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A new approach to special relativity

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A New approach to Special Relativity
Gordon Liu
a
(Received 9 April, 2013)
Abstract
: The success of Special Relativity (SR) comes from the requirement of Lorentz covariance to all physical equations. The explanation with regard to the Lorentz covariance is based on two hypotheses, namely the principle of special relativity and the constancy of the speed of light. However, the state-ments of the principle of special relativity are various and confusing. The co-variance of physical equations and the equality of inertial frames of reference are mixed up. The equality of inertial frames of reference is obvious, but the covariance of the physical equations is a more advanced requirement. Addi-tionally, the way that the propagation property of light is placed in a central position of SR has caused people misunderstandings towards space-time, and also there is a logical circularity between the measurement of speed of light and the synchronization of clocks. These have obstructed to correctly extend the theory of space-time from an inertial frame of reference to a non-inertial frame of reference. These are the main reasons why many people criticize SR. In present paper, the two hypotheses have been discussed in detail and a new requirement to the equations of Physics has been proposed. The requirement is the Requirement of Special Completeness, namely, the physical equations used to describe the dynamics of matter and/or fields should include the de-scriptions that not only the matter and/or fields are at rest relative to an iner-tial frame of reference, but also they move relative to this frame. Basing on this requirement and the equality of the inertial frames of reference, we can approach to SR. Thereby let the theory of Lorentz covariance has a clear and solid foundation. The constancy of the speed of light is just a deduction, not a premise. The Lorentz covariance is just a characteristic of the Special Com-plete equations. Maxwell equations automatically satisfy the Lorentz trans-formations without any modification, while Newton law of gravity does not, because Newton law of gravity is not Special Complete and Maxwell equa-tions are. The new approach has paved a road leading towards the generaliz-ing of the theory of space-time from the inertial frame of reference to non-inertial frame of reference without considering gravitation.
Résumé
: Le succès de la relativité restreinte (RR) provient de l'exigence de la covariance de Lorentz à toutes les équations physiques. L'explication en ce qui concerne la covariance de Lorentz est fondée sur deux hypothèses, à sa-
a
gordonliu168@gmail.com
2
voir le principe de la relativité et la constance de la vitesse de la lumière. Ce-pendant, les déclarations de principe de la relativité restreinte sont diverses et confus. La covariance des équations physiques et l'égalité des référentiels inertiels sont mélangées. L'égalité des référentiels inertiels est évidente, mais la covariance des équations physiques est une exigence plus avancée. En outre, la facon dont la propriété de propagation de la lumiére dans une posi-tion centrale dans la RR a provoqué des malentendus vers l'espace-temps, et il ya aussi une circularité logique entre la mesure de la vitesse de la lumière et de la synchronisation des horloges. Celles-ci ont obstrué d'étendre correcte-ment la théorie de l'espace-temps à partir d'un repère inertiel à un cadre non-inertiel. Ce sont les principales raisons pour lesquelles beaucoup de gens critiquent RR. En présent document, les deux hypothèses ont été examinées en détail et une nouvelle exigence pour les équations de la physique a été proposée. L'exigence est l'exigence de complétude spécial, à savoir, les équa-tions physiques utilisés pour décrire la dynamique de la matière et / ou les champs devraient inclure les descriptions que non seulement la matière et / ou les champs sont au repos par rapport à un référentiel inertiel, mais aussi ils se déplacent par rapport ce cadre. S'appuyant sur cette exigence et l'égalité des référentiels inertiels, nous pouvons nous approcher de RR. Ainsi la théorie de Lorentz covariance a une base claire et solide. La constance de la vitesse de la lumière est juste une déduction, pas une prémisse. La covariance de Lorentz est juste une caractéristique des équations spéciales complètes. Les équations de Maxwell satisfont automatiquement les transformations de Lorentz sans aucune modification, tandis que leloide Newton sur la gravité ne fait pas, parce que les équations decette loi ne sont pas spéciales complètes mais celles de Maxwell les sont. La nouvelle approche a ouvert une route menant vers la généralisation de la théorie de l'espace-temps du cadre de référence inertiel à cadre non inertiel sans tenir compte de la gravitation.
Keywords: Special Relativity; Principle of Relativity; Completeness; Constancy of the Speed of Light; Lorentz Transformations; Covariance; Minkowski Space-time
Special Relativity (SR) has already been published for over 100 years. In the last 100 years, it has already permeated through the extensive realm of the modern physics, and has become one of the corner stone of the modern physics. However, the arguments with respect to SR have not ceased since its publishing
1-12
. Why could such scene appear? The reasons that SR can obtain plenteous achievements
I.
Introduction
3
are because it has inherited and extended the Galilean principle of relativity from single Me-chanical phenomenon to all physical phenomena and it has also correctly adopted Lorentz Transformations (LT)
13
. However, Einstein
’s
derivation and explanation of LT rely upon a procedure of clock of synchronization in which light signals are used. One easily thinks that the property of space-time is determined by the propagation property of light
14,15
. Also there is the logical circularity between the measurement of the speed of light and the synchroniza-tion of clocks. Other less well-known problem is the problem of statement regarding to the principle of special relativity or the problem how to comprehend the covariance of physical equations and the equality of inertial frames of reference. People always mix up the equality of inertial frames of reference with the covariance of physical equations. The equality of iner-tial frames of reference is obvious. The physical laws obtained by doing physical experiments in an inertial frame of reference will be identical with the laws obtained in another inertial frame of reference by doing the same experiment. But the covariance of physical equations is a more advanced requirement. The equality does not equal to the covariance! Actually, in or-der to make the physical equations covariant, we must bring some certain requirements to bear on the physical equations. We know that not all of physical equations automatically ful-fill the covariance. For instance, Gauss law of Electrostatics does not satisfy Lorentz covari-ance. In those years, Einstein had found Newton law of gravity did not satisfy Lorentz covar-iance. We also know, if the physical laws can be formulated into tensor form, the physical laws must be covariant. Why? Since the equality is not equal to the covariance, we have no reason to regard the equality of inertial frames of reference as the requirement of the covari-ance to use. However, if not adopting the requirement of the covariance, only using the equality of inertial frames of reference, we cannot establish the theory of Lorentz covariance. Actually, we need to bring a requirement to bear on physical laws or equations to let them have the covariance. We will discuss these in details in present paper. All above problems make the foundations of SR is in fact no better than
Thomson’s
16
and Lo
rentz’
s
17
explana-tions through the use of the hypotheses of the "length contraction" and "time dilatation". These are the main reasons why many people criticize SR. These have also obstructed Ein-stein correctly to extend his theory of space-time from inertial frames of reference to noninertial frames of reference, because it is difficult to synchronize the clocks by using light signals in accelerated frames of reference. In order to give the theory of Lorentz covariance or SR a solid foundation, a number of authors have paid their efforts.
4,7,8,15,18-31
Thereinto, the theory of
“
Special Relativity without
the postulate of constancy of light” has
achieved a great progress. They have obtained the Lorentz-like transformations without light. But they still realize that the equality of inertial frames of reference equals to the covariance of physical equations or laws. They believe they can propose the requirement of the covariance to the physical equations according to the equality of inertial frames of reference. In present paper, we will discuss the two principles of SR, and put forward a new foundation to approach to the theory of Lorentz covariance or SR. It is not necessary to put
4
the propagation property of light in the centre position of the theory, just adopt the equality of inertial frames of reference and a requirement of completeness to the physical equations or laws (see Chapter III, d). The paper is organized as follow: In Chapter II, we discuss two basic principles of SR. In Chapter III, we reiterate the definitions of space, time and inertial frame of reference, rename the principle of special relativity and give it the certain content, and propose a new requirement to the physical equations. In Chapter IV, we deduce that the physical equations that fulfill the new requirement have the universal applicability and the covariance. In Chapter V, we introduce the Lorentz-like transformations. In Chapter VI, we elaborate relationship of the tensor expression of the physical equations and the new re-quirement.
a.
The Problems of the Statements on Principle of Special Relativity
By and large, there are two statements with respect to the principle of special relativity.
The first, “
all physical laws are identical in all inertial frames of reference
”,
we will call this statement the physical statement. Einstein wrote in his book :
14
“
If K is an inertial system, then every other system
K'
which moves uniformly and without rotation relatively to K is also an inertial system; the laws of nature are in concordance for all inertial systems. This statement we shall c
all the “
principle of special relativity
””
.
The second, “The
physical equa-tions are covariant under the transformations of space-time
”,
we will call this statement the mathematical statement. Weinberg wrote in his well-known book
32
“Gravitation and Co
s-
mology”: “The new physics, consisting of Maxwell’s electrodynamics and Einstein’s m
e-chanics, then satisfied a new principle of relativity, the Principle of Special Relativity, which says that all physical equations must be invariant under LT
”
. The physical statement is the generalization of the Galilean Principle of Relativity. The sentence can be obviously understood in a way which states that the physical laws obtained by doing physical experiments in an inertial frame of reference will be identical with the laws obtained in another inertial frame of reference by doing the same experiment. According to this comprehension, we can deduce that one can obtain completely identical Gauss law of Electrostatics in all inertial frames of reference (if Gauss law is a precise law of experiment in one inertial frame of reference). Therefore, Gauss law of Electrostatics satisfies the physical statement of the principle of special relativity. We do not think that Gauss law is only suited for the frame of reference in which the ether exist, but in all other inertial frames of reference. However, Gauss law alone (suppose that we still do not know other laws of Electromag-netism) cannot satisfy the mathematical statement of the principle of special relativity, name-ly, does not satisfy the Lorentz covariance. But we cannot deny that Gauss law of Electro-statics is a correct physical law.
II.
On Discussions of Two Basic Principles of SR
5
In the past, people always thought the two statements were identical and had not real-ized these two statements were in fact different. The physical statement emphasizes the equality of all inertial frames of reference. No any inertial frame of reference is special. The absolute space-time does not exist. The physical statement does not relate to the relationships of the physical quantities observed in different inertial frames of reference. In other words, it does not relate to the transformations of the physical quantities between different inertial frames of reference. The mathematical statement extends the meaning of the physical state-ment and emphasizes the transformations of physical quantities observed the same one series of motion of matter and/or fields from different inertial frames of reference. Thus this is so different with the srcinal meaning of the physical statement. Just as the previous analysis about Gauss law of Electrostatics, the physical statement of the relativity principle can assure that the physical laws or equations are identical in different inertial frames of reference, but cannot ensure that the physical laws or equations are covariant under the transformations of space-time. In two relatively moving inertial frames of reference, the physical quantities measured separately by observers to the same one series of motion of matter and/or fields are different. For example, a charge, in an inertial frame of reference which is at rest relative to the charge, there is an electric field; but in another inertial frame of reference which is mov-ing relative to the charge, there is an electric field and a magnetic field. In fact, when we talk about the covariance of physical equations, we must make clear what meaning the transfor-mations of physical quantities (or equations) are and what meaning the transformations of space-time are! The transformations of physical quantities mean the relationships of the physical quantities measured the same one series of motion of matter and/or fields in different inertial frames of reference. The transformations of physical quantities can finally be come down to the transformations of space-time (we will discuss this in detail in Chapter IV, sec-tion c). The transformations of space-time actually are the transformations of the physical quantities measured the same one series of motion of matter and/or fields in different inertial frames of reference. However, the adoption of the mathematical statement is the important factor which allowed SR to obtain many correct results. We will see that the mathematical statement is a stronger requirement than the physical statement and is a deduction of more basic premise and requirement. But if the mathematical statement is used directly as the basic premise of SR, this premise is not obvious and cannot easily be intuitively understood. It is not appropriate to use an obscure requirement as a basic premise of a theory without any fur-ther explanation. In doing so, it will usually cover up many essential meanings and hinder our understanding of physical concepts. The hypotheses of the "length contraction" and the "time dilation" respectively proposed by Thomson
16
and Lorentz
17
are just like so. This is one of the reasons that lead SR encounters criticism.
b.
Adopting the Constancy of the Speed of Light as a Basic Premise Is Inap-propriate
Einstein put the propagation property of light in the centre position of his theory of rel-ativity. His derivation and explanation of LT relies upon a procedure of clock of synchroniza-

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