Translation:Is the Michelson Experiment Conclusive?

by Max von Laue

Is the Michelson experiment conclusive?

by M. Laue.

Since the interference experiment of Michelson – as the foundation of the theory of relativity – has achieved a unique importance for the development of physics, it is necessary to ensure its meaning against all doubts, thus also against the reservations expressed by Kohl^[1]. It lies in the nature of things, that only well-known theorems from the theory of interference phenomena can be debated here; the intention, which is to establish clarity, may excuse it when we should become somehow long-winded here.

We start with stating the essential parts of Michelson's interferometer; in neglecting the other parts of the much more complicated apparatus, we are in agreement with Kohl. Those are: the (half-permeably silvered) plane glass plate $P$ (which we can assume as very thin against the wavelength of light), at which the light-ray incident under ca. 45° is split into a reflected and a passing ray; furthermore two mirrors $I$ and $II$, upon which the latter impinge approximately perpendicular, so that they come back to plate $P$ after reflection. A repeated splitting of them at $P$, gives (besides other things) two rays propagating to telescope $F$, whose interference can be observed in $F$.

The interference image now can be part of two totally different types; one studies them best, when one asks after the location of the mirror image casted from $P$ by mirror $II$ (indicated in the Fig. by $II'$).

I. That mirror image is exactly parallel to mirror $I$. Then one sees (in the focal plane of the telescope objective) the interference phenomenon at the plan-parallel layer of air $I$ $II'$.

The physical difference between the front- and the back-surface of a real plan-parallel plate, causing the phase shift $\pi$ at one of the interfering rays, is replaced here by the difference of the reflections from the silvered surface of plate $P$; because one of the rays becomes reflected in glass, the other one in air.

These interference curves of same inclination are known to be concentric circles^[2]. The rate-difference amounts

$(a-b)\cos \beta \,$

when we denote by $\beta$ the angle between the ray direction and the normal of the plates, by $a$ and $b$ the distances of mirrors $I$ and $II$ from $P$, measured along the midmost ray. From that it follows first, that the rate-difference cannot be zero for any of the fringes, except when $a=b$, but in this case no interference fringes occur at all. In white light these fringes are therefore, if at all, only to be seen as blurred. If one also changes the difference $a-b$, then every fringe is moving, so that

$(a-b)\cos \beta \,$

remains constant, thus in the vicinity of the center of the ring systems it is moving, so that

$(a-b)\left(1-{\frac {\beta ^{2}}{2}}\right)$

doesn't change. The fringe displacement is thus not even approximately proportional to the change of $a-b$.

II. The mirror image $II'$ forms a certain angle with $I$. Then in the air-wedge between $II'$ and $I$, interference fringes occur that are complementary to the Newtonian fringes of equal thickness. They are located in that plane, in which the telescope objective is projecting the wedge, and they are lines parallel to the edge of the wedge. If $a$ is chosen as equal to $b$, then $II'$ intersects mirror $I$. Then we have a fringe of rate-difference zero in the middle of the visual field, thus a minimum of intensity being the only one that is not displaced when changing the wave length, and remains clear and colorless under illumination by white light. If one of the arm lengths $a$ or $b$ is slightly changed, then this is the cause for the displacement of the mentioned double-wedge in its plane, and the interference fringes are traveling along with it perpendicular to their direction, by a distance proportional to the difference $a-b$. Another difference of these fringes with respect to the plan-parallel rings is the circumstance, that this whole system also occurs under the illumination by a single wave, while as regards the former one, only one point of the visual field is illuminated by that.

As to the relevant experiment concerning the influence of Earth's motion, interference fringes of the second kind have been used without doubt. This can be unequivocally seen from Michelson's representation in his book "Light waves and their uses"^[3]. From the later papers of Morley and Miller, I literally quote as references: "In one of the usual adjustments of distances and angles, parallel fringes are seen when the eye or the telescope is made to give distinct vision of one of the mirrors I or II. The fringes apparently coincide with these surfaces. A central fringe is black; on either side are coloured fringes, less and less distinct till they fade away into uniform illumination. If the path of either ray is shortened, the fringes move rapidly to one side. If we engrave a scale on I or II, we can, after any alteration of one of the paths, restore with great accuracy and ease the former relations by bringing the central dark fringe to its original place on this scale." Furthermore: "A telescope magnifying thirty-five diameters gave distinct vision of mirror 8, at whose surface the interference-fringes are apparently localized."^[4] From this quotation it also emerges, that the authors have seen, that a change of the arm lengths causes the fringe displacement required by the theory. Since the influence of motion changes the traversing times of $a$ or $b$, it is thus acting as a mechanical change of the distances, thus by this alone the conclusiveness of the Michelson experiment is proven without respect to all other considerations.

Now to the work of Kohl. In its first part, it deals with the influence of Earth's motion upon the geometric law of reflection; namely it proves, that the images produced by reflection at $I$, $II$ and $P$ fall apart into $F$ visible images of the light source in consequence of motion, so that the light source is thus apparently doubled for the observer. This contradicts the relativity principle, yet it is a necessary consequence of the theory of the stationary aether. If the distance of both mirror images wouldn't be unobservably small, then one would have another criterion for experimentally deciding between both theories. This appears to be paradox at first sight, because the geometric law of reflection (shared by all theories) is entirely based on the assumption of linearity of the limiting condition, thus it is common to all of them. Though the agreement is missing between them with respect to the form, that has to be ascribed to the moving interferometer. According to the theory of stationary aether, the angle between plate $P$ and the central fringe, is 45° in motion as well as at rest. According to relativity theory, however, due to Lorentz contraction it deviates in the case of motion by magnitudes of second order from that value, when it amounts 45° in the state of rest. Conversely, if it would have exactly this value in the case of motion, then it would be different with respect to the co-moving system. One recognizes without further ado, that in this case two mirror images must occur, and thus the discussed result is confirmed in this way. Moreover, this question is of minor importance, since the distance of both images amounts ca. $10^{-5}$, thus it (at best) could be observed only by the best microscope with the greatest aperture; as regards the telescope of the interferometer, however, the circles of diffraction (of which the image consists) have diameters that exceed the geometric projection of such a distance by a large multiple.^[5]

The error of Kohl lies at another place. The author namely assumes, that it is about interferences of same inclination at a plan-parallel plate; namely, according to him the boundaries of the telescope- and the collimator objective should constrict the ray path, so that only the midmost spot of the whole ring-system should be visible, so that $(a-b)\cos \beta$ has (with optical precision) the same value everywhere in the visual field.^[6] This emerges with full certainty from his Fig. 3 and 4; because while discussing them on the basis of the doubling of the light source under consideration of smallest angles ($10^{-8}$), the angles between the mirrors $I$ and $P$ as well as $II$ and $P$ are always assumed as being exactly 45°; this also follows from the fact, that he always sees the arm lengths $a$ and $b$ as slightly different, and eventually from the fact, that (neglecting the difference $a-b$) the inclination $i$ of the rays against the central ray occurs as the only variable that determines the intensity. To explain the occurrence of rectilinear parallel rays, he makes the assumption not supported by any comment in the publications of Michelson or Morley, that light experiences diffraction at a slit being fixed at the collimator lens.^[7] As one can see, according to him there is a interference phenomenon of constant phase difference between two equal diffraction images. Now, that under these assumptions no fringe displacement arises as the consequence of a change of arm lengths or as the effect of motion, but (with homogeneous light) only an uniform change of brightness in the entire image, can be seen without calculation.^[8] Though it is near at hand, that it has nearly nothing to do with the real Michelson experiment any more, whose conclusiveness is in no way affected by that.

With authorization of Kohl I shall add, that after taking notice of these considerations, he completely agrees with the given explanations regarding the formation of the system of interference fringes.

Munich, Institute for theoretical physics, May 1910.

(Received May 12, 1910)

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1. E. Kohl, Ann. d. Phys. 28. p. 259. 1909
2. See e.g. E. Gehrcke, Die Anwendung der Interferenzen in der Spektroskopie und Metrologie. Braunschweig 1906, Fig. 31.
3. A. A. Michelson, Chicago 1903.
4. E. W. Morley and C. L. Miller, Phil. Mag. (6) 9. p. 669 a. 680. 1905. Esp. p. 670 a. 681. Mirror 8 corresponds to $I$.
5. See Kohl's paper, p. 285, section e.
6. See p. 285, section d; 287, line 5 from below until 288, line 9 from above; 289, line 11 ff; p. 306, summary 1.
7. p. 292, line 12 from above.
8. p. 298 a. 307, section 2.