D O C . 3 1 I D E A S A N D M E T H O D S 1 2 9
ity, it is . The latter quantity approaches infinity when approaches the
speed of light; the first quantity does not. This is connected to the fact that, accord-
ing to the theory of relativity, it is impossible, however large the forces or for how-
ever long they act, to raise the velocity of a material point 〈body〉 up to the speed of
light (or beyond it). The speed of light, quite generally, plays within the theory the
role of a physically infinitely large and insurmountable velocity; i.e., for projec-
tiles, signals (waves) etc. This is already seen by glancing at the Lorentz transfor-
mation and the consequences previously drawn from it about measuring rods and
clocks.
For the energy of a moving mass point, one finds the expression
(7)
or, developed in powers of ,
(7a)
The second term in this expansion is the “kinetic energy” of classical mechanics.
The third, fourth, etc. terms vanish against the second one when becomes neg-
ligible compared to the unit. The first term in (7a) deserves special attention. It has
to be remembered that the energy of the masspoint itself does not follow from the
equation of motion, so that the energy is defined only up to a constant which we
omitted in (7). However, the first term in (7a), to which the expression of
reduces in case , is formally so closely connected to the terms following (as
a glance upon [7] shows) that we are inclined to attribute physical meaning to it.
We can look at as the energy of the masspoint when it is at rest .
This interpretation receives mighty support from a theoretical investigation
based upon the following
consideration.[25]
According to the theory of special rel-
ativity, the theorem of the conservation of energy must hold not only relative to one
coordinate system but also relative to a system in uniform motion relative to
. From this one can derive—without here detailing how—the theorem:
mq
1
q2-
c2
---- –
------------------
q
L〉E 〈
L〉E 〈
mc2
1
q2-
c2
---- –
------------------ =
q2
L〉E 〈 mc2
1
2
--mq2 -
3
8c2
------q4…. -
m
- + + =
q2
c2
---- -
mc2 E
q 0 =
mc2 q 0) = (
[p. 15]
K K′
K