Derivation of Relativistic Acceleration Relations without Time Dilation

Transcription

1 Derivation of Relativistic Acceleration Relations without Time Dilation Drs. N. Hamdan, S. Baza Department of Physics University of Aleppo, Aleppo, Syria Abstract We prove and extend our hypotheses that a different set of postulates, one consisting of the relativity principle and the Lorentz force law (Lfl), enables us to cancel the Lorentz transformation (LT) and its time dilation from the main body of special relativity theory (SRT). We use the well-known acceleration transformation relations to confirm that time dilation is not required to derive these relations. Key words: Lorentz force law, special relativity, time dilation. 1 - Introduction: Einstein s [1], in his two postulates, presented a derivation of LT, which forms the basis of the SRT, and he reached to the ideas of kinematics length contraction and time dilation by interpreting the physical significance of Lorentz transformation equations. So he derived and interpreted the LT in a way different from Lorentz:,, (1) Where As it s well known Einstein s method [1] in deriving the LT aimed to an interpretation of the invariance of light speed, which was not included in Galilean transformation. He considered the Cartesian points in the frame be the same in the frame providing that we maintain the constancy of light speed for the movement of this point in both frames. Then the particle with rest mass replaced the engineering point, and the SRT succeeded in connected between the three quantities space, time and light speed through new velocity addition formulas. And to remove the contradiction concerning the symmetrical properties of space-time between classical mechanics and electrodynamics, he altered classical mechanics to make it compatible with LT. These alterations have led to the well-known relativistic mechanics. However, the LT and its kinematics effects played an essential role in SRT formalism. This led directly to the demonstration of the essential role for LT and its kinematics effects in deriving the relativistic dynamical quantities and in the interpretation of relativistic phenomena. But the equations of LT that appear to describe the transformation of dynamical properties of particle were based on the paradoxical fluctuations of time and length [2,3,4]. And this way of interpreting the LT and its kinematics effects has long been questioned and misunderstood. Up to date, the criticisms [5] as well as paradox [6] still continue to receive attention. So many physicists are motivated to think that alternatives to SRT and a new interpretation [7], or even a theory alternative to SRT may be needed [8]. In electrodynamics, the paradoxical fluctuations between SRT and particle dynamics can be eliminated if we have to

2 the primacy of physical laws over the LT [9a,9b]. 2 Relativistic Transformation Relations: In contrast to Einstein's LT, the relativistic relations and relativistic transformation relations are produced by an alternative method [9a]. We obtained the relativistic transformation relations fore force, velocity and mass as well as the relativistic factor : (2a),, (2b,2c) and (3a) ), (3b), (3c), as well as As demonstrated also in my papers [9a,9b], contrary to what is often claimed in SRT presentation[10], the relativistic mass relations were derived starting from the conventional definition of momentum in frames and,i.e. and. So we deduced, without using the time dilation, the relativistic mass relations: (a), (b) (5) Bei this approach [9b], one get also the Lorentz contracted charge density containing, defined as This was done without using the hypothesis proposed by Lorantz-Fitzgerlad as well as without using the length contraction. In the present paper we continue this approach to derive the 3-vector relativistic acceleration without using time dilation.

3 3 The 3- Vector Relativistic Acceleration Relations The 3-vector relativistic acceleration in SRT formalism is a consequence of LT and its time dilation. Therefore, to derive these transformation relations without LT, we will depend on the form of the force transformation i.e., Eqs. (2), and from it we deduce the3-vector relativistic acceleration. Let us take now Eq.(2c),i.e.: (6) Since the external electromagnetic force is equal to the rate of change in momentum with time in frames, then according to the relations describing the movement of a charged particle in an external electromagnetic field, i.e.: and, It is possible to replace the Cartesian components of Lorentz force with their equivalent in relations (6). Hence, according to the definition of we could write (8a) Could be written similarly, i.e.: Now, starting from Eq.(8a) and adding, at the same time, subtracting the term, we then have

4 Or, Now if we take Eq.(9) and multiplying, at the same time, dividing it by result in Eq.(6) we obtain, and then substituting the It could be noticed that Eqs. (3) are equivalent to the following : Multiplying, at the same time, dividing the last three terms in Eq.(10) by (11), and then using Eq.(11) we get

5 Using Eqs.(3) and (4) in the last relation, we have Comparing the last equation with Eq.(8b), we have The same result can be derived from the z and x forces. Conclusion It seems unacceptable to keep the LT and its role in SRT, especially if the LT is not necessary and can easily be canceled. As we have shown, the postulates chosen [9a,9b], (relativity principle and Lorentz force law (LFL)) enable us to cancel the LT and its kinematical effects from the main body of SRT. References: [1] A. Einstein, "On the Electrodynamics of Moving Bodies" Ann. Phys. 17, pp. 891 (1905). [2] M. Harada and M. Sachs, "Reinterpretation of the Fitzgerald-Lorentz contraction", phys. Ess. 11, (1998). [3] O. D. Jefimenko, "On the Experimental Proofs of Relativistic Length Contraction and Time Dilation", Z. Naturforsch. 53a, (1998). [4] S. Golden, "Non-Kinematicity of the Dilation of time Relation of Einstein for Time intervals", Z.

6 Naturforsch. 55a, (2000). [5] Ludek Nerad, "A Critical Analysis of Special Relativity Theory", Galilean Electrodynamics, 8, (1997). [6] C. K. Whitney, "How Can Paradox Happen?", prepared for Seventh Conference on Physical Interpretations of Relativity Theory - London, September (2000). [7] S. J. Prokhovnik, The Physical Interpretation of Special Relativity a Vindication of Hendrik Lorentz, Z. Naturforsch. 48a, (1993). [8] H. E. Wilhelm, "Physical Problematic of Einstein s Relativety Theories" Hadronic J. 19, pp (1996). [9a] N. Hamdan, "Abandoning the Ideas of Length Contraction and Time Dilation", "Galilean Electrodynamics", Vol. 14, p (2003). [9b] N. Hamdan, "Abandoning the Ideas of Length Contraction in Relativistic Electrodynamics", "Galilean Electrodynamics", Vol. 154, p (2004). [10] L. B. Okun, Note on the Meaning and Terminology of Special Relativity, Eur. J. Phys., 15 (1998)