Odnos jednadžbi gibanja ispitne i efektivne čestice u relativističkom gravitacijskom problemu dva tijela

Abstract

The correspondence between the test- and effective-particle equations of motion of a non-relativistic gravitational two-body system is well understood. But the same is not true for a relativistic two-body system. This is because the effective one-body approach to a relativistic two-body problem is not yet fully elucidated. Among the known two effective one-body approaches to relativistic two-body problem, we follow up the one addressed through a constraint Hamiltonian. We investigate the correspondence of the resulting effective one-body equation of motion with the geodetical equation of motion of a test body in the Schwarzschild field. Next, we extend the two-body problem by endowing a spin to the central body, and examine again the correspondence between the effective one-body equations of motion of such a problem with the test-body description. In particular, we show the relation between the Carters equations of geodetical motion in the Kerr field with the equations indicated by the effective one-body approach of the two-body problem. In both the Schwarzschild and the Kerr fields, we determine the location of the innermost stable circular orbit (ISCO), which is an important key for the study of astrophysical binary stars. Subsequently, we examine the correspondence between the ISCO in the test-particle orbit and in the effective-particle orbit.Odnos jednadžbi gibanja ispitne i efektivne čestice u nerelativističkom problemu dva tijela dobro je poznat. To ne vrijedi za relativistički problem dva tijela. Razlog tome je što pristup efektivnog jednog tijela još nije razjašnjen. Od dvaju poznatih pristupa efektivnog jednog tijela relativističkom problemu dva tijela, mi slijedimo pristup preko uvjetovanog hamiltonijana. Ispitujemo odnos postignute jednadžbe efektivnog jednog tijela i geodetske jednadžbe gibanja ispitnog tijela u Schwarzschildovom polju. Nadalje, proširujemo problem dva tijela razmatrajući i vrtnju središnjeg tijela, i opet ispitujemo odnos jednadžbi gibanja efektivnog jednog tijela tog problema i opisa za ispitno tijelo. Posebno, pokazujemo odnos Carterovih jednadžbi za geodetsko gibanje u Kerrovom polju i jednadžbi u pristupu efektivnog jednog tijela. U Schwarzschildovom i u Kerrovom polju određujemo najmanju stabilnu kružnu stazu (ISCO) koja je važna za proučavanje astrofizičkih binarnih zvijezda, te zatim ispitujemo odnos ISCO staze ispitnog tijela i staze efektivnog tijela

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