4088 lines
208 KiB
Plaintext
4088 lines
208 KiB
Plaintext
Open Questions in Relativistic Physics
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Edited by Franco Selleri
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Open Questions in Relativistic Physics
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Edited by Franco Selleri Università di Bari Bari, Italy
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Apeiron Montreal
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Published by Apeiron 4405, rue St-Dominique Montreal, Quebec H2W 2B2 Canada http://redshift.vif.com
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© C. Roy Keys Inc.
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First Published 1998
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Canadian Cataloguing in Publication Data
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Main entry under title:
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Open questions in relativistic physics
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Includes index. Proceedings of an international conference on Relativ-
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istic Physics and Some of its Applications, June 25-28, 1997, Athens, Greece ISBN 0-9683689-1-3
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1. Special relativity (Physics). I. Selleri, Franco
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QC173.5.O64 1998 530.11 C98-900498-8
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Contents
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Preface ................................................................................................... i
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Velocity of Light
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Patrick Fleming: An Explanation of the Sagnac Effect Based on the Special Theory of Relativity, the de Broglie/Bohm Interpretation of Quantum Mechanics, and a NonZero Rest Mass for the Photon................................................ 3
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F. Goy: On Synchronisation of Clocks in Free Fall Around a Central Body .............................................................................. 7
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Andrzej Horzela: Remarks on Clock Synchronization ..................................... 19
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A. G. Kelly: Synchronisation of Clock-Stations and the Sagnac Effect ......................................................................................... 25
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Joseph Lévy: Is Simultaneity Relative or Absolute?................................... 39
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Adolphe Martin: Reception of Light Signals in Galilean Space-Time............ 47
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J.Ramalho Croca: Experiments on the Velocity c............................................... 57
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Ramón Risco-Delgado: Inertial Transformations from the Homogeneity of Absolute Space......................................................................... 65
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F. Selleri: On a Physical and Mathematical Discontinuity in Relativity Theory..................................................................... 69
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Tom Van Flandern: What the Global Positioning System Tells Us about Relativity................................................................................... 81
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History and Philosophy
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M. Barone: Some Almost Unknown Aspects of Special Relativity Theory ..................................................................... 93
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Jenner Barretto Bastos Filho: Correspondence and Commensurability in Modern Physics (a Study of the Compton Effect) ........................... 103
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Eftichios Bitsakis: Space and Time: Who was Right, Einstein or Kant? ........ 115
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A. Jannussis: Einstein and the Development of Physics ......................... 127
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Ludwik Kostro: The Physical and Philosophical Reasons for A. Einstein’s Denial of the Ether in 1905 and its Reintroduction in 1916 ......................................................... 131
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N.A. Tambakis: On the Question of Physical Geometry ............................. 141
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G. Tarozzi: Nonlocality, Relativity, and Two Further Quantum Paradoxes ............................................................................... 149
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Alexei A.Tyapkin: On the History of the Special Relativity Concept............. 161
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Structures in Space and Time
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A. K. T. Assis & J. I. Cisneros: The Problem of Surface Charges and Fields in Coaxial Cables and its Importance for Relativistic Physics .................................................................................... 177
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in Relativistic Physics
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5
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A.M. Awobode: A New Appraisal of the Relativistic Quantum Theories of the Electron ....................................................... 187
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V. S. Barashenkov: Nature of Relativistic Effects and Delayed Clock Synchronization .................................................................... 195
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Mirjana Bozic′ : On a Relativistic Magnetic Top ........................................... 201
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Henrik Broberg: The Interface between Matter and Time: a Key to Gravitation ............................................................................. 209
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W. A. Hofer: Internal Structures of Electrons and Photons and some Consequences in Relativistic Physics ....................... 225
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Edward Kapuscik: Generally Covariant Electrodynamics in Arbitrary Media ...................................................................................... 241
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Marek Pawlowski: On Weyl’s Extension of the Relativity Principle as a Tool to Unify Fundamental Interactions ........................... 247
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J. P. Wesley: Evidence for Newtonian Absolute Space and Time......... 257
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Cosmology and Astrophysics
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Halton Arp: Evolution of Quasars into Galaxies and its Implications for the Birth and Evolution of Matter.......... 267
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J. Brandes: A Lorentzian Approach to General Relativity: Einstein’s Closed Universe Reinterpreted......................... 275
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Zbigniew Jacyna-Onyszkiewicz & Bogdan Lange: The Simplest Inflationary Scenario in Relativistic Quantum Cosmology........................................................... 283
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D. F. Roscoe: An Analysis of 900 Rotation Curves of Southern Sky Spiral Galaxies: Are the Dynamics Constrained to Discrete States? ...................................................................... 291
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Quantum Theory and Relativity
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A.P. Bredimas: Schrödinger’s “Aether” Unifies Quantum Mechanics and Relativistic Theories................................... 305
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Augusto Garuccio: Entangled States and the Compatibilty Between Quantum Mechanics and Relativity................................... 311
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Luis Gonzalez-Mestres: Lorentz Symmetry Violation, Vacuum and Superluminal Particles.......................................................... 321
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A. Jannussis & S. Baskoutas: On Superluminal Velocities................................................. 339
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José L. Sánchez-Gómez: Are Quantum Mechanics and Relativity Theory really Compatible? ................................................................ 347
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Caroline H Thompson: Behind the Scenes at the EPR Magic Show............................................................................ 353
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M.A.B. Whitaker: Relativistic Physics and Quantum Measurement Theory..................................................................................... 363
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Preface
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The true conceptual background of the two relativistic theories (special and general) has been re-emerging in recent times, after more than half a century of domination of the neopositivist conception of science. Einstein himself was strongly influenced by positivism in his youth, and admitted that the special theory of relativity was based on a direction of thought conforming with Mach’s ideas [1]. The hegemony of logical empirism had, as a first consequence, that Einstein’s role was somewhat inflated, while the contributions of other authors (Lorentz, Larmor, Poincaré, etc.) were generally underestimated. More than experimental evidence, this was the reason why the typically realistic conjectures, such as that of ether, were eliminated in favour of more abstract conceptions. At the present time the domination of positivism appears to have come to an end, and a new era may be opening for realism.
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A correct understanding of the true history of relativity has recently produced several surprises, the first one being the realisation that such important scientists as Lorentz [2], Poincaré [3], Larmor [4], Fitzgerald [5] took a fully realistic approach to relativistic physics, though they did not necessarily consider this to be in contradiction with Einstein’s theory. For example Poincaré, often described as a conventionalist, repeatedly stated that ether was a necessary ingredient of physics. He did so, for example, in the same page of his famous 1904 St. Louis paper [6], where the first precise modern formulation of the relativity principle is given. It has also become fully evident that Einstein came back to the idea that a physical vacuum must exist, which he called “ether” in several papers from 1918 to 1955 [7]. He confessed to Popper that the greatest mistake of his scientific life was the acceptance of positivistic philosophy in his youth [8].
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It has been firmly established that certain fundamental ingredients of the two relativistic theories are basically arbitrary, the main one being the introduction of the so-called “Einstein clock synchronisation.” This conclusion surfaced at first in the works of philosophers and historians of physics (Reichenbach [9], Grünbaum [10], Jammer [11]), and then influenced the works of physicists as well (Sexl [12], Sjödin [13], Cavalleri [14], Ungar [15], etc.). But Einstein clock synchronisation is based on the assumed invariance of the one-way speed of light. Since a statement whose conventional nature has been recognized cannot be a necessary consequence of a true property of nature, it follows that invariance of one-way speed of light is not a law of nature! Accordingly, the general-relativistic invariance of the ds2 should also
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Open Questions in Relativistic Physics
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Edited by Franco Selleri (Apeiron, Montreal, 1998)
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i
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ii
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Open Questions
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be considered as a basically human choice, rather than a property of the physical world [16].
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In the fifties and early sixties, Herbert Dingle, professor of History and Philosophy of Science in London, fought a battle against some features of the relativity theory, in particular against the asymmetrical aging present in the twin paradox argument. He believed that the slowing down of moving clocks was pure fantasy. This idea has of course been demolished by direct experimental evidence, collected after his time [17]. Nevertheless, his work has left posterity a rare jewel: the syllogism bearing his name. Given that a syllogism is a technical model of perfect deduction, its consequences are absolutely necessary for any person accepting rational thinking in science.
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Dingle’s syllogism is the following [18]:
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1. (Main premise) According to the philosophy of relativism, if two bodies (for example two identical clocks) separate and reunite, there is no observable phenomenon that will show in an absolute sense that one rather than the other has moved.
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2. (Minor premise) If upon reunion, one clock were retarded by a quantity depending on its relative motion, and the other not, that phenomenon would show that the first clock had moved (in an observer independent “absolute” sense) and not the second.
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3. (Conclusion) Hence, if the postulate of relativity were universally true, as required by the philosophy of relativism, the clocks must be retarded equally or not at all: in either case, their readings will concord upon reunion if they agreed at separation. If a difference between the two readings were to show up, the postulate of relativity cannot be always true.
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Today it can be said that the asymmetrical behaviour of the two clocks is empirically certain (muons in cosmic rays, experiment with the CERN muon storage ring [19], experiments with linear beams of unstable particles, Hafele and Keating experiment [20]). Therefore, as a consequence of point 3. above, the postulate of relativity must somehow be negated. Actually, in recent years it seems to be almost normally accepted in the scientific milieu that the “theory of relativity” is just a name, not to be taken too literally. The total relativism which the theory could seem to embody is now perceived to be only an illusion. One can conclude that not all is relative in relativity, because this theory also contains some features that are observer independent, i.e. features which are absolute! As Dingle wrote: “It should be obvious that if there is an absolute effect which is a function of velocity, then the velocity must be absolute. No manipulation of formulae or devising of ingenious experiments can alter that simple fact.” [18]
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From the new point of view that the theory relativity does not embody a complete relativism the so called “twin paradox” is not a real paradox of the theory, but only a huge problem for the few remaining believers in the philosophy of relativism. In fact, the twin paradox is discussed in many
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in Relativistic Physics
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iii
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works, and they can be divided into two groups: (a) Those which recognize the velocity of the travelling twin as the cause of the slowing down of his biological processes; (b) Those which instead seek to attribute the same effect to the accelerations felt by the traveler at departure, arrival, and the instant in which the direction of velocity is reversed. Obviously, the followers of the second line of thought try to save the perfect symmetry between the rectilinear uniform motions required by the relativity principle, but their position is really impossible to save, as was shown by Builder [21]. His argument is very simple: in physics one can recognise the cause of a phenomenon by varying it, and verifying that corresponding variations of the effect exist. In short, in the case of the twins, if the traveler doubles the length of the paths described with rectilinear uniform motion and travels in them with the same velocity, leaving unchanged the accelerations, he will find that his age difference from the stationary twin is also doubled: therefore velocity, and not acceleration is responsible for asymmetrical ageing. Accelerations as such have no effect on clocks, as shown very convincingly also by the CERN experiment [19], where accelerations as large as 1018 g did not have any effect
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on muon lifetimes. For the reasons cited here, the new trends in relativistic research are
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based on: (1) Overcoming of positivistic limitations to the conceptions to be used in scientific research; (2) Awareness of the limited applicability of the relativity principle itself; (3) Conventionality of the invariance of the one way velocity of light; (4) Probable existence in nature of absolute velocities; (5) Possibility of re-introducing the luminiferous ether. This highly interesting situation has brought new life in a field that many considered finally settled. Particularly remarkable has been the revival of the Lorentzian approach defended by Jánossy [22], Erlichson [23], Prokhovnik [24], Bell [25] and Brandes [26], the study of synchronisation procedures different from the usual one [16], the new discussion of Thomas rotation [27]. Very interesting perspectives are now opening up. At the present time there are several physicists active in the foundations of relativity, and every second year an international conference devoted to these matters is organised in London by the British Society for the Philosophy of Science [28].
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The most general space-time transformations leading both to invariance of the two way velocity of light (not necessarily of the one way velocity!) and to the usual time dilation effects have been obtained [16]. Sets of such transformations differ from one another only for the value of a coefficient e expressing the dependence on space of the transformation of time. At our conference the present writer reported that there is necessarily an unacceptable discontinuity between the physics of accelerated frames and the physics of inertial frames, unless e = 0 . In this way we obtain something different from the Lorentz transformations. A new theory of space, time and motion is clearly starting to emerge. Evidence that the standard theory has great difficulties in explaining the Sagnac effect has been presented [29].
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iv
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Open Questions
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In atomic physics there is has been an indication of superluminal crossing of potential barriers [30]. It has, however, been suggested that this could be a spurious effect: the photonic particles in the first arriving part of the wave packet cross the barrier, those in the final part are reflected. In this way, every photon moves with the speed of light, while the “centre of energy” of the wave packet crossing the barrier looks as if it had propagated superluminally. Astrophysical evidence has been reported of superluminal propagations in jets emerging from galactic nuclei and in active clouds emitted from quasars. These can often be explained away if quasars are indeed associated with nearby galaxies and their redshifts are not due to expansion [31]. There remain the M87 ejections (blue knots propagating at a velocity 5-6 c!) whose distance does not depend on redshift, but was obtained from Cepheids, planetary nebulae, apparent size of galaxy, etc. This distance is somewhere around 50 million light years. The M87 evidence was reported immediately after our conference. The so called “leading model” could perhaps get rid of this superluminality as well [32].
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The Global Positioning System (GPS) is a set of 24 satellites moving around the Earth in such a way that from any point of our planet at any instant at least four of them are visible. It was built for military applications and is said to allow a ballistic precision such that an intercontinental missile can enter from the chosen window of a building 20 000 Km away. All these satellites have atomic clocks on board. Timing and distances are continuously transmitted to the ground. Some very partial results from the GPS were reported at our conference, the startling news being that there is some evidence of a very weak absorption of the solar gravitational field from the mass of the Earth. Rumors are circulating of evidence for other unexpected physical effects which are difficult to explain with the standard theories (relativity, special and general). The highly frustrating situation is that a military secrecy protects the data, so that only very few privileged physicists have access to them.
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A critical reconsideration of the published experiments on Bell’s inequality (especially of the Orsay experiments) was been reported at this conference [33], with the conclusion that their meaning is rather doubtful as a consequence of several logical and practical ambiguities which emerge when one considers the experiments carefully. It appears that much more detailed investigations are needed before any conclusion against the validity of local realism in nature can eventually be reached. This new result agrees with older studies of the role of the so called “additional assumptions” in the deduction of Bell type inequalities [34]. In another talk the possibility has been discussed of transmitting superluminal signals by using EPR pairs of photons and unusual reflections on nonlinear crystals [35]. Recent work suggests that a solution of the EPR paradox could come from research on the two neutral kaons arising in the decay of φ mesons, e.g. produced at rest in the laboratory in e+ e− collisions at a φ factory accelerator [36]. The reported discrepancy
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in Relativistic Physics
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v
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between local realistic and quantum theoretical predictions for EPR correlated neutral kaon pairs is numerically very impressive. The full validity of quantum theory could be at stake in these new researches.
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References
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[1] A. Einstein, Ernst Mach, Phys. Zeits. 17, 103 (1916); reprinted in: E Mach, Die Mechanik in ihrer Entwicklung: Historisch-Kritisch Dargestellt, R. Wahsner and H.H. Borzeszkowski, eds., Akademie Verlag, Berlin (1988), pp. 683-689.
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[2] E. Zahar, Brit. J. Phil. Sci. 24, 95 and 223 (1973). [3] H. Poincaré, Les rapports de la matière et de l’éther, Jour. Phys. théor. appl. 2, 347
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(1912). [4] J. Larmor, Aether and Matter, Cambridge Univ. Press (1900). [5] J. S. Bell, George Francis FitzGerald, Physics World, September 1992, pp. 31-34. [6] Henri Poincaré, The Monist, 15, 1 (1905). [7] L. Kostro, An Outline of the History of Einstein’s Relativistic Ether Conception,
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in: Studies in the History of General Relativity, J. Eisenstaedt & A.J. Kox, eds., Birkhäuser, Boston (1992). [8] K. R. Popper, Unended Quest. An Intellectual Autobiography, Fontana/Collins, Glasgow (1978), pp. 96-97. [9] H. Reichenbach, The Philosophy of Space & Time, Dover, New York (1958). [10] A. Grünbaum, Philosophical Problems of Space And Time, Reidel Dordrecht (1973). [11] M. Jammer, Some foundational problems in the special theory of relativity, in: Problems in the Foundations of Physics, G. Toraldo di Francia ed., North Holland, Amsterdam (1979), pp. 202-236. [12] R. Mansouri and R. Sexl, General Relat. and Grav., 8, 497 (1977). [13] T. Sjödin, Nuovo Cim. 51 B, 229 (1979). [14] G. Cavalleri, Nuovo Cim. 104 B, 545 (1989). [15] A.A. Ungar, Found. Phys. 21, 691 (1991). [16] F. Selleri, Found. Phys. 26, 641 (1996); Found. Phys. Lett. 9, 43 (1996). [17] H.C. Hayden, Galilean Electrodynamics, 2, 63 (1991). [18] H. Dingle, Nature, 179, 866 and 1242 (1957); H. Dingle, Introduction, in: Henri Bergson, Duration and Simultaneity, pp. xv-xlii, The Library of Liberal Arts, Indianapolis (1965). [19] J. Bailey, K. Borer, F. Combley, H. Drumm, F. Krienen, F. Lange, E. Picasso, W. von Ruden, F.J.M. Farley, J.H. Field, W. Flegel and P.M. Hattersley, Nature, 268, 301 (1977). [20] J.C. Hafele and R.E. Keating, Science, 177, 166 (1972). [21] G. Builder, Austral. Jour. Phys., 11, 279 (1958); ibid. 11, 457 (1958). [22] L. Jánossy, Theory of Relativity Based on Physical Reality, Akadémiai Kiadó, Budapest (1971). [23] H. Erlichson, Am. J. Phys., 41, 1068 (1973). [24] S. Prokhovnik, Light in Einstein's Universe, Kluwer, Dordrecht (1985). [25] J. S. Bell, How to teach special relativity, in: Speakable and Unspeakable in Quantum Mechanics, Cambridge Univ. Press (1987). [26] J. Brandes, Die Relativistischen Paradoxien und Thesen zu Raum und Zeit, VRI, Karlsbad (1995). [27] C.I. Mocanu, Galilean Electrodynamics, 2, 67 (1991).
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Open Questions
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[28] M.C. Duffy, ed., Physical Interpretations of Relativity Theory, British Society for the
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Philosophy of Science, London (1996).
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[29] A.G. Kelly (HDS Energy, Celbridge): Synchronisation of clock-stations & the
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Sagnac effect, in this book.
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[30] G. Nimtz, A. Enders and H. Spieker, J. Phys. I France, 4, 565 (1994).
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[31] H. Arp, Quasars, Redshifts and Controversies, Interstellar Media, Berkeley (1987).
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[32] H. Arp, private communication.
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[33] C. H. Thompson, Behind the Scenes at the EPR Magic Show, in this book.
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[34] V.L. Lepore and F. Selleri, Found. Phys. Letters, 3, 203 (1990).
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[35] A. Garuccio, Entangled states and the compatibility between quantum
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mechanics and relativity, in this book.
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[36] F. Selleri, Phys. Rev. A 56, 3493 (1997).
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Franco Selleri
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Bari, May 1998
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Velocity of Light
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This page intentionally left blank.
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An Explanation of the Sagnac Effect Based on the Special Theory of Relativity, the de Broglie/Bohm Interpretation of Quantum Mechanics, and a Non-Zero Rest Mass for the Photon
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Patrick Fleming Dublin 3 E-mail: flemingp@nsai.ie
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If a beam of light (photons) is split by means of a combined beam splitter/interferometer and sent in opposite directions around the circumference of a stationary disc using mirrors or optical fibres, an interference pattern is observed on the interferometer. The disc is capable of being rotated, and the apparatus is fixed in the laboratory. If the disc is now rotated the interference fringe is shifted on the interferometer relative to the stationary disc position. If the disc is now rotated in the other direction the fringe moves to the other side of the stationary disc fringe position. The effect was first observed by the French scientist G. Sagnac in 1910, and is named after him.
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The effect is seen irrespective of whether the observer rotates with the disc on its periphery, or is stationary in the laboratory. Subsequent tests have established that the effect is also observed with neutrons [1], and electrons [2].
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Over the intervening years many explanations of the phenomenon have been suggested e.g., Anandan [3] gives an explanation based on Special Relativity, and Selleri [4] gives an explanation in terms of inertial transformations. Kelly [5] and this Conference, concludes that the speed of light is not, in all circumstances, independent of the speed of its source. A comprehensive list is given in [6].
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I wish to put forward an explanation based on the Special Theory of Relativity, the de Broglie/Bohm interpretation of quantum mechanics, and a massive photon.
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The Copenhagen interpretation of quantum mechanics describes particles as either a particle or a wave depending on the mode of observation. It cannot accept the simultaneous presence of particle and wave. The de
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Open Questions in Relativistic Physics
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Edited by Franco Selleri (Apeiron, Montreal, 1998)
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3
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4
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Open Questions
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Broglie/Bohm model states that all particles, including photons, are always
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accompanied by a (pilot) wave.
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If the tangential velocity of the disc is V1 and the velocity of the particle (the special situation of the photon is discussed later) is V2 then, when the particle and the disc are moving in the same direction, the velocity of the
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particle is V2 – V1 (say V3) relative to an observer on the periphery of the disc (in practice a photographic plate). The Lorentz transformation of the
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addition/subtraction of velocities is not shown for reasons of simplicity of
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presentation. As V2 is far greater than V1 the transformation does not affect the analysis.When the particle and disc are moving in opposite directions the
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relative velocity of the particle is V2 + V1 (say V4). V3 and V4 are relativistic. The Special Theory of Relativity states that time for the two particles will be
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dilated to different extents according to the formula:
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γ= 1
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1
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−
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v2 c 2
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where γ is the gamma factor, v is the velocity of the particle relative to the
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observer; c is the Einstein assumption of a unique limiting velocity for all
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phenomena, achieved by a massless particle.
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In dealing with the Sagnac effect there are three aspects to be explained:
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i) the fringe shift; ii) the direction of the shift in relation to the rotation of the disc; iii) the fact that the same fringe shift is seen on board the rotating disc and
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in the laboratory.
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Time dilation is greater for the particle travelling against the rotation of the disc (V4). Therefore, to an observer on the disc, for a given time, this particle will have travelled a distance greater than the other particle causing a fringe shift. The shift is in a direction against the rotation of the disc. This explains the displacement of the particle circumferentially. One is using the de Broglie/Bohm interpretation of quantum mechanics and its model of particles accompanied by a wave. The displaced particle carries its wave with it, causing the fringe and its shift.
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All these velocities appear the same to a fixed observer in the laboratory. Therefore, he sees the same fringe shift.
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This analysis shows that the photon is behaving exactly like electrons and neutrons in respect of fringe shift. It would seem that one may add to its velocity, and that it obeys the conventional laws of addition and subtraction, and is not, therefore, absolute. It never reaches c, and must, therefore, have a rest mass.
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A rest mass for the photon has been suggested by many authors. Vigier [7] states that Einstein, Schrödinger and de Broglie suggested a rest mass of ~10–65 gr. Goldhaber et al. [8] discuss an upper limit of ~10–44 gr. Barrow et al.
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in Relativistic Physics
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5
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[9] also discuss limits and some of the implications. The Particle Data Group [10] give an upper bound of 3 × 10–33 MeV/c2.
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This time dilation is the same effect as that observed in the CERN experiments of 1976 when muons were accelerated to a speed approaching c, in a circular orbit, in an accelerator ring. This produced the same effect as the Sagnac rotating disc, the ring being fixed in the same frame of reference as the laboratory. Bailey et al. [11] report that only the effects of special relativity are relevant even under an acceleration of 1018 g. This increased their half-life from 2.2 μs to an observed 64.5 μs, i.e. by a factor of 29.3. The speed of the muons in the accelerator ring was 0.9994. Substituting this in the above formula gives a γ factor of 28.9, a near perfect agreement between theory and experimental result. The effect must be used by CERN engineers in designing their particle accelerators.
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When similar experiments are carried out on the surface of the earth (which, of course, can be considered as rotating disc at a particular latitude) the same effect is noted. Michelson and Gale [12] carried out an experiment on the effect of the earth’s rotation on the velocity of light. They recorded the difference in time taken for the light signals to travel clockwise and anticlockwise. They got a fringe shift of 0.230 on an interferometer, indicating a time difference.
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Saburi et al. [13] sent electromagnetic signals around the Earth between standard clock stations. The results showed that the signals travelled slower eastwards than westwards. One predicts that if the tests were carried out in a north-south direction, with the particles not being affected by the rotation of the earth, one would not see a time difference or fringe effect.
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Bilger et al. [14] carried out tests, using a ring-laser fixed to the earth. The objective was to determine the effect on the laser light of the rotational effect of the earth. The tests were carried out in New Zealand, in the Southern Hemisphere. The light was sent in opposing directions around a circuit of 0.75 m2. A fringe shift was observed, but in the opposite direction to that of tests carried out in the Northern Hemisphere.
|
||
The above analysis, if correct, indicates the validity of the fundamental physical assumption of the de Broglie/Bohm theory of the objective (co)existence of the quantum wave and the particle it guides. It also indicates a non-zero mass for the photon.
|
||
|
||
References
|
||
[1] Werner, 1979, Phys. Rev. Lett. 42 No. 17, 1103-1106 [2] Hasselbach, F., Nicklaus, M., 1993 Sagnac experiment with electrons:
|
||
Observation of the rotational phase shift of electron waves in vacuum, Phys. Rev. A. Vol. 48, No. 1, 143-151. [3] Anandan, J. 1981. Sagnac effect in relativistic and non-relativistic physics. Physical Review D, Vol. 24, No. 2, 338-346 [4] Selleri, F., 1996, Noninvariant one -way velocity of light. (to be published).
|
||
|
||
6
|
||
|
||
Open Questions
|
||
|
||
[5] Kelly, A.G., 1996, A New Theory on the Behaviour of Light, The Institution of Engineers of Ireland, Monograph No. 2.
|
||
[6] Post, E.J., 1967, Sagnac effect, Rev. Mod. Phys., Vol 39, No. 2, 475-494. [7] Vigier, J.P., 1997, Relativistic interpretation (with non-zero photon mass) of the
|
||
small aether drift velocity detected by Michelson, Morley and Miller. Apeiron, Vol 2, Nos.2-3, 71-76. [8] Goldhaber, A.S., et al., 1971, Terrestrial and Extraterrestrial Limits on The Photon Mass, Rev. Mod. Phys., Vol 43, No. 2, 277-296. [9] Barrow, J.D., et al., 1984, New light on heavy light, Nature, Vol, 307, 14-15. [10] Particle Data Group, “Review of Particle Properties,” Phys. Rev. D 50, 1173-1826 (1994). [11] Saburi, Y., Yamamoto, M., Harada, K., 1976, IEEE Trans, IM25 No. 4, 473-477 [12] Bailey, J., et al., 1977, Measurements of relativistic time dilatation for positive and negative muons in a circular orbit, Nature, Vol. 268, 301-305. [13] Michelson, A.A., Gale, H.G., 1925, The effect of the earth’s rotation on the velocity of light, Astroph. J., Vol. LXI No. 3, Part II, 140-145. [14] Bilger, H.R., Stedman, G.E., Screiber, W., Schneider, M., 1995 IEEE Trans., 44 IM No. 2, 468-70
|
||
|
||
On Synchronisation of Clocks in Free Fall Around a Central Body
|
||
|
||
F. Goy Dipartimento di Fisica Universitá di Bari Via G. Amendola, 173 I-70126 Bari, Italy E-mail: goy@axpba1.ba.infn.it
|
||
The conventional nature of synchronisation is discussed in inertial frames, where it is found that theories using different synchronisations are experimentally equivalent to special relativity. On the other hand, in accelerated systems only a theory maintaining an absolute simultaneity is consistent with the natural behaviour of clocks. The principle of equivalence is discussed, and it is found that any synchronisation can be used locally in a freely falling frame. Whatever the synchronisation chosen, the first derivatives of the metric tensor disapear and a geodesic is locally a straight line. But it is shown that only a synchronisation maintaining absolute simultaneity makes it possible to define time consistently on circular orbits of a Schwarzschild metric. Keywords: special and general relativity, synchronisation, one-way velocity of light, ether, principle of equivalence.
|
||
|
||
1. Introduction
|
||
|
||
In the last few decades there has been a revival of so-called “relativistic ether theories.” This revival is partly due to the parametrised test theory of special relativity by Mansouri and Sexl [1], which unlike the test theory of Robertson [2], makes explicit allowance for the problem of synchronisation of distant clocks within an inertial frame. Even though it is of vital importance for the definition of time in special relativity, most modern texbooks on relativity treat the question of synchronisation of clocks only briefly, or do not even mention it. The problem of synchronisation of distant clocks arose at the end of the 19th century from the decline of Newtonian mechanics, in which time was absolute and was defined without any reference to experience, and in particular clock synchronisation procedures. The nature of Newtonian time, transcending any experimental definition, was severely criticized by
|
||
|
||
Open Questions in Relativistic Physics
|
||
|
||
Edited by Franco Selleri (Apeiron, Montreal, 1998)
|
||
|
||
7
|
||
|
||
8
|
||
|
||
Open Questions
|
||
|
||
Mach. However, for the synchronisation procedure one had to take into account that no instantaneous action at distance exists in nature. In his 1905 [3] article in which he expounded the theory of relativity, Einstein, influenced by Mach’s epistemological conceptions, gave an operational definition of time:
|
||
|
||
“It might appear possible to overcome all the difficulties attending the definition of “time’’ by substituing “the position of the small hand of my watch’’ for “time.” And in fact such a definition is satisfactory when we are concerned with defining a time exclusively for the place where the watch is located; but is no longer satisfactory when we have to connect in time series of events occuring at different places, or—what comes to the same thing— to evaluate the times of events occurring at places remote from the watch.”
|
||
|
||
Further, he notes:
|
||
|
||
“If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with these events. If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate neighbourhood of B. But it is not possible without further assumption to compare, in respect of time, an event at A with an event at B. We have so far defined only an “A time’’ and a “B time.” We have not defined a common “time’’ for A and B, for the latter cannot be defined at all unless we establish by definition that the “time’’ required by light to travel from A to B equals the “time’’ it requires to travel from B to A. Let a ray of light start at the “A time’’ tA from A towards B, let it at the “B time’’ tB be reflected at B in the direction of A, and arrive again at A at the “A time’’ t’A.
|
||
In accordance with definition, the two clocks synchronize if
|
||
|
||
tB − tA = t’A − tB
|
||
|
||
(1)
|
||
|
||
We assume that this definition of synchronism is free from contradictions,
|
||
|
||
and possible for any number of points; and that the following relations are
|
||
|
||
universally valid:-
|
||
|
||
1. If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B.
|
||
|
||
2. If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks at B and C also synchronize with each other.’’
|
||
|
||
As Einstein himself stresses, the time required by light to travel from A to B and from B to A is equal by definition. This means that the one-way velocity of light is given by a convention, and not by experiment. What is known with great precision is the (mean) two-way velocity of light, which obviously can
|
||
|
||
in Relativistic Physics
|
||
|
||
9
|
||
|
||
be measured with only one clock and a mirror. This is known with a precision of Δc/c = 10–9 [4] and has always been found to be constant in any direction throughout the year, despite the Earth’s motion. The one-way velocity of light, on the other hand, cannot be determined experimentally. Let us imagine that someone tries to measure it: he might send a light ray from a clock located at A to a clock located at B, at a distance d from A, and would obtain the one-way velocity of light from A to B by dividing the distance d by the difference between the time of arrival in B and the time of departure from A. But in order to compute this time difference, he first needs clocks which are synchronised, by means of light rays—whose one-way velocity is postulated. Thus the concepts of simultaneity and one-way velocity of light are bound together logically in a circular way.
|
||
One may, of course, wonder whether other conventions which are not in contradiction with experiment are possible. First we rewrite equation (1) such that the “B time’’ is defined as a function of the “A time.” That is:
|
||
|
||
tB = tA + ½(t’A−tA)
|
||
|
||
(2)
|
||
|
||
Reichenbach commented [5]:
|
||
|
||
“This definition is essential for the special theory of relativity, but is not epistemologically necessary. If we were to follow an arbitrary rule restricted only to the form
|
||
|
||
tB = tA + ε(t’A − tA), 0 < ε < 1
|
||
|
||
(3)
|
||
|
||
it would likewise be adequate and could not be called false. If the special theory of relativity prefers the first definition. i.e., sets ε equal to ½, it does so on the ground that this definition leads to simpler relations.’’
|
||
Among the “conventionalists’’, who agree that one can choose ε freely, are Winnie [6], Grünbaum [7], Jammer [8], Mansouri and Sexl [1], Sjödin [9], Cavalleri and Bernasconi [10], Ungar [11], Vetharaniam and Stedman [12] and Anderson and Stedman [13]. Clearly, different values of ε correspond to different values of the one way-speed of light.
|
||
A slightly different position was developed in the parametric test theory of special relativity by Mansouri and Sexl [1]. Following these authors, we assume that there is at least one inertial frame in which light behaves isotropically. We call it the priviledged frame Σ and denote space and time coordinates in this frame by the letters: (x0,y0,z0,t0). In Σ, clocks are synchronised with Einstein’s procedure. We also consider another system S moving with uniform velocity v < c along the x0-axis in the positive direction. In S, the coordinates are written with lower case letters (x,y,z,t). Under rather general assumptions as to initial and symmetry conditions on the two systems (S and Σ are endowed with orthonormal axes, which coincide at time t0 = 0, [1,14]) the assumption that the two-way velocity of light is c and furthermore that the time dilation factor has its relativistic value, one can derive the following transformation:
|
||
|
||
10
|
||
|
||
Open Questions
|
||
|
||
a f x =
|
||
|
||
1 1−β2
|
||
|
||
x0 − vt0
|
||
|
||
y = y0
|
||
|
||
(4)
|
||
|
||
z = z0
|
||
|
||
a f t = s x0 − vt0 + 1 − β 2 t0 ,
|
||
|
||
where β = v/c. The parameter s, which characterizes the synchronisation in the S frame, remains unknown. Einstein’s synchronisation in S involves:
|
||
|
||
s = −v c2 1 − β 2 and (4) becomes a Lorentz boost. For a general s, the inverse
|
||
|
||
one-way velocity of light is given by [15]:
|
||
|
||
FG IJ 1 = 1 + β + s 1 − β 2 cosΘ , a f H K c→ Θ c c
|
||
|
||
(5)
|
||
|
||
where Θ is the angle between the x-axis and the light ray in S. c→(Θ) is in general dependent on the direction. A simple case is s = 0. From (4), this means that at t0 = 0 in Σ we set all clocks in S at t = 0 (external synchronisation), or that we synchronise the clocks by means of light rays with velocity c→(Θ) = c/1+β cosΘ (internal synchronisation). We obtain the transformation:
|
||
|
||
a f x =
|
||
|
||
1 1−β2
|
||
|
||
x0 − vt0
|
||
|
||
y = y0
|
||
|
||
(6)
|
||
|
||
z = z0
|
||
|
||
t = 1 − β 2 t0.
|
||
|
||
This transformation maintains an absolute simultaneity (as in Σ) between all inertial frames. It should be stressed that, unlike the parameters of length contraction and time dilation, this parameter s cannot be tested, but its value must be assigned in accordance with the synchronisation choosen in the experimental setup. This means, as regards experimental results, that theories using different s are equivalent. Of course, they may predict different values of physical quantities (for example the one-way speed of light). The difference lies not in nature itself, but in the convention used for the synchronisation of clocks. In other words, two transformations (4) with different s represent the same transformation but relative to different time coordinates. For a recent and comprehensive discussion of this subject, see [16]. A striking consequence of (4) is that the negative result of the Michelson-Morley experiment does not rule out an ether. Only an ether with Galilean transformations is excluded, because the Galilean transformations do not lead to an invariant two-way velocity of light in a moving system.
|
||
Strictly speaking, the conventionality of clock synchronisation has only been shown to hold in inertial frames. The derivation of equation (4) is done in inertial frames and is based on the assumption that the two-way velocity of light is constant in all directions. This last assumption is no longer true in
|
||
|
||
in Relativistic Physics
|
||
|
||
11
|
||
|
||
accelerated systems. However, special relativity is not used just in inertial frames. Many textbooks give examples of calculations done in accelerated systems, using infinitesimal Lorentz transformations. Such calculations use an additional assumption: the so-called Clock Hypothesis, which states that, seen from an inertial frame, the rate of an accelerated ideal clock is identical to a clock in the instantaneously comoving inertial frame. In other words, the rate of a clock is not influenced by acceleration per se. This hypothesis, first used implicitely by Einstein in his 1905 article, was superbly confirmed in the famous time decay experiment on muons at CERN, where the muons had an acceleration of 1018g, but where their time decay was due only to their velocity [17]. We stress here the logical independence of this assumption from the structure of special relativity as well as from the assumptions necessary to derive (4). The author’s opinion is that the Clock Hypothesis, added to special relativity in order to extend it to accelerated systems, leads to logical contradictions when the question of synchronisation is brought up. This idea has also been expressed by Selleri [18]. The following example (see [19]) demonstrates this point: imagine that two distant clocks are secured to an inertial frame (say a train at rest) and synchronised using Einstein’s synchronisation. We call this rest frame Σ. The train accelerates for a certain period. After that, the acceleration stops and the train again has inertial motion (sytem S). During acceleration, the clocks are subjected to exactly the same influences, so they have the same rate at all times, and remain synchronous relative to Σ. Due to the relativity of simultaneity in special relativity, where an Einstein’s procedure is applied to the synchronisation of clocks in all inertial frames, they are no longer Einstein synchronous in S. So the Clock Hypothesis is inconsistent with the clock setting of relativity. On the other hand, the Clock Hypothesis has been tested with a high degree of accuracy [20] and cannot be rejected; consequently, we must reject the clock setting of special relativity. The only theory which is consistent with the Clock Hypothesis is based on transformations (4) with s = 0.
|
||
This is an ether theory. The fact that only an ether theory is consistent with accelerated motion provides strong evidence that an ether exists, but does not inevitably imply that our velocity relative to the ether is measurable. The author’s opinion is that it cannot be measured, because (6) represents another coordinatisation of the Lorentz transformation (obtained by clock resynchronisation). In principle, this prevents any detection of uniform motion through the ether. By changing the coordinate system, one cannot obtain a physics in which new physical phenomena appear. But we can obtain a more consistent description of these phenomena.
|
||
In all the above considerations, space-time was flat and no gravitational forces were present. In the following, we want to treat the question of synchronisation of clocks in the framework of general relativity, were special relativity is only valid locally. In section 2, we calculate the equations of motion for circular orbits in a Schwarzschild metric. In section 3, we treat the
|
||
|
||
12
|
||
|
||
Open Questions
|
||
|
||
problem of synchronisation of clocks on these orbits, and discuss the compatibility of different synchronisations with the principle of equivalence.
|
||
|
||
2. Circular orbits in a Schwarzschild metric
|
||
|
||
In a reference system R with coordinates S, (x0,x1,x2,x3) = (ct,r,ϕ,θ) (θ is the azimuthal angle) the spherical symmetric solution of Einstein’s equations in vacuum, with the boundary condition that the metric becomes Minkowskian at infinity is the Schwarzschild metric:
|
||
|
||
FG IJc h FG IJ c h ds2 = − 1 − α
|
||
|
||
dx0 2 +
|
||
|
||
1−α
|
||
|
||
−1
|
||
dr 2 + r 2 sin2 θ d2 + dθ 2
|
||
|
||
,
|
||
|
||
H K H K r
|
||
|
||
r
|
||
|
||
(7)
|
||
|
||
where α = 2GM/c2 is the Schwarzschild radius of the field of total energy Mc2 and G the gravitational constant. In the following we will consider only geodesics of test particles of mass m with r > α, so that we are not concerned here with the breakdown of the coordinate system at r = α. A Lagrangian function can be written as:
|
||
|
||
L = −m c
|
||
|
||
gij
|
||
|
||
dxi dτ
|
||
|
||
dx j dτ
|
||
|
||
(8)
|
||
|
||
and the Lagrange equations by
|
||
|
||
d i ∂ L = d ∂ L
|
||
|
||
∂ xi
|
||
|
||
dτ
|
||
|
||
∂
|
||
|
||
dxi dτ
|
||
|
||
,
|
||
|
||
i = 0,…,3.
|
||
|
||
(9)
|
||
|
||
The variables x0, θ, φ are cyclic and their conjugate momentum is conserved. Without loss of generality we can take θ = π/2, i.e., equatorial orbits only. The energy E and angular momentum L per unit of mass are conserved quantities:
|
||
|
||
L = r 2 dϕ
|
||
|
||
dτ
|
||
|
||
FHG IKJ E
|
||
|
||
=
|
||
|
||
c
|
||
|
||
dx o dτ
|
||
|
||
1− α r
|
||
|
||
.
|
||
|
||
(10)
|
||
|
||
From (9) and (10) the equation for the variable r can be written
|
||
|
||
F IF I dr HG KJHG KJ a f dτ
|
||
|
||
=
|
||
|
||
E2 c2
|
||
|
||
−
|
||
|
||
1−α r
|
||
|
||
c
|
||
|
||
2
|
||
|
||
+
|
||
|
||
L2 r2
|
||
|
||
=
|
||
|
||
1 c2
|
||
|
||
E2 −V2
|
||
|
||
r
|
||
|
||
(11)
|
||
|
||
where V(r) is an effective potential. This effective potential has a local minimum; thus we have stable circular orbits. From (10), we then find for these circular orbits:
|
||
|
||
dr dτ
|
||
|
||
=0⇒r
|
||
|
||
=
|
||
|
||
cst
|
||
|
||
a f a f dϕ
|
||
dτ
|
||
|
||
=
|
||
|
||
L r2
|
||
|
||
⇒ϕ τ
|
||
|
||
=ϕ τ =0
|
||
|
||
+ cst1τ
|
||
|
||
(12)
|
||
|
||
a f a f dt = E
|
||
|
||
c h dτ
|
||
|
||
c2
|
||
|
||
1−
|
||
|
||
α r
|
||
|
||
⇒ τ t = τ t = 0 + cst2t,
|
||
|
||
in Relativistic Physics
|
||
|
||
13
|
||
|
||
where cst1 =
|
||
|
||
c r
|
||
|
||
α 2r − 3α and cst2 =
|
||
|
||
2−
|
||
|
||
3α r
|
||
|
||
.
|
||
|
||
3. Two clocks in orbit
|
||
|
||
We now consider a clock A in event-point A(x0A,rA,ϕA), and do all calculations in 1 + 2 dimensional space-time, since we treat equatorial orbits only. On a circular orbit, its velocity is given by
|
||
|
||
a f U = c,0,ω 1 − α rA − rA2ω 2 c 2 . We have UiUjgj = −c2 and ω = dϕ/dt, and is
|
||
|
||
given by the Kepler law ω2 = GM/rA3 for circular orbits [21]. The principle of equivalence assures us that we can find a system of
|
||
|
||
o
|
||
|
||
o
|
||
|
||
reference R , with a coordinate system S such that at event-point A,
|
||
|
||
o
|
||
|
||
o
|
||
|
||
FH IKa f gij (A) = ηj and ∂ gij ∂ xk A = 0 , where ηj = diag(−1,1,1). In particular, it is
|
||
|
||
possible to choose a set of three mutually orthogonal unit vectors ei(a) such that ei(0) = Ui/c and e(1) and ei(2) fulfil the orthonormality conditions: gikei(a)eik(b) = ηab. Indices without parenthesis of ei(a) are lowered with gik, while indices with parenthesis are raised with ηab. We can choose e(1) radial and e(2) tangential to
|
||
the orbit:
|
||
|
||
FHG IKJ ea1f =
|
||
|
||
0,
|
||
|
||
1 − α ,0 rA
|
||
|
||
FHG IKJ ea2f =
|
||
|
||
1 1 − α rA − rA2ω 2 c 2
|
||
|
||
rAω ,0, 1 − α rA .
|
||
|
||
c 1 − α rA
|
||
|
||
rA
|
||
|
||
(13)
|
||
|
||
o
|
||
The following transformation from coordinate system S to S is such that the metric tensor in the new coordinates is Minkowskian and its first derivatives disapear at point A [§9.6][22]:
|
||
|
||
c h a fc h oi
|
||
|
||
x = eri
|
||
|
||
xr
|
||
|
||
−
|
||
|
||
x
|
||
|
||
r A
|
||
|
||
a f +
|
||
|
||
1 2
|
||
|
||
eri
|
||
|
||
Γsrt
|
||
|
||
A
|
||
|
||
xs
|
||
|
||
−
|
||
|
||
x
|
||
|
||
t A
|
||
|
||
,
|
||
|
||
i = 0,1,2 .
|
||
|
||
(14)
|
||
|
||
In the case of (7), the Christofell symbols Γ at A are given by:
|
||
|
||
b g b g Γ010
|
||
|
||
=
|
||
|
||
1α 2
|
||
|
||
1−α rA2
|
||
|
||
rA
|
||
|
||
Γ001
|
||
|
||
=
|
||
|
||
1 2
|
||
|
||
α
|
||
|
||
α 1−α
|
||
|
||
rA
|
||
|
||
rA2
|
||
|
||
b g b g Γ111
|
||
|
||
=−1 2
|
||
|
||
2rA2
|
||
|
||
α 1−α
|
||
|
||
rA
|
||
|
||
Γ212 = −r 1 − α rA .
|
||
|
||
Γ122
|
||
|
||
=
|
||
|
||
1 rA
|
||
|
||
(15)
|
||
|
||
o
|
||
We obtain for the transformation between S and S :
|
||
|
||
14
|
||
|
||
Open Questions
|
||
|
||
LMFG IJc h b g OP o0 NMMMH cK ha f a fQPPP x =
|
||
|
||
1 1 − α rA − rA2ω 2 c 2
|
||
|
||
1− α rA
|
||
|
||
x0
|
||
|
||
−
|
||
|
||
x
|
||
|
||
0 A
|
||
|
||
− ω rA2 c
|
||
|
||
ϕ −ϕA
|
||
|
||
+
|
||
|
||
1 2
|
||
|
||
α rA2
|
||
|
||
x0
|
||
|
||
−
|
||
|
||
x
|
||
|
||
0 A
|
||
|
||
r − rA
|
||
|
||
+1α 4
|
||
|
||
1 − α rA rA2
|
||
|
||
r − rA
|
||
|
||
a f c h o 2
|
||
x=
|
||
|
||
1 1 − α rA
|
||
|
||
r − rA
|
||
|
||
+1α 4
|
||
|
||
1 − α rA rA2
|
||
|
||
x0
|
||
|
||
−
|
||
|
||
x
|
||
|
||
0 A
|
||
|
||
2
|
||
|
||
(16)
|
||
|
||
a f b g − 1 b g 4 rA2
|
||
|
||
α 1−α
|
||
|
||
rA
|
||
|
||
3 2
|
||
|
||
r − rA
|
||
|
||
2−1 2
|
||
|
||
1 − α rA ϕ − ϕ A 2
|
||
|
||
LM c h b g OP o 2 NMMM b gc ha f a fb gQPPP x =
|
||
|
||
1 − α rA 1 − α rA − rA2ω 2 c 2
|
||
|
||
− ω rA c
|
||
|
||
x0
|
||
|
||
−
|
||
|
||
x
|
||
|
||
0 A
|
||
|
||
+ rA ϕ − ϕ A
|
||
|
||
−1 ωα 2 crA 1 − α rA
|
||
|
||
x0
|
||
|
||
−
|
||
|
||
x
|
||
|
||
0 A
|
||
|
||
r − rA
|
||
|
||
+ r − rA
|
||
|
||
ϕ −ϕA
|
||
|
||
This transformation looks like the Lorentz transformation at first order, in
|
||
|
||
o
|
||
particular, two distant events which are simultaneous in S are not
|
||
|
||
simultaneous in S. We now imagine that a clock B is located at B (x0A+dx0, rA, ϕA + dϕ) and we want to synchronise it with A at A using Einstein’s
|
||
|
||
o
|
||
procedure. Since the metric is Minkowskian in S , the velocity of light is c in
|
||
|
||
this (local) frame. The two clocks will be Einstein synchronised when:
|
||
|
||
o
|
||
|
||
o
|
||
|
||
x
|
||
|
||
0 A
|
||
|
||
=
|
||
|
||
xB0
|
||
|
||
=
|
||
|
||
0
|
||
|
||
.
|
||
|
||
Using
|
||
|
||
(16)
|
||
|
||
we
|
||
|
||
obtain
|
||
|
||
that
|
||
|
||
the
|
||
|
||
infinitesimal
|
||
|
||
time
|
||
|
||
difference
|
||
|
||
in
|
||
|
||
S
|
||
|
||
dx0 between these events is given by:
|
||
|
||
b g dx0
|
||
|
||
=
|
||
|
||
c
|
||
|
||
ω rA2dϕ 1 − α rA
|
||
|
||
(17)
|
||
|
||
We generalise this synchronisation procedure all along the circular orbit. This means that we synchronise A in (rA,ϕA), with B in (rA,ϕB = ϕA + dϕ), and then B with C located at (rA,ϕC = ϕB+dϕ), etc. If we do a whole round trip, we find a
|
||
time lag Δx0 given by:
|
||
|
||
z b g b g Δx0 = ω rA2dϕ = 2π ω rA2 c 1 - α rA c 1 − α rA
|
||
|
||
(18)
|
||
|
||
This means that A is not synchronisable with itself, when we extend the
|
||
synchronisation procedure spatially out of a local domain; this is clearly absurd. The problem occurs because dx0 is not a total differential in r and ϕ,
|
||
thus the synchronisation procedure is path-dependent. In general, one can say that if A is synchronized with B, then B does not synchronise with A. The
|
||
same remark is valid for the transitivity of the relation “is synchronous with’’ in the case of three clocks A, B and C
|
||
According to Einstein in the citation quoted above, the definition of
|
||
synchronism given by (1) which is free from contradictions in the case of
|
||
inertial frames in flat space is no longer free from contradictions when we
|
||
|
||
in Relativistic Physics
|
||
|
||
15
|
||
|
||
want to define time globally in a curved space. One might think that this difficulty is insuperable, and that it is not possible to:
|
||
|
||
1. find a local inertial system such that the equivalence principle is respected
|
||
2. define time in this system in such a way that extending the synchronisation procedure out of a local domain is self consistent: “is synchronised with’’ is an equivalence relation.
|
||
|
||
A similar problem occurs in the case of a rotating disk in flat space. It has been shown that only the transformation (6) allows a consistent definition of time on the rim of a rotating disk, while an Einstein synchronisation leads to the impossibility of defining time without contradictions on the rim of this disk [23].
|
||
Guided by the experimental equivalence of relativistic ether theories and
|
||
o
|
||
special relativity, we are looking for another synchronisation of clocks in R such that the conditions 1 and 2 above are fullfilled. The spatial part of transformation (16) is not changed by a resynchronisation of clocks, and we can again choose the vectors e(1), and e(2) as they can be read out from (16). We are looking for a transformation from coordinate system S to local coordinate
|
||
|
||
system S such that the time transformation does not depend on the space
|
||
|
||
variables at first order. This means that e(0) is of the type e(0) = (y,0,0). In order
|
||
|
||
o
|
||
to find y, we postulate that the sychronisation only is different in S and S . In
|
||
|
||
o
|
||
other words, the rate of a clock at rest at the origin of S and S is the same
|
||
|
||
when seen from S. From (16) we easily calculate that:
|
||
|
||
o
|
||
|
||
o
|
||
|
||
δ x0 = 1 − α rA − ω 2rA2 c2 δ x0 , where δ x0 is the coordinate time difference
|
||
|
||
o
|
||
between two ticks of the clock in S and δ x0 is the same quantity in S. We find
|
||
|
||
that y = 1 − α rA − ω 2rA2 c 2 . Thus the transformation of the time coordinate
|
||
|
||
from S to S is now given by:
|
||
|
||
c h b g c ha f x0 =
|
||
|
||
1−α
|
||
|
||
rA − ω 2rA2
|
||
|
||
c2
|
||
|
||
x0
|
||
|
||
−
|
||
|
||
x
|
||
|
||
0 A
|
||
|
||
+1α 2
|
||
|
||
1 − α rA − ω 2rA2 c 2 1 − α rA rA2
|
||
|
||
x0
|
||
|
||
−
|
||
|
||
x
|
||
|
||
0 A
|
||
|
||
r − rA
|
||
|
||
(19)
|
||
|
||
1. Are we sure that S is a local inertial system of coordinates? Yes. The
|
||
|
||
o
|
||
proof is indeed the same as it would be for S . From (14) and using the fact that e(r)ierj = δij, we have:
|
||
|
||
c h a fc hc h eairfxr =
|
||
|
||
xi
|
||
|
||
−
|
||
|
||
x
|
||
|
||
i A
|
||
|
||
+
|
||
|
||
1 2
|
||
|
||
Γsit
|
||
|
||
A
|
||
|
||
xs
|
||
|
||
−
|
||
|
||
x
|
||
|
||
s A
|
||
|
||
xt
|
||
|
||
−
|
||
|
||
x
|
||
|
||
t A
|
||
|
||
i = 0,1,2. .
|
||
|
||
(20)
|
||
|
||
Differentiating two times with respect to xk and xl gives:
|
||
|
||
16
|
||
|
||
Open Questions
|
||
|
||
a fLNM c h OQP 0
|
||
|
||
=
|
||
|
||
∂2xi ∂ xl∂ xk
|
||
|
||
+ Γsit
|
||
|
||
A
|
||
|
||
∂2xs ∂ xl∂ xk
|
||
|
||
xt
|
||
|
||
−
|
||
|
||
x
|
||
|
||
t A
|
||
|
||
+
|
||
|
||
∂ ∂
|
||
|
||
xs xk
|
||
|
||
∂ xt ∂ xt
|
||
|
||
(21)
|
||
|
||
Thus at point A:
|
||
|
||
a fLNM OQP 0
|
||
|
||
=
|
||
|
||
∂
|
||
|
||
∂2xi xl∂ xk
|
||
|
||
+ Γsit
|
||
|
||
A
|
||
|
||
∂xs ∂xt ∂xk ∂xt
|
||
|
||
(22)
|
||
|
||
Because of the law of transformation of Christoffel symbols, this mean that:
|
||
a f Γkil A = 0 . So in S at A, a geodesic becomes a straight line:
|
||
|
||
d2xk dλ2
|
||
|
||
+ Γikl
|
||
|
||
dxl dλ
|
||
|
||
dxi dλ
|
||
|
||
=
|
||
|
||
d2xk dλ2
|
||
|
||
=0
|
||
|
||
(23)
|
||
|
||
2. Can time be defined consistently on the whole circular orbit? Yes. We
|
||
|
||
treat again the problem of synchronising a clock A at A (x0A, rA, ϕA) and a clock B at B (x0A + dx0, rA, ϕA+dϕ) The two clocks are synchronised in the system of
|
||
|
||
coordinates
|
||
|
||
S
|
||
|
||
if
|
||
|
||
x
|
||
|
||
0 A
|
||
|
||
= xB0
|
||
|
||
=0.
|
||
|
||
Then
|
||
|
||
the
|
||
|
||
time
|
||
|
||
difference
|
||
|
||
dx0
|
||
|
||
between
|
||
|
||
these
|
||
|
||
events in S calculated with (19) gives: dx0 = 0. A similiar calculation as in (18)
|
||
|
||
shows that Δx0 = 0 for a whole round trip. Thus the time can be defined
|
||
|
||
consistently on the orbit with such a synchronisation.
|
||
|
||
The metric in system S at A is given by ei(a) gijej(b) = η ab . We find
|
||
|
||
F G
|
||
|
||
−1
|
||
|
||
0
|
||
|
||
I rAω J c 1−α rA
|
||
|
||
HGG KJJ ηab =
|
||
|
||
0
|
||
rAω c 1−α rA
|
||
|
||
1
|
||
|
||
0
|
||
|
||
b g 0 1 − rA2ω 2 c 2 1−α rA
|
||
|
||
(24)
|
||
|
||
In the case where the vector potential ηoα ; α = 1,2 is different from zero, the
|
||
|
||
spatial part of the metric is given by the space-space coefficients of the metric
|
||
|
||
as
|
||
|
||
well
|
||
|
||
as
|
||
|
||
by
|
||
|
||
γ αβ
|
||
|
||
= ηαβ
|
||
|
||
− η oαη oβ η oo
|
||
|
||
.
|
||
|
||
In
|
||
|
||
our
|
||
|
||
case
|
||
|
||
we
|
||
|
||
have
|
||
|
||
γ αβ
|
||
|
||
= δ αβ .
|
||
|
||
Thus
|
||
|
||
the
|
||
|
||
spatial system of coordinates is orthonormal. The velocity of light c(Θ) is
|
||
|
||
found by solving the equation ds2 = ηabdxadxb = 0 . We find that:
|
||
|
||
a fc Θ
|
||
|
||
=
|
||
|
||
1+
|
||
|
||
c
|
||
rAω cosΘ c 1−α rA
|
||
|
||
(25
|
||
|
||
where Θ is the angle between the light ray and the x2 −axis
|
||
|
||
4. Remarks
|
||
|
||
1. The transformation of the time variable can easily be generalised to all
|
||
|
||
synchronisations with a parameter s like in (4):
|
||
|
||
a f c h LNM b g c hOQP c h x0 s =
|
||
|
||
1−α
|
||
|
||
rA − ω 2rA2
|
||
|
||
c2
|
||
|
||
x0
|
||
|
||
−
|
||
|
||
x
|
||
|
||
0 A
|
||
|
||
+ s rA ϕ − ϕ A
|
||
|
||
− rAω c
|
||
|
||
x0
|
||
|
||
−
|
||
|
||
x
|
||
|
||
0 A
|
||
|
||
+ O xi − xiA 2
|
||
|
||
(26)
|
||
|
||
in Relativistic Physics
|
||
|
||
17
|
||
|
||
The transformation (19) is given by s = 0 and the transformation (19) by
|
||
|
||
s=−
|
||
|
||
ω rA
|
||
|
||
A similar argument as in section 3 shows that only s = 0
|
||
|
||
c 1−α r A −ω 2rA2 c 2
|
||
|
||
lead to Δx0 = 0 for a whole round trip of synchronisation around the orbit.
|
||
|
||
o
|
||
2. The inertial coordinate systems S and S are different
|
||
|
||
o
|
||
|
||
o
|
||
|
||
coordinatisations of the same reference frame R . The transformation from S
|
||
|
||
to S does not involve time in the transformation of space variables, and thus
|
||
is what Møller [p. 267, 316][22] calls a linear gauge transformation. 3. If a clock A at A (x0A,rA,ϕA) and a clock B at B (x0A+dx0,rA,ϕA+dϕ) are
|
||
o
|
||
Einstein synchronised in the system S of section 3 [i.e dx0 is given by (17)],
|
||
they remain Einstein’s synchronised during their trip around the orbit. From the equation of motion (12) one sees that they will be at point ~A and ~B at a
|
||
|
||
d i d i later time with coordinates in S: x~0A , r~A ,ϕ ~A and x~B0 + dx0 , r~A ,ϕ ~A + dϕ . We
|
||
can take a local inertial system at ~A and from (16) one sees that: ~x~A0 = ~x~B0 = 0 .
|
||
|
||
5. Conclusion
|
||
In flat space, a whole set of theories equivalent to special relativity can be constructed. These theories are obtained by adopting another convention on the synchronisation of clocks. In accelerated systems, only the theory maintaining an absolute simultaneity is logically consistent with the natural behaviour of clocks.
|
||
In general relativity, the principle of equivalence tells us that at every space-time point one can choose a local coordinate system such that the metric is Minkowskian and its first derivatives disapear. Thus, the laws of special relativity are locally valid in general relativity. In this local frame, we can choose another synchronisation of clocks different from Einstein’s. The frame is the same but the coordinatisation is different. All these coordinatisations are locally equivallent. The transformation between them is a linear gauge transformation. The spatial part of the metric is orthonormal and the derivates of the space-time metric disapear at the point in question. Thus, a freely falling body has uniform motion in a straight line, and theses local coordinate systems are locally inertial.
|
||
An Einstein synchronisation leads to a contradictory definition of time when extended out of a local domain. It was shown in this article that in the case of circular orbits, only a transformation maintaining absolute simultaneity is able to define time globally and consistently on the orbit. An observer moving around a central body, who does not want to adopt a contradictory definition of time (when extended spatially out of his local domain) must then conlude that the velocity of light is not constant.
|
||
|
||
18
|
||
|
||
Open Questions
|
||
|
||
Acknowledgement
|
||
I wish to thank the Physics Departement of Bari University for hospitality, and Prof. F. Selleri for his kind suggestions and criticisms.
|
||
|
||
References
|
||
[1] R. Mansouri and R. U. Sexl, Gen. Rel. Grav. 8, 497-513 (1977); 8, 515-524 (1977); 8, 809-813 (1977).
|
||
[2] H. P. Robertson, Rev. Mod. Phys. 21, 378-382 (1949). [3] A. Einstein, Ann. Phys. 17, 891-921 (1905). Translated in The Principle of Relativity
|
||
(Dover, USA, 1923). [4] H. E. Bates, Am. J. Phys. 56, 682-687 (1988). [5] H. Reichenbach, The Philosophy of Space and Time (Dover, New-York, 1958). [6] J. A. Winnie, Phil. Mag. 37, 81-99 and 223-238 (1970). [7] A. Grünbaum, Philosophical Problems of Space and Time (Reidel, Dodrecht, 1973). [8] M. Jammer, Some Fundamental Problems in the Special Theory of Relativity, in:
|
||
Problems in the Foundation of Physics (G. Toraldo di Francia ed., North Holland, Amsterdam, 1979). [9] T. Sjödin, Nuov. Cim. 51B, 229-245 (1979). [10] G. Cavalleri and C. Bernasconi, Nuov. Cim. 104B, 545-561 (1989). [11] A. A. Ungar, Found. Phys. 6, 691-726 (1991). [12] I. Vetharaniam and G. E. Stedman, Found. Phys. Lett. 4, 275-281 (1991). [13] R. Anderson and G. E. Stedman, Found. Phys. Lett. 5, 199-220 (1992); Found. Phys. Lett. 7, 273-283 (1994). [14] F. Selleri, Phys. Essays 8, 342-349 (1995). [15] F. Selleri, in Frontiers of Fundamental Physics, Ed. by M. Barone and F. Selleri (Plenum Press, New-York, 1994), pp. 181-192. [16] I. Vetharaniam and G. E. Stedman, Phys. Lett. A 183, 349-354 (1993). [17] J. Bailey et al., Nature 268, 301-304 (1977). [18] F. Selleri, Found. Phys. 26, 641-664 (1996). [19] S. R. Mainwaring and G. E. Stedman, Phys. Rev. A 47, 3611-3619 (1993). [20] A. M. Eisle, Helv. Phys. Act. 60, 1024-1037 (1987). [21] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (W. H. Freeman and Company, New-York, 1973). [22] C. Møller, The Theory of Relativity, 2nd ed. (Oxford University Press, Oxford, 1972). [23] F. Goy and F. Selleri, Found. Phys. Lett. 10, 17-29 (1997).
|
||
|
||
Remarks on Clock Synchronization
|
||
|
||
Andrzej Horzela Department of Theoretical Physics H. Niewodniczanski Institute of Nuclear Physics ul. Radzikowskiego 152, 31 342 Kraków, Poland
|
||
|
||
Introduction
|
||
Space-time coordinates, like any other physical quantity, should be given by their own operational definitions. This means that we should point out a physical process which can be used to measure them by comparison with standard phenomena defining the system of units. In particular, in order to be able to measure time at distant points in a unique way we must synchronize all clocks present in the system. The choice of synchronization method defines the model of space-time obtained and significantly influences its properties, physical as well as mathematical, including the structure of the space-time symmetry group [1,2].
|
||
|
||
Properties of coordinate systems
|
||
|
||
A general definition of the space-time coordinates may be obtained provided we can specify [1]:
|
||
|
||
1. the class of observers, 2. the class of elementary events, 3. the class of signals used to communicate between observers and
|
||
elementary events, 4. the interaction of the elementary events with signals used for
|
||
communication.
|
||
|
||
The set of concepts listed above generalizes Einstein’s fundamental
|
||
|
||
assumptions of special relativity theory [3]. Within the latter the classes of
|
||
|
||
observers and elementary events coincide and each of them consists of an
|
||
|
||
identical set of clocks equipped with light emitters and detectors and
|
||
|
||
reflecting mirrors. Communication between observers and events is
|
||
|
||
accomplished by light pulses travelling in space with universal, constant and
|
||
|
||
isotropic velocity. The light pulses interact only with mirrors which are
|
||
|
||
supposed to be ideal and the reflection is described by the laws of geometrical
|
||
|
||
optics. To achieve synchronization of all clocks Einstein’s procedure needs
|
||
|
||
Open Questions in Relativistic Physics
|
||
|
||
Edited by Franco Selleri (Apeiron, Montreal, 1998)
|
||
|
||
19
|
||
|
||
20
|
||
|
||
Open Questions
|
||
|
||
only one clock placed at the origin of the reference frame. If at the instant of time T1 it emits a light pulse towards an event A and if it detects at the instant of time T2 the same pulse coming back after reflection at A then the instant of
|
||
time T where reflection has occurred equals
|
||
|
||
a f T = 1 2
|
||
|
||
T1 + T2
|
||
|
||
(1)
|
||
|
||
while the distance between the origin and the event A is given by
|
||
|
||
a f X =
|
||
|
||
c 2
|
||
|
||
T2 − T1
|
||
|
||
(2)
|
||
|
||
where c denotes the velocity of light. In Einstein’s synchronization there is no
|
||
|
||
distinction between one-way and two-way velocity of light. They are equal by
|
||
|
||
definition and, as a consequence, no preferred reference frame is permitted to
|
||
|
||
exist. If we change the coefficient in (1) and replace this formula, according to
|
||
|
||
[4], by
|
||
|
||
a f T = T1 + ε T2 − T1
|
||
|
||
(3)
|
||
|
||
we arrive at models of space-time in which signals obeying frame dependent velocity are used in order to synchronize distant clocks. The example of such an approach are models which distinguish among one-way velocities of light and allow an ether to exist and to be the only medium where the electromagnetic waves propagate isotropically with the velocity c [2]. Also, within Newtonian mechanics it is possible to construct a synchronization procedure based on (3) where observers communicate using signals obeying all the laws of Galilean physics, in particular velocity addition rule [1].
|
||
Any recipe postulated to give the time shown by a distant clock should be completed by a prescription which enables us to calculate the distance how far the investigated event is located from the observer. It is obvious that it should be expressed in terms of time (T − T1) spent by the signal in its travel to the event, as well as in terms of the time of duration of the return trip, (T2 – T). If it will be unique, we should have
|
||
|
||
X = f→ (T – T1) = f← (T2 – T)
|
||
|
||
(4)
|
||
|
||
which may be rewritten as
|
||
|
||
X = f→[ε(T2 – T1)] = f←[(1 – ε) (T2 – T1)]
|
||
|
||
(5)
|
||
|
||
with functions f→ and f← in general different each from another and depending on the details of the model. They become the same when ε = ½
|
||
|
||
which means the situation when the motion towards the event and
|
||
|
||
backwards does not change its properties. It is true for (1) and (2) valid, that is
|
||
|
||
for the standard special relativity.
|
||
|
||
Does light move uniformly?
|
||
In the following it is our aim to continue the analysis performed in [5] where we proposed clock synchronization with the use of signals moving with constant acceleration, and studied possible consequences of such an
|
||
|
||
in Relativistic Physics
|
||
|
||
21
|
||
|
||
assumption. The next step in this direction consists in considering a more general situation where (1) holds, but on the right hand side of (5), instead of the linear expression used in special relativity, we are going to investigate a nonlinear, strongly increasing function. We assume this function to be the same in both directions and form-invariant with respect to the choice of a class of reference frames called inertial frames. Introducing distance in such a way, we allow the communication signal to move nonuniformly, always in the forward direction and with all properties of its motion independent on the reference frame as long as it is an inertial one. Within such a model the reflection can not be described by geometrical optics. To be consistent with the assumption above we must exclude classical reflection which preserves instant velocity, and replace it by the process when the mirror at first absorbs the incoming signal and then emits a new one, always with the same initial velocity. Within such a process it is also necessary to consider the possibility of delayed emission of the signal, which, as it is shown in [6], does not influence the form of space-time transformations.
|
||
We will derive generalized Lorentz transformation rules between inertial frames for new T and X basing our considerations on the principles of Bondi’s k-coefficient method [7]. For the times T1 and T2 which are the only directly measurable quantities in radiolocation method their transformation rules when one passes from one inertial reference frame to another read
|
||
|
||
T ’1 = kT1
|
||
|
||
(6)
|
||
|
||
T2′
|
||
|
||
=
|
||
|
||
1 k
|
||
|
||
T2
|
||
|
||
(7)
|
||
|
||
Because the function f in (5) is strongly monotonic we can invert the relation
|
||
|
||
(5) and next use it, together with (1), in order to express T1 and T2 uniquely in terms of T and f–1(X). In this case, the transformation rules (6) and (7) contain
|
||
enough information to write down the transformation rules for T and f–1(X)
|
||
|
||
with the form invariance of the latter explicitly taken into account. They are
|
||
|
||
c h c h a f k2 + 1 T − k2 − 1 f −1 X
|
||
|
||
T′ =
|
||
|
||
(8)
|
||
|
||
2k
|
||
|
||
a f c h a f c h f −1 X′ = k2 + 1 f −1 X − k2 − 1 T
|
||
|
||
(9)
|
||
|
||
2k
|
||
|
||
which, if needed, may be rewritten as
|
||
|
||
F c h a f c h I X′ = f k2 + 1 f −1 X − k2 − 1 T HG KJ 2k
|
||
|
||
(10)
|
||
|
||
The physical meaning of reference frames connected by transformations
|
||
|
||
(6)−(7) or (8)−(9),(10) may be found if one looks for the motion of the origin of
|
||
|
||
the primed frame X’ = 0 observed in the unprimed frame. If the physically
|
||
|
||
obvious relation
|
||
|
||
f(0) = f–1(0) = 0
|
||
|
||
(11)
|
||
|
||
22
|
||
|
||
Open Questions
|
||
|
||
is satisfied then we immediately obtain its trajectory
|
||
|
||
Xatf
|
||
|
||
=
|
||
|
||
f
|
||
|
||
FHG
|
||
|
||
k2 k2
|
||
|
||
− +
|
||
|
||
1 1
|
||
|
||
tIKJ
|
||
|
||
(12)
|
||
|
||
which means that within the proposed model all inertial reference frames move nonuniformly with respect to another. The characteristics of their motion are defined by those of synchronizing signal scaled by factors
|
||
c h k 2 − 1 k 2 + 1 N respectively. Absolute values of all these factors are less than
|
||
|
||
1 which means that characteristics of the synchronizing signal take the biggest values allowed in the model. This generalizes the case of special relativity theory where inertial frames move uniformly with the velocity v connected to k of (6) and (7) by
|
||
|
||
v c
|
||
|
||
=
|
||
|
||
k2 k2
|
||
|
||
−1 +1
|
||
|
||
(13)
|
||
|
||
The transformation rules in the form (8) and (9) possess the same
|
||
|
||
structure as the usual Lorentz transformations, and they become these
|
||
|
||
transformations for f(X) as an identity function. We can therefore ask what are
|
||
|
||
the forms of the time dilation and the length contraction in the model
|
||
|
||
proposed, and compare them with the standard ones following from the
|
||
|
||
ordinary Lorentz transformations. If (11) holds then the formula (8) implies
|
||
|
||
the same relation between the time intervals measured by moving and resting
|
||
|
||
clocks as the special relativity does. It reads
|
||
|
||
ΔT ′ = k 2 + 1 ΔT
|
||
|
||
(14)
|
||
|
||
2 k
|
||
|
||
The analogy of the Lorentz-Fitzgerald contraction of moving rods may be derived from (8) and (9) according to the standard construction. The difference of time intervals passed between emission and detection of signals
|
||
|
||
seen in a moving reference frame as reflected simultaneously at the ends of the rod is
|
||
|
||
a f a f a f a f f −1
|
||
|
||
X 2′
|
||
|
||
− f −1
|
||
|
||
X1′
|
||
|
||
=
|
||
|
||
2k k2 +
|
||
|
||
1
|
||
|
||
f −1 X 2
|
||
|
||
− f −1 X1
|
||
|
||
(15)
|
||
|
||
with primed quantities denoting results of measurements obtained by a
|
||
|
||
moving observer. It is clear that
|
||
|
||
f–1(X’2) – f–1(X’1) ≤ f–1(X2) – f–1(X1)
|
||
|
||
(16)
|
||
|
||
c h because of the factor 2k k 2 + 1 which implies that moving observers see
|
||
|
||
rods being shorter. However (15) does not determine L’ = X’2 − X’1 uniquely if distances are not proportional to time intervals. This points out the case of special relativity theory, where in order to calculate length defined as above, we do not need to know an instant of time when the observation has been done because a light pulse has a velocity c wherever it reaches a rod. For linear f’s there is always
|
||
|
||
in Relativistic Physics
|
||
|
||
23
|
||
|
||
L′
|
||
|
||
=
|
||
|
||
2k k2 +1
|
||
|
||
L
|
||
|
||
(17)
|
||
|
||
which we understand as a consequence of a particularly chosen definition of
|
||
|
||
length being additional assumption in the radiolocation method.
|
||
|
||
Conclusions
|
||
Einstein’s synchronization which leads to the definition of time shown by a distant clock in a form (1) has been discussed from many years and we understand now that it is a consequence of a priori assumed invariance of oneway velocity of light. This is an attractive hypothesis because it enables us to synchronize all clocks with respect to only one chosen clock, but in fact it never has been checked as precisely as other fundamental assumptions of physics. The main problem is that in order to make such a measurement we must not perform an experiment which uses distant clocks being synchronized in advance according to Einstein’s method and we should know the distance between them given independently from (2) which assumes the properties of the light motion. Such an experiment is not easy to imagine, and is sometimes even considered to be impossible, but there exist physical phenomena which known explanations forbid from replacing oneway velocity of light by an accurately measured two-way velocity and suggest that the former is frame dependent.
|
||
Our consideration shows that not only the clock synchronization formula (1) may be treated as a matter of convention. Also coordinatization of space dimensions may be different from (2) provided the definitions used are independent from the choice of the reference frame. It is possible to do this in agreement with the transformation rules (6) and (7) which give mutual relations between inertial reference frames and reflect the physically observed Doppler effect. The proposed approach enlarges the class of reference frames considered as inertial ones and connects their definition to the properties of the signal used for synchronization. The parameters of its motion give limits for allowed velocities, accelerations and other properties of any motion generated by higher order derivatives, which agrees with physical intuition and our experience from special relativity. Elimination of the definition of inertial reference frames as those which move only uniformly may be also useful in considerations where the presence of the gravitational field must be taken into account, or it is necessary to pass to rotating reference frames.
|
||
Acknowledgments
|
||
The author is very grateful to the Organizing Committee of the conference “Relativistic Physics and Some of Its Applications” for the support which made possible his participation in the conference. He also expresses his gratitude to Prof. E. Kapuscik for discussions and fruitful remarks as well as for information on results presented in reference [6].
|
||
|
||
24
|
||
|
||
Open Questions
|
||
|
||
References
|
||
[1] E. Kapuscik, Acta Phys. Pol. B17, 569 (1986). [2] F. Selleri, Theories equivalent to special relativity, in Frontiers of Fundamental
|
||
Physics, pp. 181-192, M. Barone & F. Selleri eds. (Plenum Press, London/New York, 1994); also Found. Phys. 26, 641 (1996). [3] A. Einstein, Ann. Phys. (Germany) 17, 891 (1905). [4] H. Reichenbach, The Philosophy of Space & Time, (Dover Publ., New York, 1958). [5] A. Horzela, E. Kapuscik, J. Kempczynski, Phys. Essays 6, 314 (1992). [6] V. S. Barashenkov, E. Kapuscik, M. V. Liablin, Nature of Relativistic Effects and Delayed Clock Synchronization, this volume. [7] H. Bondi, Brandeis Lectures, 1964, vol. 1, (Englewood Cliffs, NJ: Prentice Hall), p. 386.
|
||
|
||
Synchronisation of Clock-Stations and the Sagnac Effect
|
||
|
||
A. G. Kelly HDS Energy Ltd. Celbridge, Co. Kildare Ireland 1
|
||
|
||
It is shown that the Sagnac correction, as applied to time comparisons upon the Earth, does not derive from the normal Relativistic corrections. It is proposed that the reason given for the application of the Sagnac correction, and the circumstances appropriate to its application, require amendment. Key words:: Clock synchronisation; Sagnac effect; Relativistic corrections.
|
||
|
||
Standards for the synchronisation of clock-stations upon the Earth are to be found in the 1990 publication of the CCIR (International Radio Consultative Committee: International Telecommunication Union) [1]. Similar rules are in the 1980 publication of the CCDS (Comité Consultatif Pour la Définition de la Seconde: Bureau International des Poids et Mesures) [2]. Two methods are used to synchronise clocks at different clock stations. The first method is physically to transport a clock from one site to the other, and thereby to compare the times recorded at the two clock stations. The second method is to send an electromagnetic signal, from one site to the other.
|
||
Three corrections to be applied, as listed in the above publications, are as follows:(a) to take account of the Special Relativistic velocity effect, caused by
|
||
carrying a portable clock at speed aboard an aeroplane, from one site to the other. (b) under General Relativity, to allow for height above sea level. (c) a correction described as being for the rotation of the earth. Correction (a) is quantified as v2/2c2. This is the slowing of time as calculated under the Special Theory of Relativity. A clock transported from one site to another will have such a correction applied, because of the ground speed v of the aeroplane; c is the velocity of light. Correction (b) is quantified
|
||
b g as g φ h c 2 where g is the total acceleration at sea level (gravitational cum
|
||
|
||
centrifugal) at a latitude of φ, and h is the height over sea level.Correction (c) is
|
||
|
||
Open Questions in Relativistic Physics
|
||
|
||
Edited by Franco Selleri (Apeiron, Montreal, 1998)
|
||
|
||
25
|
||
|
||
26
|
||
|
||
Open Questions
|
||
|
||
quantified as 2AE ω c 2 , where AE
|
||
|
||
is the equatorial projection of the
|
||
|
||
area enclosed by the path of
|
||
|
||
travel of the clock being
|
||
|
||
transported from one site to
|
||
|
||
another (or of the electromagnetic
|
||
|
||
signal) and the lines connecting
|
||
|
||
the two clock-sites to the centre of
|
||
|
||
the Earth; ω is the angular
|
||
|
||
velocity of the Earth. As the area
|
||
|
||
AE is swept, it is taken as positive
|
||
|
||
when the projection of the path
|
||
|
||
of the clock (or signal), on to the
|
||
|
||
Figure 1 - Sagnac Test
|
||
|
||
equatorial plane, is Eastward.
|
||
|
||
Both reports include all three
|
||
|
||
terms under the umbrella
|
||
|
||
description of being “of the first order of general relativity.” The first two
|
||
|
||
corrections are clearly the result of the Special Theory and the General Theory
|
||
|
||
of Relativity respectively. But, what is the third? This paper examines the
|
||
|
||
precise meaning and derivation of the third correction.
|
||
|
||
To understand the meaning of the third term, we must study the Sagnac
|
||
|
||
effect. Sagnac (1914) showed that light took different times to traverse a path,
|
||
|
||
in opposite directions, upon a spinning disc [3]. Figure 1 shows a schematic
|
||
|
||
representation of the test that was done by Sagnac. A source at A sends light
|
||
|
||
to a half-silvered mirror at C. Some of the light goes from C to D, E, F and C
|
||
|
||
and is reflected to a photographic plate at B. Some of the light goes the other
|
||
|
||
way around. The whole apparatus (including A and B), can turn with an
|
||
|
||
angular velocity of ω.When the apparatus is set spinning, a fringe shift occurs
|
||
|
||
at an interferometer, indicating a difference (dt) in the time taken by the light
|
||
|
||
to traverse the path in opposite directions. For this difference in time Sagnac
|
||
|
||
derived the formula
|
||
|
||
δt
|
||
|
||
=
|
||
|
||
4Aω c2
|
||
|
||
(1)
|
||
|
||
where A is the area enclosed by the path of the light signals, and ω is the
|
||
|
||
angular velocity of spin in R/s.
|
||
|
||
Sagnac also showed that the centre of rotation can be away from the
|
||
|
||
geometric centre of the apparatus, without affecting the results, and that the
|
||
|
||
shape of the circuit was immaterial. He also proved that the tilting of the
|
||
|
||
mirrors, as they spin, caused an insignificant alteration in the overall effect.
|
||
|
||
In order to derive the Sagnac equation, consider the theoretical circular
|
||
|
||
model shown in Figure 2. Light is emitted at S; a portion of the signal goes
|
||
|
||
clockwise (denoted by the inner line), and some goes anti clockwise, around a
|
||
|
||
circular disc of radius r. The light source at S and the photographic recorder,
|
||
|
||
also situated at S, rotate with the disc. The disc is rotating with an angular
|
||
|
||
in Relativistic Physics
|
||
|
||
27
|
||
|
||
velocity ω in a clockwise direction. The anti clockwise beam is going against the rotation of the equipment, and will return to Point S when it has moved to S’. The second beam, travelling clockwise, will return when S has moved to S.” As viewed by an observer on the spinning platform, the light signals return to the same point, but at different times.
|
||
Taking to as the time observed when the disc is stationary, i.e. the path length divided by the speed of light
|
||
|
||
to
|
||
|
||
=
|
||
|
||
2π r c
|
||
|
||
(2)
|
||
|
||
Let δ s’ be the distance SS’ and δ s” be the distance SS.” Let t’ be the time for
|
||
|
||
the light to go from S to S’ in the anti clockwise direction.
|
||
|
||
t′ = 2πr − δs′
|
||
|
||
(3)
|
||
|
||
c
|
||
|
||
But, t’ is also the time taken for the disc to move the distance δs’ in the
|
||
b g clockwise direction. Therefore t′ = δs′ v , and δs′ = t′v ; δs′ = 2πr − δs′ v c ;
|
||
|
||
a f δs′ v = 2πr c + v , and
|
||
|
||
a f t′ = 2πr c+v
|
||
|
||
(4)
|
||
|
||
Similar calculations give the time (t”) for the light to go from S to S” in a
|
||
|
||
clockwise direction,
|
||
|
||
a f t′′ = 2πr c−v
|
||
|
||
(5)
|
||
|
||
Subtracting equation (4) from (5), the difference (δt) between the times for the
|
||
|
||
light to go clockwise (t”) and anti clockwise (t’) is
|
||
|
||
LM OP δt = 2πr 1 − 1 = 4πrv
|
||
|
||
Na f a fQ c h c − v c + v
|
||
|
||
c2 − v2
|
||
|
||
(6)
|
||
|
||
This is the same as equation (1), because v2 is negligible. From the point of view of the observer in the fixed laboratory the disc moves a distance δ s’ while the light completes a distance of 2π r − δ s′ around
|
||
|
||
in the other direction from S to S’. Equation (3) describes the time interval, as it would be discerned by the observer in the laboratory. From the point of view of the moving observer, upon the spinning disc, the light has, relative to that observer, completed one revolution of the disc (2π r) at velocities of c ± v in the two opposing directions. Equations (4) and (5) describe this.
|
||
|
||
28
|
||
|
||
Open Questions
|
||
|
||
In the above calculation, the light
|
||
|
||
is assumed to travel at a constant
|
||
|
||
velocity of c in relation to the fixed
|
||
|
||
laboratory. But, the fringe shift
|
||
|
||
measured solely aboard the spinning
|
||
|
||
disc, and which is a record of the time
|
||
|
||
difference for the light beams to
|
||
|
||
complete a circuit in opposing
|
||
|
||
directions, corresponds exactly to the
|
||
|
||
time difference in equation (1). How
|
||
|
||
can this be? The only possible
|
||
|
||
explanation is that the time in the
|
||
|
||
Figure 2 - Circular Sagnac Test: Whole apparatus turning at ω clockwise
|
||
|
||
fixed laboratory, and that upon the spinning disc are precisely the same. This fact is at the core of the postulates
|
||
|
||
being put forward in this paper.
|
||
|
||
The Sagnac effect shows that the velocity of the light is not affected by
|
||
|
||
the movement of the source of that light (Point S); this accords with Special
|
||
|
||
Relativity theory. It also shows that the light travels at the velocity c solely
|
||
|
||
relative to the laboratory. Assuming that the light travels at the velocity c,
|
||
|
||
relative to the laboratory, gives the correct result. The light does not adapt to
|
||
|
||
the movement of the disc.
|
||
|
||
To get a fringe shift of one fringe, the velocity of Point S in Figure 2,
|
||
|
||
relative to the laboratory, has to be about 13 m/s per meter of radius. This is a
|
||
|
||
very low velocity. Fringe shift is got from time difference by multiplying by
|
||
|
||
c/λ. Where, for example, λ = 5500 × 10–10 m, this gives v = 13 m/s, per meter of
|
||
|
||
radius, from 1 = (4Aω)/(cλ) = (4π rv)/(3 × 108 × 5,500 × 10–10). In equation (4), as
|
||
|
||
v approaches c, t’ becomes to/2, and the speed relative to the observer is now 2c. In equation (5), as the speed v approaches c, t” becomes infinite, because
|
||
|
||
the light and the Point S are travelling in the same direction, and the time for
|
||
|
||
the light signal to gain one complete circuit on the Point S is infinite; the
|
||
|
||
speed of the light, relative to the
|
||
|
||
observer, becomes zero.
|
||
|
||
Dufour & Prunier (1942)
|
||
|
||
repeated the Sagnac test, and got
|
||
|
||
the same result [4]. They then
|
||
|
||
did a variation of that test. A
|
||
|
||
practical example of a case
|
||
|
||
where the signal is not solely in
|
||
|
||
the plane of the disc is their test,
|
||
|
||
in which the path of the light
|
||
|
||
was partly on the spinning disc,
|
||
|
||
and partly in the fixed
|
||
|
||
laboratory. The light signal was
|
||
|
||
introduced (Figure 3) from C out
|
||
|
||
Figure 3 - Dufour & Prunier Test
|
||
|
||
in Relativistic Physics
|
||
|
||
29
|
||
|
||
to Point 1, and sent from there in opposite directions. As shown schematically, the light went firstly on a path on the spinning
|
||
disc (Point 1 to Point 2), then went vertically up to a mirror fixed to the laboratory overhead the disc (Point 3). It then traversed linear paths 3 to 4 to 5 in the fixed laboratory, and came vertically back down to the disc at Point 6, whereupon it finished the trajectory on the disc back to the starting point at Point 1. The reverse beam went the other way. The plane of the path, of the portion that was fixed in the laboratory, was parallel to the plane of the disc. Lines 3-4 and 4-5 are directly overhead 2-C and C-6. The two short connections, 2-3 and 5-6 (shown exaggerated here for clarity) were 10 cm each. The mirrors at 2 & 6 rotated with the disc. The fringe shifts were the same as in their repeat of a test with the light path solely upon the spinning disc (on the circuit 1-2-C-6-1). This test by Dufour & Prunier confirms that the light does not adapt to the movement of the disc, and that it is travelling relative to the fixed laboratory.
|
||
A young German student Harress (1911) had done a test on the refraction of light [5]. This test was later shown by von Laue (1920) to have produced the Sagnac effect, but Harress was not aware of this [6]. Harress had both the photographic equipment and the light source fixed in the laboratory, whereas Sagnac had both on the spinning disc. This shows that the photographic record of the fringe shift and/or the origin of the light may be made on or off the disc, without affecting the result; this is because it is the behaviour of the light relative to the spinning disc that is being measured. Dufour & Prunier also did tests with the light source fixed in the laboratory and with the photographic plate fixed in the laboratory; the results were the same as in a traditional Sagnac test. The fringe shift occurs, whether there is any observer (camera) present on the disc, or in the fixed laboratory. There is a slight Doppler effect in the case where the photographic equipment is in the fixed laboratory, because the disc is moving past the viewing lens. Post (1967) discusses the magnitude of the distortion introduced, and correctly dismisses the effect as too small to have any observable effect, being “v/c times smaller than the effect one wants to observe.” [7].
|
||
Michelson & Gale (1925) showed that electromagnetic signals sent around the Earth did not travel at the same speed in the East-West direction [8]. They constructed a large rectangular piping system fixed to the Earth, and sent light signals in opposite directions around the circuit. The signals did not arrive back at the same time, as evidenced by the resulting fringe shift. That test was a Sagnac test on a disc of radius equal to that of the Earth at the Latitude concerned, and rotating at the angular velocity of the Earth. The results were within 3% of the forecast and were also in the correct direction (signal retarded in the direction of the spin of the Earth). Tests by Bilger et al. (1995) using a ring-laser, confirmed the Sagnac effect to better than one part in 1020. This was a Michelson & Gale type test with the ring laser fixed to the Earth; the retardation of the signal was also in the direction of the spin of the
|
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|
||
30
|
||
|
||
Open Questions
|
||
|
||
Earth (as this was done in the Southern hemisphere, the retardation was in the opposite sense to the Michelson & Gale test)[9].
|
||
Saburi et al. (1976) transported a clock from Washington (USA) to Tokyo (Japan), and compared the difference in the time displayed by the two clocks on the arrival of the transported clock, with the time relayed from one station to the other, via an electromagnetic signal.[10] The two sites were almost at the same latitude. They calculated from the Sagnac effect that there should be a difference of +0.333 ms (Japan ahead of Washington, DC, because of the direction of rotation of the Earth). The Sagnac correction, on its own, applied solely to the electromagnetic signal (and not to the time displayed by the clock that was physically transported from one site to the other), bridged the gap to a very close agreement with the test results (to –0.02 μs). The Relativistic effects applied solely to the portable clock, which was physically transported from one site to the other, amounted to +0.08 μs. The uncertainty of the reading being recorded by the portable clock was ±0.2 μs. This test could nowadays be repeated to greater accuracy.
|
||
Special Relativity has no role in trying to explain the Sagnac effect. Post (1967) states that the Sagnac effect and the Special Relativity effect are of very different orders of magnitude. He says that the alteration to be applied to the Sagnac effect under Special Relativity is a v2/c2 effect which is “indistinguishable with presently available equipment” and “is still one order smaller than the Doppler correction, which occurs when observing fringe shifts.” Post derives the Sagnac formula as given above in equation (1) and then applies the Special Relativity γ factor to that formula; in this he distinguishes clearly between the two. Post says that “for all practical purposes we may accept as adequate for the time interval in the stationary as well as in the rotating frame, the formula” as in equation (1). This confirms that the difference in the time recorded in a Sagnac test is the same in the laboratory and upon the spinning disc. Post also says that “the time interval between the consecutive positions of the beam splitter is observed in the stationary frame and is therefore dilated by a factor γ .” Here again Post distinguishes between the Sagnac effect and the Relativistic time dilation.
|
||
The basis of timekeeping by the CCIR is time at the non-rotating centre of the Earth. It defines that the “TAI is a coordinate time scale defined at a geocentric datum line.” The unit of time is defined as “one SI second as obtained on the geoid in rotation.” The time scale and the unit of time are not measured at the same place; the unit of time is based upon the spinning Earth, which has motion in relation to the geocentre where the time scale is measured. The CCIR report recommends that “for terrestrial use a topocentric frame be chosen.” It continues “when a clock B is synchronised with a clock A (both clocks being stationary on the Earth) by a radio signal travelling from A to B, these two clocks differ in coordinate time by” the Sagnac effect. These statements make it clear that the time upon the rotating Earth is viewed as differing from that at the geocentre. This assumption is in contradiction of the analysis in this paper, and of the conclusions of Post [7].
|
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|
||
in Relativistic Physics
|
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31
|
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|
||
The CCIR report states that “the time of a clock carried eastward around the earth at infinitely low speed at h = 0 at the equator will differ from a clock remaining at rest by –207.4 ns.” That amount is the Sagnac one-way effect. The significance of the h = 0 is that there would be no effect under the General Theory of Relativity. The infinitely low speed eliminates any effect from the Theory of Special Relativity. The CCIR report here assumes that when a clock is physically transported around the globe, a Sagnac-type correction has to be applied. Because the area is taken as “positive if the path is traversed in a clockwise sense as viewed from the South Pole,” a clock transported around the Earth in a Westward direction would gain time by +207.4 ns, relative to the stationary clock. Consider two clocks that are sent, in opposite directions, around the globe at the equator at the same time; when they have completed one revolution each, there would be a supposed time difference of 414.8 ns between them, and they would each differ from a clock that remained at the starting place by 207.4 ns. They have had no effect from Special Relativity (velocity infinitely slow) or from General Relativity (at sea level). We then would have the strange situation where we have three clocks at the same spot on the Earth recording different times; we could repeat the circumnavigation as often as we wish and get clocks, at the same spot, which have had zero corrections under normal Relativity theory, recording times which are different from each other by larger and larger amounts. All times here are coordinate times as earlier defined.
|
||
Both the CCIR and CCDS reports make it clear that considering time upon the Earth, from the point of view of “a geocentric non-rotating local inertial frame,” requires no Sagnac correction. But, when considering time upon the rotating Earth, they apply a Sagnac correction.. Langevin (1937) proposed that, to explain the Sagnac effect, one had to assume that either (a) the velocity of the signal was c ± v in the two directions or, (b) the time aboard the spinning disc was altered by 2Aω/c2 [11]. The CCIR and CCDS reports assume that (b) is true. As we saw above, it is (a) that is the correct explanation.
|
||
Special Relativistic time dilation does nor contribute very much towards the Sagnac effect. Taking an example, where the surface velocity of the Earth at a particular latitude is v = 300 m/s and a portable clock is transported at, say, x = 10 m/s (the CCIR defines the transportation as”slowly”). In this case
|
||
a f the difference between the v2 2c 2 and v + x 2 2c 2 , which is
|
||
c h 2vx + x2 2c 2 , gives a difference of 4 × 10–14 s/s. An electromagnetic signal
|
||
circumnavigates the Earth in about 0.1 s. The Sagnac one-way difference 2Aω/c2 for a light signal to circumnavigate the Earth is about 2 × 10–7s/s, as calculated in the CCIR Report. The ratio of the two is thus 107. Thus, the two effects are not at all of the same magnitude. This agrees with the analysis by Post [7]. Another basic difference between the Relativistic and Sagnac effects, as calculated for movements measured upon the spinning Earth, is that the former is non-directional, whereas the latter is ± depending upon the
|
||
|
||
32
|
||
|
||
Open Questions
|
||
|
||
direction of sending the signal West or East respectively, and zero in a NorthSouth direction.
|
||
In the CCIR analysis, the starting point is time at the “local non-rotating geocentric reference frame.” This is done “to account for relativistic effects in a selfconsistent manner.” If we assume that the speed of light upon the rotating Earth must be the constant value c, then perforce we must vary the time upon the Earth, by the Sagnac formula, as compared with time measured from the geocentric reference frame. Special Relativity theory is designed specifically to alter the time upon the moving object in direct accord with the requirement that the speed of light must be the constant value c.
|
||
The application of the γ factor correction, under Special Relativity, to time on the spinning Earth, as compared to time at the centre of the Earth, ensures that
|
||
|
||
a) the speed of light is c as calculated in all directions upon the spinning Earth
|
||
b) time upon the Earth is consequently calculated to vary by precisely the amount necessary to agree with the constant value for the speed of light.
|
||
|
||
If this were not so, the velocity of the electromagnetic signals upon the Earth would remain unchanged as c ± v in the opposing directions. This method arrives at a solution that conforms with Relativity theory. Is this method justifiable?
|
||
It is convenient to start with time at a geocentric datum line. This conforms with the fact that electromagnetic signals do not adapt to the spin of the Earth; this datum corresponds to the ‘laboratory’ in a Sagnac bench test. The speed of the signals can confidently be taken as c as measured in that frame of reference. The orbital movement of the Earth around the Sun. and its other movements in the Universe, can be ignored, and assumed to have no effect upon the results being calculated.
|
||
Allan et al., compare the Sagnac correction as applied to (a) slowly moving portable clocks upon the Earth and (b) electromagnetic signals, used for clock synchronisation [12]. They state that “the Sagnac effect has the same form and magnitude whether slowly moving portable clocks or electromagnetic signals are used to complete the circuit.” They say that the Sagnac correction applies in both cases, and that it has the same magnitude. In case (a) they define the Sagnac effect as “being due to a difference between the second-order Doppler shift (time dilation) of the portable clock and that of the master clock whose motion is due to the Earth’s motion” as “viewed from a local nonrotating geocentric frame.” Petit & Wolf also state that the correction 2Aω/c2 is applied equally “if the two clocks are compared by using portable clocks or electromagnetic signals in the rotating frame of the Earth.” If we take the time measured at the geocentric datum line as to, and the time upon the spinning Earth as t’, Special Relativity Theory requires that to = t’γ. Applying a correction of v2/2c2 to the time taken for two clocks, which move at speeds of v relative to the ground, to circumnavigate the Earth in opposing directions, as viewed from the geocentre, gives the following result.
|
||
|
||
in Relativistic Physics
|
||
|
||
33
|
||
|
||
The moving clocks have the speeds of ω r + v and ω r – v in the opposing
|
||
|
||
directions, relative to the geocentric time frame; r is the radius of the Earth,
|
||
|
||
and ω its angular velocity. The time dilations of the two clocks are
|
||
|
||
z b g z b g 1 2
|
||
|
||
ωr+v
|
||
|
||
2
|
||
c dt ,
|
||
|
||
and
|
||
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||
1 2
|
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|
||
ωr−v
|
||
|
||
2
|
||
c dt
|
||
|
||
respectively. The difference
|
||
|
||
z b g between these two time dilations is therefore 2ω rv c2 dt .
|
||
|
||
When the two clocks have gone right around the equator (a distance of
|
||
z 2π r) the cdt = 2πr , and the difference between the time dilations is
|
||
|
||
(4π r2ω)/(c2), which is the same as equation (1) (Burt, 1973) [13]. The result is independent of v, so the speed of transportation of the clocks will not affect this result. A similar analysis using electromagnetic signals to circumnavigate the globe Eastward and Westward (that is substituting c for v in the above equations) also gives the same result. In this way this analysis gives the Sagnac formula as a supposed correction for the difference in the time taken by two electromagnetic signals sent in opposing directions around the globe, or for the time correction to be applied to clocks that are physically transported around the globe in an East-West direction. This is the correction published in the CCIR and CCDS reports. It also shows that the application of the γ factor to time as measured upon a moving object agrees with the speed of light being measured upon that object as c in all directions. All of this scheme is consistent.
|
||
There is one problem. The Sagnac tests, done with ever increased accuracy down the years, show a difference in the time taken by electromagnetic signals to circumnavigate any spinning disc (including a cross-section of the Earth) and consequently a difference in the speed of the signal in the opposing directions. No difference in time for activities upon the spinning disc is required, when viewed from the stationary laboratory. This difference in the speed of the signal contradicts a basic assumption of the scheme of synchronisation that is used.
|
||
It is assumed by the CCIR that the time upon the spinning Earth is altered by the γ factor of Special Relativity in all calculations carried out on time durations upon the Earth, from the viewpoint of the geocentric nonrotating system. If no difference in time was measured in a Sagnac test, then the speed of the signal would have been measured as c in the opposing directions, upon the spinning disc. It can be argued that the rotating disc is not an Inertial Frame, and that therefore the matter is not relevant. As larger and larger discs are considered, we approach the situation where the movement is tantamount to that in a straight line at constant velocity. In this case the matter applies to an Inertial frame. There does not seem to be any plausible solution which shows the fringe shift measured aboard the spinning disc to be caused by other than a difference in the speed of light relative to that disc.
|
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||
34
|
||
|
||
Open Questions
|
||
|
||
The application of the CCIR correction to time upon the spinning Earth gives a correct answer, whenever electromagnetic signals are used to compare the time being recorded at two clock stations upon the Earth. However, where the physical transportation of a clock around the globe is concerned, it introduces an error.. The CCIR report works out an example where the three specified corrections are applied to the frequency of a clock that is physically transported from one site to another. However, as discussed earlier, Saburi et al. showed that the physical transportation of a clock does not require the application of any Sagnac correction to the time being recorded upon that travelling clock [10]. They also confirmed that it is the electromagnetic signal speed Eastward and Westward that varies, and that requires a Sagnac correction to its speed of transmission.
|
||
Allan et al. (1985) did a Sagnac-type test between standard time-keeping stations in USA, Germany and Japan [12]. These tests confirm the Sagnac effect, as applied to electromagnetic signals, sent right around the Earth in opposing directions, to an accuracy of 1% over a period of 3 months. There were no further corrections made to the results (got by sending electromagnetic signals between the clock-stations) on the basis of Special Relativity or General Relativity; in this case, where electromagnetic signals are used to synchronise the clock stations, no measurable effect under Special Relativity or General Relativity are to be expected. Saburi et al. state that “in a comparison experiment via a satellite, it is considered that the effect of gravitational potential on the light path is small and cancelled out by the two-way method, and that other relativistic effects are negligibly small.”
|
||
In the CCIR report, the corrections to be applied are listed as three viz. “the corrections for difference in gravitational potential and velocity and for the rotation of the Earth.” In describing these corrections the report names them as “corrections of the first order of general relativity.” We now see that the third one is the Sagnac effect (2Aω/c2). By naming the Sagnac correction as a separate item from the other two factors, the CCIR report tacitly accepts that it is not a Special Relativity or a General Relativity effect. This paper shows that no such Sagnac correction should be applied to the case where a clock is physically transported from one site to another. However, in all cases of synchronising clocks by electromagnetic signal comparison, the Sagnac correction is properly quantified in the CCIR report, and thus the timekeeping authorities are applying it correctly, even if they assume that it is derived from the Theory of General Relativity. The Sagnac correction is nowadays automatically applied to all electromagnetic signals used in the synchronisation of clock stations. The CCIR report gives an incorrect value for the angular velocity of the Earth (7.992 R/s instead of 7.292 R/s); this error was not carried forward into the calculations given in examples in the report.
|
||
Winkler (1991), in a paper on the subject of the synchronisation of clocks around the world, ascribed the Sagnac effect to the General Theory of Relativity [14]. He explained the effect by saying that “accelerations have an effect on timekeeping and on the propagation of light.” He also stated that “on a
|
||
|
||
in Relativistic Physics
|
||
|
||
35
|
||
|
||
rotating system, the velocity of light must be added to (or subtracted from) the speed due to rotation, an effect that produces a time difference for two rays that travel in opposite directions around a closed path.” Here he has accepted that the velocity of the signal is different in the opposing directions, and that the signals take different times to complete the circuit, relative to an observer upon the rotating Earth.
|
||
Other publications also purport to show that the Sagnac effect is part of the General Relativity Theory. An example is the paper by Petit & Wolf (1994), which begins by assuming that the light travels relative to the stationary frame (in their case the “geocentric ‘non-rotating frame’”) [15]. They assume that the light velocity relative to the spinning object is not c. They take it as “c + s where s represents the time taken for the signal to travel the extra path due to the motion of b in the non-rotating frame during transmission”: “b” is the clock moving on a rotating disc. This is the same as the analysis of the Sagnac effect, given earlier in this paper, where the extra distance travelled by the Point S in Figure 2, while the signal is travelling around the circuit, yields a speed of the light of c – v in one direction. But, they then assume that the time aboard the spinning Earth alters by the equivalent of the Sagnac effect; this is, as seen above, not sustainable.
|
||
Two clocks upon the Earth at the same Latitude have no relative motion in respect to each other, as considered in a geocentric Earth-fixed system. It is only when we attempt to compare the time being kept by the two clocks that we have to employ either an electromagnetic signal or a physical transportation of a comparison clock. The time keeping of those two clocks does not alter because of the measuring process. The Sagnac correction has to be applied to the time taken by the electromagnetic signal to get from one clock site to the other. No corrections apply to the time being kept by the clocks in relation to each other. By shifting the time base to the geocentre, the CCIR introduce a supposed Sagnac effect alteration to the time difference measured between the two clocks when transporting a portable clock or sending an electromagnetic signal between the two sites.
|
||
There are various reasons that can be advanced to answer the apparent contradictions between Relativistic theory and the Sagnac effect. One could say that it is correct to state that the Sagnac effect is not relativistic; but it comes out naturally if one writes the equations of time transfer, from the geocentric frame to the spinning Earth, in the context of general relativity, with some very small additional terms that are genuinely relativistic. It can be claimed that Newtonian Mechanics are not relativistic, but that General Relativity includes all terms of Newtonian theories of motion plus additional corrections. So we could claim that it is not wrong to say that the Sagnac effect is also relativistic in the sense that it also appears in the solution in a general relativity theory. Such an argument would agree that the Sagnac effect is a first order effect that cannot have any explanation purely by Relativistic theory.
|
||
|
||
36
|
||
|
||
Open Questions
|
||
|
||
It could be debated that we have to (a) adopt a relativistic model, because the classical treatment leads to contradictions with experiment, and (b) have a convention for the meaning of clock comparison. As a model, we use Einstein’s General Relativity because this theory is the simplest which, up to now, agrees with all observed facts. The convention for clock comparison is based on the convention of coordinate simultaneity; the readings of the clocks take place at the same value of some specified coordinate time (geocentric in metrology on the Earth). The question, it could be claimed, is not to distinguish in the theory of clock comparison some classical terms, some terms due to Special Relativity, and some gravitational terms. General Relativity, it can be said, is a self-contained theory and provides all the terms we need, as a consequence of its basic postulates. The separation of the various terms is a consequence of the choice of coordinates we have made and of the low level of approximation which is accepted.
|
||
General Relativity theory is required to make corrections to the time keeping of the clocks. It includes the corrections for height over sea level, and also the corrections under Special Relativity (velocity effects). The setting of the atomic clocks that are to be placed aboard a satellite are made, in advance of launching the satellite, to allow for both of those corrections. These corrections anticipate the increased reading that will emerge as a result of height over sea level, and also the decreased reading that will emerge because of the higher velocity of the satellite as compared with the velocity of the surface of the Earth. The clocks are set before launch, and will then be correct in keeping time the same as upon the surface of the Earth, when they are in orbit. These alterations are appreciable, and are a precise confirmation of these two corrections. Without making these corrections, the clock on the satellite would not keep an unaltered time, as compared with a clock upon the surface of the Earth.
|
||
However, there is another correction to be made and that is the Sagnac correction, whenever one has to compare the time upon such a satellite with the time being recorded by a clock on another satellite or upon the Earth. It is this quite separate correction that is the dichotomic problem being addressed here.
|
||
Some publications try to avoid the problem of the Sagnac effect by declaring that the Theory of Special Relativity is not applicable to a rotating Frame of Reference. But, some precise explanation of the effect is required. It is not sufficient to say that the Sagnac effect is not explained by Special Relativity theory, and to leave the matter at that. Einstein (1905) seems to have accepted, in his first paper on Relativity Theory, that movement on a circular path had the same result as movement in a straight line, when considering the question of measurement of distance or time.[16]. Having derived his formula for straight line movements, he said “it is at once apparent that this result still holds good if the clock moves from A to B in any polygonal line” and “if we assume that the result proved for a polygonal line is also valid for a continuously curved line, we arrive at this result: If one of two synchronous clocks at
|
||
|
||
in Relativistic Physics
|
||
|
||
37
|
||
|
||
A is moved in a closed curve with constant velocity until it returns to A, the journey lasting t seconds, then by the clock which has remained at rest the travelled clock on its arrival at A will be ½tv2/c2 second slow.” An observer riding upon the moving clock B will not be measuring time in an Inertial Frame with respect to clock A, but in a Rotating Frame of Reference. The argument that Special Relativity Theory is not applicable to movement in a circuit, such as that of circumnavigation of the Earth, is thus open to different interpretations. Even though the effect Einstein described is much smaller than the Sagnac effect (as shown above), it is the application, from a straight path to a curved path, that is of interest here.
|
||
The Sagnac corrections applied by the CCIR and the CCDS is not a Relativistic correction. It is not a continuing correction, such as are the Relativistic corrections. It is necessary when comparing the time being recorded at different clock stations, because the velocity of electromagnetic signals, travelling in an East-West direction, as measured upon the Earth, is not constant, but c ± v, where v is the spin velocity of the surface of the Earth at the particular Latitude. The outstanding problem is to devise a theory that will fit both the Relativistic corrections, that are vindicated in everyday use, and the Sagnac correction.
|
||
The Sagnac effect is proof that light travels at a constant velocity, in relation to the fixed laboratory, and does not adapt to the movement of a spinning disc. This requires that time aboard a spinning disc is the same as time in the fixed laboratory. The Sagnac correction is being correctly applied to the sending of electromagnetic signals between standard clock stations on the Earth; the reason given (relativistic correction) is incorrect. It is proposed that the Sagnac correction should not be applied to the physical transportation of clocks between sites, as is presently done in the CCIR and CCDS rules; it is solely the Relativistic corrections, due to velocity of travel and height over sea level, that should be applied in such a case.
|
||
The CCIR report concludes by saying that “additional definitions and conventions are under consideration.” These are awaited with interest. An amendment to relativistic theory to accommodate the true application of the Sagnac correction would give a more precise solution to the problem of clock synchronisation.
|
||
|
||
References
|
||
[1] CCIR Internat. Telecom. Union Annex to Vol. 7 1990, No 439-5, 150-4 [2] CCDS Bureau Internat. Poids et Mesures, 1980, 9th Sess., 14-17 [3] Sagnac M G J. de Phys. 1914, 4, 177-95 [4] Dufour A & Prunier F J. de Phys. 1942, 3, No 9, 153-61 [5] Harress F Thesis (Unpublished) Jena 1911 [6] von Laue M Ann. der Phys. 1920, 62, 448-63 [7] Post E J Rev. Mod. Phys. 1967, 39, No 2, 475-93 [8] Michelson A A & Gale H G Astroph. J.,1925, 61, 137-45 [9] Bilger H R et al. IEEE Trans. 1995, 44 No 2, 468-70
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||
|
||
38
|
||
|
||
Open Questions
|
||
|
||
[10] Saburi Y et al. IEEE Trans. 1976, IM25, 473-7 [11] Langevin P Compt. Rend. 1937, 205, 304-6 [12] Allan D W et al. Science 1985, 228, 69-70 [13] Burt E. G. C. Nature Phys. Sci. 1973, 242, 94-5 [14] Winkler G M R et al. Meterologia, 1970, 6, No 4, 126-33 [15] Petit G & Wolf P Astron Astrophys. 1994, 286, 971-4 [16] Einstein A, Lorentz H et al. The Principle of Relativity ( Metheun, 1923)
|
||
|
||
Is Simultaneity Relative or Absolute?
|
||
|
||
Joseph Lévy 4, Square Anatole France 91250 Saint-Germain les Corbeil France
|
||
Due to their conviction that the laws of nature must be identical in any inertial frame, the physicists of the beginning of the twentieth century were led to extend the relativity of Galileo to the electromagnetism of Maxwell, but this seemed to imply the abandonment of universal time and absolute simultaneity. In a previous paper1 we have criticised, from a logical viewpoint, the criteria intended to demonstrate the relativity of simultaneity and we have proposed replacing them by other criteria. According to these, the relativity of simultaneity was called into question.
|
||
We propose here a rigorous experimental method intended to verify the simultaneity of two events. By means of this device, we demonstrate that one can define an absolute simultaneity. The method also permits an exact synchronization of clocks. We then demonstrate that the relativity of time of Einstein’s theory must be discarded. On the other hand, the relativity principle appears as an approximation only, valid for bodies moving at low speeds relative to one another.
|
||
On the contrary, the slowing of the clocks moving with respect to the ether can be maintained. But this does not mean that Lorentz’s theory can be retained without change.
|
||
|
||
I. Position of the Problem
|
||
One of the concepts which has most drastically changed our vision of the world since the origin of philosophical thinking is the idea of “relativity of time.” Until the beginning of this century, time was considered as absolute, flowing uniformly, and identical for all observers.
|
||
Certainly, if one supposes that different stars are inhabited, one could suppose that the people living there use units of time different from our own,
|
||
|
||
1 J. Lévy, Some important questions regarding Lorentz Poincaré’s theory and Einstein’s
|
||
|
||
relativity II. Proceedings of the P.I.R.T. Conference, supplementary papers, Imperial College
|
||
|
||
London, 6-9 September 1996.
|
||
|
||
Open Questions in Relativistic Physics
|
||
|
||
Edited by Franco Selleri (Apeiron, Montreal, 1998)
|
||
|
||
39
|
||
|
||
40
|
||
|
||
Open Questions
|
||
|
||
adjusted to the rhythm of their central sun. Nevertheless, time by itself would not be affected. An appropriate conversion would be enough to come to an agreement.
|
||
Until the end of the nineteenth century this conception was the object of a large consensus. Better still, the idea that it could have been called into question looked deprived of any meaning. It was not before the advent of Maxwell’s electromagnetic theory that the problem began to be addressed. In effect, by using the Galilean transformations, (although these were universally accepted), one could realize that Maxwell’s equations, which crown this theory, were not invariant under a change of inertial reference frame. In other words, the electromagnetic laws were different for an observer on earth than for another travelling inside a rocket, or for an inhabitant of another solar system.
|
||
The physicists could not content themselves with such a disparity. Knowing that their obsession was the discovery of the universal laws of nature, this result conflicted with their firm convictions. But, on the other hand, to call into question the Galilean relationships seemed to imply a heartbreaking revision of the concept of universal time.
|
||
The end of the nineteenth century and the beginning of the twentieth century were periods of deep reappraisal. The notions which appeared most firmly established seemed to collapse. The experiment of Michelson and Morley came at the right moment to increase this perplexity.
|
||
Different physicists, all around the world, set themselves the task of putting in order the sum of the newly appeared disparate notions. Voigt, Larmor, Lorentz, Poincaré, in turn, showed great concern about them. Lorentz was particularly shocked when he knew the null result of the Michelson experiment, which called into question the concept of ether. In order to save this, he formulated (at the same time as Fitzgerald) the hypothesis of a contraction of the lengths moving through the ether. Alas, the hypothesis could never be verified experimentally; the experiments of Rayleigh and Brace, those of Chase and Tomashek, those of Trouton and Rankine, and those of Wood, all proved negative. Lorentz was then compelled to formulate other hypotheses in order to explain such negative results: at first, the variation of mass with speed, but this was not sufficient. It was necessary to postulate the existence of a local time needed by the consistency of the theory.
|
||
The sum of these ad hoc hypotheses finally ended in the formulation of a set of equations, improved and modified by Poincaré, and named by him “Lorentz transformations.” This was the first breach against universal time. Nevertheless, the latter was not really abolished since, for Lorentz, the idea of an absolute space (absolute inertial frame) was not called into question, and the time measured in this privileged frame was exclusively considered as the real time. The local time was described as fictitious, which implied that the measurement of the time is distorted. That is the reason why it is more
|
||
|
||
in Relativistic Physics
|
||
|
||
41
|
||
|
||
appropriate to define the process as ‘slowing of clocks’ rather than relativity of time.
|
||
It is at that moment that Poincaré took part in the debate. Poincaré acknowledged the Lorentz transformations but, at the same time, showing great concern about the fact that the laws of nature ought to be universal, he believed that they should take the same form in any inertial frame. The final form he gave to them, seemed to satisfy this requirement: in effect, the Lorentz transformations assume a group structure. He then formulated his relativity principle. Although he was convinced that a preferred inertial system supporting the ether should exist, Poincaré was persuaded of the impossibility of distinguishing it from the other inertial frames.
|
||
What Poincaré did not realize is that, if one takes for granted the postulates of Lorentz (existence of an ether at rest in the absolute inertial frame, contraction of moving lengths absolute and not reciprocal), the inertial transformations take the form of those of Lorentz exclusively in the absolute frame. They take a different form in all other inertial frames2 and, as a consequence, the whole of these inertial transformations do not really constitute a group. From a strictly mathematical viewpoint Poincaré was right, but the point of view of physics is different, since we must make allowance for the systematic errors carried out during the measurements inherent in Lorentz’s theory.
|
||
The approach of Einstein proved very different. Einstein also showed great concern about the requirement of the universal character of the laws of nature. His relativity principle is based on hypotheses different from those of Lorentz and Poincaré. Since it assumes a total reciprocity of the observations (contraction observational and reciprocal of moving lengths), it is not affected by the same difficulties as that of Poincaré.
|
||
Einstein’s transformations constitute a fully fledged group. In other words, Einstein succeeded in deriving a set of transformations that maintain invariant the laws of nature. His system was universally adopted because, at the time when the relativity theory was published, the universality of the laws of nature appeared to be an unquestionable requirement.
|
||
At first sight, Einstein’s transformations look completely identical to those of Lorentz, but when we look at them more accurately, we note two essential differences: first, the constant C appears universal whereas, for Lorentz, the speed of light is constant exclusively in the privileged inertial frame. On the other hand, contrary to Lorentz’s approach, there is no
|
||
|
||
2 J. Lévy, Relativity and Cosmic Substratum. Precirculated proceedings of the P.I.R.T. Conference p. 231 Imperial College London 6-9 September 1996. Some important questions regarding Lorentz-Poincaré’s theory and Einstein’s relativity I. Proceedings of the PIRT 1996 Late papers p. 158. Relativité et substratum cosmique, a book of 230 p. Dist. Lavoisier, 14 rue de Provigny 94 236 Cachan Cedex France Tél.: 0147406700.
|
||
|
||
42
|
||
|
||
Open Questions
|
||
|
||
absolute time and fictitious times; the different local times assume the same status.
|
||
It is this aspect of Einstein’s transformations that we propose talking about here.
|
||
The question asked is the following: what is the price to pay for maintaining the universality of the laws of nature, and should they be kept? This question, which could not be asked some decades before without raising a general outcry, is seriously envisaged today by numerous physicists.
|
||
In other words, is the relativity principle really unquestionable? For certain physicists3 it has no absolute meaning but only a conventional character, in connection with a method of synchronization of clocks questionable and relative.
|
||
The relativity principle seems to require the abandonment of the universal time in favour of local times having identical properties. We will try to see, in what follows, if these two concepts are mutually compatible, and if Einstein’s local time is in agreement with logic.
|
||
In order to make us understand the relativity of time, Einstein takes the classical example of the train and the two flashes of lightning that we have studied in detail in a previous paper4. Let us recall it briefly here: two flashes of lightning break at the two ends of a railway platform at the very instant when the two ends of the train meet them. The two flashes reflect against mirrors and then run in opposite direction towards the middle of the platform. According to Einstein, the two flashes are considered simultaneous if they reach the middle of the platform at the same instant. The same definition is also valid for the train, but this one moves towards one of the flashes and moves away from the other and, consequently, the middle of the train will be reached at different instants by the two flashes. Einstein concludes that two events which are simultaneous for an observer, are not simultaneous for another moving with respect to the first. From which the relativity of simultaneity.
|
||
In our previous article5, we demonstrated that the assertion was incorrect, because it implied a confusion between the instantaneous flashes breaking at the ends of the platform, and the light issuing from them. We have also defined differently the simultaneity after correction of the errors of judgement generated by the non instantaneous translation of light.
|
||
|
||
3 F. Selleri, Inertial systems and the transformations of space and time, Phys Essays 8, 342, 1995,
|
||
section 3. & Le principe de relativité et la nature du temps, Fusion, Paris, 66, 50, June 1997. 4 J. Lévy, Some important questions regarding Lorentz Poincaré’s theory and Einstein’s
|
||
relativity II. Proceedings of the P.I.R.T. Conference, supplementary papers, Imperial College London, 6-9 September 1996. 5 Ibid.
|
||
|
||
in Relativistic Physics
|
||
|
||
43
|
||
|
||
A'
|
||
|
||
B'
|
||
|
||
•• •• •• ••••
|
||
|
||
A
|
||
|
||
B
|
||
|
||
Figure 1
|
||
|
||
Train
|
||
|
||
II. A New Experimental Test of Simultaneity
|
||
We now propose another experimental test of simultaneity deprived of the defects of that of Einstein. Of course, it is a matter of a thought experiment that may be difficult to carry out, but this does not deprive it of its character of logical foundation of thinking.
|
||
Let us consider a precision balance (of great sensitivity) longer than an ordinary one, and suppose that two rubber spheres fall down and bounce instantaneously on its pans. If the beam of the balance remains steady during the experiment, one will be authorized to conclude that the spheres have met the pans at the same instant.6
|
||
Let us suppose now that a train passes alongside the beam of the balance, and that the two ends of the train meet the pans at the very instant when the spheres bounce. Let us designate by A the rear of the train, by B its front, and by A’ and B’ the places of the corresponding pans (figure 1).
|
||
After the spheres have bounced, a recording device situated at the middle of the train, will receive the light issuing from B’ before that issuing from A’. From this, one could conclude that two events simultaneous for the earth, are not simultaneous for the train.
|
||
Nevertheless, an observer inside the train will realize that the beam of the balance has not moved. In consequence, he will deduce that the two spheres have fallen down at the same instant. There is no doubt that this criterion of simultaneity is better than that of Einstein, because it allows an instantaneous appreciation which does not need the mediation provided by the photons. It permits the absolute simultaneity independent of the motion of the observer, to be rediscovered.
|
||
Moreover the method should permit the clocks to be exactly synchronized. For that it could suffice placing two clocks in proximity to the two pans of the balance; if the spheres bounce without making the beam move, then it is the same time at the two ends of the balance: for example 8
|
||
|
||
6 Such a schematic device could be replaced by a more sophisticated one implying beams of electrons and coincidence circuits. We entrust the engineers with the task of imagining an appropriate device.
|
||
|
||
44
|
||
|
||
Open Questions
|
||
|
||
o’clock. It will also be possible to synchronize the clocks of the train identically. It will suffice for that to make them indicate 8 o’clock when they pass in front of A’ and B’.
|
||
Now, with the help of several identical closely related balances, it is theoretically possible to synchronize clocks distributed on the whole surface of the Globe.
|
||
Let us now consider a train equipped at its two ends with clocks synchronized with the clocks situated at A’ and B’. Suppose now that the train continues on its way and, after a certain time, meets two other clocks synchronous with A’ and B’, aligned with them, and separated from one another with the same distance. The question asked now is to know if the clocks of the train will be synchronous with these two new clocks.
|
||
According to the relativity principle, this should be the case, because there is no reason to favor the terrestrial frame rather than that of the train.
|
||
This can be easily understood with the help of the following reasoning: Let A and B be the clocks of the train, A’ and B’ the terrestrial clocks met first and A” and B” the terrestrial clocks met secondly. Let us recall that A’ and B’ and A” and B” have been synchronized beforehand. On the other hand A and B are synchronized with A’ and B’ when they pass in front of A’ and B’. So, at this instant, the six clocks are synchronous. If one supposes that the earth frame is an inertial frame (which is only approximately true, but that we will consider absolutely true for the purpose in hand) the frame of the train and the earth frame are equivalent. Therefore, when A and B meet A” and B” they will indicate the same time. This fact demonstrates that the relativity principle is not compatible with the relativity of time (contrary to what Einstein’s postulates suggest), and that the relativity principle also excludes the slowing down of moving clocks. Now, knowing that clock retardation is an experimental fact, the relativity principle cannot be maintained. This result calls into question all the derivations of the inertial transformations which assume the relativity principle, or are in agreement with it, including our own.7 (However, the arguments in the same paper, according to which the speed of light must be different from a limiting velocity, remain unchanged.) Conversely, the existence of a privileged inertial frame could generate a dissymmetry responsible for a slowing of a pair of clocks relative to the other. So the notion of local time (slowing of clocks), proposed by Lorentz, looks possible. But it is a matter of a physical effect concerning the clocks rather than an effect regarding the time itself. This probably explains the experiments of Hafele and Keating, and those regarding the pions.8
|
||
|
||
7 J. Levy, Invariance of light speed: Reality or fiction? Phys. Essays 6:241, 1993. 8 B. Rossi, D.B. Hall, Phys. Rev., 59, 223, 1941.
|
||
D.H. Frish, J.H. Smith, Am J. Phys, 31, 342, 1963 J. Hafele, R. Keating, Science, 177, 166, 1972.
|
||
|
||
in Relativistic Physics
|
||
|
||
45
|
||
|
||
But, as demonstrated in previous papers9, the existence of a privileged frame is not compatible with the relativity principle (Contrary to the opinion of Poincaré).
|
||
The conclusion of this study is that we are compelled to make a choice: we either consider the relativity principle as true and abandon the relativity of time, or we take for granted the slowing of moving clocks and renounce the relativity principle.
|
||
It is easy to see that the first option is contradictory to logic. The second option implies that the laws of nature are different in the different inertial frames. This option looks more likely today. Nevertheless, although in agreement with the theory of Lorentz, it does not imply a total support for it for the reasons previously mentioned10.
|
||
|
||
Acknowledgements
|
||
I would like to thank Professor Franco Selleri for having given me the opportunity to participate in the debate raised around Special Relativity, at the occasion of this Conference.
|
||
|
||
Note added in proofs:
|
||
In fact, the relativity principle remains approximately true for bodies moving with respect to one another, and with respect to the ether frame, at low speeds (v/c << 1). For this reason, the conclusions of Galilei can be retained as a good approximation.
|
||
|
||
9 J. Lévy, Relativity and Cosmic Substratum. Precirculated proceedings of the P.I.R.T. Conference p. 231 Imperial College London 6-9 September 1996. Some important questions regarding Lorentz-Poincaré’s theory and Einstein’s relativity I. Proceedings of the PIRT 1996 Late papers p. 158. Relativité et substratum cosmique, a book of 230 p. Dist. Lavoisier, 14 rue de Provigny 94 236 Cachan Cedex France Tél.: 0147406700.
|
||
10 Ibid.
|
||
|
||
This page intentionally left blank.
|
||
|
||
Reception of Light Signals in Galilean Space-Time
|
||
|
||
Adolphe Martin 2235 Brebeuf, Apt. 3 Longueuil, Quebec, Canada J4J 3P9
|
||
By interpreting Relativity in Galilean space and time, it was found that the time of light reception by an observer moving relative to a source is a different event from the reception of the same light by an observer at rest. The Einstein viewpoint considers these two events to be the same, thus introducing the Special Relativity paradoxes. Using only Einstein’s own coordinates and speed parameter as given by the Lorentz equations but introducing his Doppler factor and reasonning as a Galilean, we arrive at a different viewpoint from Einstein’s with light speeds c, relative to the source, and c’, relative to the observer, different from Einstein’s constant co. This viewpoint becomes an isomorphism of a truly Galilean viewpoint with relative velocity V given by
|
||
c h v co = tanh V co and light speeds C = C’ + V and C’ = C – V, where
|
||
angles remain the same as in Einstein space-time, and all corresponding distances have the same ratio sinhB/B, the Galilean coordinates being given by both the Lorentz equations and Galilean transformation. This eliminates the Special Relativity paradoxes, though respecting the relativity principle and invariance of space and time of Galilean Relativity. Keywords: Special Relativity, Galilean Relativity, Light speeds.
|
||
|
||
Introduction
|
||
|
||
A study was started as early as 1960 to determine the possibility of interpreting Special Relativity in invariant Galilean space and time. By using the Doppler factor and reasoning as a Galilean, the reception of light by an observer moving relative to a source was found to be a different event from the reception of light by an observer at rest relative to the source. As a result, entirely different viewpoints from Einstein’s were obtained for Einstein speed v and proper speed v/L, L being the Lorentz factor:
|
||
|
||
L=
|
||
|
||
1
|
||
|
||
−
|
||
|
||
v2
|
||
|
||
c
|
||
|
||
2 o
|
||
|
||
Open Questions in Relativistic Physics
|
||
|
||
Edited by Franco Selleri (Apeiron, Montreal, 1998)
|
||
|
||
47
|
||
|
||
48
|
||
|
||
Open Questions
|
||
|
||
Figure 1
|
||
In 1970, the speed V was introduced as a purely Galilean speed, by making the celerity B of the Einsteinians equal to V/co. Equations for Galilean light speeds C relative to the source and C’ relative to the observer were obtained before December 1981.[4] These light speeds and V add vectorially, as they should in Galilean space and time: C = C’ + V and C’ = C – V. Four related viewpoints on Relativity are thus possible.[3] Einstein viewpoint with speed v and three Galilean viewpoints with Einstein speed v, proper speed v/L and Galilean speed V.
|
||
The present paper is restricted to the arguments leading to a differentiation between Einstein and Galilean viewpoints at speed v to emphasize the fact that the reception of light by the moving observer is not the same event as the reception of light by the equivalent fixed observer. Again for ease of comprehension, the arguments are restricted to one space dimension and the time dimension.
|
||
|
||
in Relativistic Physics
|
||
|
||
49
|
||
|
||
For ease of understanding, Einstein values are shown in lower case while Galilean values are shown in UPPER CASE.
|
||
|
||
Einstein viewpoint
|
||
|
||
Einstein considers two postulates [2]:
|
||
|
||
a) The principle of relativity. b) The principle of constancy of light speed.
|
||
|
||
In Figure 1, light speed co = 1 is represented by lines at 45° from time axis, speed zero is parallel to time axis. The example is for an observer moving at 0.8co passing at M at a distance 10 ahead of a source of light emitted at time t = 0. Time t = 50 of reception of light by m is calculated by Einstein and all others by:
|
||
|
||
t = AM = 10 = 50, co − v 1 − 0.8
|
||
|
||
x = cot = 50 .
|
||
|
||
(1)
|
||
|
||
The distance 50 covered by light also corresponds to the position m of a fixed
|
||
|
||
object receiving light at time 50. Light would be reflected back at same speed
|
||
|
||
toward A at time tr = 100.
|
||
|
||
tr = 100;
|
||
|
||
tr − te = 50; 2
|
||
|
||
tp
|
||
|
||
=
|
||
|
||
AM v
|
||
|
||
=
|
||
|
||
−12.5 .
|
||
|
||
(2)
|
||
|
||
This corresponds to a radar shot: dividing the time interval taken by light to go and come back by 2, distance 50 of object m is obtained. In practice, to
|
||
|
||
localize the trajectory of a moving object a minimum of two radar shots is required. Time of passage of M at world line of A is tp – 12.5, equation (2). This corresponds to a second radar shot with zero time interval: trajectory and speed are then determined.
|
||
|
||
Galilean viewpoint
|
||
The Galilean viewpoint also considers two postulates:
|
||
a) The principle of relativity. b) The principle of space and time invariance.
|
||
A Galilean frame at rest with the source is same as Einstein rest frame. A moving Galilean frame utilizes the same scales as the rest frame.
|
||
|
||
Time dilation
|
||
|
||
The experimentally proven Doppler factor value, given by the Einstein formula but inverted for wavelengths, is used to determine the time of light reception by the moving observer:
|
||
|
||
β = v = 0.8, co
|
||
|
||
1−β2 Dλ = 1 − β cosθ ,
|
||
|
||
Dλ
|
||
|
||
=
|
||
|
||
1 + β cosθ ′ 1−β2
|
||
|
||
(3)
|
||
|
||
50
|
||
|
||
Open Questions
|
||
|
||
a f cos 0 = 1,
|
||
|
||
Do =
|
||
|
||
1+β 1−β
|
||
|
||
= 3 in one space dimension’
|
||
|
||
a f cos 180 = −1,
|
||
|
||
D180 =
|
||
|
||
1−β 1+β
|
||
|
||
=
|
||
|
||
1 in one space dimension’ 3
|
||
|
||
Subscripts 0 and 180 refer to the angle in degrees between relative velocity and light velocity. The light wavelength at observer M, moving away from A is 3 times the wavelength at the source, and the frequency is decreased 3 times (as shown in Fig. 1).
|
||
If the source sent light toward M from time of passage tp to time t = 0 during 12.5 time units, the moving observer receives the same number of waves during a 3 times longer interval, or 37.5 units. The reception would cease at time: T = 37.5 – 12.5 = 25.
|
||
According to the relativity principle, an observer at A also receives light back from M at a 3 times reduced frequency and longer time. Emission from M lasts 37.5 time units; the reception at A lasts 3 times 37.5 or 112.5 units measured from tp to tr = 112.5 – 12.5 = 100, exactly the return time of a radar shot as given by Einstein and observed!
|
||
Since the Doppler effect is a fact of experience, all viewpoints have to respect it, including Einsteinians. No matter what speed, equivalent to Einstein’s v, is assigned to the moving observer passing at (0,tp), the observer receives light emitted at A(0,0) at time T = 25, in order to have the same Doppler factor.
|
||
Similarly light reflected back to A from the moving observer is received at time tr = 100, the observed return time. These are experimental facts to be respected by all.
|
||
At velocity 0.8co the Lorentz factor L is 0.6.
|
||
|
||
β= v , co
|
||
|
||
L=
|
||
|
||
1
|
||
|
||
−
|
||
|
||
v2
|
||
|
||
c
|
||
|
||
2 o
|
||
|
||
= 0.6 .
|
||
|
||
(4)
|
||
|
||
Therefore, with x = 50, t = 50
|
||
|
||
a f c h x′ = x − vt = 16.66,
|
||
|
||
t′ =
|
||
|
||
t − vx
|
||
|
||
c
|
||
|
||
2 o
|
||
|
||
= 16.66 .
|
||
|
||
L
|
||
|
||
L
|
||
|
||
and with x = 0, tp = –12.5
|
||
|
||
c h x′ = x − vt = 16.66, L
|
||
|
||
tp′ =
|
||
|
||
t − vx
|
||
|
||
c
|
||
|
||
2 o
|
||
|
||
L
|
||
|
||
= −20.833 ,
|
||
|
||
d i d i and t′ − tp′ = 37.5 , T − tp = t′ − tp′ = 0.6 t − tp = L t − tp .
|
||
|
||
Using coordinates of m (50,50) in the Lorentz equations, x’ = t’ = 16.66 is obtained. Similarly, using coordinates of tp (0, –12.5) gives the same x’ = 16.66 and tp’ = –20.83. Notice that time intervals (t’ – tp’) and (T – tp) are strictly equal to 37.5—proof that there is no time dilation, since the Galilean time interval (T – tp) is invariant.
|
||
|
||
in Relativistic Physics
|
||
|
||
51
|
||
|
||
Time intervals (T – tp) and (t’ – tp’) are smaller than (t – tp) by exactly the Lorentz factor. Einsteinians then say: the moving frame time is dilated by the inverse of the Lorentz factor. However, it only appears so. In fact, the reception of light by an observer moving away from the source preceeds the reception by the observer at rest whose coordinates are given by the Lorentz equations.
|
||
If the origin of time is taken at tp in both reference frames, to agree with the canonical Lorentz equations to make x, t, x’, t’ = 0 then tp and tp’ = 0. Einstein time t’, proper time (τ) and the Galilean invariant time T (same in both reference frames in relative motion) become numerically equal, showing again there is no time dilation.
|
||
|
||
T = t′ = t
|
||
|
||
1
|
||
|
||
−
|
||
|
||
v2
|
||
|
||
c
|
||
|
||
2 o
|
||
|
||
=τ
|
||
|
||
(5)
|
||
|
||
This is extremely important, as it proves beyond a doubt, using only the Lorentz equations with the Einstein coordinates and his Doppler factor, that there is no time dilation.
|
||
|
||
Space contraction
|
||
|
||
Einstein says t’ is the time of light reception on the moving observer clock. It is then positioned on the right side of Figure 1, at light reception time T = 25 and tp’ = –20.833 at tp = –12.5. Zero of t’ = 0 is at T – t’ = 25 – 16.66 = 8.33. This corresponds exactly to the time required by the moving observer starting at x’L to reach x’:
|
||
|
||
T − t′ = x′ − x′L = 8.33
|
||
|
||
(6)
|
||
|
||
v
|
||
|
||
Now x’ is the coordinate of the moving observer in his own rest frame. At
|
||
|
||
time t’ = 0, the origin of the moving frame coincides with origin of the source
|
||
|
||
frame at time t = 8.33.
|
||
|
||
Figure 1 shows that at time T, the moving observer is at distance 30
|
||
|
||
(exactly smaller by the Lorentz factor 0.6 than x = 50) where Einstein says he
|
||
|
||
receives the light. The position of the moving observer at T = 25 is then
|
||
|
||
a f AM + vT = xL = x′ + vt′ = co + v t′ = 1.8 × 16.66
|
||
|
||
(7)
|
||
|
||
We see why, for Einsteinians, assuming that events m’ and m coincide in the source frame, the moving frame appears contracted by the Lorentz factor L. Looking at the Lorentz equations in Galilean form, we recognize, in Figure 1, distances AM = x’L and of m’ = xL given by
|
||
|
||
x′L = x − βcot, xL = x′ + βcot′ cot′L = cot − βx, cotL = cot′ + βx′
|
||
|
||
(8)
|
||
|
||
and when x = cot, x′ = cot′ in one space dimension. The difficulties with Special Relativity stem entirely from the Einsteinian
|
||
assumption that the reception of a light flash by an observer at rest relative to the source is the same event as reception of the signal by a moving observer
|
||
|
||
52
|
||
|
||
Open Questions
|
||
|
||
whose coordinates are related by the Lorentz equations. In Galilean spacetime the two events, definitively different, are (x,t) and (xL,T) in the source and fixed observer frame, and (x’,t’ + 25) and (x’,t’) respectively in the moving observer frame. Thus x’L and xL are not contractions of x’ and x, but positions of the moving observer in the source frame at times of light emission t = 0 and reception T = 25 respectively.
|
||
We immediately see that there is no space contraction. The postulate of space and time invariance is respected. We are living in Galilean space and time!
|
||
|
||
Velocity addition
|
||
|
||
Einsteinians use equation (1) to calculate t, which is very different from Einstein’s own formula for addition of velocities—a glaring inconsistency! This would mean that light from A takes 50 time units to travel a distance 10 in the moving frame at a relative speed 0.2, not 1, contrary to Einstein’s second postulate. To respect his postulate in the moving frame, Einstein has to postpone light emission to 8.33 corresponding to his time t’ = 0 and bring the reception back from t = 50 to t’ = 16.66 corresponding to Galilean time T = 25. Only then can he write, following Lorentz, equation (8).
|
||
Costa de Beauregard [1] wrote the Lorentz equations in hyperbolic form:
|
||
|
||
x′ = x cosh B − cot sinh B x = x′ cosh B + cot′ sinh B cot′ = co cosh B − x sinh B cot = cot′ cosh B + x′ sinh B
|
||
|
||
(9)
|
||
|
||
These formulae give the same numerical results as the canonical Lorentz equations. The relationships of the hyperbolic functions to Einstein values is given by
|
||
|
||
β = v = tanh B cosh B = 1 = 1 sinh B = β = β (10)
|
||
|
||
co
|
||
|
||
1−β2 L
|
||
|
||
1−β2 L
|
||
|
||
The equations of Galilean time T in relation to Einstein values or to hyperbolic values are given by
|
||
|
||
T
|
||
|
||
=
|
||
|
||
tLNMM1
|
||
|
||
−
|
||
|
||
cosθ
|
||
|
||
FHG
|
||
|
||
1 β
|
||
|
||
−
|
||
|
||
L β
|
||
|
||
IKJ
|
||
|
||
OQPP
|
||
|
||
=
|
||
|
||
t
|
||
|
||
′LNMM1
|
||
|
||
+
|
||
|
||
cos
|
||
|
||
θ
|
||
|
||
′FHG
|
||
|
||
1 β
|
||
|
||
−
|
||
|
||
L β
|
||
|
||
IKJ
|
||
|
||
OQPP
|
||
|
||
T
|
||
|
||
=
|
||
|
||
t
|
||
|
||
LNM1
|
||
|
||
−
|
||
|
||
cosθ
|
||
|
||
FHG
|
||
|
||
cosh B − sinh B
|
||
|
||
1IKJ
|
||
|
||
OQP
|
||
|
||
=
|
||
|
||
t
|
||
|
||
′LNM1
|
||
|
||
+
|
||
|
||
cos
|
||
|
||
θ
|
||
|
||
′FHG
|
||
|
||
cosh B − sinh B
|
||
|
||
1
|
||
|
||
IKJ
|
||
|
||
OQP
|
||
|
||
=
|
||
|
||
T
|
||
|
||
′
|
||
|
||
(11)
|
||
|
||
The hyperbolic argument B is called celerity by Einsteinians. Jacques Trempe [4,5] assumed that B = V/co, V being the Galilean velocity corresponding to Einstein’s v. These speeds are related by v/co = tanhV/co. When v = co, V = infinite.
|
||
In Galilean space-time, due to the invariance of space and time, velocities add vectorially in three dimension space, or velocity components add arithmetically along each space dimension. Galilean velocity V meets these requirements. In Einstein–Minkowski space-time, the velocity addition
|
||
|
||
in Relativistic Physics
|
||
|
||
53
|
||
|
||
formula is really the addition of hyperbolic tangents of B, while proper
|
||
|
||
velocities add as hyperbolic sines of B.
|
||
|
||
a f β = β1 + β 2 1 + β1β2
|
||
|
||
tanh B1 + B2
|
||
|
||
= tanh B1 + tanh B2 1 + tanh B1 tanh B2
|
||
|
||
a f β
|
||
L
|
||
|
||
=
|
||
|
||
v coL
|
||
|
||
=
|
||
|
||
sinh
|
||
|
||
B1
|
||
|
||
+ B2
|
||
|
||
= sinh B1 cosh B2 + cosh B1 sinh B2
|
||
|
||
(12)
|
||
|
||
Therefore, Einstein velocity v and proper velocity v/L cannot be Galilean,
|
||
|
||
since they do not add according to the Galilean prescription, as V does.
|
||
|
||
Light velocity
|
||
This Galilean viewpoint uses the same coordinates used in Einstein space-time. One must be careful not to confuse this viewpoint using Einstein velocity v in Galilean space-time where light speeds c and c’ are variable, with the Einstein viewpoint in Einstein–Minkowski space-time where light speed co is constant. This is not an idle remark, as the great majority of critics of Einstein, claiming to be Galileans, use the speeds co + v and co – v in their arguments. These speeds, when divided by co, are hyperbolic tangents of speed parameter additions, and thus are not Galilean speeds[6]. The variable speeds c and c’ are truly Galilean in one space dimension and, if considered as vectors, in three space dimensions. Hence, c’ + v = c and c – v = c’ are vectors mapping Galilean vectors C’ + V = C and C’ – V = C,. C and C’ being light speeds relative to source and to observer respectively. The Galilean light speeds are obtained by equating the Lorentz equations for Galilean coordinates to the Galilean transformations, with T = T’:
|
||
X ′ = X cosh B − CT sinh B = X − VT X = X ′ coshB + C′T ′ sinh B = X ′ + VT ′ (13)
|
||
|
||
C′T ′ = CT cosh B − X sinh B CT = C′T ′ cosh B + X ′ sinh B
|
||
|
||
(14)
|
||
|
||
a f a f C
|
||
|
||
=
|
||
|
||
sinh B
|
||
|
||
−
|
||
|
||
coB cosθ cosh B
|
||
|
||
−1
|
||
|
||
C′
|
||
|
||
=
|
||
|
||
sinh
|
||
|
||
B
|
||
|
||
+
|
||
|
||
coB cosθ ′
|
||
|
||
cosh B
|
||
|
||
−
|
||
|
||
1
|
||
|
||
(15)
|
||
|
||
Moreover, the ratio of light speed c or C relative to the source to the light
|
||
|
||
speed c’ or C’ relative to the observer is equal to the Doppler factor, even in
|
||
|
||
three space dimensions, for all angles between light and relative speeds.
|
||
|
||
b g c = C = c L = D b g c′ C′ c′ L
|
||
|
||
(16)
|
||
|
||
According to Einstein, light emitted by the moving observer at M (10,0) should arrive with speed co at A at time 10, while light emitted at A (0,0) should arrive at the moving observer at time 50. This is a flagrant breach of the relativity principle! On the contrary, in Galilean space-time, with light velocities dependent on the source–observer relative velocity, light emitted at A and moving M at instant t = 0 is received respectively at moving M and fixed A at the same time T = 25, in full compliance with the relativity
|
||
|
||
54
|
||
|
||
Open Questions
|
||
|
||
principle. Similarly, a light flash reflected from them at T = 25 is received by both at the same return time tr = 100.
|
||
The ratio of all distances or lengths between Galilean and Einstein values are B/sinh B:
|
||
|
||
b g X = Y = Z = X′ = Y′ = Z′ = CT = C′T = VT = B
|
||
x y z x′ y′ z′ cot cot′ v L T sinh B
|
||
|
||
(17)
|
||
|
||
Since angles are the same in both space-times, and the Galilean coordinates are also related by the Lorentz equations, the Einstein–Minkowski space-time with speed v or v/L, as viewed by a Galilean, is a mathematical mapping of Galilean space-time with speed V! This makes it possible to solve problems in one (e.g. Einstein-Minkowski) space-time and convert the results into the other (e.g. Galilean.
|
||
Consequently, the Galilean viewpoint is also in accordance with the experimentally proven Doppler effect, the vectorial composition of velocities and the relativity principle.
|
||
|
||
Twins paradox
|
||
|
||
The Galilean viewpoint eliminates the Einsteinian concepts of space contraction and time dilation. Furthermore, it removes the twin paradox. In this example, by placing event m’ at m, with Einstein, a sizeable interval of time (equal to 2 times 25 or 50) is forgotten in the Einsteinian calculation of the moving observer’s age. This brings it down to 75, for a total time span of 125, the age of the stay-at-home twin, in a ratio of 0.6, the Lorentz factor, according to the time dilation concept.
|
||
In Galilean space-time, the two twins age at the same rate; both would be 125 at the return. This may be a disappointment for science fiction authors. However, trips would be shorter in time since Galilean velocities greater than co are attainable for all Einstein velocities above 0.7616 co or at Doppler D > 2.71828!
|
||
Hence, for B = 1 D = e = 2.71828
|
||
|
||
β
|
||
|
||
=
|
||
|
||
D2 D2
|
||
|
||
−1 +1
|
||
|
||
=
|
||
|
||
e2 e2
|
||
|
||
−1 +1
|
||
|
||
=
|
||
|
||
0.7616
|
||
|
||
v ≥ 0.7616 co
|
||
|
||
(18)
|
||
|
||
Moreover, there is only one Universe with one universal time. Time travel is absolutely physically impossible as it would entail the existence of an infinity of simultaneous past and future universes.
|
||
|
||
References
|
||
[1] Costa de Beauregard, O., 1968. Relativité et quanta. Les grandes théories de la Physique Moderne, Masson et Cie Éditeurs, Paris.
|
||
[2] Einstein, A., 1905.On the electrodynamics of moving bodies. Annalen Der Physik 17(1905), Doc. 23, The Collected Papers of Albert Einstein, Vol. 2, Princeton University Press.
|
||
[3] Martin, Adolphe, 1994. Light signals in Galilean Relativity. Apeiron 18(1994)
|
||
|
||
in Relativistic Physics
|
||
|
||
55
|
||
|
||
[4] Trempe, Jacques, 1981. Einstein aurait–il pris des vessies pour des lanternes? Spectre, decembre 1981, Montreal.
|
||
[5] Trempe, Jacques, 1990. Laws of light propagation in Galilean space-time, Apeiron 8:1,Montreal.
|
||
[6] Trempe, Jacques, 1992. Light kinematics in Galilean space-time, Physics Essays 5:1
|
||
|
||
This page intentionally left blank.
|
||
|
||
Experiments on the Velocity c
|
||
|
||
J.Ramalho Croca Departamento de Física Faculdade de Ciências, Universidade de Lisboa Campo Grande, Ed C1, 1700 Lisboa Portugal email: croca@fc.ul.pt
|
||
|
||
1. Introduction
|
||
|
||
One of the cornerstones of contemporary physics is the invariance of the
|
||
|
||
velocity of light. As a natural consequence of this postulate comes the fact that
|
||
|
||
the velocity of light does not depend, in any way, on the velocity of the
|
||
|
||
emitting source.
|
||
|
||
Unfortunately, the experimental evidence confirming this corollary of
|
||
|
||
the basic postulate of the relativity is not very accurate. It comes mainly from
|
||
|
||
astronomical sources, such as star doublets, see Fig.1.
|
||
|
||
In 1913 De Sitter, as quoted by Wesley[1], showed that if the velocity of
|
||
|
||
light depended on the velocity of the source, then stellar binaries revolving
|
||
|
||
on a mass center would be very irregular and in certain cases should show
|
||
|
||
ghosts. The light from the approaching star would arrive before light from the
|
||
|
||
receding star. It is known that no such phantom stars are seen.
|
||
|
||
There are still other
|
||
|
||
possibilities discussed in the book
|
||
|
||
of Wesley[1]. Yet most of them are
|
||
|
||
astronomical scale experiments
|
||
|
||
where the errors of measurement
|
||
|
||
are so great that they are not even
|
||
|
||
quantified.
|
||
|
||
Here a laboratory scale
|
||
|
||
experiment is proposed to remedy
|
||
|
||
this situation. It is a one-way
|
||
|
||
experiment where the light travels
|
||
|
||
Fig.1. Star doublet. If the velocity of light
|
||
|
||
a single independent path.
|
||
|
||
depended on the velocity of the source,
|
||
|
||
The absolute temporal
|
||
|
||
then the orbit of the star doublet would be precision of the measurement is of
|
||
|
||
very irregular.
|
||
|
||
subfemtosecond order, that is of
|
||
|
||
Open Questions in Relativistic Physics
|
||
|
||
Edited by Franco Selleri (Apeiron, Montreal, 1998)
|
||
|
||
57
|
||
|
||
58
|
||
|
||
Open Questions
|
||
|
||
about of 10–17 s. This high temporal precision will allow the detection of an hypothetical change in the one-way velocity of the light, due to the movement of the source, of the order of one meter per second or even better. The temporal precision of the measurement is possible thanks to recently developed technology in the production of pulsed laser beams, associated with interferometric detection techniques of second or fourth order.
|
||
|
||
2. Second Order Interference
|
||
|
||
It known that when two coherent gaussian pulses overlap in an
|
||
|
||
interferometer a steady, in time, interference pattern is observed.
|
||
|
||
In this interferometer, the mixing region of which is shown in Fig.2,
|
||
|
||
interference is observed when the phase shifting device Ps moves slightly, changing the relative phase difference between the two overlapping beams.
|
||
|
||
Consider incident gaussian pulses of the form
|
||
|
||
z z+∞ +∞
|
||
|
||
V=
|
||
|
||
g(k,ω )e i(kx − ωt) dk dw ,
|
||
|
||
−∞ −∞
|
||
|
||
(1)
|
||
|
||
with
|
||
|
||
e j g(k,ω) =
|
||
|
||
e
|
||
|
||
−
|
||
|
||
(k−k0 )2
|
||
|
||
2σ
|
||
|
||
2 k
|
||
|
||
−
|
||
|
||
(ω −ω 0 )2
|
||
|
||
2σ
|
||
|
||
2 ω
|
||
|
||
,
|
||
|
||
(2)
|
||
|
||
which by substitution and integration of expression (1) becomes
|
||
|
||
V
|
||
|
||
=
|
||
|
||
Ae
|
||
|
||
−
|
||
|
||
( x−ct)2 2σ 2
|
||
|
||
e i(k0 x − ω 0t)
|
||
|
||
.
|
||
|
||
(3)
|
||
|
||
The generic intensity at the output port of the interferometer is given by
|
||
|
||
I =<|V1 + V2|2 >=<|V1|2 > + <|V2|2 > + <|V1V2*|> + <|V1*V2|> ,
|
||
|
||
(4)
|
||
|
||
which for the gaussian pulse of the form given by (3) transforms, for each
|
||
|
||
output port, into
|
||
|
||
R|I1
|
||
|
||
=
|
||
|
||
2I0 (1
|
||
|
||
+
|
||
|
||
e
|
||
|
||
−
|
||
|
||
ε2 2σ 2
|
||
|
||
ST|I2
|
||
|
||
=
|
||
|
||
2I0 (1 −
|
||
|
||
e
|
||
|
||
−
|
||
|
||
ε2 2σ 2
|
||
|
||
cosδ ) ,
|
||
cosδ )
|
||
|
||
(5)
|
||
|
||
Fig.2. Overlapping region of a second order interferometer. Ps phase shifting device.
|
||
|
||
in Relativistic Physics
|
||
|
||
59
|
||
|
||
P12 1 0.8 0.6 0.4 0.2
|
||
|
||
-4
|
||
|
||
-2
|
||
|
||
2
|
||
|
||
Fig.3. Plot for the joint probability detection in the case of coherent overlapping.
|
||
¶ 4
|
||
|
||
where ε represents the time delay between the two pulses, σ the length of
|
||
|
||
the pulses and δ the minute relative phaseshift.
|
||
|
||
The joint probability detection P12 = I1I2 is given by
|
||
|
||
FHG IKJ P12
|
||
|
||
=1−e
|
||
|
||
−
|
||
|
||
ε2 2σ 2
|
||
|
||
cos2 δ ,
|
||
|
||
which, for a relative phase shift δ = 0,...,2n π becomes
|
||
|
||
FHG IKJ P12 = 1 − e
|
||
|
||
−
|
||
|
||
ε2 2σ 2
|
||
|
||
,
|
||
|
||
(6)
|
||
|
||
a plot of which is shown in Fig.3 As can be seen, from the plot, when the two pulses arrive at the same
|
||
time, meaning that ε = 0, the normalised coincidence detection goes to zero. In this case coincidence never occurs, the photons go either to one output port or to the other. If the time delay between them is much greater than the length of the pulse, ε >> σ , then all happens as if each pulse arrives alone at the overlapping region. In this situation the pulse is split into two parts, each going to a different detector, and hence the probability of coincidence is one. For the case of partial overlapping, the probability of coincidence lies between zero and one. What is important in this measure is that from the actual value of the joint probability detection, given by expression (6), one can deduce the time delay between the two pulses.
|
||
|
||
3. Fourth Order Interference
|
||
When two overlapping waves come from mutually incoherent sources, no interference is to be seen in a usual second order interferometer. Nevertheless, even in this case it is possible to detect intensity correlation in
|
||
|
||
60
|
||
|
||
Open Questions
|
||
|
||
Fig.4. Fourth order interferometer.
|
||
|
||
the field resulting from the superposition of two incoherent waves. This phenomenon, usually known as fourth order interference, since it relates four fields, was experimentally discovered by Brown and Twiss[2] and is now widely applied in stellar interferometry.
|
||
The mathematical treatment of the effect can be found in many works that now are classic, such as those of Paul[3], Mandel[4] and others. An interferometer of this type is shown in Fig.4, where the interference is observed when the detectors scan the distance x1 – x2. The joint probability detection for the experimental set-up is given by
|
||
|
||
P12 ∝ (I A + IB )2 − 2 I A IB cosδ
|
||
|
||
(7)
|
||
|
||
which can be written
|
||
|
||
P12
|
||
|
||
∝1−
|
||
|
||
<
|
||
|
||
I
|
||
|
||
2 A
|
||
|
||
>
|
||
|
||
2 < IAIB >
|
||
|
||
+
|
||
|
||
<
|
||
|
||
I
|
||
|
||
2 B
|
||
|
||
>
|
||
|
||
+2
|
||
|
||
<
|
||
|
||
IAIB
|
||
|
||
>
|
||
|
||
cos δ
|
||
|
||
(8)
|
||
|
||
where δ = 2 π (x1 + x2)/L, and the separation between the fringes L = λ sinθ ≅ λ θ . By substitution in expression (8) of the gaussian pulses, of
|
||
the form described previously, the joint probability detection for this case turns out to be
|
||
|
||
d d i i P12
|
||
|
||
=1−
|
||
|
||
e 1+
|
||
|
||
− ε2
|
||
2σ 2
|
||
|
||
e
|
||
|
||
−
|
||
|
||
ε2 2σ 2
|
||
|
||
cos δ
|
||
|
||
(9)
|
||
|
||
which for the minimum coincidence rate, δ = 2 π (x1+x2)/L = 2n π is
|
||
|
||
P12
|
||
|
||
=1−
|
||
|
||
e−
|
||
|
||
ε2 2σ 2
|
||
|
||
1 + e−
|
||
|
||
ε2 2σ 2
|
||
|
||
.
|
||
|
||
(10)
|
||
|
||
The plot for this case is shown in Fig. 5, together with the one for
|
||
|
||
coherent overlapping, dashed line. The difference between them is that for
|
||
|
||
complete overlapping when the pulses are coherent, the probability of
|
||
|
||
coincidence is zero while in the incoherent case it is one half.
|
||
|
||
in Relativistic Physics
|
||
|
||
61
|
||
|
||
P12 1 0.8 0.6 0.4 0.2
|
||
|
||
-4
|
||
|
||
-2
|
||
|
||
2
|
||
|
||
Fig.5. Plot for the probability of coincidence. Solid line incoherent overlapping. Dashed line coherent superposition.
|
||
¶ 4
|
||
|
||
4. Proposed Experiment
|
||
The set-up representing the experimental arrangement is shown in Fig.6. It is composed of a laser able to emit short pulses of femtosecond order, a turning wheel plus an interferometer.
|
||
The pulsed light from the laser enters the axis of the turning wheel, and is split into two and directed by means of an optical fiber, or some other device, to the points S1 and S2, which act as the moving sources.
|
||
Based on suggestions from the experimentalists[5] the experiment is best done using the alternate set-up shown in Fig.7.
|
||
This alternate device differs from the previous one in the way the moving sources are made. The laser pulse is split outside and the two beams are directed to the turning wheel made of stainless steel, which acts as a moving mirror.
|
||
In either case the light from S1 and S2 follows two independent parallel paths of the same length and is mixed in the in the interferometer set for the optimal conditions.
|
||
If the wheel is not turning, since the two paths are equal, the two pulses arrive in complete overlapping conditions corresponding to the situation
|
||
|
||
Fig.6. Set-up to test the independence of C on the velocity of the emitting source through single independent paths.
|
||
|
||
62
|
||
|
||
Open Questions
|
||
|
||
ε = 0 which results in the minimum in the coincidence rate. When the wheel is set in motion, the sources S1 and S2 start moving in
|
||
opposite directions with a velocity V. Assuming that the velocity of light depends hypothetically on the
|
||
velocity of the moving source, them the pulse from S2 shall arrive at the interferometer a certain time in advance of the pulse from S1, therefore increasing the joint detection probability. The probability of coincidence P12 may increase to reach its maximum when no overlapping occurs.
|
||
From the actual value of the joint detection probability P12, one derives the hypothetical relative delay ε between the two pulses when the wheel is turning.
|
||
If the arriving pulses are totally coherent, the distribution would be given by the expression (6). In the case of perfect incoherent overlapping, the second expression (10) should fit the data. The real situation is probably a mixture of the two cases. Only the concrete experimental conditions would allow us to determine the right distribution.
|
||
The absolute precision, in time, of this experiment depends on the time width of the laser pulse. For steady laser pulses of about 3 fs, currently operating, it is possible, in principle, to determine fractions (say one thousandth) of this value, corresponding to differences in time up to 10–18 s.
|
||
As is well known, there is a certain controversy surrounding the possibility of measuring the one-way velocity of light[6] due to the problem of clock calibration. Some authors even deny the possibility of measuring the one-way velocity of light. Hence, in order to measure any hypothetical change in the one-way velocity of light due to the motion of the source, it must be assumed, of course, that this property of light is measurable, even if only for the sake of logical coherence.
|
||
On this problem my opinion coincides with Selleri who says[7] “After all, light goes from one point to another in a well-defined way and it would be very strange if its true velocity were forever inaccessible to us.” Concrete experiments may eventually show that this working hypothesis is wrong.
|
||
Since we do not know the law for the change in the velocity of light, due to the motion of the emitting source, for simplicity, we assume that it will
|
||
increase by a certain hypothetical amount δ v when the emitting source moves in direction of the light
|
||
|
||
Fig.7. Alternate set-up to test the independence of c of the velocity of the emitting source.
|
||
|
||
in Relativistic Physics
|
||
|
||
63
|
||
|
||
c+ = c + δ v,
|
||
|
||
(11)
|
||
|
||
where c+ represents the velocity of light when the emitting source moves in
|
||
|
||
direction of the light, c is the usual velocity of light and δ v is hypothetical
|
||
|
||
amount of velocity needed to add to c in order to attain c+.
|
||
|
||
Symmetrically, we shall assume that when the velocity of the emitting
|
||
|
||
source is contrary to the light its velocity would be given by
|
||
|
||
c– = c – δ v.
|
||
|
||
(12)
|
||
|
||
It must be stressed, in order to avoid misinterpretations, that δ v is not the velocity of the emitting source. It only represents the hypothetical change in the velocity of light due to the motion of the source.
|
||
No attempt is made to propose an expression for the hypothetical dependence of the velocity of light on the velocity of the emitting source.
|
||
According to those assumptions we shall write, for the hypothetical time difference between the two paths
|
||
|
||
FHG IKJ ε
|
||
|
||
= t2
|
||
|
||
− t1
|
||
|
||
=
|
||
|
||
c−
|
||
|
||
+
|
||
|
||
c+
|
||
|
||
=
|
||
|
||
c−δv
|
||
|
||
−
|
||
|
||
c+δv
|
||
|
||
≈2 δv
|
||
|
||
δv c
|
||
|
||
2
|
||
.
|
||
|
||
(13)
|
||
|
||
That is
|
||
|
||
δv ≈ ε c2 ,
|
||
|
||
(14)
|
||
|
||
2
|
||
|
||
where t1, t2 are the time the laser pulses take to go through paths one and two and δ v is the hypothetical change in the velocity of light due to the motion of
|
||
|
||
the sources.
|
||
|
||
Assuming a moderated minimal time resolution ε ≈ 10−16 s and that the
|
||
|
||
experiment is done with an interferometer with arms of about 3m, one obtains
|
||
|
||
a lower limit of δ v ≈ 1.5 m/s
|
||
|
||
5. Conclusion
|
||
It was shown, under reasonable experimental conditions, I believe, that it is possible, in principle, to determine, in one-way independent trip, if the velocity of the light is independent or not on the velocity emitting source to an hypothetical change of about 1.5 m/s in 300 000 km/s (3 × 108m/s).
|
||
References
|
||
[1] J.P. Wesley, Selected Topics in Advanced Fundamental Physics, (Benjamin Wesley, Blumberg, 1991)
|
||
[2] R.H. Brown and R.Q. Twiss, Nature (London) 177(1956) 27; 178(1956) 1046 [3] H. Paul, Rev. Mod. Phys. 58(1986) 209 [4] Z.Y. Ou and L. Mandel, Phys. Rev. Lett. 62(1989) 2941. [5] A. Melo, as suggested this change in the planning of a concrete experiment to be
|
||
done with J.P. Ribeiro et al. at Lisbon.
|
||
|
||
64
|
||
|
||
Open Questions
|
||
|
||
[6] F. Selleri, Found Phys. Letters, 9(1996) 43. [7] F. Selleri, Found. Phys. 26(1996) 641.
|
||
|
||
Inertial Transformations from the Homogeneity of Absolute Space
|
||
|
||
Ramón Risco-Delgado Dpto. de Física Aplicada, E. Superior de Ingenieros Universidad de Sevilla (Spain) e-mail: ramon@cica.es
|
||
A transformation law for the space and time coordinates between two reference systems is deduced from the hypothesis of a privileged frame. No experiment performed up till now can distinguish them from Lorentz transformations.
|
||
|
||
Introduction
|
||
After the work of Mansouri and Sexl [1], it is well known that there exist many theories equivalent to special relativity from the experimental point of view that negate the relativity principle. This is possible because the experiments have only established the constancy of the two-way velocity of light. Accordingly, the one-way velocity of light is a degree of freedom in the space-time transformations that allows one to build a infinite set of transformations, indistinguishable from those of special relativity until now.
|
||
The existence of absolute affects, like the acceleration in the clock paradox or the length contraction in prerelativistic physics due to the deformation of the electromagnetic field of moving charges, allows us to change the starting point, and take the homogeneity of absolute space and time, instead of the relativity principle, as the basis of our argument. Homogeneity of space and time is a very plausible characteristic, one on which well established conservation laws are based.
|
||
|
||
Space-time transformations
|
||
|
||
If absolute space and time are homogeneous, then the transformation laws between a reference system fixed in this space and time (S0) and any other inertial one (S) must be linear (for the sake of simplicity we shall only consider one spatial axis):
|
||
|
||
x = a1x0 + a2t0, t = b1x0 + b2t0,
|
||
|
||
(1)
|
||
|
||
Open Questions in Relativistic Physics
|
||
|
||
Edited by Franco Selleri (Apeiron, Montreal, 1998)
|
||
|
||
65
|
||
|
||
66
|
||
|
||
Open Questions
|
||
|
||
supposing that the origins of S0 and S coincide at time t = t0 = 0. If the systems move in the standard configuration, in such a way that the origin of S is seen from S0 to move with velocity v, then a2 = −a1v.
|
||
Now let us postulate the well classical fact that due to the deformation of
|
||
the electromagnetic fields of moving atomic charges, a material body contracts itself just by the usual factor γ–1 [2]:
|
||
|
||
x = γ (x0 − vt0), t = b1x0 + b2t0
|
||
|
||
(2)
|
||
|
||
The next step will be to introduce the well established experimental
|
||
|
||
result that the two-way velocity of light must be c in all reference systems. In
|
||
|
||
order to do that we shall first compute the velocity transformation law from
|
||
|
||
Eq. (2). Next, we shall apply this velocity transformation law for light going
|
||
|
||
forward and backward with velocity c and −c in the S0 system respectively. Then we shall require that the two-way velocity of light be c in S as well.
|
||
|
||
By taking differentials in (2)
|
||
|
||
a f dx = γ dx0 − vdt0
|
||
dt = b1dx0 + b2dt0
|
||
|
||
,
|
||
|
||
(3)
|
||
|
||
dividing by one another and calling u = dx/dt and u0 = dx0/dt0, we obtain
|
||
|
||
a f u = γ u0 − v .
|
||
|
||
(4)
|
||
|
||
b2 + b1u0
|
||
|
||
If Eq. (4) is applied to light that travels forward (with velocity c) and
|
||
|
||
backward (with velocity −c) in S0, then the corresponding velocities in S are:
|
||
|
||
c+
|
||
|
||
=γ
|
||
|
||
c−v b2 + b1c
|
||
|
||
;
|
||
|
||
c−
|
||
|
||
=γ
|
||
|
||
−c − v b2 − b1c
|
||
|
||
.
|
||
|
||
(5)
|
||
|
||
When light in S makes a two-way trip, travelling a distance 2d, then the total time spent in the journey must be the sum of the times in the forward and backward trips. Requiring the two-way velocity of light to be c in S, then
|
||
|
||
2d = d + d .
|
||
|
||
(6)
|
||
|
||
c c+ −c−
|
||
|
||
By introducing (5) in (6) it is possible to find a relation between b1 and b2:
|
||
|
||
b2 = γ–1 − vb1,
|
||
|
||
(7)
|
||
|
||
and the transformations read
|
||
|
||
a f x = γ x0 − vt0 c h t = b1x0 + γ −1
|
||
|
||
− vb1
|
||
|
||
t0 .
|
||
|
||
(8)
|
||
|
||
The former equations are the most general transformations compatible with experimental knowledge until now.
|
||
|
||
Clock synchronization
|
||
Eqs. (8) still have a parameter to be found. This freedom in the transformations is fixed by the value of the one-way velocity of light, a quantity that has not been measured previously. This fact is translated into
|
||
|
||
in Relativistic Physics
|
||
|
||
67
|
||
|
||
different clock synchronization criteria. The only thing we can do for the moment is to follow our physical intuition in order to fix its value. Although in principle the choices can be many, there are two that seem specially justified: the Einstein approach and absolute approach.
|
||
|
||
Einstein approach
|
||
|
||
By denying the existence of an absolute space and time, Einstein arrives at the Lorentz transformations. Denying an absolute space means the equivalence of all inertial systems. This implies the validity of Maxwell’s equations in all frames and, therefore, the constancy of the one-way velocity of light. In this section we shall obtain the particular element of (8) corresponding to the Lorentz transformations. We shall do so by denying the existence of an absolute space; in particular by imposing on (8) one of the most amazing consequences of this idea, the constancy of the velocity of light.
|
||
From (8), if we compute the velocity transformation laws between S0 and S we obtain
|
||
|
||
a f u = γ
|
||
|
||
γ
|
||
|
||
−1
|
||
|
||
u0 − v + b1 u0
|
||
|
||
−
|
||
|
||
v
|
||
|
||
.
|
||
|
||
(9)
|
||
|
||
Let us now impose that when u0 ≡ c, then also u ≡ c.
|
||
|
||
a f c
|
||
|
||
=γ
|
||
|
||
γ
|
||
|
||
−1
|
||
|
||
c−v + b1 c − v
|
||
|
||
,
|
||
|
||
(10)
|
||
|
||
and from here it is straightforward to obtain b1:
|
||
|
||
b1
|
||
|
||
= −γ
|
||
|
||
v c2
|
||
|
||
.
|
||
|
||
(11)
|
||
|
||
Obviously if (11) is carried into (8) we obtain Lorentz transformations. These are the only transformations that follow from denying the existence of an absolute space.
|
||
|
||
Absolute approach
|
||
Instead of denying the absolute standard, here we admit its existence. We shall see how, by assuming homogeneity, we can arrive at a set of equations, different from Lorentz transformations, but still compatible with experience.
|
||
Let us imagine the following ideal experiment. Two spacecraft, each carrying a clock, are at rest in S0 at a certain distance from one another. At the same time, as measured from S0, they start to move with the same acceleration until a certain preassigned time, and then they move with constant velocity. If S0 is homogeneous the two clocks have acquired the same delay with respect to those at rest in S0. In other words, this delay cannot depend on the spacecraft we chose, if space is homogeneous. This means that in (8) the transformation for time cannot depend on the position x0. The only way to achieve this is by taking b1 = 0. The transformations are then
|
||
|
||
68
|
||
|
||
Open Questions
|
||
|
||
a f x = γ x0 − vt0
|
||
t = γ −1t0
|
||
|
||
.
|
||
|
||
(12)
|
||
|
||
Using a similar approach, these equations were obtained first by Tangherlini
|
||
|
||
[3] and generalized by Selleri [4].
|
||
|
||
For the moment there is no experimental reason to prefer the Lorentz
|
||
|
||
transformations to (12). On the contrary, the behavior of clocks reported in
|
||
|
||
this section seems quite plausible. The existence of a crucial test is still an
|
||
|
||
open question.
|
||
|
||
I acknowledge Prof. Selleri for encouragement. I also thank La Revista
|
||
|
||
Espanola de Física for the permission to reproduce here a revised version of the
|
||
|
||
manuscript published there in Spanish in December 1997.
|
||
|
||
References
|
||
[1] R. Mansouri and R. Sexl, General Relativity and Gravitation 8, 497 (1977). [2] J. S. Bell, How to Teach Special Relativity, in Speakable and Unspeakable in
|
||
Quantum Mechanics (Cambridge University Press, 1988). [3] F. R. Tangherlini, Nouvo Cim. Suppl. 20, 1 (1961). [4] F. Selleri, Found. Phys. 26, 641 (1996).
|
||
|
||
On a Physical and Mathematical Discontinuity in Relativity Theory
|
||
|
||
F. Selleri Università di Bari - Dipartimento di Fisica INFN - Sezione di Bari I 70126 Bari, Italy
|
||
An isotropic inertial reference frame (“stationary”) is considered, in which a circular disk of radius R rotates uniformly. The velocity of light ~c relative to the rim of the disk is calculated under very general assumptions and found to satisfy ~c ≠ c . This ~c ≠ c remains the same if R is increased but the peripheral velocity of the disk is kept constant. Since by so doing any small part of the circumference can be considered (for a short time) better and better at rest in a (“moving”) inertial system, there is a discontinuity between accelerated reference frames with arbitrarily small acceleration and inertial frames, if the velocity of light is assumed to be c in the latter. Elimination of the discontinuity is shown to imply for inertial systems a velocity of light ~c ≠ c , necessarily equal to that obtained from recently published “inertial transformations.”
|
||
|
||
1. Space and time on a rotating platform
|
||
|
||
In this paper it is shown that the relativistic description of inertial
|
||
|
||
reference systems contains a basic difficulty. It is well known that no perfectly
|
||
|
||
inertial frame exists in practice, e.g., because of the Earth’s rotation around its
|
||
|
||
axis, orbital motion around the Sun and Galactic rotation. Therefore, all
|
||
|
||
knowledge about inertial frames has necessarily been obtained in frames
|
||
|
||
having small but nonzero acceleration a. For this reason the mathematical
|
||
|
||
limit a → 0 taken in the theoretical schemes should be a smooth limit, and no
|
||
|
||
discontinuities should arise between systems with small acceleration and
|
||
|
||
inertial systems. From this point of view the existing relativistic theory will be
|
||
|
||
shown to be inconsistent.
|
||
|
||
Consider an inertial reference frame S0 and assume that it is isotropic.
|
||
|
||
Therefore, the one-way velocity of light relative to S0 can be taken to have
|
||
|
||
the usual value c in all directions. In relativity, the latter assumption is true in
|
||
|
||
all inertial frames, while in other theories only one such frame exists [1]. A
|
||
|
||
Open Questions in Relativistic Physics
|
||
|
||
Edited by Franco Selleri (Apeiron, Montreal, 1998)
|
||
|
||
69
|
||
|
||
70
|
||
|
||
Open Questions
|
||
|
||
laboratory in which physical experiments are performed is assumed to be at rest in S0 , and in it clocks are assumed to have been synchronised with the Einstein method, i.e., by using light signals and assuming that the one-way velocity of light is the same in all directions.
|
||
In this laboratory there is a perfectly circular platform having radius R, which rotates around its axis with angular velocity ω and peripheral velocity v = ω R . On its rim, consider a clock CΣ and assume it to be set as follows: When a laboratory clock momentarily very near CΣ shows time t0 = 0 , then CΣ is also set at time t = 0 . When the platform is not rotating, CΣ constantly shows the same time as the laboratory clocks. When it rotates, however,
|
||
|
||
motion modifies the rate of CΣ , and the relationship between the times t and
|
||
|
||
t0 is taken to have the general form
|
||
|
||
a f t0 = t F1 v, a
|
||
|
||
(1)
|
||
|
||
where F1 is a function of velocity v, acceleration a = v2 R , and eventually of higher derivatives of position (not shown).
|
||
|
||
The circumference length is assumed to be L0 and L, measured in the laboratory S0 and on the platform, respectively. Since motion can modify length as well, the relationship
|
||
|
||
a f L0 = L F2 v, a
|
||
|
||
(2)
|
||
|
||
is assumed, where F2 is another function of the said arguments. Notice that the assumed isotropy of the laboratory frame implies that F1 ( F2 ) does not depend on the point on the rim of the disk where the clock is placed (where the measurement of length is started). It depends only on velocity, acceleration, etc., and these are the same at all points of the edge of the rotating circular platform.
|
||
|
||
One is of course far from ignorant about the nature of the functions F1 and F2 . In the limit of small acceleration and constant velocity, they are expected to become the usual time dilation and length contraction factors, respectively:
|
||
|
||
a f F1 v,0 =
|
||
|
||
1 1− v2 / c2
|
||
|
||
a f ; F2 v,0 = 1 − v2 / c 2
|
||
|
||
(3)
|
||
|
||
There are even strong indications, at least in the case of F1(v, a) , that the dependence on a is totally absent [2]. All this is, however, unimportant for present purposes, because the results obtained below hold for all possible functions F1 and F2 .
|
||
|
||
2. Velocity of light on rotating platforms
|
||
On the rim of the platform, in addition to clock CΣ there is a light source Σ, placed in a fixed position very near CΣ . Two light flashes leave Σ at the
|
||
|
||
in Relativistic Physics
|
||
|
||
71
|
||
|
||
time t1 of CΣ , and are forced to move on a circumference, by “sliding” on the internal surface of a cylindrical mirror placed at rest on the platform, all around it and very near its edge. Mirror apart, the light flashes propagate in
|
||
|
||
the vacuum. The motion of the mirror cannot modify the velocity of light,
|
||
|
||
because the mirror is like a source (a “virtual” one), and the motion of a source never changes the velocity of light signals originating from it. Thus, relative to the laboratory, the light flashes propagate with the usual velocity c.
|
||
|
||
The description of light propagation given by the laboratory observers is
|
||
|
||
as follows: two light flashes leave Σ at time t01 . The first one propagates on a circumference, in the sense opposite to the platform rotation, and comes back to Σ at time t02 after a full rotation around the platform. The second one propagates on a circumference, in the same rotational sense of the platform, and comes back to Σ at time t03 after a full rotation around the platform. These laboratory times, all relative to events taking place in a fixed point of
|
||
|
||
the platform very near CΣ , are related to the corresponding platform times via (1):
|
||
|
||
a f t0i = ti F1 v, a
|
||
|
||
(i = 1,2,3)
|
||
|
||
(4)
|
||
|
||
Light propagating in the direction opposite to the disk rotation, must cover a
|
||
a f distance smaller than L0 by a quantity x = ω R t02 − t01 equalling the shift of
|
||
|
||
Σ during the time t02 − t01 taken by light to reach Σ . Therefore
|
||
|
||
L0 − x = c ( t02 − t01 ) ; x = ω R( t02 − t01 )
|
||
|
||
(5)
|
||
|
||
From these equations one gets:
|
||
|
||
t02
|
||
|
||
− t01
|
||
|
||
=
|
||
|
||
L0 c (1 + β)
|
||
|
||
(6)
|
||
|
||
Light propagating in the rotational direction of the disk, must instead cover a
|
||
|
||
distance larger than the disk circumference length L0 by a quantity y = ω R (t03 − t01 ) equalling the shift of Σ during the time t03 − t01 taken by light to reach Σ. Therefore
|
||
|
||
L0 + y = c (t03 − t01 ) ; y = ω R ( t03 − t01 )
|
||
|
||
(7)
|
||
|
||
One now gets
|
||
|
||
t03
|
||
|
||
− t01
|
||
|
||
=
|
||
|
||
L0 c (1 −
|
||
|
||
β )
|
||
|
||
(8)
|
||
|
||
By taking the difference between (8) and (6) one sees that the delay between
|
||
|
||
the arrivals of the two light flashes back in Σ is observed in the laboratory to
|
||
|
||
be
|
||
|
||
c h t03
|
||
|
||
−
|
||
|
||
t02
|
||
|
||
=
|
||
|
||
c
|
||
|
||
2 L0 β 1− β2
|
||
|
||
(9)
|
||
|
||
Incidentally, this is the well known delay time for the Sagnac effect [3], calculated in the laboratory (our treatment is, however, totally independent of
|
||
|
||
72
|
||
|
||
Open Questions
|
||
|
||
this effect). It is shown next that these relations to some extent fix the velocity of light relative to the disk. In fact (2) and (4) applied to (6) and (8) give
|
||
|
||
FHG t 2
|
||
|
||
− t1 IKJ
|
||
|
||
F1
|
||
|
||
=
|
||
|
||
L F2 c (1 + β)
|
||
|
||
;
|
||
|
||
FHG t 3
|
||
|
||
− t1 IKJ
|
||
|
||
F1
|
||
|
||
=
|
||
|
||
L F2 c (1 − β)
|
||
|
||
(10)
|
||
|
||
whence
|
||
|
||
a f1
|
||
~c π
|
||
|
||
= t2 − t1 =
|
||
|
||
F2
|
||
|
||
L
|
||
|
||
F1 c (1 + β)
|
||
|
||
;
|
||
|
||
a f1
|
||
~c 0
|
||
|
||
= t3 − t1 =
|
||
|
||
F2
|
||
|
||
L F1 c (1 − β)
|
||
|
||
(11)
|
||
|
||
a f a f if ~c 0 and ~c π are the light velocities, relative to the rim of the disk, for
|
||
|
||
the flash propagating in the direction of the disk rotation, and in the opposite
|
||
|
||
direction, respectively. From (11) it follows that the velocities of the two light
|
||
|
||
flashes relative to the disk must satisfy
|
||
|
||
~c (π ) ~c (0)
|
||
|
||
=
|
||
|
||
1 1
|
||
|
||
+ −
|
||
|
||
β β
|
||
|
||
(12)
|
||
|
||
Notice that the unknown functions F1 and F2 cancel from the ratio (12). The
|
||
|
||
consequences of (12) will be discussed in the next section.
|
||
|
||
Now comes an important point: Clearly, Eq. (12) gives us not only the
|
||
|
||
ratio of the two global light velocities for full trips around the platform in the
|
||
|
||
two opposite directions, but the ratio of the instantaneous velocities as well. In
|
||
|
||
fact, we have assumed that the “stationary” inertial reference frame in which
|
||
|
||
the centre of the platform is at rest is isotropic, and isotropy of space ensures
|
||
|
||
that the instantaneous velocities of light are the same at all points of the rim of
|
||
|
||
the circular disk, and, therefore, that the average velocities coincide with the
|
||
|
||
instantaneous ones.
|
||
|
||
The result (12) holds for platforms having different radii, but the same
|
||
|
||
peripheral velocity v. Let a set of circular platforms with radii
|
||
|
||
R1 , R2 , ... Ri , ... ( R1 < R2 < ... < Ri < ... ) be given, and made to spin with angular velocities ω 1 , ω 2 , ... ω i , ... such that
|
||
|
||
ω 1R1 = ω 2 R2 = ... = ω i Ri = ... = v
|
||
|
||
(13)
|
||
|
||
where v is constant. Obviously then, (12) applies to all such platforms with
|
||
b g the same β β = v / c . The respective centripetal accelerations are
|
||
|
||
v2 , v2 , ... v2 , ...
|
||
|
||
(14)
|
||
|
||
R1 R2
|
||
|
||
Ri
|
||
|
||
and tend to zero with increasing Ri . This is so for all higher derivatives of position: if r0 (t0 ) identifies a point on the rim of the i-th platform seen from the laboratory, one can easily show that
|
||
|
||
dn r 0 d t0n
|
||
|
||
=
|
||
|
||
vn Rni −1
|
||
|
||
an ≥ 2f
|
||
|
||
(15)
|
||
|
||
This tends to zero with increasing Ri for all n ≥ 2 . Therefore, a very small part AB of the rim of a platform, having peripheral velocity v and very large
|
||
|
||
in Relativistic Physics
|
||
|
||
73
|
||
|
||
radius, for a short time is completely equivalent to a small part of a “comoving” inertial reference frame (endowed with a velocity only). For all practical purposes the “small part AB of the rim of a platform” will belong to that inertial reference frame. But the velocities of light in the two directions AB and BA must satisfy (12), as shown above. It follows that the velocity of light relative to the co-moving inertial frame cannot be c.
|
||
|
||
3. Speed of light relative to inertial frames
|
||
|
||
As shown in [1], one can always choose Cartesian co-ordinate systems in two inertial reference frames S and S0 and assume:
|
||
|
||
(1) that space is homogeneous and isotropic, and that time is also homogeneous;
|
||
(2) that relative to S0 the velocity of light is the same in all directions, so that Einstein’s synchronisation can be used in this frame and the velocity v of S relative to S0 can be measured;
|
||
(3) that the origins of S and S0 coincide at t = t0 = 0 ; (4) that planes (x0,y0) and (x,y) coincide at all times t0; that also planes (x0, z0)
|
||
and (x,z) coincide at all times t0; but that planes (y0,z0) and (y,z) coincide at time t0 = 0 only.
|
||
|
||
It then follows [1] that the transformation laws from S0 to S are necessarily
|
||
|
||
RS|T|
|
||
|
||
x y z t
|
||
|
||
= = = =
|
||
|
||
f1 (x0 − vt0 ) g2 y0 g2 z0 e1 x0 + e4t0
|
||
|
||
(16)
|
||
|
||
where the coefficients f1 , g2 , e4 and e1 can depend on v. If at this point one
|
||
assumes the validity of the relativity principle (including invariance of light velocity) these transformations reduce necessarily to the Lorentz ones. It was shown in [1] that the most general transformations (16) satisfying the well established experimental conditions of constant two-way velocity of light and of time dilation according to the usual relativistic factor, are such that
|
||
|
||
f1
|
||
|
||
=
|
||
|
||
1 R( β )
|
||
|
||
;
|
||
|
||
g2 = 1 ;
|
||
|
||
e4 = R(β) − e1 β c
|
||
|
||
where β = v / c , and
|
||
|
||
R(β) = 1 − β 2
|
||
|
||
(17)
|
||
|
||
so that
|
||
|
||
74
|
||
|
||
Open Questions
|
||
|
||
R| x
|
||
|
||
=
|
||
|
||
x0 − β ct0 R( β )
|
||
|
||
| y = y0
|
||
|
||
S| z = z0
|
||
|
||
T|| t = R(β) t0 + e1(x0 − β ct0 )
|
||
|
||
(18)
|
||
|
||
In (18) only e1 remains unknown. Length contraction by the factor R(β) is
|
||
|
||
also a consequence of (18): This is natural, because in [4] I showed that space-
|
||
|
||
time transformations from a “stationary” isotropic inertial system S0 to any
|
||
|
||
other inertial system S imply complete equivalence between the three
|
||
|
||
possible pairs of assumptions chosen among the following: A1. Lorentz
|
||
|
||
contraction of bodies moving with respect to S0 ; A2. Larmor retardation of
|
||
|
||
clocks moving with respect to S0 ; A3. Two-way velocity of light equal to c in all
|
||
|
||
inertial systems and in all directions. The inverse speed of light compatible
|
||
|
||
with (18) was shown to be [1]:
|
||
|
||
L O 1 NM QP ~c (θ
|
||
|
||
)
|
||
|
||
=
|
||
|
||
1 c
|
||
|
||
+
|
||
|
||
β c
|
||
|
||
+
|
||
|
||
e1
|
||
|
||
R( β )
|
||
|
||
cosθ
|
||
|
||
(19)
|
||
|
||
where θ is the angle between the direction of propagation of light and the
|
||
|
||
absolute velocity v of S. The transformations (18) represent the complete set
|
||
|
||
of theories “equivalent” to the Special Theory of Relativity (STR): if e1 is varied, different elements of this set are obtained. The Lorentz transformation
|
||
|
||
is found as a particular case with e1 = −β / c R(β) . Different values of e1 are
|
||
|
||
obtained from different clock-synchronisation “conventions.” In all cases
|
||
|
||
except that of STR, such values exclude the validity of the relativity principle,
|
||
|
||
and imply the existence of a privileged frame [1]. For all these theories only
|
||
|
||
subluminal motions are possible.
|
||
|
||
In the previous sections we found a ratio of the one-way velocities of
|
||
|
||
light along the rim of the disk, and relative to the disk itself, different from 1
|
||
|
||
and given by Eq (12). Our principle of local equivalence between the rim of
|
||
|
||
the disk and the “tangent” inertial frame requires (12) to apply in the latter
|
||
|
||
frame as well. Eq. (19) applied to the cases θ = 0 and θ = π yields
|
||
|
||
L O L O 1 NM QP NM QP ~c (0)
|
||
|
||
=
|
||
|
||
1 c
|
||
|
||
+
|
||
|
||
β c
|
||
|
||
+
|
||
|
||
e1 R( β )
|
||
|
||
;
|
||
|
||
1 ~c (π )
|
||
|
||
=
|
||
|
||
1 c
|
||
|
||
−
|
||
|
||
β c
|
||
|
||
+
|
||
|
||
e1 R( β )
|
||
|
||
(20)
|
||
|
||
This gives
|
||
|
||
~c (π ) ~c (0)
|
||
|
||
=
|
||
|
||
1+β 1−β
|
||
|
||
+ −
|
||
|
||
c e1R(β) c e1R(β)
|
||
|
||
which can agree with (12) only if
|
||
|
||
e1 = 0
|
||
|
||
(21)
|
||
|
||
The space-dependent term in the transformation of time is thus seen to
|
||
|
||
disappear from (18). The same result (21) was obtained in [1] by requiring that
|
||
|
||
the Sagnac effect be explained on the rotating disk also, and not only in the
|
||
|
||
in Relativistic Physics
|
||
|
||
75
|
||
|
||
laboratory. See also Ref. [5] for a detailed discussion of that effect both from the special and the general relativistic point of view.
|
||
|
||
4. The inertial transformations
|
||
|
||
In the previous section it was shown that the condition e1 = 0 has
|
||
|
||
necessarily to be used. This generates transformations different from the
|
||
|
||
Lorentz ones [1]:
|
||
|
||
R x = x0 − βct0 | R(β) S| y = y0
|
||
|
||
(22)
|
||
|
||
| z = z0 T|| t = R(β)t0
|
||
|
||
The velocity of light predicted by (22) can be found by taking e1 = 0 in (19):
|
||
|
||
1 ~c (θ )
|
||
|
||
=
|
||
|
||
1
|
||
|
||
+
|
||
|
||
β cosθ c
|
||
|
||
(23)
|
||
|
||
The transformation (22) can be inverted and gives:
|
||
|
||
R| LM OP x0
|
||
|
||
=
|
||
|
||
R( β )
|
||
|
||
x+
|
||
|
||
βc R2(β) t
|
||
|
||
|| N Q y0 = y
|
||
|
||
S| z0 = z
|
||
|
||
T|| t0
|
||
|
||
=
|
||
|
||
1 R( β )
|
||
|
||
t
|
||
|
||
(24)
|
||
|
||
Note the formal difference between (22) and (24). The latter implies, for example, that the origin of S0 (x0 = y0 = z0 = 0) is described in S by y = z = 0 and by
|
||
|
||
x=
|
||
|
||
−
|
||
|
||
βc 1−β2
|
||
|
||
t
|
||
|
||
(25)
|
||
|
||
This origin is thus seen to move with speed β c/(1−β2), which can exceed c, but cannot be superluminal. In fact, a light pulse seen from S to propagate in the same direction as S0 has θ = π, and thus [using (23)] has speed ~c (π ) = c / (1 − β) , which can easily be checked to satisfy
|
||
|
||
c ≥ cβ 1−β 1−β2
|
||
|
||
One of the new features of these transformations is, of course, the presence of relative velocities exceeding c. This is not a problem, in practice, because oneway velocities have never been measured. Absolute velocities are, instead, always smaller than c [1]. It is clear from (25) that the velocity of S0 relative to S is not equal and opposite to that of S relative to S0. In STR one is used to relative velocities that are always equal and opposite, but this symmetry is a
|
||
|
||
76
|
||
|
||
Open Questions
|
||
|
||
consequence of the synchronisation used, and cannot be expected to hold more generally [1].
|
||
Now consider a third inertial system S', moving with velocity β'c and its transformation from S0, which of course is given by (18) with β ′ replacing β.
|
||
By eliminating the S0 variables, one can obtain the transformation between the two moving systems S and S':
|
||
|
||
R| LM OP x′
|
||
|
||
=
|
||
|
||
R( β ) R(β ′)
|
||
|
||
x
|
||
|
||
−
|
||
|
||
β′−β R2 (β)
|
||
|
||
ct
|
||
|
||
|| N Q y′ = y
|
||
|
||
S| z′ = z
|
||
|
||
(26)
|
||
|
||
| t′ = R(β ′) t T| R(β)
|
||
|
||
Equations (22)-(24)-(26) will be called “inertial transformations.” In its most general form (26) the inertial transformation depends on two velocities (v and v'). When one of them is zero, either S or S' coincide with the privileged system S0, and the transformation (26) becomes (22) or (24), respectively.
|
||
One feature that characterizes the transformations (22)-(24)-(26) is absolute simultaneity: two events taking place at different points of S but at the same t are judged to be simultaneous also in S' (and vice versa). The existence of absolute simultaneity does not imply that time is absolute: on the contrary, the β-dependent factor in the transformation of time gives rise to timedilation phenomena similar to those of STR. Time dilation in another sense is, however, also absolute: a clock at rest in S is seen from S0 to run slower, but a clock at rest in S0 is seen from S to run faster, so that both observers agree that motion relative to S0 slows down the pace of clocks. The difference with respect to STR exists because a clock T0 at rest in S0 must be compared with clocks at rest at different points of S, and the result is, therefore, dependent on the “convention” adopted for synchronising the latter clocks.
|
||
Absolute length contraction can also be deduced from (25): all observers agree that motion relative to S0 leads to contraction. The discrepancy with the STR is due to the different conventions concerning clock synchronisation: the length of a moving rod can only be obtained by marking the simultaneous positions of its end points, and therefore depends on the very definition of simultaneity of distant events.
|
||
|
||
5. Further comments on the discontinuity problem
|
||
|
||
Our choice of synchronisation (called “absolute” by Mansouri and Sexl
|
||
|
||
[6]) was made by considering rotating platforms. The main result of this paper
|
||
|
||
is Eq. (12): the ratio
|
||
|
||
ρ
|
||
|
||
≡
|
||
|
||
~c (π ) ~c (0)
|
||
|
||
in Relativistic Physics
|
||
|
||
77
|
||
|
||
ρ
|
||
|
||
1
|
||
|
||
acceleration
|
||
Figure 1. The ratio ρ = ~c (π ) / ~c (0) plotted as a function of acceleration for
|
||
rotating platforms of constant peripheral velocity and increasing radius
|
||
(decreasing acceleration). The prediction of SRT is 1 (black dot on the ρ axis) and is not continuous with the ρ value for the rotating platforms.
|
||
has been calculated along the rims of the platforms and shown, under very general conditions, to have the value (12), which in general is different from unity. Therefore, the velocities of light parallel and anti-parallel to the disk peripheral velocity are not the same. For SRT, this is a serious problem, because a set of platforms with growing radius, but all with the same peripheral velocity, locally approaches better and better an inertial frame. To say that the radius becomes very large with constant velocity is the same as saying that the centripetal acceleration goes to zero with constant velocity. The logical situation is shown in Fig. 1: SRT predicts a discontinuity for ρ at
|
||
zero acceleration, a sudden jump from the accelerated to the “inertial” reference frames. While all experiments are performed in the real physical world ( a ≠ 0, ρ = (1 + β) / (1 − β) ), our theoretical physics seems to have gone out of the world ( a = 0, ρ = 1 )!
|
||
The above discontinuity probably is the origin of the synchronisation problems encountered by the Global Positioning System: after all, Earth is also a kind of rotating platform.
|
||
It should be stressed that non-invariant velocity of light is required for all (but one!) inertial systems. In fact, given any such system, and a small region of it, it is always possible to conceive a large and rotating circular platform, a small part of which is locally at rest in that region, and the result (12) must then apply. Therefore, the velocity of light is non-isotropic in all inertial reference frames, with the exception of one ( S0 ) where isotropy can be postulated.
|
||
|
||
78
|
||
|
||
Open Questions
|
||
|
||
Finally, one must also conclude that the famous synchronisation problem [7] is solved by nature itself: it is not true that the synchronisation procedure can be chosen freely, because the convention usually adopted leads to an unacceptable discontinuity in the physical theory.
|
||
It was pointed out by T. Van Flandern [8] that an “orthodox” approach to dealing with the rotating platform problem is to consider a positiondependent desynchronization, with respect to the laboratory clocks, as an objective phenomenon, concretely applicable to the clocks set at different points of the rim of the platform. The Lorentz transformation of time
|
||
t′ = t − xβ / c 1−β2
|
||
|
||
can in fact be written
|
||
|
||
t′ 1− β2 − t = −xβ / c
|
||
|
||
(27)
|
||
|
||
and can be read as follows: the difference between the time t′ of the
|
||
|
||
“moving” frame S′ (corrected by a 1 − β 2 factor in order to cancel the time
|
||
|
||
dilation effect) and the time t of the “stationary” frame S has a linear
|
||
|
||
dependence on the x coordinate of S as given by (27). This difference is
|
||
|
||
called “desynchronization.”
|
||
|
||
The “orthodox idea” would be to apply the previous approach to the
|
||
|
||
rotating platform. Assume that rotation is from left to right in Fig. 2. Two
|
||
|
||
flashes of light are emitted at laboratory time t = 0 by the source Σ in
|
||
|
||
opposite directions along the platform rim. When the right-moving (left-
|
||
|
||
moving) flash reaches point A (point B ) at laboratory time tA ( tB ) after
|
||
|
||
covering a distance xA ( xB ) [measured in the laboratory from Σ along the
|
||
|
||
platform border], it will find a local clock which is desynchronized by an
|
||
|
||
amount ΔtA ( ΔtB ) with respect to the laboratory clocks given by
|
||
|
||
ΔtA = xAv / c 2 + α
|
||
|
||
;
|
||
|
||
ΔtB = − xBv / c 2 + α
|
||
|
||
(28)
|
||
|
||
where α represents whatever desynchronization the clock placed in Σ
|
||
|
||
might have with respect to the laboratory clocks.
|
||
|
||
It is at once possible to see that Eq. (28) is in sharp contradiction with the
|
||
|
||
rotational invariance assumed above, which leads to the existence of the
|
||
|
||
discontinuity in Fig. 1: if the inertial system S0 (in which the centre of the
|
||
|
||
perfectly circular platform is at rest) is isotropic, and if the platform is set in
|
||
|
||
motion in a regular way, no difference between clocks on its rim can ever
|
||
|
||
arise, and Eq. (28) represents a logical impossibility. It is impossible to
|
||
|
||
understand why the clock in A should be desynchronized differently from
|
||
|
||
the clock in B, unless this is achieved artificially by some observer.
|
||
|
||
in Relativistic Physics
|
||
|
||
79
|
||
|
||
P
|
||
|
||
A B
|
||
Σ
|
||
Figure 2. Two flashes of light emitted at laboratory time t = 0 by Σ in opposite directions along the platform border overlap in a point P where the local clock on the platform should show two incompatible times.
|
||
|
||
Furthermore, the whole argument culminating in Fig. 1 was based on the existence of a single clock, arbitrarily retarded with respect to the laboratory clocks, so that even if the previous strange desynchronization were assumed true, it would still be possible to use one of the clocks satisfying (28) to repeat the argument and again deduce the existence of the same discontinuity!
|
||
It should also be stressed that the experimental evidence is that many little clocks (muons) injected in different positions of the CERN Muon Storage Ring behave in exactly the same way, independently of their position in the ring [2]. And in the present case, it is not possible to conceive a human intervention that desynchronizes the muons. Therefore, not only common sense, but also direct experimentation, shows that a position-dependent desynchronization is out of the question.
|
||
Finally, it is possible to show that the desynchronization in (28), which is unable to cure the discontinuity of Fig. 1, as just shown, introduces a further discontinuity in the time shown by clocks set all around the platform rim, so that far from simplifying the problem, it introduces new complications.
|
||
From the point of view of the laboratory observer, the space between Σ and the right-moving (left-moving) flash widens at a rate c − v ( c + v ), so that from (28) we get
|
||
|
||
a f a f c − v tA = xA
|
||
|
||
;
|
||
|
||
c + v tB = xB
|
||
|
||
Therefore
|
||
|
||
ΔtA = FHG c − vIKJ tA v / c 2 + α ; ΔtB = −FHG c + vIKJ tB v / c 2 + α
|
||
|
||
(29)
|
||
|
||
There will be a time tP when the two flashes overlap at a point P , while moving in opposite directions. When this happens
|
||
|
||
80
|
||
|
||
Open Questions
|
||
|
||
tA = tB = tP
|
||
|
||
(30)
|
||
|
||
The problem is that the Eqs. (29) should both be valid for the same clock at the
|
||
|
||
common time tP , but they are instead incompatible if (30) is satisfied, as their equality is easily shown to reduce to c = −c .
|
||
|
||
References
|
||
[1] F. Selleri, Found. Phys. 26, 641 (1996). [2] J. Bailey et al., Nature 268, 301 (1977) . [3] G. Sagnac, Compt. Rend. 157, 708 (1913); ibid. 1410; P. Langevin, Compt. Rend. 173,
|
||
831 (1921); 205, 304 (1937); E. J. Post, Rev. Mod. Phys. 39, 475 (1967). [4] F. Selleri, Apeiron, 4, 100 (1997). [5] F. Goy and F. Selleri, Found. Phys. Lett. 10, 73 (1997). [6] R. Mansouri and R. Sexl, Gen.Relat. Gravit. 8, 497 (1977). [7] A. Einstein, Relativity, the Special, the General Theory, (Chicago, 1951); H.
|
||
Reichenbach, The Philosophy of Space & Time (Dover Publ., New York, 1958). M. Jammer, Some fundamental problems in the special theory of relativity, in: Problems in the Foundations of Physics, G. Toraldo di Francia, ed., (Società Italiana di Fisica, Bologna, and North Holland, Amsterdam, 1979). [8] Tom Van Flandern, private communication.
|
||
|
||
What the Global Positioning System Tells Us about Relativity
|
||
Tom Van Flandern Univ. of Maryland & Meta Research P.O. Box 15186, Chevy Chase, MD 20825, USA
|
||
1. What is the GPS?
|
||
The Global Positioning System (GPS) consists of a network of 24 satellites in roughly 12-hour orbits, each carrying atomic clocks on board. The orbital radius of the satellites is about four Earth-radii (26,600 km). The orbits are nearly circular, with a typical eccentricity of less than 1%. Orbital inclination to the Earth’s equator is typically 55 degrees. The satellites have orbital speeds of about 3.9 km/s in a frame centered on the Earth and not rotating with respect to the distant stars. Nominally, the satellites occupy one of six equally spaced orbital planes. Four of them occupy each plane, spread at roughly 90degree intervals around the Earth in that plane. The precise orbital periods of the satellites are close to 11 hours and 58 minutes so that the ground tracks of the satellites repeat day after day, because the Earth makes one rotation with respect to the stars about every 23 hours and 56 minutes. (Four extra minutes are required for a point on the Earth to return to a position directly under the Sun because the Sun advances about one degree per day with respect to the stars.)
|
||
The on-board atomic clocks are good to about 1 nanosecond (ns) in epoch, and about 1 ns/day in rate. Since the speed of light is about one foot per nanosecond, the system is capable of amazing accuracy in locating anything on Earth or in the near-Earth environment. For example, if the satellite clocks are fully synchronized with ground atomic clocks, and we know the time when a signal is sent from a satellite, then the time delay for that signal to reach a ground receiver immediately reveals the distance (to a potential accuracy of about one foot) between satellite and ground receiver. By using four satellites to triangulate and determine clock corrections, the position of a receiver at an unknown location can be determined with comparable precision.
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Open Questions in Relativistic Physics
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Edited by Franco Selleri (Apeiron, Montreal, 1998)
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2. What relativistic effects on GPS atomic clocks might be seen?
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General Relativity (GR) predicts that clocks in a stronger gravitational field will tick at a slower rate. Special Relativity (SR) predicts that moving clocks will appear to tick slower than non-moving ones. Remarkably, these two effects cancel each other for clocks located at sea level anywhere on Earth. So if a hypothetical clock at Earth’s north or south pole is used as a reference, a clock at Earth’s equator would tick slower because of its relative speed due to Earth’s spin, but faster because of its greater distance from Earth’s center of mass due to the flattening of the Earth. Because Earth’s spin rate determines its shape, these two effects are not independent, and it is therefore not entirely coincidental that the effects exactly cancel. The cancellation is not general, however. Clocks at any altitude above sea level do tick faster than clocks at sea level; and clocks on rocket sleds do tick slower than stationary clocks.
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For GPS satellites, GR predicts that the atomic clocks at GPS orbital altitudes will tick faster by about 45,900 ns/day because they are in a weaker gravitational field than atomic clocks on Earth’s surface. Special Relativity (SR) predicts that atomic clocks moving at GPS orbital speeds will tick slower by about 7,200 ns/day than stationary ground clocks. Rather than have clocks with such large rate differences, the satellite clocks are reset in rate before launch to compensate for these predicted effects. In practice, simply changing the international definition of the number of atomic transitions that constitute a one-second interval accomplishes this goal. Therefore, we observe the clocks running at their offset rates before launch. Then we observe the clocks running after launch and compare their rates with the predictions of relativity, both GR and SR combined. If the predictions are right, we should see the clocks run again at nearly the same rates as ground clocks, despite using an offset definition for the length of one second.
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We note that this post-launch rate comparison is independent of frame or observer considerations. Since the ground tracks repeat day after day, the distance from satellite to ground remains essentially unchanged. Yet, any rate difference between satellite and ground clocks continues to build a larger and larger time reading difference as the days go by. Therefore, no confusion can arise due to the satellite clock being located some distance away from the ground clock when we compare their time readings. One only needs to wait long enough and the time difference due to a rate discrepancy will eventually exceed any imaginable error source or ambiguity in such comparisons.
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3. Does the GPS confirm the clock rate changes predicted by GR and SR?
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The highest precision GPS receiver data is collected continuously in two frequencies at 1.5-second intervals from all GPS satellites at five Air Force
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in Relativistic Physics
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monitor stations distributed around the Earth. An in-depth discussion of the data and its analysis is beyond the scope of this paper. [1] This data shows that the on-board atomic clock rates do indeed agree with ground clock rates to the predicted extent, which varies slightly from nominal because the orbit actually achieved is not always precisely as planned. The accuracy of this comparison is limited mainly because atomic clocks change frequencies by small, semi-random amounts (of order 1 ns/day) at unpredictable times for reasons that are not fully understood. As a consequence, the long-term accuracy of these clocks is poorer than their short-term accuracy.
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Therefore, we can assert with confidence that the predictions of relativity are confirmed to high accuracy over time periods of many days. In ground solutions with the data, new corrections for epoch offset and rate for each clock are determined anew typically once each day. These corrections differ by a few ns and a few ns/day, respectively, from similar corrections for other days in the same week. At much later times, unpredictable errors in the clocks build up with time squared, so comparisons with predictions become increasingly uncertain unless these empirical corrections are used. But within each day, the clock corrections remain stable to within about 1 ns in epoch and 1 ns/day in rate.
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The initial clock rate errors just after launch would give the best indication of the absolute accuracy of the predictions of relativity because they would be least affected by accumulated random errors in clock rates over time. Unfortunately, these have not yet been studied. But if the errors were significantly greater than the rate variance among the 24 GPS satellites, which is less than 200 ns/day under normal circumstances, it would have been noticed even without a study. So we can state that the clock rate effect predicted by GR is confirmed to within no worse than ±200 / 45,900 or about 0.7%, and that predicted by SR is confirmed to within ±200 / 7,200 or about 3%. This is a very conservative estimate. In an actual study, most of that maximum 200 ns/day variance would almost certainly be accounted for by differences between planned and achieved orbits, and the predictions of relativity would be confirmed with much better precision.
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12-hour variations (the orbital period) in clock rates due to small changes in the orbital altitude and speed of the satellites, caused by the small eccentricity of their orbits, are also detected. These are observed to be of the expected size for each GPS satellite’s own orbit. For example, for an orbital eccentricity of 0.01, the amplitude of this 12-hour term is 23 ns. Contributions from both altitude and speed changes, while not separable, are clearly both present because the observed amplitude equals the sum of the two predicted amplitudes.
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4. Is the speed of light constant?
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Other studies using GPS data have placed far more stringent limits than we will here. However, our goal here is not to set the most stringent limit on
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possible variations in the speed of light, but rather to determine what the maximum possible variation might be that can remain consistent with the data. The GPS operates by sending atomic clock signals from orbital altitudes to the ground. This takes a mere 0.08 seconds from our human perspective, but a very long (although equivalent) 80,000,000 ns from the perspective of an atomic clock. Because of this precision, the system has shown that the speed of radio signals (identical to the “speed of light”) is the same from all satellites to all ground stations at all times of day and in all directions to within ±12 meters per second (m/s). The same numerical value for the speed of light works equally well at any season of the year.
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Technical note: Measuring the one-way speed of light requires two clocks, one on each end of the path. If the separation of the clocks is known, then the separation divided by the time interval between transmission and reception is the one-way speed of the signal. But measuring the time interval requires synchronizing the clocks first. If the Einstein prescription for synchronizing clocks is used, then the measured speed must be the speed of light by definition of the Einstein prescription (which assumes the speed of light is the same in all inertial frames). If some other non-equivalent synchronization method is used, then the measured speed of the signal will not be the speed of light. Clearly, the measured signal speed and the synchronization prescription are intimately connected.
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Our result here merely points out that the measured speed does not change as a function of time of day or direction of the satellite in its orbit when the clock synchronization correction is kept unchanged over one day. As for seasonal variations, all satellite clocks are “steered” to keep close to the U.S. Naval Observatory Master Clock so as to prevent excessive build up of errors from random rate changes over long time periods. So we cannot make direct comparisons between different seasons, but merely note that the same value of the speed of light works equally well in any season.
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5. What is a “GPS clock”?
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Cesium atomic clocks operate by counting hyperfine transitions of cesium atoms that occur roughly 10 billion times per second at a very stable frequency provided by nature. The precise number of such transitions was originally calibrated by astronomers, and is now adopted by international agreement as the definition of one atomic second.
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GPS atomic clocks in orbit would run at rates quite different from ground clocks if allowed to do so, and this would complicate usage of the system. So the counter of hyperfine cesium transitions (or the corresponding phenomenon in the case of rubidium atomic clocks) is reset on the ground before launch so that, once in orbit, the clocks will tick off whole seconds at the same average rate as ground clocks. GPS clocks are therefore seen to run slow compared to ground clocks before launch, but run at the same rate as ground clocks after launch when at the correct orbital altitude.
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in Relativistic Physics
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We will refer to a clock whose natural ticking frequency has been precorrected in this way as a “GPS clock”. This will help in the discussion of SR effects such as the twins paradox. A GPS clock is pre-corrected for relativistic rate changes so that it continues to tick at the same rate as Earth clocks even when traveling at high relative speeds. So a GPS clock carried by the traveling twin can be used to determine local time in the Earth’s frame at any point along the journey—a great advantage for resolving paradoxes.
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6. Is acceleration an essential part of resolving the “twins paradox”?
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If the traveling twin carries both a natural clock and a GPS clock on board his spacecraft, he can observe the effects predicted by SR without need of any acceleration in the usual twins paradox. That is as it should be because cyclotron experiments have shown that, even at accelerations of 1019 g (g = acceleration of gravity at the Earth’s surface), clock rates are unaffected. Only speed affects clock rates, but not acceleration per se.
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Suppose that the traveling twin is born as his spaceship passes by Earth and both of his on-board clocks are synchronized with clocks on Earth. The natural on-board clock ticks more slowly than the GPS on-board clock because the rates differ by the factor gamma that SR predicts for the slowing of all clocks with relative speed v. [gamma = 1/√(1 – v2/c2)] But everywhere the traveling twin goes, as long as his speed relative to the Earth frame does not change, his GPS clock will give identical readings to any Earth-synchronized Earth-frame clock he passes along the way. And his natural clock will read less time elapsed since passing Earth by the factor gamma. His biological processes (including aging), which presumably operate at rates comparable to the ticking of the natural clock, are also slowed by the factor gamma.
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Since this rate difference is true at every instant of the journey beginning with the first, there are no surprises if the traveling twin executes a turnaround without change of speed and returns to Earth. He will find on journey’s completion what he has observed at every step of the journey: His natural clock and his biological age are slower and younger by the factor gamma than that of his Earth-frame counterparts everywhere along his journey, including at its completion. The same would have been true if he had not turned around, but merely continued ahead. He would be younger than his peers on any planet encountered who claim to have been born at the same time that the traveler was born (i.e., when he passed Earth) according to their Earth-frame perspective.
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Clearly, acceleration or the lack thereof has no bearing on the observed results. If acceleration occurs, it is merely to allow a more convenient comparison of clocks by returning to the starting point. But since the traveler can never return to the same point in space-time merely by returning to the same point in space, the results of a round-trip comparison are no different in kind from those made anywhere along the journey. The traveler always
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judges that his own aging is slower than that in any other frame with a relative motion.
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Then why isn’t the traveler entitled to claim that he remained at rest and the Earth moved? The traveler is unconditionally moving with respect to the Earth frame and therefore his clocks unconditionally tick slower and he ages less as judged by anyone in the Earth frame. However, if the traveler makes the same judgment, the result will depend on whether he values his natural clock or his GPS clock as the better timekeeper. If he takes readings on the GPS clock to represent Earth time, his inferences will always agree with those of Earth-frame observers. If he instead uses the results of the exchange of light signals to make inferences of what time it is at distant locations, he will conclude that the Earth-bound twin is aging less than himself because of their relative motion. But on the occasion of any acceleration his spaceship undergoes, the traveler will infer a discontinuity in the age of his Earth-bound counterpart, which can be either forward or backward in time depending on which direction the traveler accelerates. At the end of any round trip after any number of such accelerations, the traveler and Earth-bound twins will always agree about who should have aged more.
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7. Does the behavior of GPS clocks confirm Einstein SR?
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To answer this, we must make a distinction between Einstein SR and Lorentzian Relativity (LR). Both Lorentz in 1904 and Einstein in 1905 chose to adopt the principle of relativity discussed by Poincaré in 1899, which apparently originated some years earlier in the 19th century. Lorentz also popularized the famous transformations that bear his name, later used by Einstein. However, Lorentz’s relativity theory assumed an aether, a preferred frame, and a universal time. Einstein did away with the need for these. But it is important to realize that none of the 11 independent experiments said to confirm the validity of SR experimentally distinguish it from LR—at least not in Einstein’s favor.
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Several of the experiments bearing on various aspects of SR (see Table 1) gave results consistent with both SR and LR. But Sagnac in 1913, Michelson following the Michelson-Gale confirmation of the Sagnac effect for the rotating Earth in 1925 (not an independent experiment, so not listed in Table 1), and Ives in 1941, all claimed at the time they published that their results were experimental contradictions of Einstein SR because they implied a preferred frame. In hindsight, it can be argued that most of the experiments contain some aspect that makes their interpretation simpler in a preferred frame, consistent with LR. In modern discussions of LR, the preferred frame is not universal, but rather coincides with the local gravity field. Yet, none of these experiments is impossible for SR to explain.
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in Relativistic Physics
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For example, Fresnel showed that light is partially dragged by the local medium, which suggests a certain amount of frame-dependence. Airy found that aberration did not change for a water-filled telescope, and therefore did not arise in the telescope tube. That suggests it must arise elsewhere locally. Michelson-Morley expected the Earth’s velocity to affect the speed of light because it affected aberration. But it didn’t. If these experimenters had realized that the aether was not a single entity but changed with the local gravity field, they would not have been surprised. It might have helped their understanding to realize that Earth’s own Moon does not experience aberration as the distant stars do, but only the much smaller amount appropriate to its small speed through the Earth’s gravity field.
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Another clue came for De Sitter in 1913, elaborated by Phipps [3], both of whom reminded us that double star components with high relative velocities nonetheless both have the same stellar aberration. This meant that the relative velocity between a light source and an observer was not relevant to stellar aberration. Rather, the relative velocity between local and distant gravity fields determined aberration. In the same year, Sagnac showed nonnull results for a Michelson-Morley experiment done on a rotating platform. In the simplest interpretation, this demonstrated that speeds relative to the local gravity field do add to or subtract from the speed of light in the experiment, since the fringes do shift. The Michelson-Gale experiment in 1925 confirmed that the Sagnac result holds true when the rotating platform is the entire Earth’s surface.
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When Ives and Stilwell showed in 1941 that the frequencies of radiating ions depended on their motion, Ives thought he had disposed once and for all of the notion that only relative velocity mattered. After all, the ions emitted at a particular frequency no matter what frame they were observed from. He was unmoved by arguments to show that SR could explain this too because it seemed clear that nature still needed a preferred frame, the motion relative to which would determine the ion frequencies. Otherwise, how would the ions know how often to radiate? Answers to Ives’ dilemma exist, but not with a comparable simplicity. Richard Keating was surprised in 1982 that two atomic clocks traveling in opposite directions around the world, when compared with a third that stayed at home, showed slowing that depended on their
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Experiment Description
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Bradley
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Discovery of aberration of light
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Y e a r
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1728
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