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ROTATING EQUILIBRIUM
CONFIGURATIONS IN EINSTEIN’S
THEORY OF GRAVITATION
Reinhard Meinel*
Friedrich–Schiller– Universität Jena
Theoretisch–Physikalisches Institut
Max–Wien–Platz 1, D-07743 Jena, Germany
Abstract The theory of figures of equilibrium of rotating fluids arose from geophysical
and astrophysical considerations. Within Newton’s theory of
gravitation, Maclaurin (1742) found the axially symmetric and stationary
solution to the problem in the case of constant mass–density: a
sequence of oblate spheroids. In the limit of maximum ellipticity a rotating
disk is obtained. By applying analytical solution techniques from
soliton theory this Maclaurin disk has been “continued” to Einstein’s
theory of gravitation [1, 2, 3].
After an introduction into these developments, the parametric collapse
to a rotating black hole and possible generalizations are discussed.
Keywords:
Rotation configurations, exact solutions.
1. FIGURES OF EQUILIBRIUM OF
ROTATING FLUID MASSES
The theory of figures of equilibrium of rotating, self–gravitating fluids
was developed in the context of questions concerning the shape of the
Earth and of celestial bodies. Many famous physicists and mathematicians
have contributed: Newton, Maclaurin, Jacobi, Liouville, Dirichlet,
Dedekind, Riemann, Roche, Poincaré, H. Cartan, Lichtenstein, Chandrasekhar,
and others. The shape of the rotating body can be obtained
from the requirement that the force arising from pressure, the gravitational
force, and the centrifugal force (in the corotating frame) be in
equilibrium, cf. Lichtenstein [4] and Chandrasekhar [5]. Newton [6]
*E–mail: meinel@tpi.uni-jena.de
Exact Solutions and Scalar Fields in Gravity: Recent Developments
Edited by Macias et al., Kluwer Academic/Plenum Publishers, New York, 2001 69