Let X ad Y be two vectors in R4, and define the inner product of X and Y as:
(X*Y) = gikXiYk (summation convention for repeated indicies)
Then we consider the 4x4 matrix g whose components are gik. I am of course interested in the case that g is NOT positive definite, because this is the situation when g represents the gravitational field in general relativity.
Let A be a 4x4 matrix which satisfies (X*Y) = (AX*AY), then I say that A is an element O(g), the orthogonal group determined by g.
I am interested in finding any sort of formula which relates the lie algebra of O(g) to the metric g.
In a previous question, it was suggested that I diagonalize the matrix g using the theorem on diagonalizing positive definite matrices. This method works nicely and gives a simple solution for the lie algebra in terms of the transformation matrix which diagonalizes g, but only when g is positive definite.
Can I still diagonalize my non positive definite g by finding the roots of the characteristic polynomial? I believe I must first somehow restrict the set of vectors I allow the inner product to work on, to avoid the case (X*Y) = 0. Nevertheless, for arguments sake let's assume that I can diagonalize g.
Let B be the transformation matrix, then I assume that I can write:
g = B-1ηB, where η is the identity matrix of signature (1,3), i.e. the metric of flat space time.
We can characterize the elements of O(g) by realizing that our inner product can be written as:
(X*Y) = XTgY
It's clear that if A is an element of O(g), then
(AX*AY) = XTATgAY = XTgY
Hence, ATgA = g
This formula can be written as gA-1g = AT, since g = g-1 explicitly.
Now applying the transformation matrix B:
gA-1g = B-1ηBA-1B-1ηB = AT, which I rearrange as:
ηBA-1B-1η = BATB-1
If by some chance (BA-1B-1)-1 = (BATB-1)T, then I can immediately conclude that A is in O(g) so long as BA-1B-1 is in O(1,3) (the group which preserves the metric η). From this step is it quite straightforward to compute the lie algebra, by taking advantage of the formula BeXB-1 = eBXB-1.
I am concerned about several steps of this procedure:
1) Is it legal to diagonalize g? I believe I need exclude any combination of vectors X, Y for which (X*Y) = 0. Since the squared norm is (X*X), then this amounts to disregarding vectors which lie along geodesic paths. Then I can deal separately for the case (X*X) > 0 and (X*X) < 0.
2) How am I supposed to deal with the condition that (BA-1B-1)-1 = (BATB-1)T? This condition seems quite restrictive.