Proving that the det of the Lorentz tensor is +1 or -1 (MTW p 89)

[TerryW]

I have spent much effort trying to prove that det|\Lambda^\mu_\upsilon| = +1 or -1 (following a successful effort to prove (3.50g) on p87 of MTW)

From the result of \Lambda^T\eta\Lambda = \eta I've produced four equations like:

\Lambda⁰₀\Lambda⁰₀ - \Lambda⁰₁\Lambda⁰₁ - \Lambda⁰₂\Lambda⁰₂ - \Lambda⁰₃\Lambda⁰₃ = 1

and six like:

- \Lambda⁰₀\Lambda¹₀ - \Lambda⁰₁\Lambda¹₁ - \Lambda⁰₂\Lambda¹₂ - \Lambda⁰₃\Lambda¹₃ = 0

I was hoping that by various combinations of products of these equations I would be able to find all the terms needed to produce det|\Lambda^\mu_\upsilon|² but I end up with a large number of terms so of which look to be OK plus lots of terms which I need to eliminate and can't. It is also very messy!

Is there an elegant way of doing this?


TerryW

 

[WannabeNewton]

$\det(\Lambda^{T}\eta \Lambda) = \det(\eta)\det(\Lambda^{T})\det(\Lambda) = \det(\eta)\\ \Rightarrow (\det(\Lambda))^{2} = 1\Rightarrow \det(\Lambda) = \pm 1$