### A convergent post-newtonian approximation for the constraint equations in general relativity

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The geodesic deviation equations, called also the Jacobi equations, describe only the first-order effects, linear in the small parameter characterizing the deviation from an original worldline. They can be easily generalized if we take into account the higher-order terms. Here we derive these higher-order equations not only directly, but also from the Taylor expansion of the variational principle itself. Then we show how these equations can be used in a novel approach to the two-body problem in...

The Wilson scheme and the Einstein dynamics are compared for binary systems. At the second post-Newtonian approximation, genuine two-body aspects are found to differ by up to 114%.

Gravitational radiation from a small mass particle orbiting a massive black hole can be analytically studied to a very high order in the post-Newtonian expansion. Thus it gives us useful information on the evolution of a coalescing compact binary star. In this talk, I report on recent progress made in the black-hole perturbation approach.

A post-Newtonian approximation scheme for general relativity is defined using the Arnowitt-Deser-Misner formalism. The scheme is applied to perfect fluids and point-mass systems. The two-body point-mass Hamiltonian is given explicitly up to the post${}^{2.5}$-Newtonian order.

The post-Newtonian (PN) hydrodynamic equations are obtained in the (3+1) formalism, which include the 2.5PN order as the reaction due to the quadrupole gravitational radiation. These equations are valid in various slice conditions, while we adopt a kind of transverse gauge condition to determine the shift vector. In particular, we describe methods to solve the 2PN tensor potential which arises from the spatial 3-metric. Our formulaton in the PN approximation using the (3+1) formalism will be useful...

Without making recourse to Newton's law of gravitation and starting from the concept of gravitational force, the concepts of active gravitational mass and of passive gravitational mass are introduced. Furthermore it is proved that they can be identified and that in Newton's law of gravitation the linear dependence on masses necessarily follows from the principle of superposition of simultaneous forces and from Newton's third law of dynamics.

Dynamics of a point-particle system interacting gravitationally according to the general theory of relativity can be analyzed within the canonical formalism of Arnowitt, Deser, and Misner. To describe the property of being a point particle one can employ Dirac delta distribution in the energy-momentum tensor of the system. We report some mathematical difficulties which arise in deriving the 3rd post-Newtonian Hamilton's function for such a system. We also offer ways to overcome partially these difficulties....