P O P U L A R P R I N C E T O N L E C T U R E S 6 1 7
body is proportional to the gravitation of that body, or, in other words, to its weight. We
measure the inertia of a body by finding out how much acceleration a definite force gives
it. We measure the weight of a body, or the gravitational mass of a body, by finding out how
much acceleration the gravitational field of the earth gives it.
It is known that these two masses, the inert mass and the gravitational mass, are strictly
proportional to each other, and that they do not change for different bodies. In other words,
it makes no difference at all whether we speak of the inert mass of a body or whether we
speak of the gravitational mass of the body. Just exactly what the significance of this fact is
has been a puzzle for a long time, which, however, the general theory of relativity quickly
clears up.
Suppose we consider a system of coördinates which we will suppose to be absolutely at
rest. And let us assume that there is no gravitational field present in this system. Consider
another system which we will say is moving upwards with accelerated motion. Now, if we
have a mass particle in our first system which was at rest, then, in the simplest case, we
would suppose that our mass particle was at rest at one instant and therefore, according to
our assumption, remains at rest forever. That same particle, however, referred to our second
system will be found to have a uniform acceleration downwards.
The principle of equivalence states this, that it makes absolutely no difference whether
we say that our first system is at rest and the second system is uniformly accelerated, or
whether we say that we have in place of a uniformly accelerated system a gravitational field
present. This principle of equivalence is the foundation of the general principle of relativity.
Now, the problem that has got to be solved is this: it must be possible if this principle is
to hold with any generality to be able to describe any kind of a gravitational field in terms
of the motion of a particular system of coördinates. That, however, is not the way in which
the general principle has been developed. There is this difficulty which we meet with at
once. It can be shown easily that if this general principle of relativity is to hold, then our
ordinary concepts of space and time have got to be profoundly modified. You can put it in
another way. We can show that the Euclidian geometry will not hold if the general principle
of relativity is to hold.
So far we have spoken only of the fact that Euclidian geometry and our concepts of space
have got to be modified if the general theory of relativity is true.
The same thing holds for our conception of time, for we also saw yesterday that a clock
on the periphery of our disk would go slower referred to a system at rest than a clock at the
centre of the disk. This is due to the fact that the periphery is in motion with reference to
the system of reference and the centre is at rest, so that also our conception of time has got
to be modified if the general theory of relativity is true.
The problem in the general theory of relativity is to find an expression for the laws of
natural phenomena so that the principle of equivalence, for one thing, will hold. That, how-
ever, we have just seen cannot be done if we are going to hold to Euclidian geometry and
to our ordinary concept of time. An analogy here, however, helps a great deal in understand-
ing exactly the mathematical method which can be employed to find out the form of the
laws of nature in the general theory of relativity.
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