Rotating Incoherent Matter in General Relativity

Abstract

Under the assumption of rotational symmetry a method is developed for the numerical integration of exact Einstein equations describing the time evolution of a rotating incoherent matter. The complete system of equations deduced earlier for the case of an ideal fluid reduces here to six Einstein field equations and to the equation of continuity. The choice of the system of comoving cylindrical coordinates (z, r, [Formula: see text], t) is restricted by further conditions so that the only allowed coordinate transformations are the translations of the origin of the coordinates z, [Formula: see text], and t, and the reflections in these three axes. To any set of the permissible initial values of the field corresponds then only one physical situation. The exact field equations are deduced. With the exception of a few terms they are invariant with respect to certain permutations. The requirements upon the field to be regular at the initial moment and its infinitesimal past and future imposes restrictions on the functions appearing in the components of the metric field. Even then the field equations contain terms that have an indeterminate form 0/0, but are finite, at the z axis. Therefore two sets of field equations must be used : one set for the z axis in which the indeterminacy is analytically evaluated, and the other set for the space with r > 0. The junction conditions for the external field are formulated and proved that the corresponding external field does exist. The problem of the Lichnerowicz initial conditions is solved. An analytical proof is given that the rotation can stop the contraction of incoherent matter and revert it to a new expansion even along the axis of rotation. In terrestrial physics the axial acceleration is negligibly small, but it may play an essential role in cosmology.