Exact Solutions for 3-body and 4-body Problems in 4-dimensional Space Hideki Ishihara Osaka City University

1944 Research Institute for Theoretical Physics, Hiroshima University was founded 1948 RITP was re-build after the world war II at Takehara, Hiroshima 1990 RITP Hiroshima University was closed and merged together with Yukawa Institute, Kyoto University

Research Institute for Theoretical Physics

Journal of Science of Hiroshima University,Series A5 (1935) P. A. M. Dirac, "Generalized Hamiltonian dynamics". Can. J. Math. 2: 129–48 (1950). R.Arnowitt, S.Deser and C.W.Misner, " Canonical variables for general relativity,'' Phys. Rev. 117, 1595 (1960). B. S. DeWitt, "Quantum theory of gravity. I. The canonical theory". Phys. Rev. 160: 1113–48 (1967).

Staff history of RITP

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Middle era

Students in the middle era

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Students in the late era

Quantum field theory in the expanding universe (H.Nariai and T.Kimura) ADM formalism in expanding universes H.Nariai and T.Kimura, PTP 28(’62) 529. [L.Abbot and S. Deser, (’82)] Quantization of gravitational wave and mater fields in expanding universes H.Nariai and T.Kimura, PTP 29(’63) 269; PTP 29(’63) 915; PTP 31(’64) [A.Penzias and R.Wilson (’63)] [L.Parker PRL 21 (’68) 562 ] [S.W.Hawking, Nature 248 (’74) 30 ]

Development Gravitational anomaly T.Kimura, PTP 42 (‘69)1191; PTP 44 (‘70)1353 Removal of the initital singularity in a big-bang universe H.Nariai, PTP 46 (‘71)433, H.Nariai and K.Tomita, PTP 46 (‘71) 776

In theoretical physics, “unrealistic and non-urgent work” happens to turn to a cardinal issue. We should not ask a physically reasonable motivation so urgently. In the special issue for 60th anniversary of prof. Nariai But, it would be also necessary to keep a sort of soundness at each stage of research. Humitaka Sato says

Exact Solutions for 3-body and 4-body Problems in 4-dimensional Space Hideki Ishihara Osaka City University Shall we start

3-dim Gravity

Introduction Gravitational phenomena depend on spacetime dimension

Kepler motion in 3-dim. v.s. 4- dim. V 3 (r) V 4 (r) Stable bound orbits appear only in the 3-dimensional gravity

Black holes in general relativity Black Ring We shoud study Kerr black hole only Myers & Perry (1986)Emparan & Reall (2002) Black Hole (4+1)-dimensions (3+1)-dimensions

N-body problem under the gravitational interaction N-body problem under the gravitational interaction

3-body problem in 3-spatial dimensions 2-body (Kepler problem) : integrable → bound orbits are given by ellipses 3-body : not integrable in general small numbers of special solutions are known 1765 Euler, 1772 Lagrange, 2000 Eight figure choreography

N-body problem in 4-dim. space Equations of motion Lagrangian, Energy Potential is homogeneous in order -2.

Bounded orbits Constant inertial moment

Examples Exact solutions for 4-body problem 3-body problem in 4-dimensional space.

4-body problem Special configuration with the same mass Lagrangian Graviational potential

Effective Lagrangian Lagrangian Effective Lagrangian Constants of motion integrable !

Bounded solutions Equations of motion For bounded orbits

Exact solutions For closed orbits

=4/1, =3/1 Closed orbits = 2/1, = 2/1

=6/5, =4/3 = 4/3, = 5/3 Closed orbits 2

=3/2, =5/3 =3/2, =5/2 Closed orbits 3

3-body problem in 4- dimensions Special configuration with the same mass Lagrangian Graviational potential Effective Lagrangian

Bounded solutions Equations of motion For bounded orbits

Exact solutions Elliptic integral of the second kind Elliptic integrals of the first kind and third kind

Condition for closed orbits

Closed orbit 1

Closed orbit 2

Closed orbit 3

Constrained system Constant of motion on the constraint System admits conformal Killing vector Killing hierarchy (T.Igata,T.Koike,and H.I.)

Conclusions We consider systems of particles interacting by Newtonian Gravity in 4-dimensional space. There exists a special class of solutions: vanishing total energy and constant moment of inertia We obtain exact special solutions for 3-body and 4-body problems