6 0 4 A P P E N D I X C
Now, the definition which we shall introduce in order to define simultaneous events de-
pends upon our assumption of the constancy of the velocity of light. Suppose that we have
two events taking place at different places, and if we consider an observer placed just half
way between the two events, we then say that the two events are simultaneous if that ob-
server notes that those two events are simultaneous for him, and that definition is to be the
definition of simultaneity of events.
Now there is another assumption which we have made in our ordinary ideas, and that has
to do with measured lengths. We ordinarily suppose that the length of an object that we
measure is wholly independent of its state of motion. That, however, is purely an assump-
tion, and whether it is so or not will have to be tested by the consequences of the special
theory of relativity.
Everything comes to this. We refer all physical phenomena to a system of coördinates
and to a way of measuring time. Suppose that we have our physical phenomena referred to
any other system of coördinates. The special theory of relativity states that physical phe-
nomena referred to any other system of coördinates which are moving relatively to our first
system with uniform and rectilinear motion must be expressed by exactly the same laws.
And that statement enables us to find out what measured lengths in one system will be when
measured with respect to our second system. It also enables us to find out how measured
time in one system will come out when measured in respect to another system.
The thing that we have to do is to express the coördinates and time measured in one sys-
tem which we can say is at rest, and another system in motion. It was Lorentz who first
found how all measurement of lengths and time referred to one system must be expressed
when we wish to refer them to another system.
And that, then, is the principle of relativity which states that the laws of all physical phe-
nomena must be of the same form when referred to two different systems which are moving
relatively to each other with uniform rectilinear velocity.
Now, the consequences of this transformation from one system to another in brief are
these: First of all, it is found that the length of a rod, if it is moving in the direction of its
length, becomes shorter in the direction of its motion. If it is moving transversely to its
length, then there is no change in its direction. In particular, a sphere in motion becomes
flattened out in the direction of motion. In other words, it becomes an ellipsoid.
Another consequence is that time—if we have a clock in one system rest and then refer
that same clock to a system which is moving with uniform velocity with respect to what we
speak of as our system at rest—that clock is found as a result of the theory to go slower in
the second system than it did in the first system.
Another important consequence of the application of this principle to physical laws is
this: That the velocity of light is a limiting velocity, above which we cannot go. In particu-
lar, if we have two systems, one system, we will say, is moving with a velocity which is very
nearly but not quite the velocity of light, and then consider a third system which, with re-
spect to the second system is also moving with a velocity which is very nearly but not quite
the velocity of light, our ordinary notion would be that the first system would be moving in
respect to the third system at a velocity which is very much greater than the velocity of light.