Environmental Engineering Reference
In-Depth Information
time be Euclidean. All that remains is to specify the top left hand entry,
g
00
.Since
it is a dimensionless constant, we can always choose the constant
c
in Eq. (13.5)
such that
g
00
=±
1 without altering the value of the distance d
s
. We have now
almost completely specified the metric of space-time. All that remains is for us to
settle on the sign of
g
00
. It is at this point that causality enters.
We state the result first and then prove it: the metric must have
g
00
=+
1if
it is to be the metric of a space-time that satisfies the demands of causality. We
already know that space-time with
g
00
1 is causal, because that space-time is
the Minkowski space-time of Special Relativity and, following the discussion in the
previous section, we know that to be a causal space-time. The core of the argument
involved showing that lines of constant proper time are hyperbolae and that any
hyperbola lying in either the future or the past light cone of some point O always
remains inside that light cone
5
. In contrast a hyperbola that lies outside of either
light cone will span times which lie both in the future and in the past of O. Since
a shift from one inertial frame to another corresponds to sliding events around on
their corresponding hyperbolae it follows that all observers always agree upon the
time ordering of causally connected events. Notice also that this argument only
works if all matter is constrained to move on timelike trajectories (which means
they must always travel with speed
c
or less) otherwise a particle could start at
O and follow a wordline outside of O's future light cone whereupon an observer
in a second inertial frame could conclude that an event on the particle's trajectory
which lies outside of O's future light cone could have occured in O's past; so
causality is also acting to constrain the laws of dynamics as well as the structure of
space-time. Our task is now to explain why
g
00
=−
=+
1 does not lead to a space-time
that respects causality. The invariant distance between two neighbouring events in
this space-time is given by
(
d
s)
2
(c
d
t)
2
(
d
x)
2
(
d
y)
2
(
d
z)
2
,
=−
−
−
−
(13.11)
which is just the metric of a four dimensional Euclidean space (recall the overall
sign is unimportant). The locus of all points in this space-time that lie a fixed
distance from the origin O is therefore, in one spatial dimension, simply a circle of
radius
(s)
2
. In two spatial dimensions we have the surface of a sphere and in
three spatial dimensions it is the three dimensional generalisation of a spherical sur-
face, often called a 'three-sphere'. Now we know that the equations of physics must
be the same for all co-ordinate systems that preserve the invariant distance between
any two events. However, as illustrated in Figure 13.7, if in a frame
S
an event is
located at A, which lies in the future of an event at O, then there always exists a
second frame of reference
S
in which that very same event occurs in O's past. The
figure shows the location B in
S
of the event located at A in
S
. The equivalent of
Lorentz transformations in this Euclidean space are simple rotations, i.e.
ct
x
−
cos
θ
ct
x
.
−
sin
θ
=
(13.12)
sin
θ
cos
θ
5
Moreover, the hyperbolae never intersect each other.
