Regular Interior Solutions to the Solution of Kerr which Satisfy the Weak and the Strong Energy Conditions

Abstract

The line element of a class of solutions which match to the solution of Kerr on an oblate spheroid if the two functions $ F(r)$ and $H(r)$ on which it depends satisfy certain matching conditions is presented. The non vanishing components of the Ricci tensor $R_{\mu\nu}$, the Ricci scalar $ R$, the second order curvature invariant $K$, the eigenvalues of the Ricci tensor, the energy density $\mu$, the tangential pressure $P_{\perp}$, and the quantity $\mu+P_{\perp}$ are calculated. A function $F(r)$ is given for which $R$ and $K$ and therefore the solutions are regular. The function $ H(r) $ should be such that the solution it gives satisfies at least the Weak Energy Conditions (WEC). Several $H(r)$ are given explicitly for which the resulting solutions satisfy the WEC and also the Strong Energy Conditions (SEC) and the graphs of their $\mu$, $P_{\perp}$ and $\mu+P_{\perp}$ for certain values of their parameters are presented. It is shown that all solutions of the class are anisotropic fluid solutions and that there are no perfect fluid solutions in the class.