I have spent much effort trying to prove that det|[itex]\Lambda[/itex](adsbygoogle = window.adsbygoogle || []).push({}); ^{[itex]\mu[/itex]}_{[itex]\upsilon[/itex]}| = +1 or -1 (following a successful effort to prove (3.50g) on p87 of MTW)

From the result of [itex]\Lambda[/itex]^{T}[itex]\eta[/itex][itex]\Lambda[/itex] = [itex]\eta[/itex] I've produced four equations like:

[itex]\Lambda[/itex]^{0}_{0}[itex]\Lambda[/itex]^{0}_{0}- [itex]\Lambda[/itex]^{0}_{1}[itex]\Lambda[/itex]^{0}_{1}- [itex]\Lambda[/itex]^{0}_{2}[itex]\Lambda[/itex]^{0}_{2}- [itex]\Lambda[/itex]^{0}_{3}[itex]\Lambda[/itex]^{0}_{3}= 1

and six like:

- [itex]\Lambda[/itex]^{0}_{0}[itex]\Lambda[/itex]^{1}_{0}- [itex]\Lambda[/itex]^{0}_{1}[itex]\Lambda[/itex]^{1}_{1}- [itex]\Lambda[/itex]^{0}_{2}[itex]\Lambda[/itex]^{1}_{2}- [itex]\Lambda[/itex]^{0}_{3}[itex]\Lambda[/itex]^{1}_{3}= 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|[itex]\Lambda[/itex]^{[itex]\mu[/itex]}_{[itex]\upsilon[/itex]}|^{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

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums - The Fusion of Science and Community**