A Meta-Physical Investigation for the Special Theory of Relativity - Preliminary Remarks- Yoshio UENO Hiroshima University The term Meta-Physics is to be explained. An adjective "meta" is used on one hand in a similar sense to the one used in the term metamathematics, and at the same time we mean by "meta-physics" a survey of the foundation of physics. In other words we will attempt a metatheoretical and philosophical investigation for the special theory of relativity. In this paper preliminary remarks for such purpose are presented. •˜ 1.Introduction. It is usually considered that Einstein's special theory of relativity is based on the following two principles: 1) The special principle of relativity, which says that "the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good" [1], and 2) the principle of constancy of light velocity which says that "light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body" [1]. And it is also considered that the Lorentz transformation can be derived from these principles. Indeed various methods for such derivation have been proposed, but it is a question whether the Lorentz transformation can be derived from these principles only. In many derivations some additional assumptions are usually proposed in the course of the deduction. These additional conditions are considered as the natural ones from the common sense of physicists, but strictly speaking the physical foundation of such conditions is not necessarily clear. Therefore we will now attempt to propose a new method of derivation for the Lorentz transformation without such "un reasonable" conditions. In the next section we will show some examples of such "unreas onable" conditions. Our assumptions for such approach base only on physically verified facts and some further conventional assumptions whose mean ings are already clear. The purpose of this paper is to clarify the foundation of the special theory of relativity and its limit of application. In other words we will verify directly the "truth" of the Lorentz transformation in opposition tothe -71- 2 Y.UENO Vol. 4 usual belief that the "truth of the Lorentz transformation" is assured indirectly by the fact that the Lorentz invariant theories fit exactly to many experimental facts. Namely, we will survey the foundation of a theory by the heuristic considerations and the logical analysis of the theory. In this sense this investigation will be regarded as a meta-physical approach for the theory of special relativity. In such approach the relation between the theoretical concepts used in the deduction and the experimental results taken as verified should be well considered. Because the special theory of relativity is quite a new theory different from the old one, therefore in this theory old ready-made theoretical concepts must not be used unconditionally. Moreover "the relativistic effects" are very small, then the accuracy of the experiments must be examined carefully. •˜ 2.Some problems in the usual derivations. As is stated in the foregoing section it is usually understood that the Lorentz transformation can be derived from the fundamental principles of the special theory of relativity. But usually in almost every derivations some assumptions independent of the fundamental principles are proposed for "a priori" reasons. We will clarify these circumstances further in the following. Generally speaking the derivation of some consequence is a deductive reason ing written schematically as follows: S1S2 ... Sk?St, (1) where St on the right hand side is the conclusion to be derived , and Si (i=1,...., k) on the left hand side is proposition assumed already verified. In this scheme the "truth" of the proposition on the left hand side of the formula is considered to be assured by the one of the following conditions: 1) The direct conclusion from the fundamental principle , 2) the truth of mathematics and logic, 3) experimentally verified facts, 4) conventions or definitions, or 5) logical conclusion derived from the propositions situated on the left side of the proposition. On the other hand when we investigate the usual derivations of the Lorentz trans formation, we find that some assumptions which are not of the above nature are used in the course of derivation. We will show some examples of such procedure in detail, but we remark some general problems beforehand . The most important of them will be how to select the propositions having the nature stated in 1). The special principle of relativity requires that physical laws should be covari ant under a certain coordinate transformation connecting the inertial frames which move uniformly each other . In this connection the problems to be considered -72- 2 No.2 A Meta-Physical Investigation 3 are ú@) the equivalence of some kind of reference systems, and úA) the kinematical relation of such reference system, i.e. ú@') the problem how to define "the inertial system", and úA') what is the relation between them . Therefore the first problem to be considered is how to lay out the coordinate system in an inertial system. After this procedure of the layout the equivalent frame can be layed out in this system by the same method. That will be at least the suffi cient condition for the first problem. Since the second problem is the kinematical one so the question will be how to connect two mutually moving systems by what kind of coordinate transformation. Laying stress on these problems we will next show in some typical papers how to treat these problems. The first paper which we will refer is written by Einstein in 1905 [1]. He bigins his paper with the heuristic considerations how to make the principle of the constancy of the light velocity consistent with such concepts as the length of the uniformly moving rigid rod or the synchronization of clocks situated at different places. After that he searches for the form of the coordinate transformation. In the course of his reasoning he put as the mathematically important assumption that the transformation is linear. He writes as the reason of this linearity assump tion as follows:"wegen Homogenitatseigenschaften, welche wir Raum and Zeit beilegen" [1]. However the meaning of the term, "Homogenitat" of the space and time is not clear and moreover it is also not clear how the linearity of the transformation originates from this homogeneous character of the space and time. From our point of view this linearity condition cannot be considered well founded. Further the obsecurity of this assumption is pointed out later by several authors [2], [3]. Therefore such condition is to be excluded. Next we will cite the paper written by Severi [4]. This paper gives us another appropriate example for our discussion. The problematic point on this paper is the assumption that the length of the uniformly moving rigid rod seems to be of constant length for the observer in the rest system. The correctness of this assumption will be verified, if we succeed the direct experimental observation of the moving rod. However in the present we have no method having enough exactness. Therefore, though the length of the moving rod may be "observable" in principle, this assumption is, in this sense, also undesirable from our point of view. Examples of the same kind also appears in the text book written by Panofsky and Phillip [5]. They assume that parallel measuring sticks oriented perpendicular to their direction of relative motion has the same length. They also put this assumption because of the homogeneity of the space. This assumption is also -73- 4 Y.UENO Vol.4 unacceptable from the same reason. Pauli [6] also takes the same kind of assump tion that the length of the uniformly moving rods have the same value independent of the directions of motion if they moves lengthwise. His reason is the symmetri city of the space. In other papers some authors assume that the transformation makes a group, or some mathematical conditions at infinity are assumed for the a priori reasons. These are equally of undesirable nature from the standpoint stated above. Hereupon we will make some comments concerning the meaning of the term "observable" . Though we said in the above that we should not assume unobserv able facts as self-evident, the inverse problem whether we can assume that fact as self-evident only because of its "observable" nature. To make the contents of the discussion precise it is necessary to clarify the meaning of the term "observ able". Generally speaking some discrepancies exist as for the intension of the term between philosophers and physicists. Detailed discussions will be reserved to the next section concerning the criterion of meaning. Among physicists the term is used in narrow sense or in wide sense. In the latter sense it is used as "observable in principle" , in this sense the results of the observation are not necessarily known. On the other hand in the narrow sense we use only when the results can be practically obtained. We will adopt the narrow sense of the term. Therefore "the length of a moving rigid scale" is not observable in our above sense, though it is "observable in principle". •˜ 3. Our standpoint concerning the criterion of the meaning. At first we will propose a question whether the theory of natural science must be unique and necessary. Further we will investigate whether such a question is the meaningful one or not. In this respect we will reconsider the theory proposed by Kant. As is well known, Kant insists that the authentic and necessary character of the theory of natural science is due to the fact that the propositions in scientific theories have synthetic a priori character.