Let x = ( ) 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 four-vector only. Then a ``matrix'' scalar product is formed in the usual way by (a,b) = ãb where is the (row vector) transpose of . The metrix tensor is just a matrix: g = ( ) and . Finally, gx = ( ) = ( ) .
In this compact notation we define the scalar product in this metric to be a ·b = (a,gb) = (ga,b) = ãgb = a^&alpha#alpha;g_&alpha#alpha;&beta#beta; b^&beta#beta;= a^&alpha#alpha;b_&alpha#alpha;.
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 x' = Ax leaves the norm of invariant, x' ·x' = x' g x' = x g x = x ·x or x Ã g A x = x g x or Ã g A = g . Clearly this last condition is sufficient to ensure this property in .
Now, Ã g A = g (A )^2 = g where the last equality is required. But , so A = ±1 is a constraint on the allowed matrices (transformations) . There are thus two classes of transformations we can consider. The
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.