Generalized Kerr spacetime with an arbitrary mass quadrupole moment: geometric properties versus particle motion
An exact solution of Einstein's field equations in empty space first found in 1985 by Quevedo and Mashhoon is analyzed in detail. This solution generalizes Kerr spacetime to include the case of matter with an arbitrary mass quadrupole moment and is specified by three parameters, the mass M, the angular momentum per unit mass a and the quadrupole parameter q. It reduces to the Kerr spacetime in the limiting case q = 0 and to the Erez–Rosen spacetime when the specific angular momentum a vanishes. The geometrical properties of such a solution are investigated. Causality violations, directional singularities and repulsive effects occur in the region close to the source. Geodesic motion and accelerated motion are studied on the equatorial plane which, due to the reflection symmetry property of the solution, also turns out to be a geodesic plane.
Exterior and interior metrics with quadrupole moment
We present the Ernst potential and the line element of an exact solution of Einstein’s vacuum field equations that contains as arbitrary parameters the total mass, the angular momentum, and the quadrupole moment of a rotating mass distribution. We show that in the limiting case of slowly rotating and slightly deformed configuration, there exists a coordinate transformation that relates the exact solution with the approximate Hartle solution. It is shown that this approximate solution can be smoothly matched with an interior perfect fluid solution with physically reasonable properties. This opens the possibility of considering the quadrupole moment as an additional physical degree of freedom that could be used to search for a realistic exact solution, representing both the interior and exterior gravitational field generated by a self-gravitating axisymmetric distribution of mass of perfect fluid in stationary rotation.
Limitations on the interpretation of biotite substitutions from chemical analyses of natural samples
Biotites, as well as other phases, are sufficiently complex chemically that there are numerous sets of linearly independent exchange components with which to describe the composition of a naturally occurring crystal. Even after complete chemical analysis, this leads to ambiguities in the correlation of exchange components with valid substitution mechanisms in complex phases. The best means of identifuing valid substitutions are by the observation of simple endmember compositions, the investigation of compositional changes in zoned crystals or petrogenetic sequences, and by direct experimental synthesis.