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Special Relativity
The Lorentz Transformations as Viewed by Einstein
Classical mechanics rests on the notions of absolute space and absolute time. Absolute space is assumed to be an affine 3-space with a Euclidean metric that is unique up to constant multiples. One can introduce Cartesian coordinates x, y, z. Absolute time is measured by a time coordinate t, unique up to transformations t â at + b, where a and b are constants. Newtonâs laws of motion are assumed valid when x, y, z, t are used as coordinates for space and time. Newton thought of absolute space as existing independently of matter in the universe, whereas Bishop Berkeley interpreted absolute space as being fixed with respect to the bulk of matter in the universe (the so-called âfixed starsâ).
An inertial (coordinate) system is a Cartesian coordinate system that moves uniformly, i.e., without acceleration, and, in particular, without rotation, with respect to absolute space. It was realized that Newtonâs laws of motion are also valid in any inertial system, since accelerations are unaffected by uniform motion. Any two inertial systems are in uniform translational motion with respect to each other.
The existence of an absolute space or absolute frame of reference became dubious much later when highly accurate optical experiments were performed. It was found, toward the end of the last century, that light is propagated isotropically, i.e., with the same speed in all directions, in each supposed inertial system. Consider two inertial systems S and SⲠpassing one another. A light pulse is emitted at their common origin at time t = 0. It is observed (essentially in the Michelson-Morley experiment) that both systems see their respective origins as the centers of the resulting spherical light pulse for all time (the âlight pulse paradoxâ)!
This, together with other electromagnetic considerations, led Einstein (among others) to reject the notion of an absolute space. He still retained, however, the notion of a distinguished (but undefined) class of inertial systems. Einstein then showed that this rejection of an absolute space and the resulting notion of absolute motion of an inertial system forces us to abandon also the idea of an absolute time! Einstein (1905) reasoned as follows.
Consider two space-time events
E1 and
E2. When viewed from an inertial system
S these events have coordinates (
x1,
y1,
z1,
t1) and (
x2,
y2,
z2,
t2). Since light is propagated isotropically in the system
S, the two events will occur simultaneously (at the same time
t) in
S if and only if light pulses emitted
E1 and
E2 reach the spatial midpoint
at the same instant. Consider, for example, two lightning bolts that strike a railway embankment in
S at
t = 0 at spatial coordinates (
x, 0, 0) and (â
x, 0, 0). These strikes are seen simultaneously at the origin of
S at time
x/c, where
c is the speed of light. Consider an inertial system
SⲠ(a train) moving along the tracks (the
x direction in
S) and suppose that the two ends of the train are struck by the same bolts. If the origin of
SⲠis at the midpoint of the train (equal number of cars forward and behind), it is clear that the forward bolt will be seen at this midpoint before the backward bolt. (It is important here that the speed of light is not infinite.)
Two inertial systems in relative motion will disagree as to whether or not certain spatially separated events are simultaneous. S and
SⲠmust be keeping different times. This simple observation by Einstein distingui...
