Some Comments on the Foundations of Physics

The following sections are included:

• CONTENTS

• Introduction

A brief review of the history of the establishment of the Lorentz transformations and the special theory of relativity is given. The properties of privileged systems in uniform motion with respect to each other are then studied under the assumptions that space and time are homogeneous, that space is isotropic and symmetric with respect to velocities, and that the superposition of two positive velocities will again be a positive velocity, but without any reference to the phenomenon of light or electromagnetic phenomena. It is shown that the space and time coordinates of two systems are connected by linear relations, which have exactly the same form as the Lorentz transformations but contain a parameter a, which has the character of a limit velocity and is a fundamental structure constant of space-time. Phenomena moving with limit velocity in one privileged system are going to move with limit velocity in all privileged systems, and it is then natural to assume that light moves with limit velocity. After introducing the concepts of a Minkowski space, the laws of dynamics are studied in greater detail and the relation E=mc² is derived without any reference to radiation or electromagnetic phenomena in general. The results are going to be used to study the behavior of classical waves in privileged systems as a preparation for the introduction of “wave mechanics”.

After a brief discussion of the de Broglie relations and the development of the wave-corpuscle parallelism, the behavior of a classical wave in a relativistic framework is studied. By utilizing the properties of the Lorentz transformations, it is shown that there are basically three types of waves, and that in two of them there are wave packets which may be interpreted as “corpuscles”. In these cases, the study provides an alternative derivation of the de Broglie relations, but since only frequencies and wave numbers are discussed, Planck's constant does not occur.

In the growth of modern quantum theory, the quantum postulates and Planck's constant of action h have played a fundamental role. After a brief review of the historical development, the wave equation for a classical wave in a relativistic framework is here studied in somewhat greater detail. It is shown that, if the wave moves in a privileged system under the influence of a classical potential, the frequencies of the standing waves become quantized due to their association with an eigenvalue problem in accordance with Schrödinger's old idea of “quantization as an eigenvalue problem” with the difference that—since only frequencies and wave numbers are considered—the derivations are here carried out without the use of Planck's constant h. It is shown that, by using natural relativistic units, the physical wave equation may be reduced to a purely mathematical eigenvalue problem. Some applications are given.

It is well-known that, in Newton's classical mechanics, the force law, the momentum law, and the energy law are associated with each other and give the same results. It is shown that, if the special theory of relativity is applied to the phenomenon of gravitation based on the equivalence principle, these three laws are no longer compatible due to the fact that the inert mass now depends on the velocity. If one studies the motion of a particle with relative mass m in the gravitational field of a fixed spherical body with mass M and radius R, the energy law including Newton's potential with the gravitational constant f has the form

where κ = f M / c² is the gravitational radius of the mass M. It is shown that, if one rather arbitrarily starts from the energy law and the additional assumption that the gravitational force is a central force, one obtains a modification of Newton's law by a factor (1 − κ/r)⁻¹, which is close to 1 that its effect is hardly experimentally verifiable. After a brief review of the Classical case, it is shown that the soultion of the relativistic energy law leads to three different types of orbits, and that “energy law leads to three different types of orbits, and that “the elliptic case” gives a rosette orbit with a perihelion motion which is half of the one predicted in Einstein's general theory of relativity. Applications to the data from the Hulse-Taylor's binary pulsar show that, whereas the general theory of relativity gives a total mass of 2.82 solar masses, the energy law in the special theory of relativity provides an alternative giving a total mass of 7.98 solar masses, which means that, if the pulsar has a mass of 1.4 solar masses, the unseen companion has a mass of 6.58 solar masses and ought to be a “black hole”, which perhaps could be observed by means of X-ray astronomy. The paper is pedagogical in nature and indicates that it may be of some interest to investigate further the various possibilities contained also in Einstein's special theory of relativity.