What is the determinant of a Lorentz transformation matrix?

[vin300]

I have been asked to prove that the determinant of any matrix representing a Lorentz transformation is plus or minus 1. I can see that the determinant of the Lorentz transformation matrix is 1, but don't know how to prove +-1 in general. How to generalise the lorentz transformation? I've also read that rotations in the spatial planes also constitute L.T., that any transformation that keeps the metric invariant is an L.T.

 

[Muphrid]

Since any arbitrary LT is either a boost between timelike and spacelike directions, a rotation between spacelike directions, or a combination of the two, I think all you need is to check that the determinant of a 3D rotation is $\pm 1$. After that, if $\underline L \underline R$ both represent LT's, then what do you know about

$$\det (\underline L \underline R) = ?$$

 

[vin300]

Reversal of the space or time axis produces a matrix of determinant -1. That proves that any LT matrix has determinant $\pm 1$. det(LR)=det(L)det(R), so again, the resulting matrix has determinant $\pm 1$.

 

[vin300]

How to prove that the matrix version of η_a'b'= A^m_a'Aⁿ_b'η_mn is η= A^TηA, where A is an LT matrix?