VI. THE RESULT OF APPLYING THE REMEDY.

In the last chapter we saw that
by starting with
two fundamental FACTS,
we reached the conclusion
expressed in the equations

which are graphically represented in the last chapter,
and we realized that these equations
are NOT contradictory,
(as they appear to be at first),
if we remember that there is
a difference in the setting of the clocks
in the two different systems.

We shall derive, now, from (6) and (7),
relationships between the measurements
of the two observers, K and K’.
And all the mathematics we need for this
is a little simple algebra,
such as any high school boy knows.

From (6) and (7) we get

Therefore

where is a constant.
Similarly, in the opposite direction,

being another constant.
By adding and subtracting (8) and (9)

where
Let us now find the values of a and b
in terms of v
(the relative velocity of K and K’),
and c, the velocity of light.

This is done in the following
ingenious manner:
From (10)
when x’ = 0,
then x = bct/a;
but x’ = 0 at the point K’:

And x in this case is
the distance from K to K’,
that is,
the distance traversed, in time t
by K’ moving with velocity v
relatively to K.
Therefore x = vt.
Comparing with (12), we get

Let us now consider the situation
from the points of view of K and K’.
Take K first:
For the time t = 0,
K gets x’ = ax (from (10)),

Hence K says that
to get the "true" value, x,
K’ should divide his x’ by a;
in particular,
if x’ = 1,
K says that K’‘s unit of length
is only 1/a of a "true" unit.

But K’,
at t’ = 0, using (11)
says

and since from (10),

(15) becomes

or
from which

And since b/a = v/c from (13),
(16) becomes

In other words,
K’ says:
In order to get the "true" value, x’,
K should multiply his x by

In particular,
if x = 1,
then K’ says that
K’s unit is really units long.
Thus
each observer considers that
his own measurements
are the "true" ones,
and advise the other fellow
to make a "correction."
And indeed,
although the two observers, K and K’,
may express this "correction"
in different forms,
still
the MAGNITUDE of the "correction"
recommended by each of them
MUST BE THE SAME,
since it is due in both cases
to the relative motion,
only that each observer attributes this motion
to the other fellow.
Hence, from (14) and (17) we may write

Solving this equation for a, we get

Note that this value of a
is the same as that of in Chapter II.
Substituting in (10)
this value of a
and the value bc = av from (13),
we get

which is the first of the set of equations
of the Lorentz transformation in Chapter II!

Furthermore,
from (18) and
we get

Or, since t = x/c,

which is
another of the equations
of the Lorentz transformation!
The remaining two equations

are of course obvious
from the conditions of the motion
as shown in the figure above.

We thus see that
the Lorentz transformation was derived
by Einstein
(quite independently of Lorentz),
NOT as a set of empirical equations
devoid of physical meaning,
but, on the contrary,
as a result of
a most rational change in
our ideas regarding the measurement of
the fundamental quantities
length, mass and time.
And so, according to him,
the first of the equations of the
Lorentz transformation,
namely,

is so written
NOT because of any real shrinkage,
as Lorentz supposed,
but merely an apparent shrinkage,
due to the differences in
the measurements made by K and K’.
And Einstein writes

NOT because it is just a mathematical trick
WITHOUT any MEANING (see Ch. II.)
but again because
it is the natural consequence of
the differences in the measurements
of the two observers.
And each observer may think
that he is right
and the other one is wrong,
and yet
each one,
by using his own measurements,
arrives at the same form
when he expresses a physical fact,
as, for example,
when K says x = ct
and K’ says x’ = ct’,
they are really agreeing as to
the LAW of the propagation of light.

And similarly,
if K writes any other law of nature,
and if we apply
the Lorentz transformation
to this law,
in order to see what form the law takes
when it is expressed in terms of
the measurements made by K’,
we find that
the law is still the same,
although it is now expressed
in terms of the primed coordinate system.

Hence Einstein says that
although no one knows
what the "true" measurements should be,
yet,
each observer may use his own measurements
WITH EQUAL RIGHT AND EQUAL SUCCESS
in formulating
THE LAWS OF NATURE,
or,
in formulating the
INVARIANTS of the universe,
namely, the quantities which remain unchanged
in spite of the change in measurements
due to the relative motion of K and K’.
Thus, we can now appreciate
Einstein’s Principle of Relativity:
"The laws by which
the states of physical systems
undergo change,
are not affected
whether these changes of state be referred
to the one or the other
of two systems of coordinates
in uniform translatory motion."

Perhaps some one will ask
"But is not the principle of relativity old,
and was it not known long before Einstein?
Thus a person in a train
moving into a station
with uniform velocity
looks at another train which is at rest,
and imagines that the other train is moving
whereas his own is at rest.
And he cannot find out his mistake
by making observations within his train
since everything there
is just the same as it would be
if his train were really at rest.
Surely this fact,
and other similar ones,
must have been observed
long before Einstein?"

In other words,
RELATIVELY to an observer on the train
everything seems to proceed in the same way
whether his system (i.e., his train)
is at rest or in uniform motion ,
and he would therefore be unable
to detect the motion.
Yes, this certainly was known
long before Einstein.
Let us see what connection it has
with the principle of relativity
as stated by him:
Referring to the diagram in Chapter IV
we see that
a bullet fired from a train
has the same velocity
RELATIVELY TO THE TRAIN
whether the latter is moving or not,
and therefore an observer on the train
could not detect the motion of the train
by making measurements on
the motion of the bullet.
This kind of relativity principle
is the one involved
in the question above
and WAS known long before Einstein.

Now Einstein
EXTENDED this principle
so that it would apply to
electromagnetic phenomena
(light or radio waves).
Thus, according to this extension of
the principle of relativity,
an observer cannot detect
his motion through space
by making measurements on
the motion of ELECTROMAGNETIC WAVES.
But why should this extension
be such a great achievement –
why had it not been suggested before?
BECAUSE
it must be remembered that
according to fact (2) – see Ch. V,

EL = c,
whereas,
the above-mentioned extension of
the principle of relativity
requires that E’L should be equal to c
(compare the case of the bullet in Ch. IV.).
In other words,
the extension of the principle of relativity
to electromagnetic phenomena
seems to contradict fact (2)
and therefore could not have been made
before it was shown that
fundamental measurements are merely "local"
and hence the contradiction was
only apparent,
as explained in Chapter V;
so that the diagram shown above
must be interpreted
in the light of the discussion in Ch. V.

Thus we see that
whereas the principle of relativity
as applied to MECHANICAL motion
(like that of the bullet)
was accepted long before Einstein,
the SEEMINGLY IMPOSSIBLE EXTENSION
of the principle
to electromagnetic phenomena
was accomplished by him.

This extension of the principle,
for the case in which
K and K’ move relatively to each other
with UNIFORM velocity,
and which has been discussed here,
is called
the SPECIAL theory of relativity.
We shall see later
how Einstein generalized this principle
STILL FURTHER,
to the case in which
K and K’ move relatively to each other
with an ACCELERATION,
that is, a CHANGING velocity.
And, by means of this generalization,
which he called
the GENERAL theory of relativity,
he derived
A NEW LAW OF GRAVITATION,
much more adequate even than
the Newtonian law,
and of which the latter
is a first approximation.

But before we can discuss this in detail
we must first see
how the ideas which we have
already presented
were put into a
remarkable mathematical form
by a mathematician named Minkowski.
This work
was essential to Einstein
in the further development of his ideas,
as we shall see.