Static gravitational fields. II. Ricci rotation coefficients

Abstract

Investigations of static gravitational fields in paper I are continued further. Instead of the system of second order partial differential equations in I, the equivalent first order system involving the Ricci rotation coefficients is dealt with here. The algebraic and the differential dependences among the equations have been sorted out to prove that the system is determinate. Then an attempt towards classifying this system of partial differential equations has been made. Except for ``the potential equation'' which is obviously elliptic, the remaining system is hyperbolic for which the characteristic surfaces have been determined. To obtain some exact solutions the lead of Newman-Penrose is followed in constructing the complex linear combinations of the equations. The class of static universes where ``gravitational lines of force'' are geodesics have been found. In this class one subclass is transformable to the conformastat metric and the remaining one reduces to a new metric involving the gravitational field of an arbitrary number of parallel infinite plates clamped at infinity. The source at infinity corresponds to that of the Newtonian potential φ =(1/2)[(m + 1/2)(x1)2 −(1/2) {m + 1/2 − (m2 − 1/4)1/2} (x2)2 − (1/2) {m + 1/2 + (m2 − 1/4)1/2} (x3)2]. This metric belongs to the nondegenerate Type I of Petrov's classification scheme. Next the class of static universes with ``shearfree lines of force'' is obtained. Here too one subclass goes over to the conformastat metric and the remaining one reduces to a Weyl-type universe.