Environmental Engineering Reference
In-Depth Information
t
A
x
O
B
Figure 13.7
Transformations in Euclidean space-time.
In Figure 13.7, the two frames are clearly related to each by a rotation through
θ
=
π
radians. There is therefore no frame-independent notion of past, future or
present in this space-time, which means that it does not support the idea of cause
and effect in the laws of physics. We must therefore reject this space-time and are
left with only one possibility: space-time must be Minkowski space-time with a
metric
10 0 0
0
10 0
00
−
g
=
.
(13.13)
10
00 0
−
−
1
Under only a few rather natural assumptions we have arrived at the conclusion
that Minkowski space-time is the only possible space-time. The constant
c
was
originally introduced only to calibrate distances in the time direction, with the
causal structure of the theory elevating it to the status of a limiting speed. Actually
we should mention that it is still possible that
c
could be infinite and this would
lead us to Galilean relativity. There is no purely theoretical argument to reject
this possibility and it is experiment that informs us that
c
is in fact finite. Armed
with the metric we can now go ahead and re-derive all of the familiar results we
have encountered so far in Special Relativity. For example, the space-time we have
introduced supports the existence of four-vectors. The displacement four-vector
=
X
(ct,x,y,z)
(13.14)
is our prototypical four-vector and the metric tensor tells us how to form the scalar
product, i.e.
(
X
)
T
(ct)
2
(x)
2
(y)
2
(z)
2
.
g
(
X
)
=
−
−
−
(13.15)
