View source for Translation:On an Experiment on the Optics of Moving Bodies

{{translation header
 | title      = On an Experiment on the Optics of Moving Bodies
 | author     = Max von Laue
 | language  = de
 | original  =
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 | year       = 1911
| portal=Relativity
| wikipedia=Sagnac effect
 | notes    =  
In German: ''Über einen Versuch zur Optik der bewegten Körper'', Münchener Sitzungsberichte: 405-412. [http://www.archive.org/details/UeberEinenVersuchZurOptikDerBewegtenKoerper Online]
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{{center|{{x-larger|On an Experiment on the Optics of Moving Bodies}}}}


{{center|By '''M. Laue.'''}}
{{center|Presented by A. {{sc|Sommerfeld}} in the session of July 1, 1911.}}


In 1904 {{sc|Michelson}}<ref>A. A. {{sc|Michelson}}, [[Relative Motion of Earth and Aether|Phil. Mag. (6) 8, 716, 1904]]</ref> made a proposal for an interference experiment related to the rotation of the earth around its axis in a similar way, as the well-known experiment for translational motion carried out several times by him, {{sc|Morley}}, and {{sc|Miller}}. The essential idea is that two coherent rays are crossing a circle on earth in the opposite sense, then the required times are compared by an interference experiment. The experiment is aimed to bring a decision to the question (related to {{sc|Stokes'}} theory) whether the aether next to earth joins its rotation or not. In the first case, no influence of rotation on the location of the middle interference fringe of same phase is expected, but in the second case we have to expect it. We want to see what can be expected as regards the various electrodynamic theories of moving bodies, whether the experiment could bring a decision between them. Primarily we think about the theory of relativity and {{sc|Lorentz}}'s theory of a stationary aether, which is perhaps for opposition's sake called the "absolute theory"; we mean the theory in their older form, because the subsequent one published in 1904 is not different from the theory of relativity even as regards the fundamental equations - as far as processes in empty space or in air (which in both theories are not considerable different even when the air is in motion) are concerned. {{pagenum|406}} We also want to allude to the theories of {{sc|Cohn}} and {{sc|Hertz}}.

==a) Theory of relativity.==

We first presuppose a simplified experimental arrangement. The experiment may take place on a plane rotating around an axis (the axis being perpendicular to the plane and at rest in the valid<ref>A system which rotates in respect to a valid System ''K''{{sup|0}} is ''not'' valid.</ref> reference frame K{{sup|0}}) with angular velocity ''ω''; the intersection of the axis with the plane is ''M''. Around ''M'' we describe a circle of radius ''r'', and in a point ''P'' of its periphery the reflecting plate lies perpendicular to it, which according to {{sc|Michelson}}'s proposal at first separates the coherent rays in order to unite them after their return, and then to reflect them into the telescope. In (''n'' - 1) additional points of the same circle forming together with ''P'' the corners of a regular ''n''-gon, we set up several mirrors which should reflect both rays so that they are united. We also note that the speed against the system ''K''{{sup|0}} for all points of the plane never changes its absolute value, so that - in relation to that system - all distances always maintain their lengths.

The first question is whether there is a setting up of mirrors, in which both rays outgoing from ''P'' are coming back to the same point ''P''. The positively (that is: in the sense of rotation) rotating ray, related to system ''K'', will pass chords of that circle which are longer than the sides of the ''n''-gon; because while it propagates from one mirror to the other mirror, the latter moves at a certain distance approximately in the direction of the ray. But since all these chords are of the same length, they all hit the circle at the same angle ''φ''. (The figure illustrates this for the case {{pagenum|407}} ''n'' = 4.)

[[Image:Laue1911.png|thumb|400px|center|The path of the positive rotating ray relative to the ''K''{{sup|0}} system.<br />P{{sub|1}} is plate P on the departure of the ray,<br />P{{sub|2}} is plate P at the return of the ray,<br /> S{{sub|1}}, S{{sub|2}}, S{{sub|3}} is the mirror at the times as they reflect the ray.]]

If we locate the mirror tangential to the circle, the incident- and reflection angles are thus the same. This is consistent with the law of reflection. Since the translation velocity of the mirror is tangential to its surface, it is thus without influence. However, in agreement with experience we can neglect the rotation when we consider the reflection from the mirrors and plate ''P''. In that position of the mirrors the positively rotating ray (which leaves ''P'' in the appropriate direction) will exactly return to ''P''. But this applies also to the negatively rotating ray; since also in this case the incident- and reflection angles are equal, so that it travels ''n'' equal chords, and when it leaves from ''P'' in the appropriate direction it again reaches ''P''. The angles at which both rays strike at the plate in ''P'' (''φ'' in the figure), are the same with which they left it; so they form together a single ray, like they emerged from a single ray before.

{{pagenum|408}} The time τ{{sub|+}} needed by the positively rotating ray, can be set in relation to the central angle ϑ{{sub|+}} corresponding to one of the chords passed by it,

{{MathForm1|1)|<math>\tau_{+}=\frac{2nr}{c}\sin\frac{\vartheta_{+}}{2};</math>}}

it means that the ray passes ''n'' chords of length <math>2r\ \sin\ \tfrac{\vartheta_{+}}{2}</math> with velocity ''c''. Now, if the rotational velocity ω would be zero, than it would be nϑ{{sub|+}} = 2π; but in this case ''nϑ''{{sub|+}} is greater by the angle ωτ{{sub|+}} passed by ''P'' during the time τ{{sub|+}} (see the figure); thus:

{{MathForm1|2)|<math>n\vartheta_{+}=2\pi+\omega\tau_{+}.</math>}}

The elimination of ϑ{{sub|+}} from 1) and 2) gives

{{MathForm1|3)|<math>\tau_{+}=\frac{2nr}{c}\sin\left\{ \frac{1}{n}\left(\pi+\frac{\omega\tau_{+}}{2}\right)\right\} </math>}}

In a similar way we find the time for the negatively traveling ray

{{MathForm1|3a)|<math>\tau_{-}=\frac{2nr}{c}\sin\left\{ \frac{1}{n}\left(\pi-\frac{\omega\tau_{-}}{2}\right)\right\}.</math>}}

The difference between the two, is according to the formula:

{{c/s}}<math>\sin\alpha-\sin\beta=2\cos\frac{1}{2}(\alpha+\beta)\sin\frac{1}{2}(\alpha-\beta)</math>{{c/e}}

{{MathForm1|4)|<math>\Delta\tau=\tau_{+}-\tau_{-}=\frac{4nr}{c}\cos\left\{ \frac{\pi}{n}+\frac{\omega\Delta\tau}{4n}\right\} \sin\left\{ \frac{\omega}{4n}\left(\tau_{+}+\tau_{-}\right)\right\} </math>}}

Thus far the calculation is strict; now, if we neglect ωΔτ against π, we can set the cosine equal to <math>\tfrac{\pi}{n}</math>, and also

{{center|ω(τ{{sub|+}} + τ{{sub|-}}) {{=}} 2ωτ{{sub|0}}}}

where τ{{sub|O}} is the orbit time in the case ω = 0; that is according to 1) and 2)

{{center|<math>\tau_{0}{{=}}\frac{2nr}{c}\sin\frac{\pi}{n}.</math>}}

{{pagenum|409}} Furthermore, if we neglect ω²τ{{sub|0}}² against 1, than we can set <math>\sin\tfrac{\omega}{2n}\tau_{0}=\tfrac{\omega}{2n}\tau_{0}</math> and find as an approximate value

{{c/s}}<math>\Delta\tau=\frac{2n\omega r^{2}}{c^{2}}\sin\frac{2\pi}{n}.</math>{{c/e}}

The interior ''F'' of the ''n''-gon is, however,

{{c/s}}<math>F=\frac{n}{2}r^{2}\sin\frac{2\pi}{n}; </math>{{c/e}}

thus

{{MathForm1|5)|<math>\Delta\tau=\frac{4}{c^{2}}\omega F.</math>}}

According to {{sc|Michelson}}'s proposal F = 10{{sup|10}} cm², while ω is of the same order as the rotational velocity Ω of the earth (10{{sup|-4}} sec{{sup|-1}}). It follows that Δτ is of order 10{{sup|-15}} sec; this can be easily confirmed by interference experiments with visible light.

Now, of course, it is to be remembered that point M is not at rest and the plane of the experiment can not be chosen perpendicular to the axis of the earth. Maybe the center ''M'' of the ''n''-gon itself has a translational velocity to a reference frame K{{sup|0}} at rest against the sun, which is composed additively of two parts. The first of these is the translation velocity of the Earth's center, the second is due to the rotation of the earth around its axis. However, for the duration of the experiment the first part may be considered as immutable and the second one will change (during the time τ of one revolution) at most about ωRτ{{sub|0}}², where ''R'' is the radius of the earth equal to 6.10{{sup|8}} cm. This amount is below the above assumptions of the order of 10{{sup|-5}} cm/sec, therefore it can be completely neglected. A constant velocity of translation is according to the principle of relativity of no influence on the position of the interference fringes in the telescope; the point ''M'' lies just in another valid system ''K''. However, the form of the ''n''-gon with respect to ''K''{{sup|0}} is therefore changed, because due to {{pagenum|410}} {{sc|Lorentz}} contraction the circle around ''M'' becomes an ellipse. The circumstance, that in addition also the position of the plane in this system is changing during time τ, causes both rays to be deflected from the plane at an angle of order ωτ{{sub|0}} (= 10{{sup|-9}}). This is well below the limit of resolution of the telescope.

It should be considered, however, that the rotational speed ω is not the rotational speed of the Earth Ω; rather, if φ is the latitude of the observing place,

{{center|ω {{=}} Ω sin φ,}}

thus according to 5)

{{MathForm1||<math>\Delta\tau=\frac{4}{c^{2}}\Omega F\ \sin\varphi.</math>}}

At the equator (φ = 0) there is thus no displacement of the fringes, and in passing from the northern to the southern hemisphere, it must reverse its sign.

==b) Absolute theory.==

If point ''M'' is at rest in the preferred reference frame of this theory, then all considerations which we have done in a) on the position of the mirror, remain unchanged; for the geometrical law of reflection at the moving mirror is valid for all electromagnetic theories. Similarly the calculation which leads to equation 5) can the transferred step by step. A difference between the theories occurs only when point ''H'' gets a translational velocity ''v'', which, however, may be considered for the same reasons as above as constant in time (also the position change of the plane is irrelevant due to the reasons stated). Namely, the circuit described around ''M'' remains unchanged according to the electron theory, while (as it was mentioned) it becomes an ellipse in the theory of relativity. On one hand, it follows that according to the electron theory both rays are not traveling the same path any more, but reach the plate ''P'' at different {{pagenum|411}} points and at different angles from which they were leaving. But these changes are at most of the second order in <math>\tfrac{v}{c}</math>, which is unobservably small. Similarly, the difference Δτ perhaps will be changed to terms of second and higher order, but this can not be observed as well. So, also in this case, equation 6) represents an adequate approximation; thus it's not possible to decide between the theory of relativity and the stationary aether theory by the proposed experiment.

==c) The electrodynamics of {{sc|Cohn}}.==

The question, which outcome of the experiment we have to expect according to this theory, was actually answered by {{sc|Cohn}} himself in one of his papers<ref>E. {{sc|Cohn}}, [[Translation:On the Electrodynamics of Moving Systems I|Berliner Berichte 1904, p. 1404]] (see. especially p. 1410 below)</ref>. If we supplement his brief remarks, his reasoning is as follows: Imagine the plane of the experiment at any point on the earth tangentially to it. The time ''dt'' required by light to traverse the distance ''ds'' (measured on earth) is extended by <math>\tfrac{\mathfrak{q}_{s}ds}{c^{2}}</math>, if ''ds'' is moving with velocity <math>\mathfrak{q}</math>. Consequently, the propagation time of the positively rotating ray is <math>\tfrac{1}{c^{2}}\int\limits _{0}\mathfrak{q}_{s}ds</math> longer than in the case of vanishing rotation, and smaller for the negatively rotating ray by the same amount. If we apply to this the line integral of {{sc|Stokes}}' theorem, then we find

{{c/s}}<math>\Delta\tau=r_{+}-r_{-}=\frac{2}{c^{2}}\int\limits _{0}\mathfrak{q}_{s}ds=\frac{2}{c^{2}}F\ \mathsf{rot}_{n}\mathfrak{q}</math>{{c/e}}

However, if Ω is the velocity of the rotation of earth, it is:

{{c/s}}<math>\left|\mathrm{rot}\ \mathfrak{q}\right|=2\Omega,</math>{{c/e}}

{{pagenum|412}} thus, if φ is the latitude,

{{c/s}}<math>\mathsf{rot}_{n}\mathfrak{q}=2\Omega\ \sin\varphi</math>{{c/e}}

and in accordance with 6)

{{c/s}}<math>\Delta\tau=\frac{4}{c^{2}}\Omega F\ \sin\varphi.</math>{{c/e}}

It may seem doubtful, indeed, whether we can use the line integral over the volume of the ''n''-gon; since the rays, with respect to the rotating area, cannot propagate in straight lines. The consideration of this circumstance would only result in a higher order correction of Δτ.

==d) Theory of {{sc|Hertz}}.==

The decision for that theory is very simple. The experiment is performed in air, which goes along with the motion of the earth. All parts of the arrangement, including air, preserve their position with each other constantly; so no influence of the motion is present, but Δτ = 0. Since in a system moving like a rigid body, all electromagnetic processes happen as if they are at rest.


{{rule|15%}}


Thus we see that all theories which seriously come into consideration for the optics of moving bodies - that of {{sc|Hertz}} never belonged to them because of {{sc|Fizeau}}'s interference experiment on the entrainment of light - agree as to the effect in question. A decision between them can not be achieved by this experiment. Nevertheless, it would be very desirable that it would be performed; because the optics of the moving body is not rich of exact tests, so any extension of its experimental basis would be beneficial.

{{center|Munich, Institute for Theoretical Physics, June 1911.}}
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[[Category:Physics]]
[[Category:Works originally in German]]
[[Category:Relativity]]