In order to compare t₁ and t₂ more easily,
let us write for
Then we get:

Assuming that v is less than c,
and c² - v² being obviously less than c²,
then is therefore less than c,
and consequently is greater than 1
(since the denominator
is less than the numerator).
Therefore t₁ is greater than t₂,
that is,
IT TAKES LONGER TO
SWIM UPSTREAM AND BACK
THAN TO SWIM THE SAME DISTANCE
ACROSS-STREAM AND BACK.

But, what has all this to do
with the Michelson-Morley experiment?

In that experiment,
a ray of light was sent from A to B:

Where c is now the velocity of light,
and t₂ is the time required for the light
to go from A to C and back to A
(being reflected from another mirror at C).
If, therefore,
t₁ and t₂ are found experimentally, -
then by dividing (1) by (2),
the value of would be easily obtained.
And since
c being the known velocity of light,
the value of v could be calculated.
That is,
THE ABSOLUTE VELOCITY OF THE EARTH
would thus become known.

Such was the plan of the experiment.

Now what actually happened?

The experimental values of t₁ and t₂
were found to be the SAME,
instead of t₁ being greater than t₂ !
Obviously this was a most disturbing result,
quite out of harmony
with the reasoning given above.
The Dutch physicist, Lorentz,
then suggested the following explanation
of Michelson's strange result:
Lorentz suggested that
matter, owing to its electrical structure,
SHRINKS WHEN IT IS MOVING,
and this contraction occurs
ONLY IN THE DIRECTION OF MOTION.
The AMOUNT of shrinkage
he assumes to be in the ratio of
(where has the value as before).
Thus a sphere of one inch radius
becomes an ellipsoid when it is moving,
with its shortest semi-axis
(now only inches long)
in the direction of motion,
thus:

Applying this idea
to the Michelson-Morley experiment,
the distance AB (=a)
becomes ,
and t₁ becomes
instead of
so that now t₁ = t₂,
just as the experiment requires.

One might ask how it is
that Michelson did not
observe the shrinkage?
Why did not his measurements show
that AB was shorter than AC
(See the earlier figure of the river)?
The obvious answer is that
the measuring rod itself contracts
when applied to AB,
so that one is not aware of the shrinkage.

To this explanation
of the Michelson-Morley experiment
the natural objection may be raised
that an explanation which is invented
for the express purpose
of smoothing out a certain difficulty,
and assumes a correction
of JUST the right amount,
is too artificial to be satisfying.
And Poincare, the French mathematician,
raised this very natural objection.

Consequently,
Lorentz undertook to examine
his contraction hypothesis
in other connections,
to see whether it is in harmony also
with facts other than
the Michelson-Morley experiment.
He then published a second paper in 1904,
giving the result of this investigation.
To present this result in a clear form
let us first re-state the argument
as follows:

Consider a set of axes, X and Y,
supposed to be fixed in the stationary ether,
and another set X' and Y',
attached to the earth and moving with it,
with velocity v as indicated above,
Let X' move along X,
and Y' move parallel to Y.

Now suppose an observer on the earth,
say Michelson,
is trying to measure
the time it takes a ray of light
to travel from A to B,
both A and B being fixed points on
the moving axis X'.
At the moment
when the ray of light starts at A
suppose that Y and Y' coincide,
and A coincides with D;
and while the light has been traveling to B
the axis Y' has moved the distance vt,
and B has reached the position shown in the figure above,
t being the time it takes for this to happen.
Then, if DB = x and AB = x',
we have x' = x - vt.      (3)
This is only another way
of expressing what was said before
where the time for
the first part of the journey
was said to be equal to a/(c - v).
And, as we saw there,
this way of thinking of the phenomenon
did NOT agree with the experimental facts.
Applying now the contraction hypothesis
proposed by Lorentz,
x' should be divided by ,
so that equation (3) becomes

Now when Lorentz examined other facts,
as stated earlier in this chapter,
he found that equation (4)
was quite in harmony with all these facts,
but that he was now obliged
to introduce a further correction
expressed by the equation

where , t, v, x, and c
have the same meaning as before -
But what is t'?!
Surely the time measurements
in the two systems are not different:
Whether the origin is at D or at A
should not affect the
TIME-READINGS.
In other words, as Lorentz saw it,
t' was a sort of "artificial" time
introduced only for mathematical reasons,
because it helped to give results
in harmony with the facts.
But obviously t' had for him
NO PHYSICAL MEANING.
As Jeans, the English physicist, puts it:
"If the observer could be persuaded
to measure time in this artificial way,
setting his clocks wrong to begin with
and then making them gain or lose permanently,
the effect of his supposed artificiality
would just counterbalance
the effects of his motion
through the ether"!

Thus,
the equations finally proposed by Lorentz
are:

Note that
since the axes attached to the earth
are moving along the X-axis,
obviously the values of y and z
(z being the third dimension)
are the same as y' and z', respectively.

The equations just given
are known as
THE LORENTZ TRANSFORMATION,
since they show how to transform
a set of values x, y, z, t
into a set x', y', z', t'
in a coordinate system
moving with constant velocity v,
along the X-axis,
with respect to the
unprimed coordinate system.
And, as we saw,
whereas the Lorentz transformation
really expressed the facts correctly,
it seemed to have
NO PHYSICAL MEANING,
and was merely
a set of empirical equations.

Let us now see what Einstein did.