Page:Benton 1959 The Clock Problem (Clock Paradox) in Relativity.djvu/28

 are equivalent for the formulation of the general laws of nature, and that this does not imply complete equivalence between all frames of reference.

The theory of uniformly accelerated motions based on the conformal group of transformations in space-time is applied to the clock problem of relativity theory. Two solutions are found, both of which are at variance with the usual theory. The bearing of the problem on the relation between mechanics and electromagnetic theory is discussed.

An examination is made of the result of restricting translations and Lorentz transformations in space-time to those with rational coefficients. This removes the major defect of Schild's model of discrete space-time by elimination of the lower bound on the relative velocities of reference systems. The theory is formulated in two stages. In the first, the energy and momentum components of a particle are restricted to a countable set satisfying the relativistic energy-momentum space in which wave functions are almost periodic functions. In the second state, the space-time variables are restricted to rational values. This leads to the theory of discrete space-time.

Note in respect to work of H.E. Ives in connection with experimental verification of one of consequences of Einstein's restricted theory of relativity.

Singer's formula for the general relativistic red-shift on an earth satellite is modified to take account of the diurnal rotation of the earth and the lack of spherical symmetry of its gravitational field. It is shown that the Singer rates of the earth and satellite clocks need slight modifications, but that these modifications tend to cancel each other except at large distances from the earth, so that when one uses a mean radius of the earth in Singer's formula, the formula is adequate for present purposes.