Abstract
A rigorous wave-theoretic approach to the Michelson-Morley (M-M) experiment is presented, with special emphasis on the Huygens' principle derivation of the laws of reflection by a moving mirror. A detailed discussion of the Lorentz-Fitzgerald contraction hypothesis (CH) is included. Several mistakes appearing in the standard textbook treatments of these issues are pointed out, and a number of related historical questions are considered.
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Notes and references
A. A. Michelson and E. W. Morley, “On the relative motion of the earth and the luminiferous ether,”Am. J. Sci., 3rd ser. 34, 333–345 (1887).
The only recent author we have found who at leastrefers to a more detailed “discussion of the theory of the experiment” [namely, the one contained in L. Silberstein,The Theory of Relativity (Macmillan, London, 1924)] is J. L. Synge,Relativity: The Special Theory (North-Holland, Amsterdam, 1972), p. 157.
For instance: G. A. Articolo, “The Michelson-Morley Experiment and the Phase Shift upon a 90‡ Rotation,”Am. J. Phys. 37, 215–216 (1969).
P. Mazur, “On the Michelson-Morley experiment,”Am. J. Phys. 37, 218–219 (1969).
V. S. Soni, “A note on the ray diagram of the Michelson-Morley experiment,”Am. J. Phys. 56, 178–179 (1988).
However, Soni claims in Ref. 3(c) that “the oblique path of rays generally shown for the orientation before 90‡ rotation of the apparatus in this experiment is correct,” which seems to us both misleading and inconsistent with the author's own discussion (the more so if taken together with the legends under the drawings).
We are supposing that the reader is acquainted with the general idea of the experiment and the standard computations as can be found, for instance, in Sec. 15-3 of R. Feynman, R. B. Leighton, and M. Sands,The Feynman Lectures on Physics (Addison-Wesley, 1963), Vol. I. But practically any current physics textbook devoting a chapter or two to special relativity would serve as an appropriate reference.
Formula (1″) isnot contained in Ref. 1. Instead we find an expression which in our symbols would read which of course gives the same second-order approximation as (1′'). The rationale for the strange (*) has been recently discussed by Soni [Michelson-Morley analysis,”Am. J. Phys. 57, 1149–1150 (1989)], who concluded that it has no physical explanation so far as the classical assumption of the stationary ether hypothesis is concerned.” As a matter of fact, M-M simply “contracted” a passage in the famous 1886 paper by H.A. Lorentz (“De l'influence du mouvement de la terre sur les phénomènes lumineux,”Archives Néerlandaises,21, 103–176, the source for the idea of Fig. 1), in which he says that for a certain lengthA we have\(T'' = (2/c)(L''^2 + A^2 )^{1/2} \) and thatA is,approximately (up to first order inΒ), “the path covered by the [semitransparent] mirror during the time that the light propagates along the distance [L′']′', that is,A ≈ L′'Β. M-M conflated rather carelessly the two parts of Lorentz's argument, perhaps because they knew (as evidenced by an explicit statement in the main text and by their “Supplement” in Ref. 1) that Fig. 1 is only approximately valid. Nonetheless the way they introduce (*) has confused quite a few authors. An instance of a relatively early victim of uncritical reading of Ref. 1 is O. W. Richardson,The Electron Theory of Matter (Cambridge University Press, Cambridge, 1916), p. 279.
Apart from Lorentz's article referred to in the previous note, and the “Supplement” in Ref. 1, see, for instance, W. M. Hicks, “On the Michelson-Morley experiment relating to the drift of ether,”Philos. Mag., 6th ser. 3, 9–36 (1902), and Silberstein's book, cited in note 2. The controversy surrounding the M-M experiment for about 30 years (with varying intensity and participation of the physical community) has been reported in general terms by L. S. Swenson in hisThe Ethereal Aether (University of Texas Press, Austin, Texas, 1972), which also includes a very useful bibliography. However, a detailed account of the technical aspects of this debate seems to be still missing in the literature.
An exception is C. MØller,The Theory of Relativity (Clarendon, Oxford, 1972), who uses this principle to deduce the formula for the relative ray velocity in a moving frame (p. 13), but then mistakenly assumes that reflection at amoving mirror obeys the same law as reflection at astationary mirror (p. 26). A. Sommerfeld (Optics, Academic Press, New York, 1954) derives the correct formula for the reflection at a moving mirror by using special relativity (Sec. 13, p. 72–75; compare with the text in this article corresponding to note 15) and also uses it atone point of the argument on which his computations for the M-M experiment rest, but fails to apply it for the reflection at the semitransparent mirror. Here is, we believe, another example of the bewitchment exerted by the “isosceles triangle” 3 diagram.
For instance (again just one reference among too many to even attempt a list), see H. Weyl,Space-Time-Matter [1920; Dover, 1952 reprint (with a new preface by the author) of 1922 edition], p. 171.
Ref. 3 (a), (b) give two different formulas forT′' which both, however, reduce for θ=0 to (1″). Articolo's treatment is a straightforward generalization of the standard one; Mazur's formula is more puzzling, since the author seems intent to give a wave-theoretic rigorous expression, an aim which is clearly not achieved.
For instance, H. Weyl (Ref. 11, p. 171) gives the followingwrong second-order approximation:\(DT(\theta = 90^ \circ ) - DT(\theta = 0^ \circ ) \approx - (L' + L'')\beta ^2 /c\) This mistake is rather instructive, insofar as it shows that one has to be very careful when building formulas on the unsafe ground of an “approximate”description, rather than approximating theformulas obtained from theexact description. Probably, uncritical reliance on Weyl's discussion is ultimately responsible for the same erroneous approximation appearing in the more recent R. Becker,Electromagnetic Fields and Interactions, F. Sauter, ed. (Blaisdell, New York, 1964), p. 311. Also Silberstein (Ref. 2, p. 75) ends with the same wrong formula.
The evolution of Lorentz's thought on the M-M experiment can be verified by consulting the collection Einsteinet al., The Principle of Relativity (1922; Dover, 1952 reprint of 1923 translation), where the relevant (translated) passage from the 1895 essay “Versuch einer Theorie der elektrischen und optischen Erscheinungen in bewegten Korpern” and the 1904 article “Electromagnetic phenomena in a system moving with any velocity less than that of light” are reproduced. Poincaré's criticism of the contraction hypothesis referred to in the text was first expressed in a report at the International Congress of Physics in Paris (1900), then republished in Chap. X ofLa Science et l'Hypothèse (1902).
For instance, see the very first paper of A. Einstein on special relativity, Sec. 8; a recent translation can be found in A. I. Miller,Albert Einstein's Special Theory of Relativity (Addison-Wesley, Reading, Massachusetts, 1981).
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Capria, M.M., Pambianco, F. On the Michelson-Morley experiment. Found Phys 24, 885–899 (1994). https://doi.org/10.1007/BF02067653
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DOI: https://doi.org/10.1007/BF02067653