I shall show in our figure that Lorentz’s hypothesis is fully equivalent
to the new conceptions about time and space. Thereby it may appear more
intelligible. Let us now, for the sake of simplicity, neglect (_y_, _z_)
and fix our attention on a two dimensional world, in which let upright
strips parallel to the _t_-axis represent a state of rest and another
parallel strip inclined to the _t_-axis represent a state of uniform
motion for a body, which has a constant spatial extension (see fig. 1).
If OA′ is parallel to the second strip, we can take _t′_ as the _t_-axis
and _x′_ as the _x_-axis, then the second body will appear to be at
rest, and the first body in uniform motion. We shall now assume that the
first body supposed to be at rest, has the length _l_, _i.e._, the cross
section PP of the first strip upon the _x_-axis = _l_^. OC, where OC is
the unit measuring rod upon the _x_-axis—and the second body also, when
supposed to be at rest, has the same length _l_, this means that, the
cross section Q′Q′ of the second strip has a cross-section _l_^· OC′,
when measured parallel to the _x′_-axis. In these two bodies, we have
now images of two Lorentz-electrons, one of which is at rest and the
other moves uniformly. Now if we stick to our original coordinates, then
the extension of the second electron is given by the cross section QQ of
the strip belonging to it measured parallel to the _x_-axis. Now it is
clear since Q′Q′ = _l_^· OC′, that QQ = _l_^· OD′.
If (_dc_/_dt_) = _v_, an easy calculation gives that
OD′ = OC √(1-(_v²_/_c²_)), therefore (PP/QQ) = (1/√(1-(_v²_/_c²_))
This is the sense of Lorentz’s hypothesis about the contraction of
electrons in case of motion. On the other hand, if we conceive the
second electron to be at rest, and therefore adopt the system (_x′_,
_t′_,) then the cross-section P′P′ of the strip of the electron parallel
to OC′ is to be regarded as its length and we shall find the first
electron shortened with reference to the second in the same proportion,
for it is,
P′P′/Q′Q′ = OD/OC′ = OD′/OC = QQ/PP
Lorentz called the combination _t′_ of (_t_ and _x_) as the _local time_
(_Ortszeit_) of the uniformly moving electron, and used a physical
construction of this idea for a better comprehension of the
contraction-hypothesis. But to perceive clearly that the time of an
electron is as good as the time of any other electron, _i.e._ _t_, _t′_
are to be regarded as equivalent, has been the service of A. Einstein
[Ann. d. Phys. 891, p. 1905, Jahrb. d. Radis. ... 4-4-11-1907.] There
the concept of time was shown to be completely and unambiguously
established by natural phenomena. But the concept of space was not
arrived at, either by Einstein or Lorentz, probably because in the case
of the above-mentioned spatial transformations, where the (_x′_, _t′_)
plane coincides with the _x_-_t_ plane, the significance is possible
that the _x_-axis of space some-how remains conserved in its position.
Public-domain text, read in full here on John Shaqi.
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