p to the density N 10 g/cm . of.nucl.ear matter, and one has an argument from causality (.speed of sound < speed of light) that no allowable modifications of the equations of s • ta.te at supra.nuclear densities can change the -critical mass by more than a. factor of the order of: two away from an estimated figure M -M = 2 A 1033gr. OWe focus here on the second question: can one discuss stability-against gravitational collapse without mentioning an equation of state at all? Sir; A. 4;EJddingt:on raised questions; about the possibility o f using an equation of taste atll and also purported to derive an e quation of state quite different inthe r elativistic domain from Chandra. sekhar's standard equations of state for a degenerate ideal F`erxrd gas Today one takes seriously Henn of his results but only his motivation. He sought some escape from the concept of the critical mass, made so vivid by the f:ir s detailed-calculation of the

An analytical scheme and a numerical method in order to study the effects of general relativity on the viscosity driven secular bar mode instability of rapidly rotating stars are presented. The approach consists in perturbing an axisymmetric and stationary configuration and studying its evolution by constructing a series of triaxial quasi-equilibrium configurations. These are obtained by solution of an approximate set of field equations where only the dominant non-axisymmetric terms are taken into account. The progress with respect to our former investigation consists in a higher relativistic order of the non-axisymmetric terms included into the computation, namely the fully three-dimensional treatment of the vector part of the space-time metric tensor as opposed to the scalar part, solely, in the former case. The scheme is applied to rotating stars built on a polytropic equation of state and compared to our previous results. The 3D-vector part turns out to inhibit the symmetry breaking efficiently. Nevertheless, the bar mode instability is still possible for an astrophysically relevant mass of M ns = 1.4 M ⊙ when a stiff polytropic equation of state with an adiabatic index of γ = 2.5 is employed. Triaxial neutron stars may be efficient emitters of gravitational waves and are thus potentially interesting sources for the forthcoming laser interferometric gravitational wave detectors such as LIGO, VIRGO and GEO600. From a numerical point of view, the solution of the three-dimensional minimal-distortion shift vector equation in spherical coordinates is an important achievement of our code.

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 γ = 2. PACS number(s): 04.25. Dm, 04.40.Dg, 97.60.Jd, 02.70.Hm

We propose a new formulation for 3+1 numerical relativity, based on a constrained scheme and a generalization of Dirac gauge to spherical coordinates. This is made possible thanks to the introduction of a flat 3-metric on the spatial hypersurfaces t = const, which corresponds to the asymptotic structure of the physical 3-metric induced by the spacetime metric. Thanks to the joint use of Dirac gauge, maximal slicing and spherical components of tensor fields, the ten Einstein equations are reduced to a system of five quasi-linear elliptic equations (including the Hamiltonian and momentum constraints) coupled to two quasi-linear scalar wave equations. The remaining three degrees of freedom are fixed by the Dirac gauge. Indeed this gauge allows a direct computation of the spherical components of the conformal metric from the two scalar potentials which obey the wave equations. We present some numerical evolution of 3-D gravitational wave spacetimes which demonstrates the stability of the proposed scheme.

We present a new approach to the problem of binary black holes in the pre-coalescence stage, i.e. when the notion of orbit has still some meaning. Contrary to previous numerical treatments which are based on the initial value formulation of general relativity on a (3-dimensional) spacelike hypersurface, our approach deals with the full (4-dimensional) spacetime. This permits a rigorous definition of the orbital angular velocity. Neglecting the gravitational radiation reaction, we assume that the black holes move on closed circular orbits, which amounts to endowing the spacetime with a helical Killing vector. We discuss the choice of the spacetime manifold, the desired properties of the spacetime metric, as well as the choice of the rotation state for the black holes. As a simplifying assumption, the space 3-metric is approximated by a conformally flat one. The problem is then reduced to the resolution of five of the ten Einstein equations, which are derived here, as well as the boundary conditions on the black hole surfaces and at spatial infinity. We exhibit the remaining five Einstein equations and propose to use them to evaluate the error induced by the conformal flatness approximation. The orbital angular velocity of the system is computed through a requirement which reduces to the classical virial theorem at the Newtonian limit. PACS number(s): 04.70.Bw, 97.60.Lf, We dedicate this work to the memory of our dear friend Jean-Alain Marck. I. BACKGROUND AND MOTIVATIONBinary black holes have been the subject of numerous studies in the past two decades, both from the analytical and numerical point of view. These studies are motivated by the fact that the coalescence of two black holes is expected to be one of the strongest sources of gravitational waves detectable by the interferometric detectors LIGO, GEO600, TAMA300 and VIRGO, currently coming on-line [1].From the analytical point of view, the most recent works are based on the post-Newtonian formalism (see e.g. Ref.[2] for a review) or on the effective one-body approach developed by . In these works, the black holes are treated as point mass particles 1 , which is a very good approximation when the black holes are far apart. For closer configurations, one may turn instead to some numerical approach. The numerical studies can be divided in two classes: (i) the initial value problem for two black holes (see Ref.[7] for a review) and (ii) the time evolution of the initial data (see Ref.[8] for a review and Refs. [9][10][11] for recent results). One of the major problems in this respect is to get physically relevant initial data. Indeed, initial data representing two black holes have been obtained long ago by Misner [12] and Lindquist [13], as well as Brill and Lindquist [14] (see also Ref. [15] or Appendices A and B of Ref. [16]). However these solutions correspond to two momentarily static black holes and are therefore far from representing some stage in the evolution of an isolated binary black hole in our universe. Based on the seminal work of Bowen and Yo...

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A multidomain spectral method for computing very high precision three-dimensional stellar models is presented. The boundary of each domain is chosen in order to coincide with a physical discontinuity ͑e.g., the star's surface͒. 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 is revealed to be of the order of 10 Ϫ10 . 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.