Author: Jerry G. Bails

Publisher:

Published: 1957

Total Pages: 76

ISBN-13:

The purpose of this paper is to consider the General Theory of Relativity. Einstein's gravitational equations are solved for the static field of a single gravitating center. The equations of motion of a particle in this field are then derived. Throughout the discussion it is emphasized that the Newtonian theory remains as a limiting case of the General Theory. Further, the deflection of a ray of light just grazing the sun's limb is shown to be a conclusion of this theory, which Einstein published in 1916 in his famous memoir, "Die Grundlage der allgmeinen Relativitätstheorie". In order to reat the General Theory, it has been necessary first to develop the fundamentals of the tensor calculus. Then the equations for the geodesics are derived and simplified by introducing the Christoffel three-index symbols, whose properties are examined. The fundamental metric tensor and the Riemann-Christoffel tensor are discussed in some detail. These tensors, which measure respectively the metric properties and the Rienmannian curvature, play important roles in the natural geometry developed by Einstein. It has been found entirely satisfactory to introduce the concept of the tensor systematically from the point of view of the definite line integral. Tensor equations are characterized by the fact that they are invariant under coordinate transformations. The fundamental postulate of General Relativity requires that the laws of physics be expressed by equations which hold independently of the coordinate system selected to describe events in Minkowski space. Hence, tensors are indispensable in the formulation of such laws of physics as the law of inertia and the law of gravitation. Just as the Special Theory is clearly limited by its failure to include gravitational phenomena, the General Theory is limited by its failure to embrace Maxwell's equations. This paper discusses three attempts to develop unified field theories that would relate gravitational and electromagnetic penomena in one geometry. Weyl's early attempt to remove a hidden assumption in Riemannian geometry, and introduce the notion of gauge-invariance is sketched for the reader. Einstein's final success in developing unified field equations from a variational principle is discussed briefly, and the present research of Fantappie and Arcidiacono into a Final Relativity is outlined. These two Italian mathematicians have approached the problem of a unified theory from a consideration of the generalization of the Lorentz thransformations of Special Relativity. Their efforts provide one of the most recent approaches to the present problem.