Abstract
In this chapter we summarise some of the standard results of conventional special relativity theory that are needed to formulate the proposed extension of Newton’s second law. The word special alludes to invariance under transformations relating constant relative velocity frames of reference, which is in contrast to general relativity which relates to invariance under arbitrary space-time coordinate transformations. The first section deals with the fundamental notion of Lorentz transformations and the importance of invariance with respect to frames that are moving with constant relative velocity. The following section highlights the Einstein addition of velocities law which is an immediate consequence of the notion of invariance under Lorentz transformations.
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References
D. Bohm. The special theory of relativity. W. A. Benjamin, Inc., New York, 1965.
B. Coleman. Special relativity dynamics without a priori momentum conservation. Eur. J. Phys., 26:647–650, 2005.
L. de Broglie. Recherches sur la theorie des quanta. PhD Thesis, Sorbonne University of Paris, France, 1924.
H. Dingle. The special theory of relativity. Methuen and Co.Ltd., London, 1961.
R. P. Feynman, R. B. Leighton, and M. Sands. The Feynman Lectures on Physics, volume 1. Addison-Wesley, 1964.
A. P. French. Special relativity. Thomas Nelson and Sons Ltd., London, 1968.
G. M. L. Gladwell. Contact problems in the classical theory of elasticity. Sijthoff and Noordhoof, Leyden, 1980.
E. Goursat. A course in mathematical analysis, volume 1. Dover Publications, New York, 1959.
J. Guemez, M. Fiolhais, and L. A. Fernandez. The principle of relativity and the de Broglie relation. Am. J Phys., 84:443–447, 2016.
J. M. Hill. On the formal origin of dark energy. Zeitschrift fur angewandte Mathematik und Physik, 69:133–145, 2018.
J. M. Hill. Some further comments on special relativity and dark energy. Zeitschrift fur angewandte Mathematik und Physik, 70:5–14, 2019.
J. M. Hill. Special relativity, de Broglie waves, dark energy and quantum mechanics. Zeitschrift fur angewandte Mathematik und Physik, 70:131–153, 2019.
J. M. Hill. Four states of matter and centrally symmetric de Broglie particle-wave mechanical systems. Mathematics and Mechanics of Solids, 26:263–284, 2020.
J. M. Hill. A review of de Broglie particle-wave mechanical systems. Mathematics and Mechanics of Solids, 25:1763–1777, 2020.
J. M. Hill. A mechanical model for dark matter and dark energy. Zeitschrift fur angewandte Mathematik und Physik, 72:56:14pp, 2021.
J. M. Hill. Einstein’s energy and space isotropy. Zeitschrift fur angewandte Mathematik und Physik, 73:65:9pp, 2022.
J. M. Hill and B. J. Cox. Einstein’s special relativity beyond the speed of light. Proc. R. Soc. A, 468:4174–4192, 2012.
J. M. Hill and B. J. Cox. Generalised Einstein mass variation formulae: I Sub-luminal relative frame velocities. Results in Physics, 6:112–121, 2016.
J. M. Hill and B. J. Cox. Generalised Einstein mass variation formulae: II Superluminal relative frame velocities. Results in Physics, 6:122–130, 2016.
J. M. Hill and A. Tordesillas. The pressure distribution for symmetrical contact of circular elastic cylinders. Quarterly Journal of Mechanics and Applied Mathematics, 42:581–604, 1989.
J. M. Houlik and G. Rousseaux. “Non-relativistic” kinematics: Particles or waves. Unpublished, available from Germain Rousseaux website University of Poitiers, page 4pp., 2010.
K. L. Johnson. Contact mechanics. University Press, Cambridge, 1984.
L. D. Landau and E. M. Lifshitz. Course of Theoretical Physics, volume 2. Addison-Wesley, 1951.
A. R. Lee and T. M. Kalotas. Lorentz transformations from the first postulate. Amer. J. Phys., 43:434–437, 1975.
J.-M. Levy-Leblond. One more derivation of the Lorentz transformation. Amer. J. Phys., 44:271–277, 1976.
W. H. McCrea. Relativity physics. Methuen and Co.Ltd., London, 1947.
H. Minkowski. Space and time, in “The Principle of Relativity”, by H. A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl. Dover, New York, 1952.
C. Moller. The theory of relativity. Clarendon Press, Oxford, 1966.
W. Perrett and Jeffery G. B. The principle of relativity. Dover Publications Inc., 2017.
R. Resnick. Introduction to special relativity. John Wiley and Sons Inc., New York, 1968.
G. Rousseaux. Forty years of Galilean Electromagnetism (1973–2013). Eur. Phys. J. Plus, 128:81–94, 2013.
D. Scott and G. F. Smoot. Cosmic microwave background mini-review. P.A. Zyla et al. Particle Data Group, Review of Particle Physics, Prog. Theor. Exp. Physics, 8:083C01, 2020.
S. Sonego and M. Pin. Foundations of anisotropic relativistic mechanics. Journal of Mathematical Physics, 50:042902, 2009.
R. C. Tolman. Relativity, thermodynamics and cosmology. Clarendon Press, Oxford, 1946.
F. G. Tricomi. Integral equations. Interscience, New York, 1957.
P. Weinberger. Revisiting Louis de Broglie’s famous 1924 paper in the Philosophical Magazine. Philosophical Magazine Letters, 86:405–410, 2006.
G. Weinstein. Variation of mass with velocity: “kugeltheorie” or “relativtheorie”. arXiv:1205.5951 [physics.hist-ph], 2012.
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Hill, J. (2022). Special Relativity. In: Mathematics of Particle-Wave Mechanical Systems. Springer, Cham. https://doi.org/10.1007/978-3-031-19793-2_2
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