The momentum of a body is different in different directions. The
ordinary way of measuring it is to take the velocity in a given
direction (as measured by the observer) and multiply it by the mass (as
measured by the observer). Now the velocity in a given direction is
the distance traveled in that direction in unit time. Suppose we take
instead the distance traveled in that direction while the body moves
through unit “interval.” (In ordinary cases, this is only a very slight
change, because, for velocities considerably less than that of light,
interval is very nearly equal to lapse of time.) And suppose that
instead of the mass as measured by the observer we take the proper
mass. These two changes increase the velocity and diminish the mass,
both in, the same proportion. Thus the momentum remains the same, but
the quantities that vary according to the observer have been replaced
by quantities which are fixed independently of the observer—with the
exception of the distance traveled by the body in the given direction.
When we substitute space-time for time, we find that the measured
mass (as opposed to the proper mass) is a quantity of the same kind
as the momentum in a given direction; it might be called the momentum
in the time direction. The measured mass is obtained by multiplying
the invariant mass by the _time_ traversed in traveling through unit
interval; the momentum is obtained by multiplying the same invariant
mass by the _distance_ traversed (in the given direction) in traveling
through unit interval. From a space-time point of view, these naturally
belong together.
Although the measured mass of a body depends upon the way the observer
is moving relatively to the body, it is none the less a very important
quantity. For any given observer, the measured mass of the whole
physical universe is constant.[8] The proper mass of all the bodies
in the world is not necessarily the same at one time as at another,
so that in this respect the measured mass has an advantage. The
conservation of measured mass is the same thing as the conservation of
energy. This may seem surprising, since at first sight mass and energy
are very different things. But it has turned out that energy is the
same thing as measured mass. To explain how this comes about is not
easy; nevertheless we will make the attempt.
[8] This is subject to the explanations given below as regards
conservation of energy.
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