Environmental Engineering Reference
In-Depth Information
Let us close this section by reflecting upon the necessity of space-time. Clearly
we can always represent events by a string of four numbers whose values depend
upon the inertial frame of reference and clearly an event is independent of inertial
frame. What does not follow however is that the string of four numbers should
constitute a vector in a space which possesses a well defined scalar product. To
illustrate the point, we could have attempted to seek a way to write down the laws
of physics such that they are invariant under Galilean transformations. However
our attempt to introduce vectors in four dimensional space-time would be plagued
by the fact that we cannot define a scalar product. This failure implies that the
invariant distance between any two points in Galilean space-time is not defined.
Thus Galilean four-vectors are not particularly useful objects in drawing up the
laws of physics. It is perhaps worth going into a little more detail. In Galilean
relativity the space-time four-vector of an event transforms according to
=
ct
x
y
z
1000
ct
x
y
z
u/c 100
0010
0001
(11.36)
as one moves between S and S . The time t of an event is a four-scalar but the
spatial interval between two points is not and there is no way to combine both
intervals into an invariant distance in space-time. However, we can still proceed,
even without the existence of a distance measure. The four-vector corresponding
to the velocity of a particle can be written
d X
d t =
U
=
(c,
x
).
(11.37)
This is a four-vector since X is a four-vector and t is a four-scalar, so the ratio
in the difference of two such quantites must itself be a four-vector. Similarly the
four-acceleration can be written
d U
d t
A
=
=
x
( 0 ,
).
(11.38)
Clearly there is little advantage in using the four-vector formalism since the time
component of these four vectors is either constant or zero. For example, Eq. (11.10)
gains nothing from being rewritten in terms of four-vectors. Even so, there is one
interesting insight we can gain using Galilean four-vectors. The conservation of
four-momentum implies that
m i (c,
x i )
=
m f (c,
x f ),
(11.39)
i
f
where the indices i and f label the particles in a closed system at two different
times. Apart from informing us that the momentum in three dimensions is conserved
this equation also informs us that the total mass of the system is conserved, i.e.
=
m i
m f .
(11.40)
i
f
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