Environmental Engineering Reference
In-Depth Information
or other has a special name: it is called a 'world line'. Also shown in Figure 13.3
are the light cones
x
2
c
2
t
2
. When our hypothetical person is at O, their
entire future must lie inside the light cone with
t>
0, since to escape from that
region would require the person to travel faster than the speed of light. If this is not
immediately clear then it might help to note that since the speed must always be
smaller than
c
it follows that the gradient of the wordline must always be steeper
than the slope of the light cone. This region of the space-time diagram, marked
'future' in Figure 13.3, is called the future light cone of the person at O. Similarly,
their entire past must lie inside the cone with
t<
0 (their past light cone) otherwise
the person would have travelled faster than light speed at some time in their past.
We are now ready to explain what is meant by causality. Let's start with a
definition. Two events A and B are said to be causally connected if event B lies
in either the future or past light cone of event A. Stated slightly more succinctly,
events A and B must lie within each-other's light cones. Conversely, if two events
lie outside of each-other's light cones then they are said to be causally disconnected.
Figure 13.4 illustrates what is going on with an example. Three events, A, B and C,
are represented on a space-time diagram. B lies in the future light cone of A and is
therefore causally connected to A. Another way of saying this is to say that A lies
in the past light cone of B, which is illustrated by the dotted lines in Figure 13.4. C
is just outside of A's future light cone and is therefore causally disconnected from
A. However, C does lie within the past light cone of B and so events B and C are
causally connected. All of this may sound rather academic but it is far from that.
Let us see why. In principle event A can influence event B but it cannot influence
event C. Indeed, event A might be the birth of our hypothetical person whilst event
B could be their graduation from university. Event C however knows nothing of
the person's birth since it occurs too far away from A for even a pulse of light to
travel in the time available. The event at C is however causally connected to B.
For example it might correspond to the launch of an extra terrestial spaceship from
some distant part of the Universe such that the spaceship finally arrives on Earth
just in time for graduation. If Einstein's Special Theory of Relativity is to satisfy the
demands of causality then we must insist that the time order of causally connected
events is agreed upon by all inertial observers, i.e. everyone agrees that birth
precedes death. It is very hard to imagine a universe without causality, never mind
y
2
+
=
t
B
C
A
x
Figure 13.4 Three events A, B and C along with the future light cone of A and the past
light cone of B.
