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FIZIKA B 13 (2004) 4, 699-710
 
CORRESPONDENCE BETWEEN TEST- AND EFFECTIVE-PARTICLE EQUATIONS OF MOTION
IN RELATIVISTIC GRAVITATIONAL TWO-BODY PROBLEM
S. B. FARUQUE
Department of Physics, Shah Jalal University of Science and Technology, Sylhet 3114,
Bangladesh
E-mail address: awsbf62@yahoo.com
Received 22 December 2003; Accepted 15 November 2004
Online 2 February 2005
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 Carter’s 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.
PACS numbers: 95.30.Sf
UDC 539.12
Keywords: test-particle, effective-particle, two-body problem, relativistic
gravitation
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