GAUSSIAN COORDINATE SYSTEMS FOR THE KERR AND SCHWARZSCHILD METRICS

Abstract

We present the whole class of Gaussian coordinate systems for the Kerr metric. This is achieved through the uses of the relationship between Gaussian observers and the relativistic Hamilton-Jacobi equation. We analyze the completeness of this coordinate system. The recognition that natural processes cannot be influenced by any choice of a representation of the events occurring in space-time conducted the covariance principle to be assumed as one of the fundamentals of modern physics. This idea was explicitly used to build a theory of gravity by Einstein, General Relativity, which gave a step forward by invoking that the MMG (Manifold Mapping Group) should be taken as an invariance principle of the theory. In practical uses however, one is obliged to select a particular language by choosing a special coordinate system to describe a given phenomenon. Among all the possible choices one can make – most of them dictated by symmetry of the given problem – there is a very special one that can bring us more physical insight about the problem to be treated, that is the Gaussian coordinate system. In fact, this coordinate system was suggested by C. Gauss in his works about curves and surfaces. In Gaussian coordinate system (GCS), a foliation of space-time is made in such a way that one separates space and time, as in pre-relativistic theories. The time-like world-line of an observer that is orthogonal to the 3-d space (which is identified to the co-moving system) is such that the proper time of such observer coincides with the coordinate time. Mathematically, a Gaussian coordinate system is constructed by the definition of a hypersurface S = S(x ), which satisfies