- #1

fluidistic

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## Homework Statement

The problem can be found in Jackson's book.

An infinitesimal Lorentz transform and its inverse can be written under the form ##x^{'\alpha}=(\eta ^{\alpha \beta}+\epsilon ^{\alpha \beta})x_{\beta}## and ##x^\alpha = (\eta ^{\alpha \beta}+\epsilon ^{'\alpha \beta}) x^{'}_\beta## where ##\eta _{\alpha \beta}## is Minkowski's metric and the epsilons are infinitesimals.

1)Demonstrate, using the definition of the inverse, that ##\epsilon ^{'\alpha \beta}=-\epsilon ^{\alpha \beta}##.

2)Demonstrate, using the conservation of the norm, that ##\epsilon ^{\alpha \beta}=-\epsilon ^{\beta \alpha}##

## Homework Equations

Not really sure, but I used some eq. found on some page earlier in the book: ##\epsilon ^{'\alpha \beta}=\frac{\partial x^{'\alpha }}{\partial x^\alpha} \frac{\partial x^{'\beta}}{\partial x^\beta} \epsilon ^{\alpha \beta}##.

## The Attempt at a Solution

1)I used the relevant equation and wrote that it's equal to ##\frac{\partial x^{'\alpha }}{\partial x^\beta} \frac{\partial x^{'\beta}}{\partial x^\alpha}\epsilon ^{\alpha \beta}##. Then I calculated the partial derivatives using the 2 equations given in the problem statement, I made an approximation (depreciated terms with epsilons multiplied together because they are "infinitesimals") and I reached that ##\epsilon ^{'\alpha \beta}\approx \frac{\eta^{\alpha \beta}}{\eta ^{\alpha \beta}+\epsilon ^{'\alpha \beta}}\cdot \epsilon ^{\alpha \beta}##. I don't see how the first term can be worth -1 here... So I guess my approach is wrong. Or if it's right, I still don't see how I can show that the first term is worth -1. Thanks for any comment.