ON ONE SIMPLE DERIVATION OF LORENTZ’S TRANSFORMATIONS
N. Chavarga Uzhgorod National University. Pidhirna Str., 46, Uzhgorod 88000, Ukraine. chavarga@mail.uzhgorod.ua
Analysis of the mathematical part of Lorentz’s transformations derivation is given. This derivation was done by Einstein in his work “On the Special and General Relativity Theory” (1917).
It is known that the Special Relativity Theory (SRT) is still raising doubts concerning its correctness. Quite recently V. Fedorov and D. Ponomarev’s work [1] appeared in the Internet. In this work, it is shown by means of strict mathematical analysis (due to general reasons) that the derivation of Lorentz’s transformations in Einstein’s work [2] cannot be correct. After recognizing this analysis as made properly, the decision was made to search for concrete place, in which errors were made, in Einstein’s work, and to try to discover the essence of these errors. In spite of the respect, which the author has towards Einstein’s personality, especially for his long-term and non-compromise struggle against probabilistic interpretation of the physical meaning of the wave equation psi-function, one of the reasons, that led to writing the present work, consisted in the refusal of the majority of journals not only from publishing, but even simply analyzing any works, in which at least a slight doubt concerning the correctness of the SRT, and also making up the image of Einstein as the greatest genius in the literature, etc. For illustrating what we have said, let us quote some places not from the popular science literature or mass media, but from a very serious textbook on physics for students of high-school physical specialties [3]. Though it sounds quite strange, but it concerns the authorship of Lorentz’s transformations. “It is interesting to note that the formulae of transformations obtained by Einstein coincide with the formulae earlier mentioned by Lorentz… …when the necessity of interpreting Michelson’s experiment made Lorentz introduce a shortening hypothesis, he came to the conclusion that the transformation formulae coinciding with (131.1) leave the equations of electrodynamics for vacuum invariant. Therefore, these formulae are quite often called Lorentz’s formulae. …From the formulae of Einstein-Lorentz’s transformations forming a significant part of the relativity theory, there follows a series of consequences giving such a specificity to the conclusions of this theory” [3, p. 458]. Similar statements can be also found in a series of other books. One of the causes, due to which errors of theoretical character appeared in scientific works, consists in the negligent attitude towards the physical meaning of the analyzed quantities, and also negligent graphical representation of the problem. It leads to the fact that different quantities can be denoted by one symbol. As it will be shown below, just such incorrect examples we observe in the work [2]. For convenience and vividness of the argumentation, later on the article will be divided into two columns. In the left column the broad quoting of the analyzed work is given, in the right one our commentary is represented. Usually, when a theory is made up, first the physical idea, representing the essence of the theory in general, is stated, the qualitative explanation of the discussed experiment is proposed, then follows the process of mathematical presentation of the proposed ideas, in order to obtain quantitative relations. When Lorentz’s transformations were derived by Einstein, no ideas, representing profound physical essence of the phenomena, were stated. The basic primary position consisted in the fact, that “the light velocity in the moving frame is equal to the light velocity in the immovable frame, irrespective of the direction of the frame motion”. Mathematically this idea is manifested in the fact that the measured value of the light velocity C in the numerical expression is the same in any inertial coordinate frame.
^ Transformations (Addition to § 11) If the coordinate frame is located as depicted on Fig. 2, axes X of both frames constantly coincide. Here we can divide the problem into two parts and first analyze only the events located on the axis X. Such an event is determined with respect to the coordinate frame K by the abscissa x and time t, and with respect to K – by the abscissa x and time t. One should calculate x and t, if x and t are given.
Fig. 2
The light signal propagating in the positive direction of the axis ^ is moving in accordance with the equation x=ct (0) or x–ct=0 (1) Since the same light signal is also propagated with respect to K with the same velocity c, then its motion with respect to frame K will be described by the equation x'–ct=0 (2) The space-time points (events) corresponding to equation (1) should also correspond to equation (2). Obviously, it will take place in case if the following relation is carried out in general: x'–ct=(x–ct) (3) where is some constant. In reality, in accordance with relation (3), transformation of the expression x–ct into zero means that x'–ct' is transformed into zero, too. Quite an analogous consideration used concerning light rays, propagating in the negative direction of the axis ^ , leads to the condition x'+ct=µ(x+ct) (4) When adding and subtracting relations (3) and (4) and introducing at that time (for convenience) new constants instead of the constants and µ,
we obtain
(5) Our problem would be solved, if the constants a and b were known; the latter are determined for the following reasons. For the coordinate origin of frame K, x=0 all the time, therefore, according to the first equation of (5), we have (5.1) Denoting the velocity, with which the coordinate origin of frame K moves with respect to K, by v, we calculate (6) The same value of v is obtained from the equations of (5), if one calculate the velocity of some other point of frame K with respect to K or the velocity of some point of frame K (directed towards negative values of x) with respect to K. Thus, the quantity v can be briefly called the relative velocity of both frames. Then it is clear from the relativity principle that from the viewpoint of frame K the length of some singular scale resting with respect to K should be exactly the same as the length of the same scale resting with respect to K from the viewpoint of K. In order to know the behavior of points X from the viewpoint of frame K, we should only make a “momentary photograph” of frame K from frame K; it means that instead of t (time of frame K) we must substitute some of its definite values, for example, t=0. Then, instead of the first equation (5), we will obtain: x=ax Therefore, two points of the axis X, the distance between which (measured in frame K) is equal to 1 (х=1), on our momentary photograph are located at the distance (7) But if the momentary photograph is made from frame K (t =0), then, through withdrawing t from equations (5) by means of the equality (6), we obtain (7.0) Thus, we make a conclusion that two points on the axis ^ , located at the distance equal to 1 (with respect to K), are divide on our momentary photograph by the distance (7а) Since, in accordance with the above-mentioned statement, both momentary photographs must be identical, then х in relation (7) should be equal to х in relation (7a), thus, we obtain (7b) The equality of (6) and (7b) is determined by the constants a and b. When substituting expressions for a and b into equations (5), we obtain the first and the fourth equations given in § 11: (8) Therefore, we have obtained Lorentz’s transformations for events on the axis ^ .” [2, p. 588, 2a, p. 223]. (Reverse translation from Russian). | Analysis of Lorentz’s Transformations Derivation Einstein’s purpose consisted in obtaining the known space and time transformations (8). In these equations, the quantities x and t should have the meaning of the space and time coordinates of an arbitrary event in frame K, and the quantities x and t should have the meaning of the coordinates (measured by means of frame K) for the same event. If the theory can transform the coordinates of an arbitrary event from frame to frame, it will be able to transform any physical law, any motion equation, since motion is a chain of events, each of which is a separate occasion of an arbitrary event.
Fig. 2
Note at once, that in equation (1) quantities x and t have the meaning of a photon’s space and time coordinates. They are not arbitrary points already, now they are interconnected by the photon’s motion equation – if the time coordinate is known, then the space one is unequivocally known, too. There is nothing terrible in using similar designations if the motion of only one object is analyzed in the problem. But it is quite another thing, if several objects are available, as in the analyzed case, when the motion of two physical objects – the photon and frame K – is considered. In order to avoid confusion, the photon’s coordinates ought to be denoted by other symbols, for example: in frame K and in frame K'. In these designations, equations (1) and (2) acquire the following form: –ct =0 (1’) –ct =0 (2’) Equation (2) is the mathematically correct writing of the physically irrational idea, that the value of the light velocity in frame K does not depend either on the velocity of this frame motion, or on the direction of the photon propagation, but in the present work we will not analyze this question – discussions on this matter have not been ceasing in the literature for a little more than a hundred years already. Let us note only that (2) does not follow from anywhere, it is a primary position, a postulate. First we draw reader’s attention that in reality (1) and (2) are the identities like 0=0, therefore, in reality, equation (3) represents an expression like 0= 0 (3’) which is valid for any numerical value of the coefficient , therefore, the value of this expression is, softly speaking, quite doubtful for proving something. Equation (4), and also system (5) as a result of transformations (3) and (4) have the same doubtful value. Now we will analyze the statement: “For the coordinate origin of frame K, x=0 all the time…”. It is quite evident that the coordinate origin of the moving frame has values of the space coordinates in frame K different from those of the photon, see Fig. 2’, thus, it should have an equation of motion with other designations of the space coordinate. Let us denote the coordinates of the origin K by the symbol in frame K and symbol in frame ^ . In such denotations the equation of motion of the K origin in frame K has the following form: =vt (4’) and in frame K it has the following form: =0 (5’) According to the primary agreement, the quantity x in the first equation of the system (5) denotes the photon’s coordinate in the primed frame, but Einstein, instead of this coordinate, substituted the coordinate of K origin, and denoted it by the same symbol as the photon’s coordinate in the moving frame. As a result, he “got rid” of x' which had bothered him. Now one cannot say any more, that the “value of a similar operation is doubtful”, since it is a gross error in quite simple calculations. One should assume, that if in (5) there were denotation instead of x, then, most probably, the mentioned error would not occur. Note that the coordinates of the photon and the origin of frame K' are equal to each other only at the time t=0, at the frame’s and photon’s starting moment, at the moment when the coordinate origins coincided. Then, since in equation (5.1) the quantities x and t have, as before, the meaning of the photon’s coordinates in frame K, then their relation x/t is nothing else but the light velocity c, however, Einstein denoted this relation by the symbol v, substituted it to (5.1) and, as a result, he obtained equation (6). The value of such an operation cannot be estimated as doubtful, either, since it is already the second gross error. Three errors on one page make further analysis of the work absolutely excessive; however, we will continue our analysis. In order to get rid of the vivid dependence of the quantity х on х, Einstein replaced the quantity х in (7) by a unit. Now if in (7), instead of a unit, one substitutes “its legal value” from (7b), an absolutely absurd result will be obtained: (6’) It will remain incorrect, even if instead of a unit in (7) its value х is returned – the right part of (6’), in accordance with the SRT, must be under a radical. Then, if in (7.0) the quantity х has the length dimensionality, in equation (7a) analogous to it, х is dimensionless already. One could think that it was the translation misprinting or the misprinting of Einstein himself, that the right part of (7a) in the manuscript had been multiplied by x. It is not difficult to see, however, that in such a case the result (7b) cannot be obtained at all… |
Is it possible (on the basis of the given analysis) to make a conclusion about Einstein’s physical and mathematical training? Probably not, however, one should not deny, either, that the presence of a series of gross errors in one work for all that testifies to something objectively. At least, it testifies to the fact that the image of Einstein as of the greatest genius (this image was imposed upon society) should be abolished. At the same time, the prohibition of criticizing the special relativity theory should be abolished, too. On the other hand, the fact, that the mentioned errors remained unnoticed for more than a hundred years, also testifies to many things. At least, the conclusion about bad state of junctions in the science of physics.
Bibliography 1. www.timeam.zaporozhye.net (Addition to Example 8). 2. Einstein A. On the Special and General Relativity Theory. – СSW. V.1. – M.: Nauka, 1965. – 670 p. 2a. Einstein A. Physics and Reality. – M.: Nauka, 1965. – 340 p. 3. Landsberg G.S. Optics. – M.: Nauka, 1976. – 928 p. |