B Jaime decides to build an accessibility ramp to help students reach the top of a set of stairs. If BI is 24 ft and
AB is 9 ft, what is the measure of the angle of elevation?

\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