The Ether of Space Part 12

_Michelson's Experiment._

We conclude, therefore, that general etherial drift does not affect either the path of a ray, or the time of its journey to and fro, or round a complete contour, to any important extent. But that taking second-order quant.i.ties into account, the time of going to and fro in any direction inclined at angle ? to a constant drift is, from the above expression,

T1 + T2 = 2T cos e / (1-a) = v(1 - asin?) / (1 - a) 2T,

where 2 T is the ordinary time of the double journey without any drift.

Hence some slight modification of interference effects by reason of drift would seem to be possible; since the time of a to-and-fro light-journey depends subordinately on the inclination of ray to drift.

The above expression applies to Michelson's remarkable experiment[10]

of sending a split beam to and fro, half along and half across the line of the earth's motion; and is, in fact, a theory of it. There ought to be an effect due to the difference between ? = 0 and ? = 90.

But none can be detected. Hence, either something else happens, or the ether near the earth is dragged with it so as not to stream through our instruments.

_Alternative Explanation._

But if the ether is dragged along near moving matter, it behaves like a viscous fluid, and all idea of a velocity-potential must be abandoned. This would complicate the theory of aberration (pp. 45 and 61), and moreover is dead against the experimental evidence described in Chapter V.

The negative result of Mr. Michelson's is, however, explicable in another way,--namely, by the FitzGerald-Lorentz theory that the linear dimensions of bodies are a function of their motion through the ether.

And such an effect it is reasonable to expect; since, if cohesion forces are electrical, they must be affected by motion, to a known and calculable amount, depending on the square of the ratio of the speed to the velocity of light. (See end of Chap. IV.)

The theory of Professor H.A. Lorentz, accordingly, shows that the shape of Michelson's stone supporting block will be distorted by the motion; its dimensions across and along the line of ether drift being affected differently. And the amount of the change will be such as precisely to compensate and neutralise the optical effect of motion which might otherwise be perceived. This theory is now generally accepted.

It is this neutralising or compensatory effect,--which acts equally on to-and-fro motion of light, to-and-fro motion of electric currents, and on the shape of material bodies,--that renders any positive result in experiments on ether-drift so difficult or impossible to obtain; so that, in spite of the speed with which we are rushing through s.p.a.ce, no perceptible influence is felt on either electrical or optical phenomena, except those which are due to relative motion of source and observer.

_Some Details in the Theory of the Doppler Effect, or Effect of Motion on Dispersion by Prism or Grating._

When light is a.n.a.lysed by a prism or grating into a spectrum, the course of each ray is deflected--refracted or diffracted--by an amount corresponding to its frequency of vibration or wave-length.

Motion of the medium, so long as it is steady, affects neither frequency nor wave-length, and accordingly is without influence on the result. It produces no Doppler effect except when waxing or waning.

Motion of the source alone crowds the waves together on the advancing side and spreads them out on the receding side. An observer therefore whom the source is approaching receives shorter waves, and one from whom the source is receding receives longer waves, than normal. At any fixed point waves will arrive, therefore, with modified frequency.

So long as a source is stationary the wave-lengths emitted are quite normal, but motion of an observer may change the frequency with which they are _received_, in an obvious way; they are swept up faster if the receiver is approaching, they have a stern chase if it is receding.

All this is familiar, and was geometrically ill.u.s.trated in Chapter III, but there are some minor and rather curious details which are worthy of brief consideration.

_Grating Theory._

For suppose a 'grating' is used to a.n.a.lyse the light. Its effect can depend on nothing kinetic; it must be regulated by the merely geometric width of the ruled s.p.a.ces on it. Consequently it can only directly apprehend wave-lengths, not frequencies.

In the case of a moving _source_, therefore, when the wave-length is really changed, a grating will appreciate the fact, and will show a true Doppler effect. But in the case of a moving _observer_, when all the waves received are of normal length, though swept up with abnormal frequency, the grating must still indicate wave-length alone, and accordingly will show no true Doppler effect.

But inasmuch as the telescope or line of vision is inclined at the angle of dispersion to the direction of the incident ray, ordinary aberration must come in, as it always does when an observer moves athwart his line of vision; and so there will be a spurious or apparent Doppler effect due to common aberration. That is to say a spectrum line will not be seen in its true place, but will appear to be shifted by an amount almost exactly imitative of a real Doppler effect--the imitation being correct up to the second order of aberration magnitude. The slight outstanding difference between them is calculated in my _Philosophical Transactions_ paper, 1893, page 787. It is too small to observe.

It is not an important matter, but as it is rather troublesome to work out the diffraction observed by a grating advancing towards the source of light, it may be as well to record the result here.

The following are the diffracted rays which require attention,--with the inclination of each to the grating-normal specified:--

The diffracted ray if all were stationary, ?0;

The real diffracted ray when grating is advancing, f;

The ray as perceived, allowing for aberration, ?;

The equivalent diffracted ray if all were stationary and the wave-length really shortened, ?1.

As an auxiliary we use the aberration angle e, such that sin e = a sin ?, where a = v/V.

Among these four angles the following relations hold; so that, given one of them, all are known.

{ ? = f - e { sin ?1 = (1 - a) sin ?0 { sin f = (1 - a vers f) sin ?0

Whence ? and ?1 are very nearly but not absolutely the same. ?1 is the ray observed by an instrument depending primarily on frequency, like a prism; ? is the ray observed by an instrument depending primarily on wave-length, like a grating.

_Prism Theory._

Now let a prism be used to a.n.a.lyse the light; its dispersive power is in most theories held to depend directly upon frequency--i.e. upon a time relation between the period of a light vibration and the period of an atomic or electronic revolution or other harmonic excursion.

Let us say, therefore, that prismatic dispersion directly indicates frequency. It cannot depend upon wave-length, for the wave-length inside different substances is different, and though refractive index corresponds to this, dispersive power does not.

In the case of a prism, therefore, no distinction can be drawn between motion of source and motion of receiver; for in both cases the frequency with which the waves are received will be altered,--either because they are really shorter, though arriving at normal speed, or because they are swept up faster, although of normal length.

_Achromatic Prism._

It must be noticed that the observation of Doppler effect by a prism depends entirely on dispersion; i.e. on waves of different length being affected differently. But prisms can be constructed whose dispersion is corrected and neutralised. Such achromatic prisms, if perfectly achromatic, will treat waves of all sizes alike; and, accordingly, the shortening of the waves from a moving source will not produce any effect. Achromatic prisms will therefore behave to terrestrial and to extra-terrestrial sources, i.e. to relatively stationary and relatively moving sources, in the same way.

This must be recollected in connexion with several of the negative results rightly obtained by some observers; such as Arago, for instance, who applied an achromatic prism to a star which the earth was approaching, without observing any effect. A Doppler effect should have been observed by a dispersive prism, but not by an achromatic one: for the refractive index of a substance is not affected by any motion of the earth.

It is not reasonable to expect that refractive index would be affected, since it depends in simple geometrical fashion on r.e.t.a.r.ded velocity, i.e. on optical etherial loading or apparent extra internal density.

An achromatic _grating_, however, is (rashly speaking) an impossibility.

EFFECT OF TRANSPARENT MATTER.

But when a ray is travelling through transparent matter, will not motion of that matter affect its course?

If the matter is moved relatively to source and receiver, as in Fizeau's experiment with running water, most certainly it will; to the full effect of the loading or extra or travelling density, (-1), compared with the total density .

This fraction of the velocity of the material medium must directly influence the velocity of light, for the waves will be conveyed in the sense of the material motion _u_, with the additional speed u(-1)/. (See also Appendix 3.)

But if the transparent matter through which the light is going is stationary with respect to source and receiver--only sharing with them the general planetary motion, i.e. being subject to the opposite all-pervading ether drift,--then no influence due to the drift can be experienced; for the free ether of s.p.a.ce behaves just as it would if the matter were not there. This can be shown more elaborately by the following calculation.