
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 10
the
motion.
*
The
tangent to the world line
at any point, being
dxl
dw
=
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 ()
=
dxl
dw
=
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
a velocity 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
x'
=
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.
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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
inertial
frame
to
another.
The
time
coordinate,
e,
of
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
is natural
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
will
enable us to focus more clearly on
the
interdependence
of
Bpac,::
and
time
and
its geometric representation.