Christoffel (1869), thereafter Klein (Erlanger Programm, Programm zum Eintritt in die philosophische Fakultät in Erlangen, 1872

Marian Apostol
2008 unpublished
Weak external forces and non-inertial motion are equivalent with the free motion in a curved space. The Hamilton-Jacobi equation is derived for such motion and the effects of the curvature upon the quantization are analyzed, starting from a generalization of the Klein-Gordon equation in curved spaces. It is shown that the quantization is actually destroyed, in general, by a non-inertial motion in the presence of external forces, in the sense that such a motion may produce quantum transitions.
more » ... ntum transitions. Examples are given for a massive scalar field and for photons. Newton's law. We start with Newton's law m dv dt = f ; (1) for a particle of mass m, with usual notations. I wish to show here that it is equivalent with the motion of a free particle of mass m in a curved space, i.e. it is equivalent with Du i ds = du i ds + i jk u j u k = 0 ; (2) again with usual notations. Obviously, the spatial coordinates of equation (1) are eu-clidean, and equation (1) is a non-relativistic limit. It follows that the metric we should look for may read ds 2 = (1 + h) c 2 dt 2 + 2cdtg 0 dx + g dx dx ; (3) where g = (=), while functions h; g 0 1 are determined such that equation (2) goes into equation (1) in the non-relativistic limit v c 1 and for a correspondingly weak force f. Such a metric, which recovers Newton's law in the non-relativistic limit, is not unique. The metric given by The geometry of the curved spaces originates probably with Gauss (1830). It was given a sense by Riemann (Uber die Hypothesen welche der Geometrie zugrunde liegen who made the connection with the physical theories. It is based on point (local) coordinate transforms, cogredient (contravariant) and contragre-dient (covariant) tensors and the distance element. It is an absolute calculus, as it does not depend on the point, i.e. the reference frame. It may be divided into the motion of a particle, the motion of the fields, the motion of the gravitational field, and their applications, especially in cosmology and cosmogony. As the curved space is universal for gravitation, so it is for the non-inertial motion, which we focus upon here. The body which creates the gravitation and the corresponding curved space is here the moving observer for the non-inertial motion, beside forces. It could be very well that the world and the motion are absolute, but they depend on subjectivity, though it could be an universal subjectivity (inter-subjectivity). See W. Pauli, Theory of Rel-ativity, Teubner, Leipzig, (1921). equation (3) can be written as g ij = 0 B B @ 1 + h g 10 g 20 g 30 g 01 1 0 0 g 02 0 1 0 g 03 0 0 1 1 C C A ; (4) (where g 0 = g 0 = g). We perform the calculations up to the first order in h; g and v c. The distance given by (3) becomes then ds = cdt 1 + h 2 and the velocities read u 0 = dx 0 ds = 1 h 2 ; u = dx ds = v c : (5) It is the Christoffel's symbols (affine connections) i jk = 1 2 g im @g mj @x k + @g mk @x j @g jk @x m (6) which require more calculations. First, the contravariant metric is g 00 = 1 h, g 0 = g 0 , g 0 = g 0 , g = , such that g im g mj = g jm g mi = j i. By making use of (6) we get 0 00 = 1 2c @h @t ; 0 0 = 0 0 = 1 2 @h @x 0 = 0 = 1 2 @g 0 @x + @g 0 @x 0 = 0 = 1 2 @g 0 @x @g 0 @x 00 = 1 2 @h @x 1 c @g 0 @t ; = 0