Advances in Theoretical and Mathematical Physics
Volume 13 (2009)
A new formulation of general relativity
Pages: 1713 – 1770
In Sections 1-5 of this paper an axiomatic formulation of general theory of rativity (GR) is given and studied. Here use is made of the concept of pre-radar charts. These charts have "infinitesimally" the same properties as the true radar charts used in space-time theory. Their existence in GR has far-reaching consequences which are discussed throughout the paper. For the sake of simplicity and convenience I consider only such material systems the state of which is defined by a velocity field, a mass density and a temperature field. But the main results hold also for more complex systems. It follows from the axiomatics that the pre-radar charts define an atlas for the space-time manifold and that, in addition, they generate the metric, the velocity field and the displacement of the matter. Therefore, they are called generating functions. They act like "potentials". In Section 6 it is shown that the existence of pre-radar charts allows to simplify the original axiomatics drasticly. But the two versions of GR are equivalent. In Sections 7 and 8 the so-called inverse problem is treated. This means the question whether it is possible to define preradar charts, i.e., generating functions in arbitrary space-times. This problem is subtle. A local general constructive solution of it is presented. Sufficient conditions for the existence of global solutions are given. The aim of the last sections is formulating GR as a scalar field theory. The basic structural elements of it are a generating function, a generalized density and a generalized temperature. One of the axioms of this theory is a generalized Einstein equation that determines the generating function directly. It is shown that basic concepts like orientation, time orientation and isometry are expressible in terms of generating functions. At the end of the paper six problems are formulated which are still unsolved and can act as a stimulant for further research.