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

1. Aug 16, 2013

### 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

2. Aug 16, 2013

### WannabeNewton

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