Tensor Algebra: Checking {u^i} = {g^{kj}} A _{kj}^i

[redstone]

Homework Statement

$${u^i} = {g^{kj}} A _{kj}^i$$
just trying to modify it, not sure of my tensor algebra. Is this right?

$${u^i} = {g^{kj}} A _{kj}^i$$
$${u^i} = g_a^j{g^{ka}} A _{kj}^i$$
$$g_j^a{u^i} = {g^{ka}} A _{kj}^i$$

Just not sure if there should have been a metric contraction, with the resulting D factor in there somewhere.

 

[dextercioby]

The 3rd equation is wrong in the sense that it doesn't imply the 2nd. In the 2nd equation both a and j are summed over, so to get there you need to sum/contract the 3rd one with $g_{a}^{j}$. When doing that, the RHS becomes the RHS from the 2nd equation, while in the first appears the term $\delta_{j}^{j}$ which is V, the dimension of the tangent/cotangent space.

 

[redstone]

Ah, yes, I think I see. That does give me a missing V factor. So would you consider this correct then:


$${u^i} = {g^{kj}} A _{kj}^i$$
$${u^i} = g_a^j{g^{ka}} A _{kj}^i$$
$$\frac{g_j^a{u^i}}{V} = {g^{ka}} A _{kj}^i$$

where,
$$V=g_j^j=\delta_j^j$$

Which when run in reverse...
$$\frac{g_j^a{u^i}}{V} = {g^{ka}} A _{kj}^i$$
$$\frac{g_a^j g_j^a{u^i}}{V} = g_a^j {g^{ka}} A _{kj}^i$$
$$\frac{\delta_j^j {u^i}}{V} = g_a^j {g^{ka}} A _{kj}^i$$
$$\frac{\delta_j^j {u^i}}{V} = {g^{kj}} A _{kj}^i$$
$$\frac{V {u^i}}{V} = {g^{kj}} A _{kj}^i$$
$${u^i} = {g^{kj}} A _{kj}^i$$

So I guess the rule is that whenever you move that mixed metric to the other side, you swap indices, and put in that V factor.

Unless I did something wrong here that needs to be corrected, thanks for the help!

 

[dextercioby]

Yes, it's correct. Just pay attention with the indices, so you don't end up with more than 2 appearances of the same index on the same side of an equality.

 

[redstone]

Maybe you can help me with the following. I I'm pretty sure the final answer I get is wrong, but every step I took looks reasonable to me. Do you know which step I did something illegal on?

Start: $$A={{g}_{ij}}{{x}^{i}}{{x}^{j}}\]$$

Step 1: $$A=g_{i}^{a}g_{j}^{b}{{g}_{ab}}{{x}^{i}}{{x}^{j}}$$

Step 2: $$\frac{1}{V}\frac{1}{V}g_{a}^{i}g_{b}^{j}A={{g}_{ab}}{{x}^{i}}{{x}^{j}}$$

Step 3: $$\frac{1}{V}\frac{1}{V}{{g}^{ab}}g_{a}^{i}g_{b}^{j}A={{g}^{ab}}{{g}_{ab}}{{x}^{i}}{{x}^{j}}$$

Step 4: $$\frac{1}{V}\frac{1}{V}{{g}^{ab}}g_{a}^{i}g_{b}^{j}A=V{{x}^{i}}{{x}^{j}}$$

Step 5: $$\frac{1}{V}\frac{1}{V}\frac{1}{V}{{g}^{ij}}A={{x}^{i}}{{x}^{j}}$$

I believe the correct answer is:
$$\frac{1}{V}{{g}^{ij}}A={{x}^{i}}{{x}^{j}}$$

but I'd like to know where I went wrong above.

 

[fzero]

Maybe you can help me with the following. I I'm pretty sure the final answer I get is wrong, but every step I took looks reasonable to me. Do you know which step I did something illegal on?

Start: $$A={{g}_{ij}}{{x}^{i}}{{x}^{j}}\]$$

Step 1: $$A=g_{i}^{a}g_{j}^{b}{{g}_{ab}}{{x}^{i}}{{x}^{j}}$$

Step 2: $$\frac{1}{V}\frac{1}{V}g_{a}^{i}g_{b}^{j}A={{g}_{ab}}{{x}^{i}}{{x}^{j}}$$

Step 2 is incorrect. In the expression for step 1, you're summing over all indices, so there's no operation that brings factors of the metric over to the LHS. If you can't reduce a step to multiplying left and right-hand sides of an equation by the same factor, chances are that you did something wrong.

 

[redstone]

Step 2 is incorrect. In the expression for step 1, you're summing over all indices, so there's no operation that brings factors of the metric over to the LHS. If you can't reduce a step to multiplying left and right-hand sides of an equation by the same factor, chances are that you did something wrong.

I left out the steps going between step 1 and step 2, but that's what I thought I did. Here are the steps I had.

Step 1: $$A=g_{i}^{a}g_{j}^{b}{{g}_{ab}}{{x}^{i}}{{x}^{j}}$$
Step 1a: $$\frac{V}{V}A=g_{i}^{a}g_{j}^{b}{{g}_{ab}}{{x}^{i}}{{x}^{j}}$$
Step 1b: $$\frac{1}{V}g_{a}^{a}A=g_{i}^{a}g_{j}^{b}{{g}_{ab}}{{x}^{i}}{{x}^{j}}$$
Step 1c: $$\frac{1}{V}g_{i}^{a}g_{a}^{i}A=g_{i}^{a}g_{j}^{b}{{g}_{ab}}{{x}^{i}}{{x}^{j}}$$
Step 1d: $$\frac{1}{V}g_{a}^{i}A=g_{j}^{b}{{g}_{ab}}{{x}^{i}}{{x}^{j}}$$

Step 1e: $$\frac{1}{V}\frac{1}{V}g_{b}^{b}g_{a}^{i}A=g_{j}^{b}{{g}_{ab}}{{x}^{i}}{{x}^{j}}$$
Step 1f: $$\frac{1}{V}\frac{1}{V}g_{b}^{j}g_{j}^{b}g_{a}^{i}A=g_{j}^{b}{{g}_{ab}}{{x}^{i}}{{x}^{j}}$$
Step 2: $$\frac{1}{V}\frac{1}{V}g_{a}^{i}g_{b}^{j}A={{g}_{ab}}{{x}^{i}}{{x}^{j}}$$

Is one of these steps illegal?

 

[fzero]

I left out the steps going between step 1 and step 2, but that's what I thought I did. Here are the steps I had.

Step 1: $$A=g_{i}^{a}g_{j}^{b}{{g}_{ab}}{{x}^{i}}{{x}^{j}}$$
Step 1a: $$\frac{V}{V}A=g_{i}^{a}g_{j}^{b}{{g}_{ab}}{{x}^{i}}{{x}^{j}}$$
Step 1b: $$\frac{1}{V}g_{a}^{a}A=g_{i}^{a}g_{j}^{b}{{g}_{ab}}{{x}^{i}}{{x}^{j}}$$

You already have an index $$a$$ on the RHS that's being summed over, so you shouldn't use the same index in the sum on the LHS. That's the source of your confusion. You should write:

$$\frac{1}{V}g_{c}^{c}A=g_{i}^{a}g_{j}^{b}{{g}_{ab}}{{x}^{i}}{{x}^{j}}$$

and then it's clear that you can't cancel $$g$$'s out on both sides.

You make the same mistake in some of the other steps. Just never use the same dummy index twice and it'll cut down on mistakes.