BAR MODE-INSTABILITY IN RELATIVISTIC ROTATING STARS - A POST-NEWTONIAN TREATMENT

We construct analytic models of incompressible, uniformly rotating sta rs in post-Newtonian (PN) gravity and evaluate their stability against nonaxisymmetric bar modes. We model the PN configurations by homogene ous triaxial ellipsoids and employ an energy variational principle to determine their equilibrium shape and stability. The spacetime metric is obtained by solving Einstein's equations of general relativity in 3 + 1 ADM form. We use an approximate subset of these equations well su ited to numerical integration in the case of strong-field, three-dimen sional configurations in quasi equilibrium. However, the adopted equat ions are exact at PN order, where they admit an analytic solution for homogeneous ellipsoids. We obtain this solution for the metric, as wel l as analytic functionals for the conserved global quantities, M, M-0, and J. We present sequences of axisymmetric, rotating equilibria of c onstant density and rest mass parametrized by their eccentricity. Thes e configurations represent the PN generalization of Newtonian Maclauri n spheroids, which we compare to other PN and full relativistic incomp ressible equilibrium sequences constructed by previous investigators. We employ the variational principle to consider nonaxisymmetric ellips oidal deformations of the configurations, holding the angular momentum constant and the rotation uniform. We locate the point along each seq uence at which these Jacobi-like bar modes will be driven secularly un stable by the presence of a dissipative agent such as viscosity. We fi nd that the value of the eccentricity, as well as related ratios such as Ohm(2)/(pi rho(0)) and T/\W\ (=rotational kinetic energy/gravitatio nal potential energy), defined invariantly, all increase at the onset of instability as the stars become more relativistic. Since higher deg rees of rotation are required to trigger a viscosity-driven bar mode i nstability as the stars become more compact, the effect of general rel ativity is to weaken the instability, at least to PN order. This behav ior is in stark contrast to that found recently for secular instabilit y via nonaxisymmetric, Dedekind-like modes driven by gravitational rad iation. These findings support the suggestion that in general relativi ty nonaxisymmetric modes driven unstable by viscosity no longer coinci de with those driven unstable by gravitational radiation.