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A review is given of recent results about the computation of irrotational Darwin-Riemann configurations in general relativity. Such configurations are expected to represent fairly well the late stages of inspiralling binary neutron stars., Comment: 20 pages, 11 PostScript figures, uses PTPTeX, to appear in the Proceedings of Yukawa International Seminar 99 "Black Holes and Gravitational Waves", edited by T. Nakamura & H. Kodama, Prog. Theor. Phys. Suppl

2. GRAVI TATION AL W AV ES . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . .... . . . . . . . . . . . . . . . . . . . . . . . .. . . 659 2 .1 The General Theory of Relativity: a Short Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 2.2 The Weak-Field Approximation 663 2.3 The Effect of Gravitational Waves on Matter . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 664 2.4 Emission Mechanisms for Gravitational Waves and the Quadrupole Formula . .... ....... .. . . . . . . .... . . . . . . . . . . . . . . . . 665 2 .5 The Design of Gravitational-Wave Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668

We present a numerical method to compute quasiequilibrium configurations of close binary neutron stars in the pre-coalescing stage. A hydrodynamical treatment is performed under the assumption that the flow is either rigidly rotating or irrotational. The latter state is technically more complicated to treat than the former one (synchronized binary), but is expected to represent fairly well the late evolutionary stages of a binary neutron star system. As regards the gravitational field, an approximation of general relativity is used, which amounts to solving five of the ten Einstein equations (conformally flat spatial metric). The obtained system of partial differential equations is solved by means of a multi-domain spectral method. Two spherical coordinate systems are introduced, one centered on each star; this results in a precise description of the stellar interiors. Thanks to the multi-domain approach, this high precision is extended to the strong field regions. The computational domain covers the whole space so that exact boundary conditions are set to infinity. Extensive tests of the numerical code are performed, including comparisons with recent analytical solutions. Finally a constant baryon number sequence (evolutionary sequence) is presented in details for a polytropic equation of state with gamma=2., Minor corrections, references updated, 42 pages, 25 PostScript figures, accepted for publication in Phys. Rev. D

We present a spectral method for solving elliptic equations which arise in general relativity, namely three-dimensional scalar Poisson equations, as well as generalized vectorial Poisson equations of the type $\Delta \vec{N} + \lambda \vec{\nabla}(\vec{\nabla}\cdot \vec{N}) = \vec{S}$ with $\lambda \not= -1$. The source can extend in all the Euclidean space ${\bf R}^3$, provided it decays at least as $r^{-3}$. A multi-domain approach is used, along with spherical coordinates $(r,\theta,\phi)$. In each domain, Chebyshev polynomials (in $r$ or $1/r$) and spherical harmonics (in $\theta$ and $\phi$) expansions are used. If the source decays as $r^{-k}$ the error of the numerical solution is shown to decrease at least as $N^{-2(k-2)}$, where $N$ is the number of Chebyshev coefficients. The error is even evanescent, i.e. decreases as $\exp(-N)$, if the source does not contain any spherical harmonics of index $l\geq k -3$ (scalar case) or $l\geq k-5$ (vectorial case)., Comment: Minor revisions. 31 pages, 13 figures, accepted for publication J. Comp. Phys

We present spectral methods developed in our group to solve three-dimensional partial differential equations. The emphasis is put on equations arising from astrophysical problems in the framework of general relativity., 51 pages, elsart (Elsevier Preprint), 19 PostScript figures, submitted to Journal of Computational & Applied Mathematics

We report on general relativistic calculations of quasiequilibrium configurations of binary neutron stars in circular orbits with zero vorticity. These configurations are expected to represent realistic situations as opposed to corotating configurations. The Einstein equations are solved under the assumption of a conformally flat spatial 3-metric (Wilson-Mathews approximation). The velocity field inside the stars is computed by solving an elliptical equation for the velocity scalar potential. Results are presented for sequences of constant baryon number (evolutionary sequences). Although the central density decreases much less with the binary separation than in the corotating case, it still decreases. Thus, no tendency is found for the stars to individually collapse to black hole prior to merger., Minor corrections, improved figure, 5 pages, REVTeX, Phys. Rev. Lett. in press

A multi-domain spectral method for computing very high precision 3-D stellar models is presented. The boundary of each domain is chosen in order to coincide with a physical discontinuity (e.g. the star's surface). In addition, a regularization procedure is introduced to deal with the infinite derivatives on the boundary that may appear in the density field when stiff equations of state are used. Consequently all the physical fields are smooth functions on each domain and the spectral method is absolutely free of any Gibbs phenomenon, which yields to a very high precision. The power of this method is demonstrated by direct comparison with analytical solutions such as MacLaurin spheroids and Roche ellipsoids. The relative numerical error reveals to be of the order of $10^{-10}$. This approach has been developed for the study of relativistic inspiralling binaries. It may be applied to a wider class of astrophysical problems such as the study of relativistic rotating stars too., Minor changes, Phys. Rev. D in press

A general relativistic version of the Euler equation for perfect fluid hydrodynamics is applied to a system of two neutron stars orbiting each other. In the quasi-equilibrium phase of the evolution of this system, a first integral of motion can be derived for certain velocity fields of the neutron star fluid including the (academic) case of co-rotation with respect to the orbital motion (synchronized binaries) and the realistic case of counter-rotation with respect to the orbital motion. The velocity field leading to this latter configuration can be computed by solving three-dimensional vector and scalar Poisson equations., Comment: 14 pages, 1 PostScript figure, RevTeX, accepted for publication in Physical Review D

It is shown how the special separability properties of the Kerr solution can be used to obtain an explicit analytic solution to the problem of constructing an orthonormal tetrad that is parallel-propagated along an arbitrary time-like geodesic. The components of the tidal tensor are calculated explicitly in terms of this tetrad. The special case of geodesics lying in the equatorial plane is examined.