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## Homework Statement

Prove that the proper orthochronous Lorentz group is a linear group. That is SOo(3, 1) = {a [itex]\in[/itex] SO(3, 1) | (ae4, e4) < 0 } where (x,y) = x[itex]^T[/itex][itex]\eta[/itex]y for [itex]\eta[/itex] = [1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 -1] (sorry couldn't work out how to properly display a matrix).

## Homework Equations

a is in SO(3,1) if det(a) = 1 and a[itex]^T[/itex][itex]\eta[/itex]a = [itex]\eta[/itex]

## The Attempt at a Solution

To show it is a linear group I need to show that the matrices in SOo(3, 1) are invertible, and that if a matrix is in SOo(3, 1) then its inverse is in there as well, and that it is closed under matrix multiplication. I have done the latter two bits already, but the thing I am struggling with is closure by matrix multiplication.

let a[itex]_{ij}[/itex] be the element in the ith row and jth column of a. For a to be in SOo(3, 1) the element a[itex]_{44}[/itex] must be positive. Therefore if a,b [itex]\in[/itex] SOo(3, 1) I need to show that given a[itex]_{44}[/itex] and b[itex]_{44}[/itex] positive, then ab[itex]_{44}[/itex] is also positive. This is what I'm stuck on - I can't work out how to manipulate the inequality to show that it is positive. I have managed to work out that a[itex]_{14}^{2}[/itex] + a[itex]_{24}^{2}[/itex] + a[itex]_{34}^{2}[/itex] = a[itex]_{44}^{2}[/itex] - 1 from a[itex]^T\eta[/itex]a = [itex]\eta[/itex]. This also tells me that a[itex]_{44}[/itex] ≥ 1.

So, assuming my workings are correct so far, I am trying to manipulate the following:

a[itex]_{14}^{2}[/itex] + a[itex]_{24}^{2}[/itex] + a[itex]_{34}^{2}[/itex] = a[itex]_{44}^{2}[/itex] - 1

b[itex]_{14}^{2}[/itex] + b[itex]_{24}^{2}[/itex] + b[itex]_{34}^{2}[/itex] = b[itex]_{44}^{2}[/itex] - 1

a[itex]_{44}[/itex] ≥ 1, b[itex]_{44}[/itex] ≥ 1

To show that:

ab[itex]_{44}[/itex] = a[itex]_{14}[/itex]b[itex]_{41}[/itex] + a[itex]_{24}[/itex]b[itex]_{42}[/itex] + a[itex]_{34}[/itex]b[itex]_{43}[/itex] > 0

Appreciate any guidance or hints in the right direction. I have been trying to use the triangle inequality so far but to no avail. Thanks.