# Inverse transformations

Mentz114

## Homework Statement

##{\Lambda_c}^b## is a Lorentz transformation and ##{\Lambda^c}_b## is its inverse, so ##{\Lambda_c}^b {\Lambda^c}_b## gives an identity matrix.

How can I write this, assuming it's possible, in terms of ##\delta##'s ?

## The Attempt at a Solution

##{\Lambda_a}^b {\Lambda_c}^a =\delta^b_c ##

Wrong.

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## Answers and Replies

##\Lambda^{a}{}{}_{c}\Lambda^{c}{}{}_{b} = \delta^{a}{}{}_{b}##; it's just regular matrix multiplication.

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Mentz114
##\Lambda^{a}{}{}_{c}\Lambda^{c}{}{}_{b} = \delta^{a}{}{}_{b}##; it's just regular matrix multiplication.
OK, thanks. I wasn't too far off.
Can I use a ##\delta## to relabel these indexes ? Like

##{\Lambda_c}^b {\Lambda^d}_a {\delta_d}^c = {\delta^b}_a\ \Rightarrow\ {\Lambda_c}^b {\Lambda^d}_a {\delta_d}^c{\delta^d}_c = {\delta^b}_a {\delta^d}_c ##

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I can't parse what you wrote down because the indices are being summed over more than twice. But if you're asking if something like ##\Lambda^{a}{}{}_{c}\delta_{ab} = \Lambda_{bc}## is true then the answer is yes. Keep in mind that this is technically not index raising and lowering but rather just applying the definition of ##\delta_{ab}##. The two are different mathematical operations in spaces that aren't Euclidean.

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Mentz114
I can't parse what you wrote down because the indices are being summed over more than twice. But if you're asking if something like ##\Lambda^{a}{}{}_{c}\delta_{ab} = \Lambda_{bc}## is true then the answer is yes. Keep in mind that this is technically not index raising and lowering but rather just applying the definition of ##\delta_{ab}##. The two are different mathematical operations in spaces that aren't Euclidean.
Yes, I edited while you were writing and took out the raising/lowering.

I can also see now that what I wrote is not sensible. I'm actually trying to establish if for a geodesic ##u^a##

##{\Lambda_c}^b {\Lambda^d}_a u^c \nabla_b u_d = 0##

Messing ##\delta##s looks like a red-herring.

Is this an exercise from a textbook?

Mentz114
Is this an exercise from a textbook?
No. It comes from

\begin{align} \dot{v}_a &=\left({\Lambda_c}^b u^c \right)\nabla_b \left( {\Lambda^d}_a u_d \right)\\ &= {\Lambda_c}^b {\Lambda^d}_a u^c \nabla_b u_d + {\Lambda_c}^b u^c u_d \nabla_b {\Lambda^d}_a \end{align}
I reason that if ##\Lambda## is an inertial boost then both terms must be zero. But the first term does not depend in any way on this distinction ( no derivatives of ##\Lambda##) so it must be identically zero. Am I wrong ?