1 2 8 D O C . 3 1 I D E A S A N D M E T H O D S

which we mentioned above as the result of experience. The result, therefore, fol-

lows from the basic principles of the theory of special relativity without any anal-

ysis of the optical process.

12. The Heuristic Meaning of the Lorentz

Transformation[24]

The principle of special relativity demands the equivalence of all those coordinate

systems that are in a rectilinear, uniform, rotation-free motion relative to one (ad-

missible) coordinate system; the same laws of nature shall be valid in reference to

all these coordinate systems. On the other hand, it follows from our two principles

that, between the coordinates of any two of these admissible coordinate systems,

there obtain those relations we have called Lorentz transformations. Combining

these two statements leads to this result:

The laws of nature must be such that the introduction of new coordinates by

means of a Lorentz transformation does not change their form (covariance of the

laws of nature under Lorentz transformations).

Maxwell-Lorentz electrodynamics satisfies this condition; Newtonian mechan-

ics, however, does not. Therefore, the latter had to be modified in order to yield to

this requirement. This could be accomplished without great difficulties; for exam-

ple, it turned out that Newton’s equations of motion for a pointlike mass,

,

which say that the change rate of momentum equals the force, has to be replaced by

.

In these equations, denotes the vector, the amount of the velocity, and the

force acting upon the material point. The two equations are not noticeably different

so long as the velocity is small compared to the speed of light. With Newton,

expresses the momentum (impetus) of the mass; according to the theory of relativ-

d-(mq)

dt

----

K

=

d-⎜

dt⎜

----

mq

1

q2⎟

----c2⎠- –

------------------⎟

⎝

⎜ ⎟

⎛ ⎞

K

=

[p. 14]

q q K

q mq