PERFECT FLUID AND ELECTROMAGNETIC DISTRIBUTIONS OF CLASS ONE AND CLASS TWO IN GENERAL RELATIVITY

Abstract

In general relativity the field is taken as a physical object and the four-dimensional Riemannian manifold is taken as the Mathematical description of this field* In Chapter I the basic facts regarding the description of physical fact in terms of geometrical properties of space-time are summarized and the restrictions imposed on the coordinate systems utilized to describe that fieldfare mentioned briefly. The meaning of class one and class two space-times is explained and some of the well known and important results in connection with the embedding of Riemannian four fold are given. The Chapter closes with a summary of main results obtained by the author in the present thesis- In Chapter II we start with a Riemannian four-fold of class one and express the energy momentum tensor of the field in .terms of the second fundamental form- The requirements of the perfect fluid distribution lead to two distinct canonical forms for the second fundamental form- One of these canonical forms is compatible with a perfect fluid distribution described by a conformally flat space - time-The other form is compatible with the perfect fluid distribution of class one but the resulting space-time is not conform-.lly flat- A new line element describing perfect fluid distribution of class one has been constructed which is not conformally flat. The metric of this fluid model is joined with an exterior metric with the property that the metric potentials are continuous ii across a certain hypersurface• It has also been demonstrated that the non-static analogues of Schwarzschildfs interior solution recently obtained by Vaidya can be re-discovered by imposing class one conditions on spherically symmetric space-time- The cases not included in Vaidya1 s investigat" ions have also been examined. In the III Chapter we start with a space-*time for which the Wyle conformal curvature tensor is identi cally zero and prove that conformally flat space.time is neceecessarily of class one, if it describes a perfect fluid distribution- Combining this with the fact that a conformally flat space time cannot describe null and non null electromag netic field we have concluded that the only physically signi ficant distribution described by a conformally flat space - time of class one is the perfect fluid- Some conformally flat space-times of class one have been obtained by imposing perfect fluid conditions on a most general conformally flat space-time • It is well known that conformally flat space time is in general of class two The class one conformally flat metric need not be des cribing a perfect fluid distribution* Such cases are dis cussed with the help of Gauss Codazzi equations- * some of the results of this chapter have been published as a pgper entitled • Space-time of class one and perfect fluid distributions" University of Roorkee Research Journal XII (1970), 10-15. It has also been presented at the annual conference of Indian Math- Society, Nagpur 1969. iii In Chapter IV it has been shown that an electromag netic distribution of energy is compatible with the class one metric if and only if the elementary divisors of second fundamental form are not simple- When the elementary divisors are not simple a canonical form involving a null vector has been obtained which gives rise to a null electro magnetic field. A null electromagnetic field of class one has been constructed and it is shown that the second funda mental tensor conforms with the canonical form obtained above- Starting with a photon fluid model of class two and imposing class one conditions a photon fluid model of class one has been constructed and it is shown that the second fundamental form conforms with the canonical form is discovered in this Chapter