Environmental Engineering Reference
In-Depth Information
13
Space-time Diagrams
and Causality
Right back in Section 6.1.3 we stated that it is sometimes possible for two observers
to disagree on the time ordering of a pair of events. As promised then, we shall now
take a closer look at this extraordinary statement. Our journey will eventually lead
us, at the end of this chapter, to a new way to think about Einstein's theory which
places the emphasis much more on space-time and the notions of past, present and
future than on the constancy of the speed of light.
We shall find it very helpful to draw 'space-time diagrams' and Figure 13.1
illustrates a particularly simple space-time diagram: an event is represented by a
point in the
x
t
plane. Of course actual events in space-time are represented
by points in a four-dimensional space but the diagrams are easier to draw if we
imagine there is only one spatial dimension. A slightly more interesting space-time
diagram is illustrated in the left pane of Figure 13.2 which shows the history of a
light-front which originates at the origin at time
t
−
=
0. As time progresses the light
spreads out such that at some time
t
the light-front is located at
x
=±
ct
. This
particular example is perhaps better visualised by going to two spatial dimensions,
in which case the light spreads out such that at some time
t
the light-front is a
circle of radius
ct
. In this case the history of the light-front is a cone
1
, as illustrated
in the right pane of Figure 13.2.
Our third example of a space-time diagram is shown in Figure 13.3. This dia-
gram shows the curve in space-time which corresponds to the entire lifetime of a
hypothetical person. The person was born at A and will die at B, and when they are
at O their future lies in the upper half-plane whilst their past lies in the lower half
plane. Such a curve through space-time corresponding to the history of some object
1
The equation of the cone is
x
2
y
2
c
2
t
2
.
+
=
