# The Relative Motion of the Earth and the Aether

by Lorentz To explain the aberration of light, Fresnel assumed that the aether does not share the annual motion of earth.

• This requires the Earth to be completely permeable to the aether.

Later, Stokes explained that the aether is dragged by the earth.

• This made the speed of the aether the same as that of the earth at each point at the earth’s surface.

I have extensively worked on these theories. .

I reject Stokes’ idea because the motion of the aether requires the existence of a velocity potential.

• This is incompatible with the equality between the velocities of the earth and the adjacent aether.

On the other side, it was possible to explain nearly all considered phenomena by Fresnel’s theory, if we assume the “dragging coefficient” for transparent ponderable substances given by Fresnel, and whose value was recently derived by me from the electromagnetic theory of light.

An interference experiment by Michelson created a great difficulty in deciding between Fresnel and Stokes.

According to Maxwell, if the aether remains at rest, then the earth’s motion must have an influence on the time required by light to travel back and forth between 2 points fixed on earth.

If `l` is the distance of the points, `V` is the velocity of light, and `p` is the velocity of earth, then the relevant time is given by the following, if the line of points is parallel to the direction of motion:

``````2 (l/V) ( 1 + p2 / V2 )
``````

and if it is perpendicular to it:

``````2 (l/V) ( 1 + ( p2 / 2V2 ))
``````

making a difference of:

``````lp2 / V3
``````

Michelson used a device with 2 equally long-standing horizontal arms perpendicular to each other.

• It had mirrors at the ends and perpendicular to their direction.

An interference phenomenon occurred, when (from the radius of the intersection) a ray traveled forth and back along one arm, and another along the other arm.

The entire device - including the light source and the observation telescope - could be rotated around a vertical axis.

• The observation time was chosen, so that one can bring, one arm or the other arm into the direction of motion of earth.

If so, then Fresnel’s theory is correct. Because of the earth’s motion, the rays that travel forth and back into the earth’s direction must have a delay determined by (3) in respect to the other ray.

When rotated by 90°, all phase shifts must be altered by an amount, which, expressed in unit time, can be given by the double of magnitude (3).

But a displacement of the interference fringes could not be observed.

One can argue that the length of the arms are just too small to obtain any observable displacement of the fringes.

But Michelson and Morley repeated the experiment on a larger scale.

The light rays were traveling forth and back in mutually normal directions several times, because they were reflected every time by mirrors.

The apparatus stood on a stone plate that swam on mercury.

• It could be rotated in horizontal direction.

However, the shift as required by Fresnel’s theory could not be observed again.

I could only reconcile the result with Fresnel’s theory.

It assumes that the line joining 2 points of a solid body does not conserve its length, when it is once in motion parallel to the direction of motion of Earth, and afterwards it is brought normal to it.

If for example the distance in the latter case is `l` and in the first case `l( 1 − α )`, then the first expressions (1) and (2) have to be multiplied by `1 − α`.

Neglecting αp2 / V2 we get:

``````2 (l/V) ( 1 + p2 / V2 − α )
``````

The difference to (2) - and thus the whole objection - would be removed when:

``````α = p2 / 2V2
``````

Such a change in length of the arms in Michelson’s first experiment, and in the size of the stone plate in the second, is really not inconceivable.

The intensity of molecular forces determines the size and shape of a solid body.

• Any cause that could modify those molecular forces could modify the shape and size as well.

We assume that the intervention of the aether acts on:

• electric and magnetic forces
• molecular forces.

But then it cannot make a difference, whether the connecting line of 2 particles, which move together through the ether, is moving parallel to the direction of motion or perpendicular to it.

An effect of order `p/v` is not expected. But an effect of order `p2 / V2` is not excluded. That is exactly what we need.

Since we know nothing about the nature of molecular forces, it is impossible to verify the hypothesis.

We only can calculate the influence of the motion of ponderable matter on electric and magnetic forces.

When the result obtained for the electric forces is transferred to molecular forces, it exactly gives the value of `α` given above.

Let:

-`A` be a system of material points, which bear certain electrical charges and which are at rest relative to the aether.

• `B` is the system of the same points, when they are moving in the direction of the `x` x-axis by the collective velocity p {\displaystyle p} p through the aether.

From my equations one can deduce, by which forces the particles in the system act on each other.

The result can be expressed in the most simple way, if one introduces a third system `C` that is at rest like `A`, but differs from the latter system by the mutual position of the points.

System `C` can be obtained from A {\displaystyle A} A by a mutual expansion, by which all dimensions in the direction of the x {\displaystyle x} x-axis are `1 + p 2 2 V 2` times larger, while the perpendicular dimensions remain unchanged.

Concerning the relation between the forces in B and C it follows, that the components in the direction of the `x-axis` are the same as in `C`, while the components perpendicular to the x-axis are `1 − p 2 2 V 2` times larger as in `C`.

We want to transfer this to the molecular forces, and imagine a solid body as a system of material points, in equilibrium by the influence of their mutual attractions and repulsions.

The system `B` shall be a body moving through the aether.

The forces acting on its material points eliminate each other. It follows, that this cannot be the case in `A`, however, in system C all force-components perpendicular to the x-axis are changed if one goes over from `B` to `C`, but the equilibrium will not be disturbed as they are changed in the same ratio.

In this way it can be seen, that when `B` is the state of equilibrium of the body during the displacement in the aether, `C` is the state of equilibrium when the displacement does not exist.

One therefore comes exactly to the influence of motion on the dimensions, which was shown before to be necessary to explain Michelson’s experiment.

Of course we cannot ascribe great importance to this result.

The transfer to molecular forces of what we have found for electrical forces, may be too risky for some.

Moreover, if we want to do this, it remains undecided whether earth’s motion shortens the dimensions in one direction - as it was supposed before - or elongates the length perpendicular to it, by which assumption we could reach the same result.

Anyway, it seems undeniable that changes of the molecular forces and consequently of the body’s size of order `1 − p 2 2 V 2` are possible.

Michelson’s experiment thus loses its verification power for the question at which it was aimed.

If one assumes the theory of Fresnel, then its meaning rather lies in the fact, that we can learn something about the change of dimensions.

As `p/V = 1/10000` then `p2 / 2V2` is the 200,000,000th.

A contraction of the diameter of the Earth by this ratio would amount 6 cm. We cannot speak about the observation of a change in length of 200,000,000th when comparing meter sticks.

Even if an observation method would allow this, then this method would be the juxtaposition of two sticks, but we would never detect the discussed changes, when they occur in the same way for both of them.

The only remedy is to compare the length of 2 sticks perpendicular to each other. If we want to do this by the observation of an interference phenomenon (with a light ray that travels back and forth along the first and the other ray along the second arm).

Then we would come back to Michelson’s experiment.

The influence of the change in length, however, would be compensated by the change of phase shift which is determined by expression (3).

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