Environmental Engineering Reference
In-Depth Information
ct
A
B
′
A′
Future
x
O
Past
B
Figure 13.5 Space-time diagram illustrating lines of constant proper time. Two events are
labelled by the points A and B in a particular inertial frame. In a different inertial frame the
same two events have the co-ordinates labelled by the points A
and B
.
progress without them. We know that the proper time interval
2
between any pair
of events is independent of inertial frame, i.e.
1
c
x
2
1
c
x
2
=
t
2
(τ )
2
(t)
2
=
−
−
.
(13.4)
Let us now imagine we are measuring the events in
S
. Graphically, we'll still
represent the events using Figure 13.5 but we should re-interpret the
x
axis as the
x
axis and the
t
axis as the
t
axis. The event at O stays at the origin, since as
always we're assuming that
S
and
S
t
=
0.
But what happens to the events located at A and B in
S
? Figure 13.5 contains the
answer. Two curves are drawn on the figure, they are such that all of the points
on a given curve are the same space-time distance away from the origin, i.e. they
are at a fixed value of
(τ )
2
. In fact such curves are necessarily hyperbolae since
(τ )
2
have their origins coincident at
t
=
(
c
x)
2
is none other than the equation of a hyperbola. Now it
follows that in moving from
S
to
S
the event at A can only move to another point
on the hyperbola passing through A. It might for example move from A to A
.
Similarly the event at B must remain on the hyperbola passing though B and in the
figure we have shown it moving to the point B
. We can now immediately see why
Einstein's theory does not violate causality. Events which lie in the future (or past)
light cone of O can only ever be transformed to another point which is also in the
future (or past) light cone of O under a Lorentz transformation. There is simply no
possibility for an event which lies in the future of O in one inertial frame to lie in
(t)
2
=
−
2
Or equivalently the distance in space-time.