The Lorentz Transformation 101Equations (2.4.9), (2.4.10), and (2.4.11) are the Lorentz
transformation equations that govern special relativity kinematics. If v is
very small compared to c, then 1 — (v^2 /c^2 ) is close to 1 and the Lorentz
equations become (approximately) x = x\ + vt, t = t. This is the tradi-
tional Galilean transformation relating two frames moving with relative
velocity v in Newtonian kinematics. We always use the Galilean transfor-
mation, especially the t = t\ part, in our investigations of the classical
Newtonian mechanics. Let us stress again that, in Newtonian mechanics,
time is the same in both frames.
Equations (2.4.11) were first proposed in 1904 by the Dutch scientist
HENDRIK ANTOON LORENTZ (1853-1928) as a means of explaining the re-
sult of the Michelson-Morley experiment. The transformation became the
main mathematical means of verifying whether a proposed equation can
describe effects at speeds comparable with the speed of light: all one needs
to check is whether the equation is invariant under the Lorentz transfor-
mation, or, in other words, whether the equation treats the space and time
variables in a manner consistent with special relativity.
The Galilean transformation goes back to Galilei's 1638 book Discorsi
e Dimostrazioni Matematiche, intorno a due nuoue scienze, known in the
English translation as Discourse on Two New Sciences.
The Lorentz transformation implies a number of important, and some-
times counter-intuitive, results of special relativity. Let us summarize some
of these results.
(i) RELATIVITY OF SPACE AND TIME INTERVALS: The length I of a rod
placed along the z-axis and moving with speed v, as measured by an ob-
server in the fixed frame O, is smaller than the length £Q of the same rod
measured in the frame 0\ that moves with the rod; see Figure 2.4.1. Sim-
ilarly, the time interval At between two events in a frame moving with
speed v, as measured in the fixed frame, is longer than the time interval
Aio between the same events, measured in the moving frame:
* = Wl ~ ("Vc^2 ), At= ° =• (2.4.12)
y/1 - {V^2 /C^2 )(ii) RELATIVITY OF MASS: the rest mass mo of an object is either its
inertial or gravitational mass measured in an inertial frame when the object
is not moving relative to that frame. The relativistic mass m and the