\u3b8
1 + tanh \u3b7\u2032 tanh \u3b8
\u3b7 = \u3b7\u2032 + \u3b8. (2.37)
Since rapidities are additive, their introduction simplifies some calculations
and they have often been used as variables in elementary particle physics.
With these new hyperbolic variables we can write the Lorentz transforma-
tion in a particularly simple way. Using Eq. (2.35) in Eqs. (2.27) and (2.30) we
find x = x\u2032 cosh \u3b8 + ct\u2032 sinh \u3b8, ct = x\u2032 sinh \u3b8 + ct\u2032 cosh \u3b8. (2.38)
Equation (2.33) may be written in a geometric form by introducing the
The Special Theory of Relativity
Figure 2.10: The interval between A and B is space-like, between C and D light-like,
and between E and F time-like.
Let two events be given. The coordinates of the events, as referred to two
different reference frames\u3a3 and\u3a3\u2032 are connected by a Lorentz transformation.
The coordinate differences are therefore connected by
\u2206t = \u3b3(\u2206t\u2032 + vc2\u2206x
\u2032), \u2206x = \u3b3(\u2206x\u2032 + v\u2206t\u2032),
\u2206y = \u2206y\u2032, \u2206z = \u2206z\u2032. (2.39)
Just like (\u2206y)2+(\u2206z)2 is invariant under a rotation about the x-axis,\u2212(c\u2206t)2+
(\u2206x)2 + (\u2206y)2 + (\u2206z)2
(\u2206s)2 = \u2212(c\u2206t)2 + (\u2206x)2 + (\u2206y)2 + (\u2206z)2
= \u2212(c\u2206t\u2032)2 + (\u2206x\u2032)2 + (\u2206y\u2032)2 +(\u2206z\u2032)2. (2.40)
This combination of squared coordinate-intervals is called the spacetime in-
terval, or the interval. It is invariant under both rotations and Lorentz trans-
formations.
Due to the minus-sign in Eq. (2.40), the interval between two events may
be positive, zero or negative. These three types of intervals are called:
(\u2206s)2 > 0 space-like
(\u2206s)2 = 0 light-like
(\u2206s)2 < 0 time-like
(2.41)
the distance between the events is purely spatial. Two events with a light-like
interval (C and D in Fig. 2.10), can be connected by a light signal, i.e. one
can send a photon from C to D. The events E and F have a time-like interval
between them, and can be observed from a reference frame inwhich they have
the same spatial position, but occur at different points of time.
Since all material particles move with a velocity less than that of light, the
points on the world-line of a particle are separated by time-like intervals. The
curve is then said to be time-like. All time-like curves through a point pass
inside the light-cone from that point.
is invariant under a Lorentz transformation, i.e.,
like interval (A and B in Fig. 2.10), then there exists a Lorentz transformation to
a new reference frame where A and B happen simultaneously. In this frame
Lorentz-invariant interval
If the velocity of a particle is u = \u2206x/\u2206t along the x-axis, Eq. (2.40) gives
\u221a
1\u2212 u
2(t)
c2
dt. (2.46)
The relativistic time-dilatation has been verified with great accuracy by obser-
vations of unstable elementary particles with short life-times [FS63].
The Special Theory of Relativity
2.9 The twin-paradox
Rather than discussing the life-time of elementary particles, we may as well
apply Eq. (2.46) to a person. Let her name be Eva. Assume that Eva is rapidly
acceleration from rest at the point of time t = 0 at origin to a velocity v along
the x-axis of a (ct, x) coordinate system in an inertial reference frame \u3a3. (See
Fig. 2.12.)
Figure 2.12: World-lines of the twin sisters Eva and Elizabeth.
At a point of time tP she has come to a position xP . She then rapidly
decelerates until reaching a velocity v in the negative x-direction. At a point
of time tQ, as measured on clocks at rest in \u3a3, she has returned to her starting
location. If we neglect the brief periods of acceleration, Eva\u2019s travelling time
as measured on a clock which she carries with her is
tEva =
(
1\u2212 v
2
c2
)1/2
tQ. (2.49)
Now assume that Eva has a twin-sister named Elizabeth who remains at rest
at the origin of \u3a3.
Elizabeth has become older by \u3c4Elizabeth = tQ during Eva\u2019s travel, so that
\u3c4Eva =
(
1\u2212 v
2.10 Hyperbolic motion
=
2.10 Hyperbolic motion
With reference to an inertial reference frame it is easy to describe relativistic
accelerated motion. The special theory of relativity is in no way limited to
describe motion with constant velocity.
Let a particle move with a variable velocity u(t) = dx/dt along the x-axis
in \u3a3. The frame \u3a3\u2032 moves with velocity v in the same direction relative to
\u3a3. In this frame the particle-velocity is u\u2032(t\u2032) = dx\u2032/dt\u2032. At every moment
the velocities u and u\u2032 are connected by the relativistic formula for velocity
addition, Eq. (2.33). Thus, a velocity change du\u2032 in \u3a3\u2032 and the corresponding
