Orbits in Static and Stationary Spacetimes

Orbits in Static and Stationary Spacetimes Claus Lämmerzahl with Eva Hackmann, Valeria Kagramanova and Jutta Kunz Centre for Applied Space Technology and Microgravity (ZARM), University of Bremen, 28359 Bremen, Germany 1st LARES workshop Rome, July 3 4, 2009 C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 1 / 90

Bremen Drop Tower of ZARM ZARM = Center for Applied Space Technology and Microgravity Tower 146 m drop tube 110 m free fall time = 4.7 s deceleration 30 g C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 2 / 90

Bremen Drop Tower of ZARM C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 2 / 90

BEC in microgravity design of capsule vacuum chamber capsule C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 3 / 90

BEC in microgravity long free evolution 50 ms 100 ms 500 ms 1000 ms 10 4 atoms, 1 s free evolution time (not possible on ground) C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 4 / 90

Activities at ZARM Combustion Microfluid dynamics Metallic foams Fundamental Physics Experiments BEC in free fall (+ theory) Atomic interferometry (+ theory) SQUIDs test of UFF Modeling Orbit determination Modeling of forces acting on satellites MICROSCOPE Pioneer anomaly Gravity and quantum theory Orbits in space time Orbits of spinning and extended objects (flyby anomaly) Quantum gravity phenomenology Influence of space time fluctuations Detection of gravitational waves Questioning/testing Newton s laws Structure of equations of motion Finsler geometry C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 5 / 90

Outline 1 Introduction C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 6 / 90

Outline 1 Introduction 2 Motion in gravitational fields C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 6 / 90

Outline 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 6 / 90

Outline 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields 4 Orbits in Schwarzschild C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 6 / 90

Outline 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields 4 Orbits in Schwarzschild 5 Orbits in Schwarzschild de Sitter C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 6 / 90

Outline 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields 4 Orbits in Schwarzschild 5 Orbits in Schwarzschild de Sitter 6 Orbits in higher dimensions 7 Orbits in Kerr C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 6 / 90

Outline 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields 4 Orbits in Schwarzschild 5 Orbits in Schwarzschild de Sitter 6 Orbits in higher dimensions 7 Orbits in Kerr 8 Orbits in NUT de Sitter C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 6 / 90

Outline 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields 4 Orbits in Schwarzschild 5 Orbits in Schwarzschild de Sitter 6 Orbits in higher dimensions 7 Orbits in Kerr 8 Orbits in NUT de Sitter 9 Orbits in Kerr de Sitter C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 6 / 90

Outline 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields 4 Orbits in Schwarzschild 5 Orbits in Schwarzschild de Sitter 6 Orbits in higher dimensions 7 Orbits in Kerr 8 Orbits in NUT de Sitter 9 Orbits in Kerr de Sitter 10 Orbits in Plebański Demiański C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 6 / 90

Outline 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields 4 Orbits in Schwarzschild 5 Orbits in Schwarzschild de Sitter 6 Orbits in higher dimensions 7 Orbits in Kerr 8 Orbits in NUT de Sitter 9 Orbits in Kerr de Sitter 10 Orbits in Plebański Demiański 11 Summary outlook C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 6 / 90

Outline Introduction 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields 4 Orbits in Schwarzschild 5 Orbits in Schwarzschild de Sitter 6 Orbits in higher dimensions 7 Orbits in Kerr 8 Orbits in NUT de Sitter 9 Orbits in Kerr de Sitter 10 Orbits in Plebański Demiański 11 Summary outlook C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 7 / 90

Introduction Introduction Gravity can only be explored through the motion of test particles Motion of test particles Orbits and clocks Massive particles and light What is gravity or what is the interaction depends on the structure of the equation of motion Existence of inertial systems (Finsler) Order of differential equation (higher order equations of motion) Dependence on particle parameters (Equivalence Principle) Here we assume validity of standard General Relativity C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 8 / 90

The present situation Introduction All predictions of General Relativity are experimentally well tested and confirmed Foundations The Einstein Equivalence Principle Universality of Free Fall Universality of Gravitational Redshift Local Lorentz Invariance Implication Gravity is a metrical theory Predictions for metrical theory Solar system effects Perihelion shift Gravitational redshift Deflection of light Gravitational time delay Lense Thirring effect Schiff effect Strong gravitational fields Binary systems Spin spin interaction Gravitational waves C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 9 / 90

The present situation Introduction All predictions of General Relativity are experimentally well tested and confirmed Foundations The Einstein Equivalence Principle Universality of Free Fall Universality of Gravitational Redshift Local Lorentz Invariance Implication Gravity is a metrical theory Predictions for metrical theory Solar system effects Perihelion shift Gravitational redshift Deflection of light Gravitational time delay Lense Thirring effect Schiff effect Strong gravitational fields Binary systems Spin spin interaction Gravitational waves General Relativity C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 9 / 90

The present situation Introduction All predictions of General Relativity are experimentally well tested and confirmed Foundations The Einstein Equivalence Principle Universality of Free Fall Universality of Gravitational Redshift Local Lorentz Invariance Implication Gravity is a metrical theory All consequences Black Holes, Big Bang,... Predictions for metrical theory Solar system effects Perihelion shift Gravitational redshift Deflection of light Gravitational time delay Lense Thirring effect Schiff effect Strong gravitational fields Binary systems Spin spin interaction Gravitational waves General Relativity C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 9 / 90

The present situation Introduction All predictions of General Relativity are experimentally well tested and confirmed Foundations The Einstein Equivalence Principle Universality of Free Fall Universality of Gravitational Redshift Local Lorentz Invariance Implication Gravity is a metrical theory All consequences Test Black Holes, Big Bang,... Predictions for metrical theory Solar system effects Perihelion shift Gravitational redshift Deflection of light Gravitational time delay Lense Thirring effect Schiff effect Strong gravitational fields Binary systems Spin spin interaction Gravitational waves General Relativity C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 9 / 90

The present situation Introduction All predictions of General Relativity are experimentally well tested and confirmed Foundations The Einstein Equivalence Principle Universality of Free Fall Universality of Gravitational Redshift Local Lorentz Invariance Implication Gravity is a metrical theory All consequences Test Black Holes, Big Bang,... Predictions for metrical theory Solar system effects Perihelion shift Gravitational redshift Deflection of light Gravitational time delay Lense Thirring effect Schiff effect Strong gravitational fields Binary systems Spin spin interaction Test Gravitational waves General Relativity C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 9 / 90

The present situation Introduction All predictions of General Relativity are experimentally well tested and confirmed Foundations The Einstein Equivalence Principle Universality of Free Fall Universality of Gravitational Redshift Local Lorentz Invariance Implication Gravity is a metrical theory All consequences Test Black Holes, Big Bang,... Predictions for metrical theory Solar system effects Perihelion shift Gravitational redshift Deflection of light Gravitational time delay Lense Thirring effect Schiff effect Strong gravitational fields Binary systems Spin spin interaction Gravitational waves st Test General Relativity C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 9 / 90

Outline Motion in gravitational fields 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields 4 Orbits in Schwarzschild 5 Orbits in Schwarzschild de Sitter 6 Orbits in higher dimensions 7 Orbits in Kerr 8 Orbits in NUT de Sitter 9 Orbits in Kerr de Sitter 10 Orbits in Plebański Demiański 11 Summary outlook C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 10 / 90

The framework Motion in gravitational fields The gravitational field Gravity = space time metric ds 2 = g µν dx µ dx ν What determines the metric Einstein equation What determines the motion of particles Geodesic equation R µν 1 2 g µνr + Λg µν R = 8πG c 2 T µν 0 = d2 x µ ds + { ρσ µ } dxρ dx σ ds ds There are justifications for that Ehlers Pirani Schild axiomatic approach PPN formalism + experiments/observations C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 11 / 90

Metric theory Motion in gravitational fields Predictions All metric theories imply gravitational redshift light deflection perihelion shift gravitational time delay Lense Thirring effect Schiff effect geodetic precession Einstein s theory is singled out by certain values for these effects How to find Einstein s theory? PPN C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 12 / 90

Outline Physics in weak gravitational fields 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields 4 Orbits in Schwarzschild 5 Orbits in Schwarzschild de Sitter 6 Orbits in higher dimensions 7 Orbits in Kerr 8 Orbits in NUT de Sitter 9 Orbits in Kerr de Sitter 10 Orbits in Plebański Demiański 11 Summary outlook C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 13 / 90

Physics in weak gravitational fields PPN formalism Physical situation Spherically symmetric metric ds 2 = g µν dx µ dx ν = g tt dt 2 g rr dr 2 r 2 dϑ 2 r 2 sin 2 ϑdϕ 2 g tt, g rr Gravitational field equations: not known Parametrization for asymptotically flat weak fields: g µν = η µν + h µν, h µν 1 g 00 = 1 + 2α U U2 2β c2 c 4, g 0i = 0 U = Newton potential g ij = (1 + 2γ) U c 2 C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 14 / 90

Physics in weak gravitational fields PPN formalism Physical situation Axially symmetric metric ds 2 = g µν dx µ dx ν = g tt dt 2 g rr dr 2 r 2 dϑ 2 r 2 sin 2 ϑdϕ 2 g t3 dtdϕ g tt, g rr, g ti Gravitational field equations: not known Parametrization for asymptotically flat weak fields: g µν = η µν + h µν, h µν 1 g 00 = 1 + 2α U U2 2β c2 c 4, g 0i = 4µ (J r) i c 3 r 3 U = Newton potential g ij = (1 + 2γ) U c 2 C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 14 / 90

Physics in weak gravitational fields Equations of motion Space time PN metric ds 2 = (1 2U) dt 2 + 4A i dx i dt (1 + 2U) dx 2 U = Newton potential = M r A = gravitometic vector potential = J r r 3 PN particle dynamics PN Lagrangian PN equation of motion L = m ( 1 v 2 2(1 + v 2 )U + 4v A ) mẍ = m }{{} U 2mv ( A) }{{} gravitoelectric field gravitomagnetic field C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 15 / 90

Physics in weak gravitational fields The gravitomagnetic fild C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 16 / 90

Physics in weak gravitational fields The gravitomagnetic field Main consequences Notions Angular momentum L = r p Runge Lenz vector The gravitomagnetic field g 0i Genuine post Newtonian gravitational field Analogue of magnetic field A = L ṙ + Gm r r Properties d L 0 dt (Newton: = 0) d A 0 dt (Newton: = 0) C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 17 / 90

Physics in weak gravitational fields Lense Thirring effect A particular orbit showing the Lense Thirring effect Observations exact solution: Weierstrass elliptic functions Observed quantities Ω = 2GJ c 2 a 3 (1 e 2 ) 3/2 ω = 6GJ cos i c 2 a 3 (1 e 2 ) 3/2 Measurement with LAGEOS satellites, together with data from CHAMP and GRACE Result: confirmation with approx 10 % error C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 18 / 90

Physics in weak gravitational fields Gravitomagnetic clock effect rotating gravitating mass Kerr solution ds 2 =... + 4a ϕ dϕdt +... geodesic equation for circular orbits in equatorial plane dϕ dt = ±Ω 0 + Ω Lense Thirring proper time difference of two counterpropagating clocks s + s = 4π J M 10 7 s is astonishingly large, but requires exact knowledge of orbit does not depend on G and on r decreases with inclination, vanishes for polar orbits s + s C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 19 / 90

Schiff effect Physics in weak gravitational fields Description Dynamics of direction of spin D v S = 0 Compared with direction given by distant stars Effective dynamics with Ṡ = Ω S Ω = g GP-B Ongoing data analysis is different measurement can in principle yield different outcome C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 20 / 90

Outline Orbits in Schwarzschild 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields 4 Orbits in Schwarzschild 5 Orbits in Schwarzschild de Sitter 6 Orbits in higher dimensions 7 Orbits in Kerr 8 Orbits in NUT de Sitter 9 Orbits in Kerr de Sitter 10 Orbits in Plebański Demiański 11 Summary outlook C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 21 / 90

Orbits in Schwarzschild Orbits in Schwarzschild space time The Schwarzschild metric ( ds 2 = 1 2M r ) dt 2 1 1 2M r dr 2 r 2 dϑ 2 r 2 sin 2 ϑdϕ 2 Equations of motion Conservation laws dt E = g 00 ds, With substitutions, e.g., u = 2M/r dϕ L = r2 ds d ϕ 2 = du du = 4u3 g 2 u g 3 P3 (u) and similar for t and s, with g 2,g 3 L,E,M C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 22 / 90

Orbits in Schwarzschild Orbits in Schwarzschild space time Solution u = u(ϕ) given through ϕ ϕ 0 = u du u 0 P3 (u ) Uniqueness of integration: u is function with 2 periods with half periods ω 1 = zero2 zero 1 u(ϕ + 2nω 1 + 2mω 2 ) = u(ϕ) du zero4 P3 (u), ω du 2 = zero 3 P3 (u) Solution for orbit (Hagihara, JJGA 1931) r(ϕ) = 2M 1 3 + ( 2M ϕ 2 ;g ), r(ϕ) = 1 2,g 3 3 + ( ϕ 2 + iω ) 2;g 2,g 3 C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 23 / 90

Orbits in Schwarzschild Orbits in Schwarzschild space time λ µ parameter plot, discriminant of P 3 (Weierstraß polynomial) λ 1 3 1 4 IV V VI III II VI VII I 8 9 1 µ C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 24 / 90

Orbits in Schwarzschild Orbits in Schwarzschild space time V eff VIa VIb r hyperbolic spiral C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 25 / 90

Orbits in Schwarzschild Orbits in Schwarzschild space time V eff Ia Ib r r 1 r 2 quasi hyperbolic Deflection angle given by complete elliptic integral as function of r min and E C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 26 / 90

Orbits in Schwarzschild Orbits in Schwarzschild space time V eff PIa PIb r quasi parabolic C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 27 / 90

Orbits in Schwarzschild Orbits in Schwarzschild space time V eff r 1 r 2 r 3 r IIa IIb quasi elliptic perihelion shift: δϕ = ω 1 2π analytic function of r min and r max C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 28 / 90

Orbits in Schwarzschild Orbits in Schwarzschild space time V eff PIIa PIIb r infinite parabolic spiral C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 29 / 90

Orbits in Schwarzschild Orbits in Schwarzschild space time V eff VIIa VIIb r spiral double spiral (Poincarés double circle limit) C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 30 / 90

Orbits in Schwarzschild Orbits in Schwarzschild space time V eff r Va Vb spiral circle (ISCO) C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 31 / 90

Orbits in Schwarzschild Light rays in Schwarzschild space time V eff LIIa LIIb r quasi hyperbolic deflection angle given by complete elliptic integral as function of r min C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 32 / 90

Orbits in Schwarzschild Light rays in Schwarzschild space time V eff LIIIa LIIIb r infinite quasi hyperbolic spiral C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 33 / 90

Outline Orbits in Schwarzschild de Sitter 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields 4 Orbits in Schwarzschild 5 Orbits in Schwarzschild de Sitter 6 Orbits in higher dimensions 7 Orbits in Kerr 8 Orbits in NUT de Sitter 9 Orbits in Kerr de Sitter 10 Orbits in Plebański Demiański 11 Summary outlook C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 34 / 90

Orbits in Schwarzschild de Sitter Schwarzschild de Sitter space time The Schwarzschild de Sitter metric ds 2 = ( 1 2M r + 1 3 Λr2 ) dt 2 1 1 2M r + 1 3 Λr2 dr2 r 2 dϑ 2 r 2 sin 2 ϑdϕ 2 Equations of motion Conservation laws dt E = g 00 ds, With substitutions, e.g., u = 2M/r dϕ L = r2 ds d ϕ 2 = du P5 (u) and similar for t and s, with g 2,g 3 L,E,M,Λ C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 35 / 90

Orbits in Schwarzschild de Sitter Geodesic equation in Schwarzschild de Sitter space time Effective equation of motion (u = r S /r) ( u du ) 2 = u 5 u 4 + ǫλu 3 + (λ(µ ǫ) + 13 ) dϕ ρ u 2 + ǫ 3 λρ =: P 5(u) with ( rs ) 2 λ =, µ = E 2, ρ = ΛrS 2 L Polynomial of 5th order beyond elliptic integral: hyperelliptic integral P 5 possesses at most 4 real positive zeros ǫ = 0 P 5 (u) = u 2 P 3 (u) elliptic function (Λ has no influence on light propagation) Λ = 0 P 5 (u) = u 2 P 3 (u) elliptic function C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 36 / 90

Orbits in Schwarzschild de Sitter Orbits in Schwarzschild de Sitter space time P 5 P 5 P 5 u u u (a) No real positive zero P 5 (b) One real positive zero P 5 (c) Two real positive zeros u u (d) Three real positive zeros (e) Four real positive zeros Five possibilities of having real positive zeros of P 5 (u) Zeros correspond to the positions for which V eff = E Bound non terminating, quasi periodic orbits (planetary orbits) exist only for three or more positive zeros C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 37 / 90

Orbits in Schwarzschild de Sitter Orbits in Schwarzschild de Sitter space time (f) Λ = 10 45 km 2 (g) Λ = 0 (h) Λ = 10 45 km 2 color # zeros > 0 types of orbits black 0 particle from infinity to singularity gray 1 bound terminating orbits green 2 bound terminating and escape orbits blue 3 bound terminating and quasi periodic bound orbits violet 4 bound terminating, escape and quasi periodic bound orbits C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 38 / 90

Orbits in Schwarzschild de Sitter Analytic solution of geodesic equation in SdS space time Separation of variables ϕ ϕ 0 = Not well defined in complex plane u u du u 0 P5 (u ) Looked for: u = u(ϕ) inversion problem Uniqueness of integration: u is function with 4 periods z 0 Iz C 1 x C 2 C 2 e 3 e 2 e 1 Rz C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 39 / 90

Orbits in Schwarzschild de Sitter Analytic solution of geodesic equation in SdS space time Iz Iz Rz Rz Iz Iz Iz Rz Rz Rz Iz Iz Iz Rz Rz Rz C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 40 / 90

Orbits in Schwarzschild de Sitter Analytic solution of geodesic equation in SdS space time Iz Iz + + + + + Rz + + + + + Rz Iz Iz Iz + + + + + Rz + + + + + Rz + + + + + Rz Iz Iz Iz + + + + Rz + + + + Rz + + + + Rz C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 40 / 90

Orbits in Schwarzschild de Sitter Analytic solution of geodesic equation in SdS space time Copy 1 of C Iz b 1 b 2 Copy 2 of C Iz e 1 e 2 e 3 e 4 e 5 e 6 a 1 a 2 Rz e 1 e 2 e 3 e 4 e 5 e 6 b 1 b 2 Rz b 1 a 1 a 2 b 2 P 5 X = pretzel C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 41 / 90

Orbits in Schwarzschild de Sitter Analytic solution of geodesic equation in SdS space time Holomorphic and associated meromorphic differentials dz 1 := dx P5 (x), dz 2 := xdx P5 (x) dr 1 := 3x3 2x 2 + λx 4 dx, dr 2 := x2 dx P 5 (x) 4 P 5 (x) Period matrices (2ω,2ω ) and (2η,2η ) 2ω ij := dz i, 2ω ij := dz i a j b j 2η ij := dr i, 2η ij := dr i a j b j Normalized differentials and their period matrix d z d v = (2ω) 1 d z, (2ω,2ω ) (1 2,τ) with τ = ω 1 ω C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 42 / 90

Orbits in Schwarzschild de Sitter Analytic solution of geodesic equation in SdS space time Preliminaries definitions Theta function ϑ : C 2 C (for construction of functions with 4 periods) ϑ( z;τ) := t (τ m+2 z) m Z 2 e iπ m Periodicity: ϑ( z + 1 2 n; τ) = ϑ( z; τ) Quasi periodicity: ϑ( z + τ n; τ) = e iπ nt (τ n+2 z) ϑ( z; τ) Theta function with characteristics g, h 1 2 Z2 ϑ[ g, h]( z;τ) := e iπ gt (τ g+2 z+2 h) ϑ( z + τ g + h;τ) Sigma function σ( z) = Ce 1 2 zt ηω 1 z ϑ Generalized Weierstrass functions [( 1/2 1/2 ), ( 0 1/2) ] ((2ω) 1 z;τ) ij ( z) = log σ( z) = iσ( z) j σ( z) σ( z) i j σ( z) z i z j σ 2 ( z) C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 43 / 90

Orbits in Schwarzschild de Sitter Analytic solution of geodesic equation in SdS space time Jacobi s inversion problem Determine u for given ϕ from ϕ = A u0 ( u) (Abel map), i.e. ϕ 1 = ϕ 2 = u1 u 0 u1 u 0 du u2 P5 (u) + du P5 (u) u 0 udu P5 (u) + u2 u 0 udu P5 (u) Solution of inversion problem Solution u of inversion problem given by u 1 + u 2 = 4 22 ( ϕ) u 1 u 2 = 4 12 ( ϕ) Our problem: two positions u, two angles ϕ requires reduction C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 44 / 90

Orbits in Schwarzschild de Sitter Analytic solution of geodesic equation in SdS space time Rewrite inversion problem in the form (based on Enolskii & Richter 2005) φ = A ( u) with φ = ϕ u 0 d z Extraction of one component of u (namely u 2 in Jacobi inv. prob.) through a limit with u 1 u 2 u 1 = lim = σ( φ ) 1 2 σ( φ ) 1 σ( φ ) 2 σ( φ ) u 2 u 1 + u 2 ( 2 σ) 2 ( φ ) σ( φ ) 2 2 σ( φ ) φ = lim u 2 ( ) φ = A u1 ( u ), u = One u but still two ϕ: Lemma (Mumford 1983 Riemann vanishing theorem): (2ω) 1 A ( u ) Theta divisor Θ K = { z ϑ( z + K ) = 0} with the vector of Riemann constants K ) = τ + ( ) 0 1/2 ( 1/2 1/2 σ( φ ) = 0, gives a relation between φ,1 and φ,2. C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 45 / 90

Orbits in Schwarzschild de Sitter Analytic solution of geodesic equation in SdS space time Define ( ) x ϕ Θ := ϕ ϕ 0 and choose x so that (2ω) 1 ϕ Θ Θ K. Solution (Hackmann & C.L. PRL 2008, PRD 2008) with ϕ 0 = ϕ 0 + dz 2 u 0 Complete analytic solution of equation of motion in Schwarzschild de Sitter space time r(ϕ) = r S u(ϕ) = r 2 σ( ϕ Θ ) S σ( ϕ Θ ) = 0 1 σ( ϕ Θ ) New result C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 46 / 90

Orbits in Schwarzschild de Sitter Schwarzschild de Sitter: Orbits 4 20 V eff 2 4 2 0 2 4 2 10 20 10 0 10 20 10 r 4 20 (i) Λ = 0, r 0 = 5.010km (j) Λ = 0, r 0 = 20.951km (k) effective potential 4 20 100 2 10 50 4 2 2 4 20 10 0 10 20 0 50 100 150 200 250 2 10 50 4 20 100 (l) Λ = 10 5 km 2, r 0 = 5.013km (m) Λ = 10 5 km 2, r 0 = 22.185km (n) Λ = 10 5 km 2, r 0 = 133.60km C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 47 / 90

Orbits in Schwarzschild de Sitter Schwarzschild de Sitter: Orbits 4 V eff 3 2 1 4 3 2 1 0 1 2 3 4 1 2 3 4 r (o) Λ = 0, r 0 = 3.625km (p) effective potential 4 3 2 1 200 100 4 3 2 1 0 1 2 3 4 1 2 3 4 0 100 200 100 200 300 400 500 (q) Λ = 10 5 km 2, r 0 = 3.625km (r) Λ = 10 5 km 2, r 0 = 237.61km C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 48 / 90

Orbits in Schwarzschild de Sitter Schwarzschild de Sitter: Orbits 40 4 20 2 80 60 40 20 0 20 20 4 2 0 2 4 2 40 4 (s) Λ = 0, r 0 = 6.776km (t) Λ = 0, r 0 = 4.642km 150 100 50 150 50 0 50 100 150 50 100 150 4 2 4 2 0 2 4 2 4 (u) Λ = 10 5 km 2, r 0 = 185.37km (v) Λ = 10 5 km 2, r 0 = 4.639km C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 49 / 90

Orbits in Schwarzschild de Sitter Application to Pioneer Influence of Λ on Pioneer satellites (orbital parameters from Nieto & Anderson 2005) Pioneer 10: µ = 1.00000000143, λ = 2.85557237382 10 9 Pioneer 11: µ = 1.00000000122, λ = 1.34074057459 10 9 For given angle ϕ: r Λ (ϕ) r Λ=0 (ϕ) 10 3 m For given distance r = 65 AU: rϕ Λ (r) rϕ Λ=0 (r) 10 5 m Form of the Pioneer orbits practically does not change. Cosmological constant cannot be origin of Pioneer anomaly. In principle we need also ϕ = ϕ(t) and Doppler tracking. C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 50 / 90

Orbits in Schwarzschild de Sitter Post Schwarzschild of perihelion shift Perihelion shift in Schwarzschild de Sitter (for bound orbit, Kraniotis & Whitehouse, CQG 2003) xdx δϕ perihelion = 2π ω 22 = 2π P5 (x) Approximation to first order in Λ x p5 (x) = 1 P3 (x) 2 x 2 + λ 3 m2 x 2 P 3 (x) P 3 (x) Λ + O(Λ2 ) with P 3 (x) = x 3 x 2 + λx + λ(µ 1) Schwarzschild polynomial This has to be integrated: gymnastics in elliptic integration C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 51 / 90

Orbits in Schwarzschild de Sitter Post Schwarzschild of perihelion shift result xdx P5 (x) a 2 = x P3 (x) 2 3 Λm2 a 2 1 = ω 1 + Λ m2 96 ( 3 +λ +O(Λ 2 ) j=1 a 2 η 1 + ω 1 z j ( (ρ j )) 2 ( 2η1 1 6 ω 1 16( (u 0 )) 2 + 6 16 x 2 + λ x 2 P 3 (x) P 3 (x) dx + O(Λ2 ) ( ) λ 1 + ( ) 4zj + 1 2 3 ) ) (u 0 ) ( (u 0 )) 5 (ζ(u 0) η 1 u 0 ) Needs introduction of r min and r max or a and e for interpretation Needs relativistic approximation for interpretation C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 52 / 90

Outline Orbits in higher dimensions 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields 4 Orbits in Schwarzschild 5 Orbits in Schwarzschild de Sitter 6 Orbits in higher dimensions 7 Orbits in Kerr 8 Orbits in NUT de Sitter 9 Orbits in Kerr de Sitter 10 Orbits in Plebański Demiański 11 Summary outlook C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 53 / 90

Orbits in higher dimensions Further space times: Higher dimensions Formalism can be further applied to some Reissner Nordström (anti-)de Sitter space times in higher dimensions with Equation of motion ( ) 2 dr = r4 dϕ = r4 L 2 ds 2 = αdt 2 α 1 dr 2 r 2 dω 2 d 2 α = 1 md 3 q2(d 3) + rd 3 r 2Λ 2(d 3) (d 1)(d 2) r2 )) 1 (E 2 L 2 g tt (ǫ + L2 g rr g tt r 2 ( E 2 (1 md 3 q2(d 3) + rd 3 r 2(d 3) 2Λ (d 1)(d 2) r2 )) )(ǫ + L2 Substitution u = m, systematic discussion and study of all subcases... r r 2 C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 54 / 90

Orbits in higher dimensions Analytically solvable cases Space time 4 5 6 7 8 9 10 11 d 12 Dimension Schwarzschild + + + Schwarzschild de Sitter + + Reissner Nordström + + Reissner Nordström de Sitter + (Hackmann, Kagramanova, Kunz & CL, PRD 2008) C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 55 / 90

Orbits in higher dimensions Reissner Nordström in 7 dimensions 1 10 0.5 1 0.5 0 0.5 1 20 10 0 0.5 1 10 Traversing different universes: measurement of Perihelion shift? C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 56 / 90

Orbits in higher dimensions Reissner Nordström in 7 dimensions 5 10 5 0 Escape orbit: escape into a different universe 0.2 0.2 0.5 5 C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 57 / 90

Orbits in higher dimensions Reissner-Nordström space time t III II I r r + r I 0 I 0 I + r = 0 r = I I + 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 = r = 0 r = I r = r = 0 I + I I + I I 0 I 0 u C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 58 / 90

Outline Orbits in Kerr 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields 4 Orbits in Schwarzschild 5 Orbits in Schwarzschild de Sitter 6 Orbits in higher dimensions 7 Orbits in Kerr 8 Orbits in NUT de Sitter 9 Orbits in Kerr de Sitter 10 Orbits in Plebański Demiański 11 Summary outlook C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 59 / 90

Kerr space time Orbits in Kerr Kerr de Sitter metric rotating axially symmetric black hole in space time with cosmological constant ds 2 = r θ a 2 sin 2 θ p 2 dt 2 + sin 2 θ ra 2 sin 2 θ (r 2 + a 2 ) 2 p 2 dφ 2 + 2asin 2 θ r2 + a 2 r p 2 dtdφ p2 dr 2 p 2 dθ 2 r r = r 2 + a 2 2mr p 2 = r 2 + a 2 cos 2 θ ϑ motion elentary integral, period ω ϑ ϑ min,ϑ max r motion Weierstraß elliptic integral, periods ω r 1, ω r 2 r min,r max Chandrasekhar 1983; O Neill 1995; Fujita & Hikida, CQG 2009 C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 60 / 90

Outline Orbits in NUT de Sitter 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields 4 Orbits in Schwarzschild 5 Orbits in Schwarzschild de Sitter 6 Orbits in higher dimensions 7 Orbits in Kerr 8 Orbits in NUT de Sitter 9 Orbits in Kerr de Sitter 10 Orbits in Plebański Demiański 11 Summary outlook C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 61 / 90

Orbits in NUT de Sitter NUT de Sitter space time NUT de Sitter metric ds 2 = r ρ 2 (dt 2l cos ϑdφ)2 ρ2 r dr 2 ρ 2 ( dϑ 2 + sin 2 θdφ 2) ρ 2 = r 2 + l 2, r = Λ 3 r4 + r 2 (1 2l 2 Λ) 2Mr + l 2 ( Λl 2 1 ) M = mass, l = NUT charge (gravitomagnetic mass), Λ = cosmological constant Equations of motion dr = R 6 with R 6 = (r 2 + l 2 ) 2 E 2 r (m 2 r 2 + L 2 + K) dτ dϑ = Θ with Θ = K m 2 l 2 1 ( (4E 2 dτ sin 2 l 2 + L 2 )cos 2 ϑ 4ElLcos ϑ ) ϑ dφ dϑ = 1 L 2lE cos ϑ Θ sin 2 ϑ dt dτ = E ρ4 L 2lE cos ϑ + 2l cos ϑ r sin 2 ϑ τ = Mino time C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 62 / 90

Orbits in NUT de Sitter NUT de Sitter: ϑ motion with ξ := cos ϑ ( ) 2 dξ = λ 2 ( A B 2 (A + 1)ξ 2 + 2Bξ ) = P 2 (ξ) dτ λ 1 c 1 0 c 2 0 c 1 > 0 c 2 > 0 1 1 ξ λ c2 1 c 1 0 c 2 0 c 1 0 c 2 0 µ c1 µ B = 0: P 2 symmetric A > 0: P 2 possesses zeros restrictions on λ and µ for having P 2 (ξ) 0 can be solved by elementary function C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 63 / 90

Orbits in NUT de Sitter NUT de Sitter: r motion Effective potential for r-motion given by R 6 λ µ diagrams 30 4 4 25 20 λ 15 λ 3 2 λ 3 2 10 5 1 1 0 2 4 6 8 10 µ (w) κ = 1.0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 µ (x) κ = 0.0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 µ (y) κ = 1.0 < κ crit1 3.5 3 2.5 2 λ 1.5 1 3 2.5 2 λ 1.5 1 3 2.5 2 λ 1.5 1 0.5 0.5 0.5 0 0.9 1 1.1 1.2 1.3 1.4 µ (z) κ = κ crit1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 µ () κ crit1 < κ < κ crit2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 µ () κ crit2 < κ Figure: λ µ diagrams for n = 0.5, Λ = 8.7 10 5. Dashed regions are forbidden from ϑ equation. Black: 4 zeros. Gray: 2 zeros, White: No zero. C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 64 / 90

Orbits in NUT de Sitter NUT de Sitter: r motion analytical solution r(τ) = f ( σ ) 1( τ ) σ 2 ( τ ) with τ,z2 = ( x1 x in dz 1 ) x in dz 1 τ τ, where τ = τ in + x in dz 2 C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 65 / 90

Orbits in NUT de Sitter NUT de Sitter: ϕ motion Solution to the ϕ equation: ( ϕ = 1 λ + 2µn 2 λ + 2µn arctan 1 ub 1 1 u 2 B1 2 1 with u = α + βx λ 2µn λ 2µn arctan 1 ub 1 u 2 B 2 1 ) C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 66 / 90

Orbits in NUT de Sitter NUT de Sitter: effective Potential n = 0.5, K > 0 V r eff 0.95 µ 2 = 0.94865 µ unst term µ un st term/unst bound µ unst terminating escape µ unst term/unst esc µ 2 = 0.94038 µ stbl circ orb µ 2 = 0.92701 0.9 r 0 20 40 60 Remark: K > 0: θ (0,π) K = 0: θ [0, π 2 ]. C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 67 / 90

Orbits in NUT de Sitter Orbits in NUT de Sitter space time terminating orbit escape orbit C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 68 / 90

Orbits in NUT de Sitter Orbits in NUT de Sitter space time bound orbit bound orbit C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 69 / 90

Orbits in NUT de Sitter Orbits in NUT de Sitter space time quasi hyperbolic orbit terminating inspiral orbit C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 70 / 90

Orbits in NUT de Sitter Orbits in NUT de Sitter space time terminating orbit (for θ [0, π 2 ]) quasihyperbolic escape orbit (for θ [0, π 2 ]) C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 71 / 90

Orbits in NUT de Sitter NUT de Sitter: effective Potential n = 0.5, K < 0 1.0 V r eff µ = 99756... µ unst terminating escape µ unst term/unst esc 0.95 µ unst bound/ unst esc µ unst esc µ = 0.97474... 0.9 0 20 40 60 C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 72 / 90

Orbits in NUT de Sitter Orbits in NUT de Sitter space time bound orbit (θ (0, π 2 )) quasihyperbolic escape orbit (θ (0, π 2 )) C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 73 / 90

Orbits in NUT de Sitter Orbits in NUT de Sitter space time terminating orbit (θ (0, π 2 )) quasihyerbolic escape orbit (θ (0, π 2 )) C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 74 / 90

Orbits in NUT de Sitter Orbits in NUT de Sitter space time (Hackmann, Kagramanova, Kunz & C.L. 2009) terminating inspiral orbit (θ (0, π 2 )) C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 76 / 90

Outline Orbits in Kerr de Sitter 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields 4 Orbits in Schwarzschild 5 Orbits in Schwarzschild de Sitter 6 Orbits in higher dimensions 7 Orbits in Kerr 8 Orbits in NUT de Sitter 9 Orbits in Kerr de Sitter 10 Orbits in Plebański Demiański 11 Summary outlook C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 77 / 90

Orbits in Kerr de Sitter Kerr de Sitter space time Kerr de Sitter metric rotating axially symmetric black hole in space time with cosmological constant ds 2 = r ϑ a 2 sin 2 ϑ χ 2 p 2 dt 2 + sin 2 ϑ ra 2 sin 2 θ ϑ (r 2 + a 2 ) 2 χ 2 p 2 dϕ 2 + 2asin 2 ϑ ϑ(r 2 + a 2 ) r χ 2 p 2 dtdφ p2 dr 2 p2 dϑ 2 r ϑ r = (1 Λ3 r2 ) (r 2 + a 2 ) 2mr ϑ = 1 + a2 Λ 3 cos2 ϑ χ = 1 + a2 Λ 3 p 2 = r 2 + a 2 cos 2 ϑ C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 78 / 90

Orbits in Kerr de Sitter Kerr de Sitter space time Equations of motion ( ) 2 dx = E 2 r 2 dτ S(χ 2 (x 2 + α 2 αd) 2 x (δ 2 x 2 + κ)) =: R 6 ( ) 2 dϑ = E 2 rs ( 2 ϑ (κ δ 2 α 2 cos 2 ϑ) χ2 ( α sin 2 dτ sin 2 ϑ D ) ) 2 ϑ dϕ dτ = χ2 Er S dt dτ = χ2 Er 2 S [ α ( (x 2 + α 2 ) αd) ) 1 ( α sin 2 x ϑ sin 2 ϑ D )] ϑ [ x 2 + α 2 ( (r 2 n + α 2 ) αd ) α ( α sin 2 θ D ) ] x θ τ = Mino time coupled to the proper time τ = s 0 dsp2 x := r r S, α := a r S, D := E Lr S, λ := 1 3 Λr2 S, κ := K, δ E 2 rs 2 2 = 1 E. 2 C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 79 / 90

Orbits in Kerr de Sitter Kerr de Sitter space time: ϑ motion With ξ = cos ϑ: ξ motion based on P 4 Weierstraß elliptic function ( ) a 3 ϑ(s) = arccos 4 (2Er S s s 0 ;g 2,g 3 ) a2 3 where s 0 = s 0 (s 0,θ 0 ) and a 2, a 3, g 2 and g 3 some constants. C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 80 / 90

Orbits in Kerr de Sitter Kerr de Sitter space time: r motion based on P 6 hyperelliptic function r(s) = r S σ 2 σ 1 (( f(ers a5 s s 0 ) Er S a5 s s 0 )) + l. where f such that σ((f(z),z) t ) = 0, s 0 = s 0 (s 0,r 0 ), l a zero of R 6 and a 5 a constant. C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 81 / 90

Orbits in Kerr de Sitter Kerr de Sitter space time: ϕ motion Solution for ϕ(s) consists of two parts ϕ(s) = χ 2 (ϕ r (s) ϕ ϑ (s)) + φ 0 requires hyperelliptic integrals of third kind ϕ θ (s) = c 0 (s s 0 ) 2 i=1 where v = 2Er S s s 0, v i and c i some constants i=1 [ c i 2Er S ζ(v i )(s s 0 ) + log σ(v v ] i) σ(v 0 v i ) ϕ r (s) = C 1 Er S (s s 0 ) + C 0 (f(w) + f( Er S a5 s 0 + s 0 )) ( 4 + C 2,i log σ((f(w),w)t ( ) ) k 4,i ) f(w) σ((f(w),w) t k 5,i ) k 1,i (k 2,i,k 3,i ) w where w = Er S a5 s s 0,, k j,i are constant (Hackmann, Kagramanova, Kunz & C.L. 2009) C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 82 / 90

Orbits in Kerr de Sitter Kerr de Sitter space time: Orbits Bound orbit for r S = 1, α = 0.1, Λ = 10 5, E = 0.98, L = 1, and K = 6. 15 15 10 10 5 5 0 5 0 5 5 10 15 20 25 15 10 10 15 x 5 5 0 5 0 5 5 15 10 10 15 y 10 z 10 15 15 Figure: Plot in the (r, θ) plane. Grey lines denote extremal θ values. Figure: 3d plot. Grey cones denote extremal θ values and grey sphere the horizon. C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 83 / 90

Outline Orbits in Plebański Demiański 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields 4 Orbits in Schwarzschild 5 Orbits in Schwarzschild de Sitter 6 Orbits in higher dimensions 7 Orbits in Kerr 8 Orbits in NUT de Sitter 9 Orbits in Kerr de Sitter 10 Orbits in Plebański Demiański 11 Summary outlook C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 84 / 90

Orbits in Plebański Demiański General statements Theorem: Separability The Hamilton Jacobi equation for the geodesic equation is separable if and only if space time is of Petrov type D without acceleration (Demianski & Francaviglia, JPA 1981) Theorem: Petrov type D The general type D Petrov space times are exhausted by the Plebański Demiański solutions Theorem: Integration The geodesic equation in a general Plebański Demiański space time without acceleration can be integrated using the method of hyperelliptic integrals All what is separable can be explicitely solved analytically! C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 85 / 90

Outline Summary outlook 1 Introduction 2 Motion in gravitational fields 3 Physics in weak gravitational fields 4 Orbits in Schwarzschild 5 Orbits in Schwarzschild de Sitter 6 Orbits in higher dimensions 7 Orbits in Kerr 8 Orbits in NUT de Sitter 9 Orbits in Kerr de Sitter 10 Orbits in Plebański Demiański 11 Summary outlook C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 86 / 90

Summary Summary outlook Analytic solution / integration of all separable cases New solution for Schwarzschild de Sitter Kerr de Sitter NUT de Sitter general Plebański Demiański Solutions given by Kleinian sigma functions restricted to the Theta divisor C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 87 / 90

Outlook Summary outlook Thorough discussion of observables Motion in gravitational multipole fields Comparison with LAGEOS, LARES and other geodesy missions Application to EMRIs generation of gravitational waves Application to effective one body formalism Orbits for particles with spin (C.L. & Schaffer 2009) Orbits in Finsler space-time (C.L. & Perlick 2009) C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 88 / 90

Summary outlook Last remark on LARES Lares = Roman God for protection LARES = Protection for Fundamental Physics in Space lares fisica fundamentale spatiale all geodesy missions lares geodesica good science in general lares sciencia... C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 89 / 90

Thanks Summary outlook Thank you! Thanks to German Research Foundation DFG DFG Center of Excellence QUEST German Aerospace Center DLR C. Lämmerzahl (ZARM, Bremen) Orbits in Static and Stationary Spacetimes Rome, 4.7.2009 90 / 90