Am I Calculating the Riemann Tensor Correctly?

[help1please]

The work I have been following has me very confused... and I am almost sure I am making a mistake somewhere!

After working up to this equation:

\delta V = dX^{\mu}\delta X^{\nu} [\nabla_{\mu} \nabla_{\nu}]V

I am asked to calculate the curvature tensor. Now the way I did it, turned out different to the way it is shown at the end of the work... it took a bit of time to understand what method I was using was different but I did work it out nonetheless, and what I want to know is which method is correct (most likely mine is wrong but I need some guidance.)

Ignoring V and just working out the commutator relationship, I expand:

(\partial_{\nu} + \Gamma_{\nu})(\partial_{\mu} + \Gamma_{\mu}) - (\partial_{\mu} + \Gamma_{\mu})(\partial_{\nu} + \Gamma_{\nu})

The first part of the calculation, gives

\partial_{\nu}\partial_{\mu} - \partial_{\mu} \partial_{\nu}

Which is just zero, because the ordinary derivatives commute, so they go to zero. Fine. Now, according to the work I am following, the next set of terms should have been:

\Gamma_{\nu} \partial_{\mu} - \partial_{\nu}\Gamma_{\mu}

But I ended up with

\partial_{\nu}\Gamma_{\mu} - \partial_{\mu} \Gamma_{\nu}

and I only arrived at this because it is well known that once you calculate the first set of terms, for instance, using this guide:

(a+b)(c+d)

ignoring that we are taking this part away from another part, the first term arises because you multiply a with c. Then you multiply a with d.

In the work I am following, it seems that in this:

(\partial_{\nu} + \Gamma_{\nu})(\partial_{\mu} + \Gamma_{\mu}) - (\partial_{\mu} + \Gamma_{\mu})(\partial_{\nu} + \Gamma_{\nu})

You get

\partial_{\nu} \partial{\mu}

first of all by following that rule (then the extra's of course - \partial_{\mu} \partial_{\nu} but according the second lot, the work has

\Gamma_{\nu} \partial_{\mu} - \partial_{\mu} \Gamma_{\nu}

My brain agrees with the -\partial_{\mu} \Gamma_{\nu} term but I do not understand how it gathers the

\Gamma_{\nu} \partial_{\mu}

Because for that to be true, it would mean using my expression again for simplicity that

(a+b)(c+d) - (a'+b')(c'+d')

It seems right to multiply (a' \cdot d') but with a steady analysis of the works example, it shows b \cdot c which would give the first term \Gamma_{\nu}\partial_{\mu}... but that isn't right is it? Or am I wrong? Am I doing it wrong?

Thanks

 

[help1please]

I've not made a massive amount of headway since I posted this question but I think my confusion exists in some parts I have missed out. There seems to be a sign change for:

-[\partial_{\mu} , \Gamma_{\nu}] = -\frac{\partial \Gamma_{\nu}}{\partial X^{\mu}}

Where I have a positive solution

[\partial_{\nu} , \Gamma_{\mu}] = \frac{\partial \Gamma_{\mu}}{\partial X^{\nu}}

Now this commutator would permit the calculation,

\partial_{\nu}\Gamma_{\mu} but I am still confused as to why it wasn't calculated straight away, if anyone is following my drift.
................Then upon saying that I think I have worked out my problem, but I need confirmation:

Ah, that was my breaking point - I think I know what is happening now.

Again:

Where I have a positive solution

[\partial_{\nu} , \Gamma_{\mu}] = \frac{\partial \Gamma_{\mu}}{\partial X^{\nu}}

Now this commutator would permit the calculation,

\partial_{\nu}\Gamma_{\mu} ---- so, in the working example, it looked like the person who calculated it was taking the \Gamma_{\nu} and multiplying it with \partial_{\mu} and that confused me because normally the next multiplication would give \partial_{\nu} \Gamma{\mu} but because of this delicate sign change, which exists because

(\partial_{\nu} + \Gamma_{\nu})(\partial_{\mu} + \Gamma_{\mu}) - (\partial_{\mu} + \Gamma_{\mu})(\partial_{\nu} + \Gamma_{\nu})

The

(\partial_{\nu} + \Gamma_{\nu})(\partial_{\mu} + \Gamma_{\mu})

Part is positive then it follows that

[\partial_{\nu} , \Gamma_{\mu}] = \frac{\partial \Gamma_{\mu}}{\partial X^{\nu}}

Which would dictate the multiplication in the form the work gives. If it had been negative, such as this part

- (\partial_{\mu} + \Gamma_{\mu})(\partial_{\nu} + \Gamma_{\nu})

Then the normal multiplication rule would apply, because as I said, I had no problem agreeing with the product

\partial_{\mu} \Gamma_{\nu}

Does this seem right now?

 

[help1please]

Can no one help?

 

[clamtrox]

I don't understand your question. Remember that derivatives don't commute, and they only operate on things on their right. So
(\partial + \Gamma)(\partial + \Gamma) = \partial \partial + \partial \Gamma + \Gamma \partial + \Gamma \Gamma

 

[help1please]

If I have:

(a+b)(c+d) -(a'+b')(c'+d')

multiplying it out, I begin with

ac - a'c'

Just as in the OP equation, for the GR curvature tensor, we have

\partial_{\nu} \partial_\mu - \partial_{\mu} \partial_{\nu}

I'm fine with that. It's what comes next is muddling my brain up. According to my simple equation above,

(a+b)(c+d) -(a'+b')(c'+d')

The next thing I compute is

ad - a'd'

but we don't seem to be doing that when calculating the curvature tensor.

What I am told I should have to calculate next is

\Gamma_{\nu}\partial_{\mu} - \partial_{\mu}\Gamma_{\nu}

Instead that looks like

bc - a'd'

are you following now?

 

[help1please]

Compare it with

(\partial_{\nu} + \Gamma_{\nu})(\partial_{\mu} + \Gamma_{\mu}) - (\partial_{\mu} + \Gamma_{\mu})(\partial_{\nu} + \Gamma_{\nu})

In other words. The first lot of computations I can agree with. I don't understand when it goes to compute the next set of terms, it seems to take

\Gamma_{\nu}

with

\partial_{\mu}

but then on the other side it would correctly take

\partial_{\mu}\Gamma_{\nu}

This is what has caused the confusion.

 

[help1please]

I don't understand your question. Remember that derivatives don't commute, and they only operate on things on their right. So
(\partial + \Gamma)(\partial + \Gamma) = \partial \partial + \partial \Gamma + \Gamma \partial + \Gamma \Gamma

This I agree with. But my work seems to be skipping the second part in your equation, the

\partial \Gamma part. Look at the way I have presented the equations from the work. It does not follow your pattern.

 

[help1please]

Let me show you the whole thing now, naturally we compute

(\partial_{\nu} + \Gamma_{\nu})(\partial_{\mu} + \Gamma_{\mu}) - (\partial_{\mu} + \Gamma_{\mu})(\partial_{\nu} + \Gamma_{\nu})

It says when multiplying out the terms, we should have

\partial_{\nu} \partial_{\mu} - \partial_{\mu} \partial_{\nu} \Gamma_{\nu \beta}^{\alpha} \partial_{\mu} - \partial_{\nu}\Gamma_{\nu \beta}^{\alpha} + \Gamma_{\nu \delta}^{\alpha}\Gamma_{\mu \beta} - \Gamma_{\nu \delta}^{\alpha}\Gamma_{\mu \beta}

http://www.youtube.com/watch?v=AC3TMizGpB8&feature=relmfu

The important bit begins at 38:30. Is he calculating this right? And if he is, what is it I am failing to understand?

 

[help1please]

It took me a while to find the video. But I found it.

 

[clamtrox]

So what happens when you try to calculate it properly? Remember also that you get two terms from the second covariant derivative (because \nabla V is a 1-1-tensor).

\nabla_{[\mu} \nabla_{\nu]} V^{\alpha} = \nabla_{\mu} \nabla_{\nu} V^{\alpha} - [\mu \leftrightarrow \nu] = \partial_{\mu} (\nabla_\nu V^\alpha) - \Gamma^{\lambda}_{\mu \nu} \nabla_{\lambda} V^{\alpha} + \Gamma^\alpha_{\mu \delta} \nabla_\nu V^\delta - [\mu \leftrightarrow \nu]

Oh, and that notation that Susskind is really confusing. I think you will be better off working it out without any shorthands

 

[help1please]

So what happens when you try to calculate it properly? Remember also that you get two terms from the second covariant derivative (because \nabla V is a 1-1-tensor).

\nabla_{[\mu} \nabla_{\nu]} V^{\alpha} = \nabla_{\mu} \nabla_{\nu} V^{\alpha} - [\mu \leftrightarrow \nu] = \partial_{\mu} (\nabla_\nu V^\alpha) - \Gamma^{\lambda}_{\mu \nu} \nabla_{\lambda} V^{\alpha} + \Gamma^\alpha_{\mu \delta} \nabla_\nu V^\delta - [\mu \leftrightarrow \nu]

Oh, and that notation that Susskind is really confusing. I think you will be better off working it out without any shorthands

Well take a look at your example:

(\partial + \Gamma)(\partial + \Gamma) = \partial \partial + \partial \Gamma + \Gamma \partial + \Gamma \Gamma

By the time he's canceled out the first terms, he should be writing what you have here:

\partial \Gamma

Compare it with

(\partial_{\nu} + \Gamma_{\nu})(\partial_{\mu} + \Gamma_{\mu}) - (\partial_{\mu} + \Gamma_{\mu})(\partial_{\nu} + \Gamma_{\nu})

In other words. The first lot of computations I can agree with. I don't understand when it goes to compute the next set of terms, he seems to take

\Gamma_{\nu}

with

\partial_{\mu}

but then on the other side it would correctly take

\partial_{\mu}\Gamma_{\nu}

This is what has caused the confusion. Shouldn't it be

\partial_{\nu}\Gamma_{\mu} - \partial_{\mu}\Gamma_{\nu}

 

[clamtrox]

You are still explaining very poorly. Can you show your work using proper indices step by step?

 

[help1please]

I'd rather not. I think that would complicate it.

Let's try this again. Look at what you have

(\partial + \Gamma)(\partial + \Gamma) = \partial \partial + \partial \Gamma + \Gamma \partial + \Gamma \Gamma

I agree with this. Now, take a look at the RHS of this equation, the second term

\partial \Gamma

This comes into play when you have calculated the first terms

\partial_{\nu}\partial_{\mu} - \partial_{\mu}\partial_{\nu}

That cancels out anyway, so it is of no consequence. I'm interested on the second term on the RHS of your equation, again, it is

\partial \Gamma

You get that from multiplying, I presume, the first partial in the first set of paranthesis with the last term in the next set of paranthesis, the Gamma. But Susskind is clearly doing something different, because after the first lot of terms vanish, he continues his multiplication but begins with the Gamma term in the first set of paranthesis and then he mutliplies that with the first partial in the next lot.

So effective what susskind is doing is

\Gamma_{\mu} \partial_{\nu}

you do see this yes?

My question is why. I thought it would have been \partial_{\mu} \Gamma_{\nu}

I can't explain this any more. Please read carefully, follow what I am saying and compare it with what susskind is doing.

 

[help1please]

Writing out my simple example,

(a+b)(c+d) - (a'+b')(c'+d')

Following what susskind is doing, he seems to be doing this:

ac - a'c' \cdot bc - a'd' \cdot bd - b'd'

 

[help1please]

My problem lies therein with the bc-term straight after the

ac - a'c'

(which cancels anyway. Not missing a part in case I am unclear)

 

[clamtrox]

As you know, summation is commutative, so it doesn't matter what order you write your sums in. A+B = B+A. Multiplication however, isn't. So AB != BA

 

[clamtrox]

Finally, I should say that I don't understand Susskinds notation, and I think you would be doing yourself a favour by doing the calculation properly

 

[help1please]

As you know, summation is commutative, so it doesn't matter what order you write your sums in. A+B = B+A. Multiplication however, isn't. So AB != BA

Oh I know that, I repeated some mu-nu indices which has caused a little confusion:

I'd rather not. I think that would complicate it.

Let's try this again. Look at what you have

(\partial + \Gamma)(\partial + \Gamma) = \partial \partial + \partial \Gamma + \Gamma \partial + \Gamma \Gamma

I agree with this. Now, take a look at the RHS of this equation, the second term

\partial \Gamma

This comes into play when you have calculated the first terms

\partial_{\nu}\partial_{\mu} - \partial_{\mu}\partial_{\nu}

That cancels out anyway, so it is of no consequence. I'm interested on the second term on the RHS of your equation, again, it is

\partial \Gamma

You get that from multiplying, I presume, the first partial in the first set of paranthesis with the last term in the next set of paranthesis, the Gamma. But Susskind is clearly doing something different, because after the first lot of terms vanish, he continues his multiplication but begins with the Gamma term in the first set of paranthesis and then he mutliplies that with the first partial in the next lot.

For a note:

(\partial_{\nu} + \Gamma_{\nu})(\partial_{\mu} + \Gamma_{\mu}) - (\partial_{\mu} + \Gamma_{\mu})(\partial_{\nu} + \Gamma_{\nu})

So effective what susskind is doing is

\Gamma_{\nu} \partial_{\mu}

you do see this yes?

My question is why. I thought it would have been \partial_{\nu} \Gamma_{\mu}

I can't explain this any more. Please read carefully, follow what I am saying and compare it with what susskind is doing.

 

[help1please]

You should be able to follow that now surely? Susskind isn't using a strange notation - he's just leaving out some indices for simplicity, whereas he'd have to include them if the had multiplied it by V^{\beta}.

 

[help1please]

I'll tell you what, Ill write it even more straightforward than that - I'll show you what I think should be there. The following is concerned only when the first set of terms are made to disappear as we have seen before:For a note:

(\partial_{\nu} + \Gamma_{\nu})(\partial_{\mu} + \Gamma_{\mu}) - (\partial_{\mu} + \Gamma_{\mu})(\partial_{\nu} + \Gamma_{\nu})

So effective what the work is doing is

\Gamma_{\nu} \partial_{\mu}

you do see this yes?

My question is why. I thought it would have been \partial_{\nu} \Gamma_{\mu} - This quantity is taken away from \partial_{\mu}\Gamma_{\nu} and I agree with that... so what it has is

\Gamma_{\nu} \partial_{\mu} -\partial_{\mu}\Gamma_{\nu}

Where I thought the next lot of terms should be

\partial_{\nu} \Gamma_{\mu}-\partial_{\mu}\Gamma_{\nu}

Now which one is right?

 

[clamtrox]

Like I said before, summation commutes. So you can write (a+b)(c+d) either as
(a+b)(c+d) = ac+bd+ad+bc or
(a+b)(c+d) = ac+ad+bd+bc, etc.
You are still not explaining it at all, you are just repeating what you said before, and I still cannot understand it. I think this is your problem but it's very difficult to tell. Try writing it in different words, not just copy paste.

 

[Mechdude]

Finally, I should say that I don't understand Susskinds notation, and I think you would be doing yourself a favour by doing the calculation properly

clamtrox, do you have an idea of a nice book that teaches this stuff in a nice easy to follow manner? i have always had trouble with this stuff.
mechdude.

 

[help1please]

Like I said before, summation commutes. So you can write (a+b)(c+d) either as
(a+b)(c+d) = ac+bd+ad+bc or
(a+b)(c+d) = ac+ad+bd+bc, etc.
You are still not explaining it at all, you are just repeating what you said before, and I still cannot understand it. I think this is your problem but it's very difficult to tell. Try writing it in different words, not just copy paste.

So you keep reminding me, but quite clearly these two expressions are different:

\Gamma_{\nu} \partial_{\mu} -\partial_{\mu}\Gamma_{\nu}

\partial_{\nu} \Gamma_{\mu}-\partial_{\mu}\Gamma_{\nu}

 

[help1please]

And I am not repeating myself, I made a very small error in the post previous to that. There is a difference... Susskind is calculating it different to me, and I want to know who is making the mistake.

 

[clamtrox]

clamtrox, do you have an idea of a nice book that teaches this stuff in a nice easy to follow manner? i have always had trouble with this stuff.
mechdude.

Carroll's notes, http://arxiv.org/abs/gr-qc/9712019, have the benefit of being freely available. There are plenty of good books depending on the strength of your mathematical background. Carroll is quite easy to read. If you want something heavier, you could try for example Wald or Krasinski&Plebanski

 

[clamtrox]

So you keep reminding me, but quite clearly these two expressions are different:

\Gamma_{\nu} \partial_{\mu} -\partial_{\mu}\Gamma_{\nu}

\partial_{\nu} \Gamma_{\mu}-\partial_{\mu}\Gamma_{\nu}

Writing the entire thing out
<br /> (\partial_{\nu} + \Gamma_{\nu})(\partial_{\mu} + \Gamma_{\mu}) - (\partial_{\mu} + \Gamma_{\mu})(\partial_{\nu} + \Gamma_{\nu}) = <br /> \partial_{\nu} \partial_{\mu} + \partial_{\nu} \Gamma_\mu + \Gamma_\nu \partial_\mu + \Gamma_\nu \Gamma_\mu - \partial_{\mu} \partial_{\nu} - \partial_{\mu} \Gamma_\nu - \Gamma_\mu \partial_\nu - \Gamma_\mu \Gamma_\nu = <br />
Rearranging the terms
<br /> (\partial_{\nu} \Gamma_\mu - \Gamma_\mu \partial_\nu) - (\partial_{\mu} \Gamma_\nu - \Gamma_\nu \partial_\mu) + \Gamma_\nu \Gamma_\mu - \Gamma_\mu \Gamma_\nu = <br />
Bracketed expressions are just the derivatives of connection
<br /> \frac{\partial \Gamma_\mu}{\partial x^{\nu}} - \frac{\partial \Gamma_\nu}{\partial x^{\mu}} + \Gamma_\nu \Gamma_\mu - \Gamma_\mu \Gamma_\nu<br />

So can you now please help me help you, and tell me where your problem is.

 

[help1please]

That's how I'd do it, and so it seems that Susskind's order of multiplication is wrong. He multiplies the wrong terms together. You should watch the video again. If I explain it again, you won't understand what I am talking about.

 

[clamtrox]

Can you point out where he multiplies them in the wrong order? To me that looks entirely correct, and he certainly gets the same answer so there's no obvious mistake from his part.

 

[help1please]

Maybe its just his notation then that is muddling me up. When I saw yours, yours is how I did mine originally. Then when I look at his, the order which he does things is confusing the hell out of me. I don't understand, why, for instance, when he multiplies the first terms which cancels, he begins to calculate \Gamma_{\nu}\partial_{\mu}. Straight after this term

\partial_{\mu}\partial_{\nu} - \partial_{\nu}\partial_{\mu}

I would have then calculated (on the RHS) where the minus sign separates two sides, I would have calculated

\partial_{\nu} \Gamma_{\mu}

Do you see even by now what I am talking about?

 

[help1please]

sorry, got my indices mixed up again but fixed it.

 

[help1please]

See, I was always taught a certain order to multiply out brackets.

This is what I do, the dashes represents the order of multiplication this time:

(a'+b)(c'+d)

(a'+b)(c+d')

(a+b')(c'+d)

(a+b')(c+d')

This is what susskind does:

(a'+b)(c'+d)

the terms cancel. Naturally.

Then for only the RHS (where the minus indicates to different sides) ---*** its this bit I don't agree with because of my own order.

(a+b')(c'+d)

then for the left he does

(a'+b)(c+d')

Which is not the kind of order I am used to. Then he goes on to multiply the last lot out like I would

(a+b')(c+d')

 

[clamtrox]

Well, like I said earlier, summation commutes, so you just need to get used to people writing down their sums in different order...

 

[help1please]

Well, like I said earlier, summation commutes, so you just need to get used to people writing down their sums in different order...

Can you give me an example please, like show me the math written out where the summation is commuting all over the place. I have used the notation

A_{[\mu}B_{\nu]} before, just never seen it in its full form. How would these indices commute, thank you.

 

[help1please]

The fact susskind has not done this has only caused more confusion for me, for anyone I'd presume that would like to follow the math...

 

[clamtrox]

For example, 1+2=2+1=3. This is what commuting means.