An approximate global solution of Einstein’s equations for a rotating finite body

Abstract

We obtain an approximate global stationary and axisymmetric solution of Einstein's equations which can be considered as a simple star model: a self-gravitating perfect fluid ball with constant mass density rotating in rigid motion. Using the post-Minkowskian formalism (weak-field approximation) and considering rotation as a perturbation (slow-rotation approximation), we find second-order approximate interior and exterior (asymptotically flat) solutions to this problem in harmonic and quo-harmonic coordinates. In both cases, interior and exterior solutions are matched, in the sense of Lichnerowicz, on the surface of zero pressure to obtain a global solution. The resulting metric depends on three arbitrary constants: mass density, rotational velocity and the star radius at the non-rotation limit. The mass, angular momentum, quadrupole moment and other constants of the exterior metric are determined by these three parameters. It is easy to check that Kerr's metric cannot be the exterior part of that metric.

Slowly rotating fluid balls of Petrov type D

“…As we will see shortly, the deviation from algebraically special cases can arise first when considering the second order terms in the rotational parameter. Here we require the interior solution to remain Petrov type D for slow rotation, thereby completing the system of field equations (3), (6), (7), (8), (9), (10) and (19) by a further condition, which in some sense plays the role of an equation of state.…”

“…For a review of relativistic rotating stars see [5]. In [6] global models for slowly rotating bodies in the post-Minkowskian approximation are treated. In a recent paper [7] second order perturbation theory for the matching of general stationary axisymmetric bodies to an asymptotically flat vacuum has been put on a more solid mathematical ground and the exterior metric is determined to second order.…”

An approximate global solution of Einstein’s equations for a differentially rotating compact body

“…As in our previous work [12,13,14,15] on the rigid rotation problem, here we introduce a post-Minkowskian parameter, λ, and a dimensionless rotation parameter, Ω = λ −1/2 ωr 0 , where r 0 is the radius of the source in the nonrotation limit. Then we can rewrite (12) as:…”

“…In some previous papers [11,12,13,14], we studied this problem for rigid rotation; and now, we will use the same approximation scheme to study a differentially rotating perfect fluid.…”

Stationary axisymmetric exteriors for perturbations of isolated bodies in general relativity, to second order

“…Yet another method used for the construction of models of slowly rotating stars has recently [20,21] produced a particular model to second order in the approximation for a rigidly rotating and constant density perfect-fluid interior. The method is based on a two-parameter perturbation scheme, combining the post-Minkowskian and slow-rotation approximations, both taken up to the first non-linear level.…”