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Let

(16.30) 
be a column vector. Note that we no longer indicate a vector by using a
vector arrow and/or boldface  those are reserved for the spatial part
of the fourvector only. Then a ``matrix'' scalar product is formed in
the usual way by

(16.31) 
where is the (row vector) transpose of . The metrix tensor is
just a matrix:

(16.32) 
and
. Finally,

(16.33) 
In this compact notation we define the scalar product in this metric to
be

(16.34) 
We seek the set (group, we hope) of linear transformations that leaves
invariant. Since this is the ``norm'' (squared) of a
four vector, these are ``length preserving'' transformations in this four
dimensional metric. That is, we want all matrices such that

(16.35) 
leaves the norm of invariant,

(16.36) 
or

(16.37) 
or

(16.38) 
Clearly this last condition is sufficient to ensure this property in .
Now,

(16.39) 
where the last equality is required. But
, so

(16.40) 
is a constraint on the allowed matrices (transformations) . There
are thus two classes of transformations we can consider. The
 proper Lorentz transformations
 with
; and
 improper Lorentz transformations
 with
.
Proper L. T.'s contain the identity (and thus can form a group by
themselves), but improper L. T.'s can have either sign of the
determinant. This is a signal that the metric we are using is ``indefinite''.
Two examples of improper transformations that illustrate this point are
spatial inversions (with
) and (space and
time inversion, with
).
In very general terms, the proper transformations are the continuously
connected ones that form a Lie group, the improper ones include one or more
inversions and are not equal to the product of any two proper transformations.
The proper transformations are a subgroup of the full group  this is not
true of the improper ones, which, among other things, lack the identity. With
this in mind, let us review the properties of infinitesimal linear
transformations, preparatory to deducing the particular ones that form the
homogeneous Lorentz group.
Subsections
Next: Infinitesimal Transformations
Up: The Lorentz Group
Previous: The Metric Tensor
Contents
Robert G. Brown
20071228