Environmental Engineering Reference
In-Depth Information
the past of O in some other frame. The picture is however strikingly different for
events which lie outside of O's light cones, such as the event at B. The hyperbolae
of constant proper time cross the
t
0 axis and hence it is perfectly possible for
events which lie in O's past in one frame (such as the event at B) to appear in O's
future in another frame (the event is at B
in this frame). In two or three spatial
dimensions none of the conclusions we have just drawn change but the hyperbolic
curves of constant proper time become hyperbolic surfaces of constant proper time.
Before we finish this section we shall introduce some common terminology.
Notice that all events that lie in either the future or the past light cone of the event
at O necessarily satisfy
(τ )
2
>
0 whereas all events that lie outside of these light
cones always satisfy
(τ )
2
<
0. If
(τ )
2
>
0 for a pair of events we say that
the events are separated by a 'timelike' interval and if
(τ )
2
<
0 we say they are
separated by a 'spacelike' interval. In the special case that
(τ )
2
=
0 the interval
is said to be 'lightlike'. Events that are timelike separated (such as the events
at O and at A) are always causally connected whereas events that are spacelike
seperated (such as the events at O and at B) are always causally disconnected. Only
something travelling at the speed of light can be present at both events if they are
separated by a lightlike interval.
=
13.2 AN ALTERNATIVE APPROACH
We have established that Einstein's theory of Special Relativity can be under-
stood in terms of an underlying four dimensional space-time continuum and that
physical laws are built out of objects such as vectors and scalars in this four dimen-
sional space. Equations built in such a way will automatically satisfy Einstein's
postulates, just as equations built out of three dimensional vectors and scalars are
automatically independent of any particular choice of co-ordinate system. Almost
as if by magic, we found that Special Relativity is also a causal theory although the
idea of causality was never mentioned when we originally formulated the theory in
Part II. In this section we would like to promote causality to a much more central
concept within the theory and at the same time we shall develop a new way of
viewing the special role played by the speed of light.
Our main aim is to reformulate the theory of Special Relativity. Rather than
start with Einstein's two postulates we shall start by boldly assuming that space
and time form a four dimensional continuum which we shall, of course, refer
to as space-time. What properties of space-time shall we assume? In the first
case we shall insist that it supports the notion of distance between two points.
More specifically, if we consider two neighbouring points in space-time located
at co-ordinates
(ct,x,y,z)
and
(ct
+
c
d
t,x
+
d
x,y
+
d
y,z
+
d
z)
then the squared
distance between these two points
(
d
s)
2
is defined by
g
00
000
c
d
t
d
x
d
y
d
z
0
10 0
00
−
(
d
s)
2
=
(c
d
t,
d
x,
d
y,
d
z)
10
000
−
−
1
g
00
(c
d
t)
2
(
d
x)
2
(
d
y)
2
(
d
z)
2
.
=
−
−
−
(13.5)
