound Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 29 / 54

Geodesics in higher dimensional spacetimes Reissner Nordström Reissner Nordström space time in 7D ) 2 du = 4u η 4 u 5 η 4 λu 4 u 3 λu 2 u λµ ) ) = 4P 6 u) dϕ 2 with u = x u 6: x dϕ) dx = 4P5 x) 2 V eff µ =.9 2 V eff µ =.9 µ =.82 µ =.6 µ =.0 µ =. 0 0 0 20 r 0 0 0 r Potential for λ = 0.35, η = 0.7 Potential for λ = 0.35, η = 0.2 Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 30 / 54

Geodesics in higher dimensional spacetimes Reissner Nordström Reissner Nordström space time in 7D λ λ µ µ Figure: η = 0.7 Figure: η = 0.2 FEATURES: Existence of two horizons, an inner Cauchy) and an outer horizon which for d = 7 are given by r inner = 2 2 2 ) 4η 4 4 router = 2 2 2 ) 4η 4 4 Antigravit. potential barrier preventing a particle from falling into the singularity: lim r 0 V eff = Degener. RN ST for η = 2. Extrema are: r = 2 /4 and r 2 = λ λ 2 5 2 V eff r ) = 0 and V eff r 2) = MaxV eff ) Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 3 / 54

Geodesics in higher dimensional spacetimes Reissner Nordström Reissner Nordström ST in 7D: Orbits η = 0.2 and η = 0.7 0.5 0 0.5 5 0.5 0.5 0 0.5 0 5 0 5 20.5 0.5 0 0.5.5 20 0 0 0.5 0.5 5 0 0.5 ) µ =.0, η = 0.2: many world periodic bound and escape orbit 2) µ =.82, η = 0.2: many world periodic bound and escape orbit 0 0 0.5 0.5 0.5 0 0.5 5 0 5 0 5 20 0.5 0 0.5 20 0 0 0.5 0.5 0 0 3) µ =., η = 0.7: many world periodic bound and escape orbit 4) µ =.6, η = 0.7: many world periodic bound and escape orbit Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 32 / 54

Geodesics in higher dimensional spacetimes Reissner Nordström Reissner Nordström ST in 7D: Orbits η = 0.2 and η = 0.7 5 5 0 5 0 0 5 0 0.2 5 0.5 5 0.2 5) µ =.9, η = 0.2, two world escape orbit 6) µ =.9, η = 0.7: two world escape orbit The perihelion shift for periodic bound orbits is perihel = 2π 2 ei e i xdx 4P5 x) = 2π 4ω 2i, where the zeros e i and e i of P 5 x) are again related to the range of motion [r min, r max]. ω 2i depends on the position of the other zeros of P 5 under consideration. Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 33 / 54

Geodesics in higher dimensional spacetimes Reissner Nordström I I 0 t I III II I r I r r I 0 r = 0 r = r = 0 r = I r = r = 0 I r = I III r = r r = r r = r r = r III r = r r = r r = r r = r III II II II II v III r = r r = r r = r r = r III r = r r = r r = r r = r III r = 0 r = I r = 0 r = I r = 0 I I 0 I u r = I r = I I 0 Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 34 / 54

Outline Geodesics in higher dimensional spacetimes Reissner Nordström de Sitter Preliminaries to geodesics 2 Topology of orbits 3 Analytical solution of geodesic equation 4 Geodesics in higher dimensional spacetimes Schwarzschild Schwarzschild in 9D Schwarzschild in D Schwarzschild de Sitter Schwarzschild de Sitter in 9D Schwarzschild de Sitter in D Reissner Nordström Reissner Nordström in 7D Reissner Nordström de Sitter Reissner Nordström de Sitter in 4D Reissner Nordström de Sitter in 7D 5 Conclusion and Outlook Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 35 / 54

Geodesics in higher dimensional spacetimes Reissner Nordström de Sitter Reissner Nordström de Sitter space time in 4D ) ) 2 d r = Λλ r 6 3λµ ) dϕ 3 Λ ) r 4 3λ r 3 3ηλ ) r 2 3 r 3η = 3 P6 r) with r = x r 6: x dx dϕ ) 2 = P 5x) 3 λ µ blue region indicates 3 positive roots, grey and red 5 Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 36 / 54

Geodesics in higher dimensional spacetimes Reissner Nordström de Sitter Reissner Nordström de Sitter space time in 4D Example for the blue region in λ µ diagram: Example for the red region in λ µ diagram: 5 V eff µ = 5.0 V eff µ = 0.888 4 3 µ = 2. 2 µ = 2.0 µ =.3 µ = 0.95 0 0 0 20 30 r 0.9 0.8 0 3 6 0.7 0 20 40 60 r λ = 0. λ = 0.3576404 Effective potential for η = 0. and Λ = 8.7 0 5 Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 37 / 54

Geodesics in higher dimensional spacetimes Reissner Nordström de Sitter Reissner Nordström de Sitter ST in 4D: Orbits 40 0 0.5 20 0.5 0.5 0 0.5 0 20 40 60 80 0.5 0 0.5 20 0 0 0.5 20 0.5 0 40 8) µ = 0.95: many world periodic bound and escape orbit 9) µ =.3: many world periodic bound and escape orbit 0 5 5 0.5 0.5 0 0.5 20 0 0 0 5 0 0 5 0 0.5 0 5 5 0) µ = 2: many world periodic bound and escape orbit ) µ = 2., two world escape orbit Figure: Orbits in RNdS ST in 4 dimensions for λ = 0. and η = 0., Λ = 8.7 0 5 2) µ = 5.0, two world escape orbit Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 38 / 54

Geodesics in higher dimensional spacetimes Reissner Nordström de Sitter Reissner Nordström de Sitter ST in 4D: 2 bound regions 4 40 2 20 4 2 0 2 4 0 20 40 60 80 2 20 40 4 ) many world periodic bound orbit 2) periodic bound orbit 3) escape orbit Figure: Three geodesics in RN de Sitter space time in 4 dimensions for the parameters λ = 0.3576404, µ = 0.888, η = 0., and Λ = 8.7 0 5 Since in general there are two periodic bound orbits we may obtain two perihelia shifts. Each is given by ei j) perihel = 2π 2 xdx P5 x)/3 = 2π 4ω 2k, e i the zeros e i and e i of P 5x) correspond to the range of motion [r j) min, rj) max ]; the path a k surrounds the interval [e i, e i]. j {} bound orbit or j {, 2} 2 bound orbits. Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 39 / 54

Geodesics in higher dimensional spacetimes Reissner Nordström de Sitter Reissner Nordström de Sitter ST in 4D: Naked Singularity V eff 3 2 η = 0.33 Horizons: NS condition: η > 2 Λ 9 Λ Λ 3 r4 r 2 r η = 0 ) 2 cos 3 ) 3 ) 8 89 Λ 2 Λ 2 arctan 880 Λ8 Λ 2 µ =.0 Figure: λ = 0.3, Λ = 8.7 0 5 η = 0.26 0 0 3 6 r 0 0 0.5 0.5 0 0.5 20 0 0 20 0 0 0.5 0 0 ) periodic bound orbit, η = 0.26 2) escape orbit, η = 0.26 3) escape orbit, η = 0.33 Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 40 / 54

Geodesics in higher dimensional spacetimes Reissner Nordström de Sitter Reissner Nordström de Sitter space time in 7D ) 2 du = 4 η 4 u 6 η 4 λu 5 u 4 λu 3 u 2 λµ ) Λ ) u Λ ) dϕ 5 5 λ = 4P 6u) with u = x u 6: x dx dϕ) 2 = 4P5 x) 8 V eff µ = 8.4 6 λ 4 µ = 4.0 2 µ =. µ 0 0 5 0 5 r blue 3 positive roots, grey ; V eff λ = 0.05 and η = 0.4, and Λ = 8.7 0 5 Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 4 / 54

Geodesics in higher dimensional spacetimes Reissner Nordström de Sitter Reissner Nordström de Sitter space time in 7D: Orbits 00 20 0.5 50 0.5 0 0.5 0 0.5 0 50 00 50 200 0.5 0 0.5 20 0 0 0 20 0.5 50 0.5 0 00 20 4) µ =.: many world periodic bound and escape orbit 5) µ = 4.0: many world periodic bound and escape orbit 40 20 0.5 0.5 0 0.5 0.5 0 20 40 7) µ = 8.4: many world periodic bound and escape orbit Figure: Orbits in RNdS ST in 7 dimensions for λ = 0.05 and η = 0.4, and Λ = 8.7 0 5 Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 42 / 54

Geodesics in higher dimensional spacetimes Light rays and Radial motion Reissner Nordström de Sitter Motion of light EOM: ) ) ) d r 2 2 Λ = dϕ r 2d4) λµ r 2d2) r 2d3) r d3 η d3 d )d 2) ) d r 2 or = P ) d r) d r 2 dϕ r d5 for η = 0 and = P 2d2) r) dϕ r 2d4) for η 0 V eff = fr) L 2 Λ r 2 = λd)d2) λ r 2 η2d3) λ r d λ r 2d2) many world periodic bound and escape orbits or two world escape orbits: light can disappear into another universe or appear from another universe no escape orbits for a large negative Λ Radial motion L = 0 ) d r 2 = µ ds r d3 2 Λ r 2 d )d 2) ηd3 r 2d3) V eff = r 2 Λ r 2 d3 d)d2) ηd3 r 2d3) V eff = d 3) rd3 4 Λ d)d2) r2d2) 2d 3)η d3 = 0 equilibrium position for charged solution for η 4 3d. V eff equilibrium position r Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 43 / 54

Outline Conclusion and Outlook Preliminaries to geodesics 2 Topology of orbits 3 Analytical solution of geodesic equation 4 Geodesics in higher dimensional spacetimes Schwarzschild Schwarzschild in 9D Schwarzschild in D Schwarzschild de Sitter Schwarzschild de Sitter in 9D Schwarzschild de Sitter in D Reissner Nordström Reissner Nordström in 7D Reissner Nordström de Sitter Reissner Nordström de Sitter in 4D Reissner Nordström de Sitter in 7D 5 Conclusion and Outlook Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 44 / 54

Conclusion and Outlook Conclusion and Outlook CONCLUSION : Analytical solution of geodesic equation in higher dimensional STs is presented Motion of test particles in these static spherically symmetric STs is analysed OUTLOOK: Analytical integration of geodesic equations in higher dimensional axial symmetric STs e.g. Myers-Perry...) Solution of geodesic equations with polynomials P n where n 7 Riemann surface of genus g 3) Results published in: Phys. Rev. D 78 2008) 2408 Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 45 / 54

Last page Conclusion and Outlook Special thanks to the UGO Universitätsgesellschaft Oldenburg) for the financial support towards the participation at the MG2 Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 46 / 54

Conclusion and Outlook go back Table: Schwarzschild de Sitter space times for Λ > 0 d = 4 d r dϕ ) 2 = r [ Λ 3 λ r5 Λ > 0 λµ ) Λ 3 ) ] r 3 λ r 2 r µ > ; λµ ) Λ 3 > 0 for µ < - tb, esc λµ ) Λ 3 d r dϕ < 0 - - tb, pb, esc ) λµ ) Λ 6 r 4 λ ) r 2 ) 2 = Λ 6 λ r6 λ > not allowed d = 5 µ > ; λµ ) Λ 6 > 0 for µ < λ < - tb, esc λ > - tb, esc λµ ) Λ 6 < 0 λ < - - tb, esc ) 2 [ ) ] d r dϕ = r d5 2 Λλ d)d2) rd 2 Λ λµ ) r d r d3 λ r 2 d)d2) d 6 2 Λ µ > ; λµ ) > 0 for µ < d)d2) - tb, esc 2 Λ λµ ) < 0 d)d2) - - tb, esc tb terminating bound, pb periodic bound, esc escape Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 47 / 54

Conclusion and Outlook go back Table: Schwarzschild de Sitter space times for Λ < 0 d = 4 d r dϕ ) 2 = r [ Λ 3 λ r5 Λ < 0 λµ ) Λ 3 ) ] r 3 λ r 2 r µ < ; λµ ) Λ 3 < 0 for µ > - - - tb, pb λµ ) Λ 3 d r dϕ > 0 - - tb, pb ) λµ ) Λ 6 r 4 λ ) r 2 ) 2 = Λ 6 λ r6 λ > - - tb d = 5 µ < ; λµ ) Λ 3 < 0 for µ > λ < - - - tb λ > - tb λµ ) Λ 6 > 0 λ < - - tb, pb ) 2 [ ) ] d r dϕ = r d5 2 Λλ d)d2) rd 2 Λ λµ ) r d r d3 λ r 2 d)d2) d 6 2 Λ µ < ; λµ ) < 0 for µ > d)d2) - - - tb 2 Λ λµ ) > 0 d)d2) - - tb, pb tb terminating bound, pb periodic bound, esc escape Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 48 / 54

Conclusion and Outlook Holomorphic and meromorphis differentials Choose a basis of holomorphic differentials d z and associated meromorphic differentials d u dz = dx P5 x), dz 2 = xdx P5 x) du = a 3x 2a 4 x 2 3a 5 x 3 4 dx, du 2 = x2 dx P 5 x) 4 P 5 x). with their matrices of a and b periods 2ω ij,2η ij and 2ω ij,2η ij 2ω ij = dz i, 2ω ij = dz i a j b j 2η ij = du i, 2η ij = du i a j b j which satisfy the Legendre relation ) ) ) ω ω 0 g ω ω t η η g 0 η η = ) 0 2 πi g g 0 where g is the g g unit matrix. Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 49 / 54

Conclusion and Outlook Normalized holomorphic differentials We also need the normalized holomorphic differentials d v ) d v = 2ω) dz d z, d z = dz 2 which satisfy a d v = a 2ω) d z = 2ω) a d z = 2ω) 2ω = g and b d v = b 2ω) d z = 2ω) b d z = ω ω = τ. The period matrix of these differentials is given by g,τ), where τ is Riemann matrix: τ is symmetric and has real eigenvalues Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 50 / 54

Conclusion and Outlook ϑ, σ and functions To construct 2g periodic functions, we need theta function ϑ ϑ z; τ) = t τ m2 z) m Z g e iπ m Periodicity: ϑ z g n; τ) = ϑ z; τ) Quasi periodicity: ϑ z τ n; τ) = e iπ nt τ n2 z) ϑ z; τ) Kleinian sigma function where C = const. σ z) = Ce 2 zt ηω z ϑ 2ω) z τ g ) h generalized Weierstrass function connected to the σ function ij z) = log σ z) = σ i z)σ j z) σ z)σ ij z) z i z j σ 2 z) σ i is derivative with respect to the ith component) Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 5 / 54

Conclusion and Outlook Jacobi inversion problem and its solution Jacobi s inversion problem Determine x for given ϕ from ϕ = A x0 x) Abel map): ϕ = ϕ 2 = x x 0 x x 0 dx x2 P5 x) dx P5 x) x 0 xdx P5 x) x2 x 0 xdx P5 x) Solution of inversion problem Solution u of inversion problem is given by x x 2 = 4 22 ϕ) x x 2 = 4 2 ϕ) Two positions x and x 2? Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 52 / 54

Conclusion and Outlook Solution of the problem Step I: get rid of extra x Considering the limit x 2 we get the solution of Jacobi s inversion problem in the form choose base point x 0 = ) x x 2 x = lim = σ ϕ )σ 2 ϕ ) σ ϕ )σ 2 ϕ ) x 2 x x 2 σ2 2 ϕ ) σ ϕ )σ 22 ϕ ) where ϕ = lim x2 ϕ = A x ) = x d z with x = x, ) t. Step II: restricting the problem to theta-divisor Theta-divisor: set of zeros θ Kx0 of theta function ϑ 2ω) ) z K x0, where Kx0 is the Riemann vector associated with the base point x 0 Riemann s vanishing theorem). For K ) ) ) )) /2 = τ 0 /2 /2 is true that ϑ 2ω) /2 z τ 0 /2 /2 = 0 σ ϕ ) = 0 Now we can write the analytic solution to the geodesic equation as rϕ) = fxϕ)) = fx ) = f σ ) ϕ ) = f σ 2 ϕ ) σ x x d z in x d z) in σ 2 x x d z in x d z) in ) Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 53 / 54

Conclusion and Outlook A homology basis of hyperelliptic curve of genus 2 go back Kagramanova Uni Oldenburg) Integrable geodesics in HD BHT2, Paris 7.07.2009 54 / 54