No 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.

In Newtonian theory, gravity inside a constant density static sphere is independent of spacetime dimension. Interestingly this general result is also carried over to Einsteinian as well as higher order Einstein-Gauss-Bonnet (Lovelock) gravity notwithstanding their nonlinearity. We prove that the necessary and sufficient condition for universality of the Schwarzschild interior solution describing a uniform density sphere for all n ≥ 4 is that its density is constant.

We use analytic perturbation theory to present a new approximate metric for a rigidly rotating perfect fluid source with equation of state (EOS) ǫ+(1−n)p = ǫ 0 . This EOS includes the interesting cases of strange matter, constant density and the fluid of the Wahlquist metric. It is fully matched to its approximate asymptotically flat exterior using Lichnerowicz junction conditions and it is shown to be a totally general matching using Darmois-Israel conditions and properties of the harmonic coordinates. Then we analyse the Petrov type of the interior metric and show first that, in accordance with previous results, in the case corresponding to Wahlquist's metric it can not be matched to the asymptotically flat exterior. Next, that this kind of interior can only be of Petrov types I, D or (in the static case) O and also that the non-static constant density case can only be of type I. Finally, we check that it can not be a source of Kerr's metric.

To compare two space-times on large domains, and in particular the global structure of their manifolds, requires using identical frames of reference and associated coordinate conditions. In this paper we use and compare two classes of time-like congruences and corresponding *

We try to lay down the foundations of a Newtonian theory where inertia and gravitational fields appear in a unified way aiming to reach a better understanding of the general relativistic theory. We also formulate a kind of equivalence principle for this generalized Newtonian theory. Finally we find the non-relativistic limit of the Einstein's equations for the space-time metric derived from the Newtonian theory.

By considering the product of the usual four dimensional spacetime with two dimensional space of constant curvature, an interesting black hole solution has recently been found for Einstein-GaussBonnet gravity. It turns out that this as well as all others could easily be made to radiate Vaidya null dust. However there exists no Kerr analogue in this setting. To get the physical feel of the four dimensional black hole spacetimes, we study asymptotic behavior of stresses at the two ends, r → 0 and r → ∞.

We use a recent result by Cabezas et al. [1] to build up an approximate solution to the gravitational field created by a rigidly rotating polytrope. We solve the linearized Einstein equations inside and outside the surface of zero pressure including second-order corrections due to rotational motion to get an asymptotically flat metric in a global harmonic coordinate system. We prove that if the metric and their first derivatives are continuous on the matching surface up to this order of approximation, the multipole moments of this metric cannot be fitted to those of the Kerr metric.