26 Special Relativity
Lorentz Transformations. Consider two linear axes푥and푥′in one-dimensional
space,푥′being at rest and푥moving with constant velocity푣in the positive푥′direction.
Time increments are measured in the two coordinate systems as d푡and d푡′using two
identical clocks. Neither the spatial increments d푥and d푥′nor the time increments d푡
and d푡′are invariants—they do not obey postulate (i). Let us replace d푡and d푡′with
the temporal distances푐d푡and푐d푡′and look for alinear transformationbetween
the primed and unprimed coordinate systems, under which the two-dimensional
space-time distancedsbetween twoevents,
d푠^2 =푐^2 d휏^2 =푐^2 d푡^2 −d푥^2 =푐^2 d푡′^2 −d푥′^2 , (2.1)is invariant. Invoking the constancy of the speed of light it is easy to show that the
transformationmustbeoftheform
d푥′=훾(d푥−푣d푡),푐d푡′=훾(푐d푡−푣d푥∕푐), (2.2)where
훾=1
√
1 −(푣∕푐)^2
. (2.3)
Equation (2.2) defines theLorentz transformation, afterHendrik Antoon Lorentz
(1853–1928). Scalar products in this two-dimensional(ct,푥)-space are invariants
under Lorentz transformations.
Time Dilation. The quantity d휏in Equation (2.1) is called theproper timeand dsthe
line element. Note that scalar multiplication in this manifold is here defined in such
a way that the products of the spatial components obtain negative signs (sometimes
the opposite convention is chosen). (The mathematical term for a many-dimensional
space is amanifold.)
Since d휏^2 is an invariant, it has the same value in both frames:
d휏′^2 =d휏^2.While the observer at rest records consecutive ticks on his clock separated by a
space-time interval d휏=d푡′, she receives clock ticks from the푥direction separated
by the time interval d푡and also by the space interval d푥=푣d푡:
d휏=d휏′=√
d푡^2 −d푥^2 ∕푐^2 =√
1 −(푣∕푐)^2 d푡. (2.4)In other words, the two inertial coordinate systems are related by a Lorentz transfor-
mation
d푡= d푡′
√
1 −(푣∕푐)^2≡훾d푡′. (2.5)Obviously, the time interval d푡is always longer than the interval d푡′, but only notice-
ably so when푣approaches푐. This is called thetime dilation effect.
The time dilation effect has been well confirmed in particle experiments. Muons
are heavy, unstable, electron-like particles with well-known lifetimes in the labora-
tory. However, when they strike Earth with relativistic velocities after having been