Infinitesimal Lorentz transform and its inverse, tensors

[fluidistic]

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.

 

[vela]

For #1, apply the transformation and followed by its inverse and use the fact that that product should yield the identity.

 

[fluidistic]

For #1, apply the transformation and followed by its inverse and use the fact that that product should yield the identity.

I see... do you mean that I should perform $x^{'\alpha}x^{\alpha}=\text{Identity}$?
Also I don't see what's wrong with what I did in my attempt.
By the way I don't understand how tensors "work" yet, I am self studying this topic right now.

Edit: Nevermind my first question, the answer is no, the expression I wrote makes no sense...

 

[vela]

There's a problem with the equation you started with. A dummy index should only appear twice in each product. You wrote
$$\epsilon ^{'\alpha \beta}=\frac{\partial x^{'\alpha }}{\partial x^\alpha} \frac{\partial x^{'\beta}}{\partial x^\beta} \epsilon ^{\alpha \beta}.$$ The indices $\alpha$ and $\beta$ appear only once on the lefthand side, so they should appear only once on the righthand side. What you probably meant was something like
$$\epsilon ^{'\alpha \beta}=\frac{\partial x^{'\alpha }}{\partial x^\gamma} \frac{\partial x^{'\beta}}{\partial x^\delta} \epsilon ^{\gamma \delta}.$$ Notice how $\gamma$ and $\delta$ appear in pairs, which implies a summation over those indices.

This relationship, however, doesn't apply here. It relates the components of a tensor in one frame to the components of the same tensor in a different frame. In this problem, $\epsilon$ and $\epsilon'$ aren't the same tensor. One generates a Lorentz transformation, and the other, the inverse Lorentz transformation.

 

[fluidistic]

I see... thanks. Yes this is what I've done and I replaced gamma and delta by alpha and beta respectively... Ok I didn't know this could not be done.

 

[vanhees71]

I'd start from the Lorentz-transformation property of the representing matrices:
$$\eta_{\mu \nu} {\Lambda^{\mu}}_{\rho} {\Lambda^{\nu}}_{\sigma}=\eta_{\rho \sigma}.$$
Now write
$${\Lambda^{\mu}}_{\rho}=\delta_{\rho}^{\mu}+{\epsilon^{\mu}}_{\rho},$$
plug this in the above defining equation for LT matrices and expand up to the 1st-order terms in ${\epsilon^{\mu}}_{\rho}$. Finally note that by definition one applies the index-dragging rule not only to tensor components but also to Lorentz matrices. Note that a Lorentz matrix does not form tensor components but the infintesimal generators do, but that doesn't matter for your problem here. The only additional thing you need to know is that
$$\epsilon_{\rho \sigma}=\eta_{\rho \mu} {\epsilon^{\mu}}_{\sigma}.$$

 

[andrien]

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.

I will say you will not get anywhere using this.Just follow the idea jackson gave to you.Like when he say use conservation of norm then use x^αx_α=x'^αx'_α.You will have to leave some second order terms here to show ε^αβ=-ε^βα.First one is simple manipulation.

 

[WannabeNewton]

Hi there fluidistic! For the first part, note that all you have to do is a simple substitution. By definition we have $x'_{\beta} = (\eta_{\beta \gamma} + \epsilon_{\beta \gamma}) x^{\gamma}$ so plug this into $x^{\alpha} = (\eta^{\alpha \beta} + \epsilon'^{\alpha\beta})x'_{\beta}$ to get the quadratic $x^{\alpha} = (\eta^{\alpha \beta} + \epsilon'^{\alpha\beta})(\eta_{\beta \gamma} + \epsilon_{\beta \gamma}) x^{\gamma}$. Expand the quadratic to $O(\epsilon^2)$ and at the every end of the calculation use the fact that $x^{\alpha}$ is arbitrary. The other parts should be very similar.

 

[fluidistic]

Thanks guys for all the help. Unfortunately I'm way too sloppy with tensors in order to manipulate them.
Here's what I reached using WBN's suggestion: $x^\alpha \approx (\delta ^\alpha _\gamma +\eta ^{\alpha \beta}\epsilon _{\beta \gamma}+\epsilon ' ^{\alpha \beta}\eta _{\beta \gamma})x^\gamma$. By looking at this equation I have a feeling that gamma must be worth alpha and that what is in parenthesis must be worth the identity. This would imply that $\eta ^{\alpha \beta}\epsilon _{\beta \gamma}=-\epsilon ' ^{\alpha \beta}\eta _{\beta \gamma}$. And I guess that it's by working on that equation that I will get the desired result.

 

[WannabeNewton]

You're pretty much there. Recall that $\eta^{\alpha\beta}$ converts covariant indices into contravariant indices, and vice versa, upon contraction (more precisely, it is what we call a musical isomorphism and serves as a map between elements of the tangent space and its dual). So $\eta^{\alpha \beta}\epsilon_{\beta \gamma} = \epsilon^{\alpha}{}{}_{\gamma}$ and similarly $\epsilon'^{\alpha \beta}\eta_{\beta \gamma} = \epsilon'^{\alpha}{}{}_{\gamma}$.

So now you have $\delta^{\alpha}{}{}_{\gamma}x^{\gamma} + \epsilon'^{\alpha}{}{}_{\gamma}x^{\gamma} + \epsilon^{\alpha}{}{}_{\gamma}x^{\gamma} + O(\epsilon^2) = x^{\alpha}$. The result should then be immediate.

 

[fluidistic]

You're pretty much there. Recall that $\eta^{\alpha\beta}$ converts covariant indices into contravariant indices, and vice versa, upon contraction (more precisely, it is what we call a musical isomorphism and serves as a map between elements of the tangent space and its dual). So $\eta^{\alpha \beta}\epsilon_{\beta \gamma} = \epsilon^{\alpha}{}{}_{\gamma}$ and similarly $\epsilon'^{\alpha \beta}\eta_{\beta \gamma} = \epsilon'^{\alpha}{}{}_{\gamma}$.

So now you have $\delta^{\alpha}{}{}_{\gamma}x^{\gamma} + \epsilon'^{\alpha}{}{}_{\gamma}x^{\gamma} + \epsilon^{\alpha}{}{}_{\gamma}x^{\gamma} + O(\epsilon^2) = x^{\alpha}$. The result should then be immediate.

I see.
So I reach that $(\delta ^\alpha _\gamma +\epsilon ^\alpha _\gamma+\epsilon '^\alpha _\gamma )x^\gamma \approx x^\alpha$. The only way for both sides to be approximately equal requires that $\gamma =\alpha$ and that what is in parenthesis is worth the identity right?
In this case I reach that $\epsilon '^\alpha _\alpha =-\epsilon ^\alpha _\alpha$. But I still don't see how to reach the final result. I guess I must contract these tensors as to get only upper scripts with alpha and beta.
If I multiply the last expression by $\eta ^{\beta \alpha}$, I reach that $\epsilon '^{\beta \alpha}=-\epsilon ^{\beta \alpha}$. Now if this tensor is symmetric then I reach the desired result but how do I know whether it is symmetric?
I'm sure I made some error(s)...Edit: Nevermind, I think I reach the final result, if I multiply both sides of $\epsilon '^\alpha _\alpha =-\epsilon ^\alpha _\alpha$ by $\eta ^{\alpha \beta}$, but a multiplication by the RIGHT and not the left. This yields the result. Is that correct?

 

[vela]

I see.
So I reach that $(\delta^\alpha{}_\gamma +\epsilon^\alpha{}_\gamma + \epsilon'^\alpha{}_\gamma) x^\gamma \approx x^\alpha$. The only way for both sides to be approximately equal requires that $\gamma =\alpha$ and that what is in parenthesis is worth the identity right?

Remember that you're summing over repeated indices. So you really have
$$\sum_\gamma \delta^\alpha{}_\gamma x^\gamma + \sum_\gamma (\epsilon^\alpha{}_\gamma + \epsilon'^\alpha{}_\gamma) x^\gamma = x^\alpha.$$ The first term just leaves you with $x^\alpha$ after the summation, so you end up with
$$\sum_\gamma (\epsilon^\alpha{}_\gamma + \epsilon'^\alpha{}_\gamma) x^\gamma = 0.$$ Now use the fact that this has to hold for arbitrary $x^\mu$.

 

[fluidistic]

Remember that you're summing over repeated indices. So you really have
$$\sum_\gamma \delta^\alpha{}_\gamma x^\gamma + \sum_\gamma (\epsilon^\alpha{}_\gamma + \epsilon'^\alpha{}_\gamma) x^\gamma = x^\alpha.$$ The first term just leaves you with $x^\alpha$ after the summation, so you end up with
$$\sum_\gamma (\epsilon^\alpha{}_\gamma + \epsilon'^\alpha{}_\gamma) x^\gamma = 0.$$ Now use the fact that this has to hold for arbitrary $x^\mu$.

Ah right, thanks!
This leaves me with $\epsilon ^\alpha _\gamma=-\epsilon ' ^\alpha _\gamma$. By multiplying each side from the right by $\eta ^{\gamma \beta}$, I do reach the final result.

 

[fluidistic]

I attempted part 2) but I get some non sense.
As andrien pointed out, conservation of norm is $x^\alpha x_\alpha=x'^\alpha x'_\alpha$.
What I did:
$x'^\alpha x'_\alpha=(\eta^{\alpha \beta }+ \epsilon ^{\alpha \beta})x_\beta x'_\alpha=(\eta^{\alpha \beta }+ \epsilon ^{\alpha \beta}) x_\beta \eta _{\alpha \gamma}x'^\gamma =(\eta^{\alpha \beta }+ \epsilon ^{\alpha \beta}) x_\beta \eta _{\alpha \gamma}(\eta^{\gamma \omega }+ \epsilon ^{\gamma \omega} ) x_\omega$ $= (\eta^{\alpha \beta } x_\beta + \epsilon ^{\alpha \beta} x_\beta )(\delta ^\omega _\alpha x_\omega + \epsilon ^\omega _\alpha x_\omega )=\eta ^{\alpha \beta}x_\beta \delta ^\omega _\alpha x_\omega +\eta ^{\alpha \beta}x_\beta \epsilon ^\omega _\alpha x_\omega +\epsilon ^{\alpha \beta} x_\beta \delta ^\omega _\alpha x_\omega +O(\epsilon ^2)$ $\approx x^\alpha \delta ^\omega _\alpha x_\omega + x ^\alpha \epsilon ^\omega _\alpha x_\omega + \epsilon ^{\alpha \beta} x_\beta \delta ^\omega _\alpha x_\omega$.
Here on my draft I rewrote that last expression with sums (2 double sums, 1 triple sum) just to simplicate the Kronecker's delta's.
Then I got rid once again of the sums. And I reached that it's worth $x^\alpha x_\alpha +x^\alpha \epsilon ^\omega _\alpha x_\omega +\epsilon ^{\alpha \beta}x_\beta x_\alpha$. Now using the conservation of the norm, I got that $x^\alpha \epsilon ^\omega _\alpha x_\omega =-\epsilon ^{\alpha \beta}x_\beta x_\alpha$.
But here I realized that this is a non sense: for if $x_\beta$ is a 1x4 covector, epsilon a 4x3 matrix, then $x _\omega$ sould be a 3x1 vector... but it is a 1x3 covector. Therefore the left hand side doesn't make any sense.
I don't see where I went wrong though.

 

[WannabeNewton]

$\epsilon^{\alpha}{}{}_{\beta}$ has a 4x4 matrix representation, not 4x3.

 

[fluidistic]

$\epsilon^{\alpha}{}{}_{\beta}$ has a 4x4 matrix representation, not 4x3.

Oh you're right.
But I still have the same problem, the left hand side of the last equation would be (1x4)x(4x4)x(1x4) where the colored digits indicate that something doesn't match. I seem to have a covector multiplied by a covector instead of a covector multiplied by a vector.

 

[WannabeNewton]

What you have is (1x4)x(4x4)x(4x1) = 1x1 on the left hand side, which is fine. The right hand side is the same thing once you raise the index of one of the $x_{\alpha}$ and lower the corresponding index on $\epsilon^{\alpha\beta}$. If you want to represent $\epsilon^{\alpha\beta}$ as a matrix then it has to be in the form $\epsilon^{\alpha}{}{}_{\beta}$.

 

[fluidistic]

What you have is (1x4)x(4x4)x(4x1) = 1x1 on the left hand side, which is fine. The right hand side is the same thing once you raise the index of one of the $x_{\alpha}$ and lower the corresponding index on $\epsilon^{\alpha\beta}$. If you want to represent $\epsilon^{\alpha\beta}$ as a matrix then it has to be in the form $\epsilon^{\alpha}{}{}_{\beta}$.

I see thank you. I'll need to digest this. I'll come back tomorrow on this.

 

[vanhees71]

I don't understand the trouble you make with this problem. Just expand
$$\eta_{\mu \nu} \left (\delta^{\mu}_{\rho} + {\epsilon^{\mu}}_{\rho} \right) \left (\delta^{\nu}_{\sigma} + {\epsilon^{\nu}}_{\sigma} \right) \stackrel{!}{=}\eta_{\rho \sigma}.$$
up to first order in $\epsilon$, and you'll find that $\epsilon_{\mu \nu}=-\epsilon_{\nu \mu}$.

 

[fluidistic]

I don't understand the trouble you make with this problem. Just expand
$$\eta_{\mu \nu} \left (\delta^{\mu}_{\rho} + {\epsilon^{\mu}}_{\rho} \right) \left (\delta^{\nu}_{\sigma} + {\epsilon^{\nu}}_{\sigma} \right) \stackrel{!}{=}\eta_{\rho \sigma}.$$
up to first order in $\epsilon$, and you'll find that $\epsilon_{\mu \nu}=-\epsilon_{\nu \mu}$.

Hmm but I wouldn't be using the conservation of the norm by doing so, or I'm wrong?

 

[vanhees71]

But this is the conservation of the "norm"!

 

[vela]

Conservation of the norm says
$$\eta_{\mu\nu} x^\mu x^\nu = \eta_{\rho\sigma}x'^\rho x'^\sigma.$$ Again, write x' in terms of x to get
$$\eta_{\mu\nu} x^\mu x^\nu = \eta_{\rho\sigma}[(\delta^\rho_\mu + \epsilon^\rho{}_\mu) x^\mu] [(\delta^\sigma_\nu + \epsilon^\sigma{}_\nu)x^\nu] = [\eta_{\rho\sigma}(\delta^\rho_\mu + \epsilon^\rho{}_\mu) (\delta^\sigma_\nu + \epsilon^\sigma{}_\nu)] x^\mu x^\nu.$$ Comparing the two sides of the equation, you should be able to see you must have
$$\eta_{\mu\nu} = \eta_{\rho\sigma}(\delta^\rho_\mu + \epsilon^\rho{}_\mu) (\delta^\sigma_\nu + \epsilon^\sigma{}_\nu).$$ Without the x's cluttering things up, it's a little easier to see where you're headed.