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Foundations of anisotropic relativistic mechanics
3.V. Lalan, Bull. Soc. Math. France 65, 83 (1937).
4.J. -M. Lévy-Leblond, Riv. Nuovo Cimento 7, 187 (1977);
4.in Problems in the Foundations of Physics, edited by G. Toraldo di Francia (North-Holland, Amsterdam, 1979), pp. 237–263;
5.H. Westman and S. Sonego, Found. Phys. 38, 908 (2008);
6.L. J. Eisenberg, Am. J. Phys. 35, 649 (1967).
7.A. A. Ungar, Am. J. Phys. 59, 824 (1991);
7.Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces (Kluwer, Dordrecht, 2001);
7.Comput. Math. Appl. 49, 187 (2005);
7.Eur. J. Phys. 27, L17 (2006).
8.J. W. Milnor, Topology from the Differentiable Viewpoint (University Press of Virginia, Charlottesville, 1965), pp. 55–57.
11.W. A. von Ignatowsky, Verh. Dtsch. Phys. Ges. 12, 788 (1910);
11.W. A. von Ignatowsky, Phys. Z. 11, 972 (1910);
11.W. A. von Ignatowsky, Arch. Math. Phys. 3, 1 (1911);
11.W. A. von Ignatowsky, Phys. Z. 12, 779 (1911).
12.P. Frank and H. Rothe, Ann. Phys. 34, 825 (1911);
12.P. Frank and H. Rothe, Phys. Z. 13, 750 (1912);
12.A. C. van Rijn van Alkemade, Ann. Phys. 38, 1033 (1912);
12.A. N. Whitehead, An Enquiry Concerning the Principles of Natural Knowledge (Cambridge University Press, Cambridge, 1919), Chap. XIII;
12.L. A. Pars, Philos. Mag. 42, 249 (1921);
12.T. Kaluza, Phys. Z. 25, 604 (1924);
12.F. Severi, in Cinquant’anni di Relatività, edited by M. Pantaleo (Giunti, Firenze, 1955), pp. 309–333;
12.H. Almström, J. Phys. A 1, 331 (1968);
12.G. Süssmann, Z. Naturforsch. A 24, 495 (1969);
12.G. A. Ramanujam, Eur. J. Phys. 4, 248 (1983);
12.N. D. Mermin, Am. J. Phys. 52, 967 (1984)
12.[reissued in N. D. Mermin, Boojums All the Way Through (Cambridge University Press, Cambridge, 1990), pp. 247–265];
12.G. Süssmann, Opt. Commun. 179, 479 (2000);
12.Z. K. Silagadze, Acta Phys. Pol. B 39, 811 (2008);
13.M. Jammer, in Problems in the Foundations of Physics, edited by G. Toraldo di Francia (North-Holland, Amsterdam, 1979), pp. 202–236;
15.Y. P. Terletskii, Paradoxes in the Theory of Relativity (Plenum, New York, 1968), pp. 17–24;
15.W. Rindler, Essential Relativity, 2nd ed. (Springer, New York, 1977), pp. 51–53;
15.R. Torretti, Relativity and Geometry (Dover, New York, 1996), pp. 76–82.
16.H. R. Brown, Physical Relativity (Clarendon, Oxford, 2005).
17.W. F. Edwards, Am. J. Phys. 31, 482 (1963);
17.F. Selleri, Found. Phys. 26, 641 (1996);
17.F. Selleri, Found. Phys. Lett. 9, 43 (1996);
17.P. W. Bridgman, A Sophisticate’s Primer of Relativity (Dover, New York, 2002);
18.H. Reichenbach, The Philosophy of Space & Time (Dover, New York, 1958), pp. 127–129;
18.A. Grünbaum, Philosophical Problems of Space and Time, 2nd enlarged edition (Reidel, Dordrecht, 1973), pp. 342–368 and 666–708;
18.D. Malament, Noûs 11, 293 (1977);
18.T. Sjödin, Nuovo Cimento Soc. Ital. Fis., B 51, 229 (1979);
18.M. F. Podlaha, Lett. Nuovo Cimento 28, 216 (1980);
18.G. Spinelli, Nuovo Cimento Soc. Ital. Fis., B 75, 11 (1983);
18.R. de Ritis and S. Guccione, Gen. Relativ. Gravit. 17, 595 (1985);
18.A. Ungar, Philos. Sci. 53, 395 (1986);
18.P. Havas, Gen. Relativ. Gravit. 19, 435 (1987);
18.A. P. Stone, Found. Phys. Lett. 4, 581 (1991);
18.S. K. Ghosal, P. Chakraborty, and D. Mukhopadhyay, Europhys. Lett. 15, 369 (1991);
18.S. K. Ghosal, K. K. Nandi, and P. Chakraborty, Z. Naturforsch., A: Phys. Sci. 46, 256 (1991);
18.R. Golestanian, M. R. H. Khajehpour, and R. Mansouri, Class. Quantum Grav. 12, 273 (1995);
18.V. Karakostas, Stud. Hist. Philos. Mod. Phys. 28, 249 (1997);
18.F. Selleri, Found. Phys. 27, 1527 (1997);
18.M. Mamone Capria, Found. Phys. 31, 775 (2001);
18.E. Minguzzi, Found. Phys. Lett. 15, 153 (2002);
18.A. Valentini, in Einstein, Relativity and Absolute Simultaneity, edited by W. L. Craig and Q. Smith (Routledge, London, 2008), pp. 125–155.
19.R. Torretti, The Philosophy of Physics (Cambridge University Press, Cambridge, 1999), Sec. 5.3.2.
20.C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, New York, 1973), pp. 23 and 26.
21.G. Yu. Bogoslovsky, Nuovo Cimento Soc. Ital. Fis., B 40, 99 (1977);
21.G. Yu. Bogoslovsky, Nuovo Cimento Soc. Ital. Fis., B 40, 116 (1977);
21.G. Yu. Bogoslovsky, Nuovo Cimento Soc. Ital. Fis., B 43, 377E (1978);
21.G. Yu. Bogoslovsky and H. F. Goenner, Phys. Lett. A 244, 222 (1998);
22.G. Yu. Bogoslovsky, Fortschr. Phys. 42, 143 (1994).
23.N. D. Mermin, Space and Time in Special Relativity (McGraw-Hill, New York, 1968) (reissued by Waveland, Long Grove, 1989, pp. 33–37).
24.L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed. (Pergamon, Oxford, 1976), pp. 6–7.
25.C. Lanczos, The Variational Principles of Mechanics, 4th ed. (University of Toronto Press, Toronto, 1970) (reissued by Dover, New York, 1986, pp. 132–133).
26.J. B. Barbour, The Discovery of Dynamics (Oxford University Press, Oxford, 2001), p. 471.
27.W. C. Davidon, Found. Phys. 5, 525 (1975);
28.J. -M. Lévy-Leblond, Lect. Notes Phys. 50, 617 (1976);
28.Institute of Physics Conference Series, edited by J. -P. Gazeau, R. Kerner, J. -P. Antoine, S. Metens, and J. -Y. Thibon (IOP, Bristol, 2004), Vol. 173, pp. 173–182.
29.D. Bao, S. -S. Chern, and Z. Shen, An Introduction to Riemann-Finsler Geometry (Springer, New York, 2000).
31.T. Budden, Stud. Hist. Philos. Mod. Phys. 28, 325 (1997).
33.D. Mattingly, Living Rev. Relativ. 8, 5 (2005),
36.G. Sigl, Lect. Notes Phys. 556, 259 (2000);
43.M. H. L. Pryce, Proc. R. Soc. London, Ser. A 195, 62 (1948);
43.W. Rindler, Essential Relativity, 2nd ed. (Springer, New York, 1977), p. 85;
43.R. Ferraro, Einstein’s Space-Time (Springer, New York, 2007), pp. 146–147;
43.A. A. Ungar, Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativity (World Scientific, Singapore, 2008), Chap. 11.
44.The existence of a (possibly infinite) invariant speed follows from these hypotheses and does not require an independent postulate. The actual value of such a speed is an experimental issue.
45.There is no fundamental reason why it should be so,4 but there is excellent experimental evidence that any difference is very small. This justifies our identification.
46.It is perhaps worth stating explicitly that the “coordinates” used in this paper (and in almost all the literature on special relativity) correspond to well-defined readings of time and distance, so they are not just arbitrary labels, but have a clear operational meaning.5 Thus, all statements about the vague notions of “time” and “space” can be unambiguously interpreted in terms of the behavior of clocks and rulers (or other physical systems used in measurement protocols).
47.In order for the notion of velocity to make sense operationally, it is obvious that some prescription must have been adopted for the synchronization of clocks in a reference frame. We do not need to specify what the prescription is — we only require it to be compatible with the relativity principle.
48.By the relativity principle, the velocities will all belong to the same open interval , independently of the reference frame in which they are measured, so .
49.Hereafter, a prime will denote the derivative of a function with respect to its argument, with only a few obvious exceptions—see, e.g., Eq. (2.11) below.
50.Also, for the limit velocities in (2.32) are both negative, and for they are both positive. This violates the condition .
51.In general, , so only when .
52.See Ref. 9 for an alternative proof of this theorem, with applications to relativistic kinematics.
53.Note that the function does not enter the velocity composition law.
54.It could correspond to a purely geometrical occurrence like, for example, the intersection between two moving straight lines, which can have an arbitrarily high speed even if the lines move rather slowly, provided the angle they form is small enough.
55.Note that the possibility for the existence of invariant speeds has been derived as a kinematical possibility only from the postulates of relativity, homogeneity, and precausality. This approach to relativistic kinematics was pioneered by von Ignatowsky in 1910 (Ref. 11) and was later rediscovered many times in different ways.3,9,12,13 See also Ref. 14 for a rigorous treatment and Refs. 15 and 16 for clear presentations at a textbook level.
56.These statements are less tautological than it may seem at first, as one can synchronize without using light.
57.See Refs. 16, 18, and 19 for the debate.
58.This operation does not destroy the group structure, and is therefore compatible with the principle of relativity. On the contrary, eliminating the Lorentz factor by a rescaling of units would not preserve the group structure, hence would imply a violation of the principle.
59.The situation is different for the transformation (2.40). When this is a rotation, anticlockwise by an angle , of orthogonal axes in a Euclidean plane, accompanied by a global dilatation by the factor . A true anisotropy cannot produce effects for , so the factor can only be due to a devious choice of units in the frame .
60.Hereafter, whenever we refer to “energy” we mean the sum of kinetic energy and a possible rest energy.
61.Basically, a straightforward generalization of an argument originally due to Huygens.26 See also Refs. 27 and 28 for similar developments.
62.Note that with this identification, linear momentum turns out to be (correctly) a one-form rather than a vector.2
63.No new conservation laws can arise at higher orders in , as argued by Lévy–Leblond.28 See also Ref. 1 for an explicit proof when is only a function of (which, however, is not the case in the presence of anisotropy).
64.The constants and have, a priori, nothing to do with those introduced in Sec. II A, but we shall soon discover that they actually coincide. This justifies using the same letters in the notation.
65.One could distinguish between the momentum and a “kinetic momentum” , just as one usually distinguishes between the energy and the “kinetic energy” .
66.It might have been logically possible that were equal to the Newtonian mass multiplied by a function of and that reduces to 1 when both these parameters tend to infinity.
67.The difference between Finslerian29 and pseudo-Finslerian30 structures on a manifold is the same as between Riemannian and pseudo-Riemannian ones.
68.In the case (i.e., ), this expression was also considered in Refs. 21, 22, and 31.
69.The indices , , run from 0 to 1.
70.Which vanishes when the rest momentum and energy are given by the expressions (4.7) and (4.8).
71.The are homogeneous functions of degree 0 of their arguments (as the are), so their definition is insensitive to the coefficients in Eq. (6.11).
72.When and , the center-of-momentum frame can equivalently be defined as the one where total momentum vanishes. This is not the appropriate characterization when anisotropy is present in some form, as one can realize considering a situation in which the system is made of a single particle. Also, note that defining a center-of-mass frame is problematic in mechanics where there is no absolute time.43
73.Not to be confused with the four coefficients in Eq. (2.1).
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