
Space. Time Diagrams
189
World line
of particle
OIL_%
Section
A.I
and waxes of frame S orthogonal (perpendicular) to one another (Fig.
A.I).
If
we wanted to represent the motion of a particle in this frame,
we would draw a curve, called a
world line,
which gives the loci of space
time points corresponding 10the motion.
*
The tangent to the world line
at any point, being
dxldw
=
1..
(dx/dt),
is always inclined at an angle
c
.
less than 45
0
with the time axis. For this angle (see Fig. A·I) is given by
tan ()
=
=
uf
c and we must have
u
<
c
for a material particle.
The world line of a light wave, for which
u
=
c, is a straight line making
a 45
0
angle with the axes.
Consider now the primed frame
(8')
which moves relative to S with
avelocity v along the common
xx'
axis. The equation of motion of
5'
relative to S can be obtained by selling
x'
=
0 (which locates the origin
of
S');
from Eq. A·I, we see that this corresponds to
x
=
f3w
(=
vt).
We
draw the line
x'
=
0 (that
IS,
x
=
f3w)
on our diagram (Fig. A.2) and note
that, smce
v
<
c
and
f3
<
1, the angle which this line makes with the
w.axis,
<!>(=
tan
1
{3), is less than 45°. Just as the waxis corresponds to
x
=
0 and is the time axis in frame 5, so the line
=
0 gives the time axis
*
Minkowski referred to spacetime as "the world." Hence, events are world points and a collection
,
/
of events giving the history of_a particle is a
worldlin~~pn:ysl.cal
laws on the Interaction of par
ticles can be thought of as the geometri.c relations between their world lines. In this sense,
Minkowski may be said to have geometrized phYSICS.
I
I
i
I
I
,I
1
" ::r;'"
fro.tv
<.~
+0
12c.l,'t:
v ,
'tr
1\
Supplementary Topic A
The
Geometric
Repvesentation
of
Space.
.
Time
188
Notice the symmetry in this form of the equations.
To represent the situation geometrically, we begin by drawing the
x
AI
Space.Time Diagrams
In classical
p'hysics~
the time coordinate ie unaffected
by
a trans
fO:J:111ClHon from one inertialframe to another. The time coordinate,
e,
one inertial 'system does not depend on.the space coordinates,
x. y,
,z
of another inertial system.
.thetransformation equation being
t;
=
t.
In
relativity.ihowever.repace
alldtimear~.::interdependent.
The time coordi
nate of one inertial system depends on both the time and the space coor
dinates of another inertial system, the transformation equation being
e
=
[t 
(vlc
2
)xJl
VI
v
21c2
,
Hence, instead of treatmg space and
time separately, as is quite properly done in classical theory, it isnatural
in relativity to treat them together.
J:I.
Minkowski [1] was first to show
clearly how this could be done.
In what follows, we shall consider only one Bpace axis, the xaxis,and
shall ignore the
y
and
z
axes. We lose
1'10
generality by this algebraic
simplification and this procedure willenable us to focus more clearly on
the interdependence of Bpac,:: and time and its geometric representation.