Environmental Engineering Reference
In-Depth Information
12
Four-vectors and Lorentz
Invariants
In the last chapter we arrived at the conclusion that physical laws should utilise
vectors in the four dimensional space-time of Minkowski. In this chapter we explore
more fully the four-vector formalism and in so doing it should become clear that
this is indeed the language of Special Relativity.
The prototype four-vector is the one which specifies the displacement between
two events located at positions
X
1
and
X
2
, namely
X
=
X
2
−
X
1
(12.1)
and the square of the invariant distance between the two events is
c
2
(t)
2
(x)
2
(y)
2
(z)
2
,
X
·
X
=
−
−
−
c
2
(τ )
2
.
=
(12.2)
The second line defines what is called the 'proper time' interval between the two
events
τ
. We will usually speak of the proper time rather than the 'invariant
distance' although the two are the same thing up to a factor of
c
. If it is possible
to find a frame in which the two events occur at the same point then the proper
time is simply the time interval between the two events in that inertial frame. For
example, in a frame of reference attached to your body the proper time interval
between any two events in your life is simply the time difference measured by
the watch on your wrist. Notice that
(τ )
2
can in principle take on the value of
any real number; positive, negative or zero. This is quite different from distances
in ordinary Euclidean space, and we shall explore the consequences in the next
chapter. Suffice to say here that if
(τ )
2
<
0 then the proper time interval is
