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:

$\Lambda$⁰₀$\Lambda$⁰₀ - $\Lambda$⁰₁$\Lambda$⁰₁ - $\Lambda$⁰₂$\Lambda$⁰₂ - $\Lambda$⁰₃$\Lambda$⁰₃ = 1

and six like:

- $\Lambda$⁰₀$\Lambda$¹₀ - $\Lambda$⁰₁$\Lambda$¹₁ - $\Lambda$⁰₂$\Lambda$¹₂ - $\Lambda$⁰₃$\Lambda$¹₃ = 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]

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