Kichiro Hiida

Substitution Law and Identity for Long-Range Potentials

We discuss the relations between two-body and many-body static potentials for long-range interactions. There is a simple substitution law to get the two-body potential V2<2nl in the 2n-th order from (n + 1) -body potential Vn+l obtained from tree diagrams for (n + 1) -body scatterings. When interaction Lagrangian density is proportional to mass in the static limit, the identity Vn+l= V2<2nl holds, Vn+l being the two-body potential obtained from Vn+l by summing it up over all elementary particles in two celestial bodies or charged spheres. These statements are proved to be right for n=2 and partly for n=3, and expected to be true for all values of n. The reason why the substitution law exists and the identity holds is discussed in detail.

Static gravitational potential and singular surface of the metric tensor for a two-body system in the time-symmetric approximation

SummaryThe properties of the gravitational potential and the singular surface of the metric tensor for a two-body system are investigated without resorting to any perturbative method, under the approximation that the transverse traceless part of the gravitational field is disregarded and the bodies are at rest momentarily. When the masses of the two bodies are different and the distance between them is restricted in some region, the lighter body is located outside the singular surface. The manybody static gravitational potential is also discussed.RiassuntoSi studiano le proprietà del potenziale gravitazionale e la superficie di singolarità del tensore metrico per un sistema di due corpi senza ricorrere a nessun metodo perturbativo, nell’approssimazione che la parte senza traccia trasversale del campo gravitazionale sia trascurata e che i corpi siano momentaneamente in stato di quiete. Quando le masse dei due corpi sono differenti e la distanza fra loro è contenuta in una certa zona, il corpo più leggero è ubicato al di fuori della superficie di singolarità. Si discute anche il potenziale gravitazionale statico di molti corpi.РезюмеИсследуются свойства гравитационного потенциала и сингулярная поверхность метрического тензора для двухчастичной системы. Рассмотрение производится без использования методов теории возмущений, но используя предположение, что поперечная с нулевым следом часть гравитационного поля не учитывается и что тела в данный момент покоятся. Когда массы рассматриваемых двух тел различны и когда расстояние между ними ограничено в некоторой области, тогда более легкое тело расположено вне сингулярной поверхности. Также обсуждается многочастичный гравитационный потенциал.

Mechanics of a particle near the center of a rotating ring

Abstract The equations of motion of a test particle moving near the center of a massive rotating ring are derived up to the post-post-Newtonian order of approximation, by using the metric tensor for many body system which is Minkowskian at spatial infinity. Logarithmic divergences due to self-interaction of the ring appear in the equations of motion. These divergences can be removed by the procedure which is similar to the renormalization method in particle physics. In the equations of motion there appears a force directing to the rotation axis and depending on the angular velocity of the ring. This force vanishes when the magnitude of the gravitational constant times the mass of the ring divided by the radius of the ring is about one tenth of the square of the velocity of light. Under this condition it is shown that the relative magnitude of the Coriolis force to the centrifugal force in the equations of motion agrees with the expected one from the equations of motion in a rotating reference frame.