3 Slowly rotating , compact fluid sources embedded in Kerr empty space - time

Spherically symmetric static fluid sources are endowed with rotation and embedded in Kerr empty space-time up to an including quadratic terms in an angular velocity parameter using Darmois junction conditions. Einstein's equation's for the system are developed in terms of linear ordinary differential equations. The boundary of the rotating source is expressed explicitly in terms of sinusoidal functions of the polar angle which differ somewhat according to whether an equation of state exists between internal density and supporting pressure. Following the publication by Kerr [1] of the metric which describes analytically , the asymptotically flat, vacuum gravitational field outside a rotating source in terms of Einstein's field equations, there has been much discussion concerning the existence of possible interior solutions which match the exterior smoothly. In an important development Hartle [2] uses a second order perturbation technique to describe the slow rotation of equilibrium configurations of cold stars having constant angular velocity. Solutions of Einstein's equations are developed in terms of Legendre polynomials but the issue of matching the results to Kerr empty space-time is not addressed. In the case of non-equilibrium configurations Kegeles [3] has applied the method to Robertson-Walker dust sources up to the first order in angular velocity parameter although, the results are somewhat restrictive and are not suitable for application to sources supported by internal pressure. In a recent work the case of the Wahlquist [4] closed form interior was shown not to fit the Kerr exterior by Bradley et al [5]. Only for the important case of thin super-massive rotating discs, supported by internal pressure have analytic sources for the Kerr metric been found (Pi-chon and Lynden-Bell [6]). This has led to an 'embarrassing hiatus' according to Bradley et al [5] in the number of potential interior solutions available for matching which in turn has contributed to a lack in the development in the theory of differentially rotating fluid bodies in general relativity. Yet it is important to develop further the relativistic theory of rotation since it has considerable potential application in astrophysics, for example, in the description 1