Environmental Engineering Reference
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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.
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