Index Notation: Understanding LHS = RHS

In summary: I would have to start explaining why a sum is commutativeIn summary, the conversation discusses the geodesic equation for four-velocity in GR and the confusion around the summation of indices. The equation that is being questioned is \partial_i g_{aj} u^i u^j - \frac{1}{2} \partial_a g_{ij} u^i u^j = \frac{1}{2} u^i u^j \partial_a g_{ij}, which is not found in the linked article. The confusion arises from the swapping of indices and the use of symmetry in the equation. The conversation also includes a demonstration of how renaming null indices can result in the same equation.
  • #1
unscientific
1,734
13
I was reading my lecturer's notes on GR where I came across the geodesic equation for four-velocity. There is a line which read:

Summing them up,
[tex]\partial_i g_{aj} u^i u^j - \frac{1}{2} \partial_a g_{ij} u^i u^j = \frac{1}{2} u^i u^j \partial_a g_{ij} [/tex]

I'm trying to understand how LHS = RHS, surely the indices ##a## and ##i## are different, how can you simply combine them?

I tried writing them out:

[tex]\partial_i g_{aj} u^i u^j - \frac{1}{2} \partial_a g_{ij} u^i u^j[/tex]
[tex] = g_{aj} \partial_i u^i u^j + \left( u^i u^j \partial_i g_{aj} - \frac{1}{2} u^i u^j \partial_a g_{ij} \right) - \frac{1}{2} g_{ij} \partial_a u^i u^j [/tex]Source: http://physicspages.com/2013/04/02/geodesic-equation-and-four-velocity/
 
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  • #2
unscientific said:
Summing them up,

[tex]\partial_i g_{aj} u^i u^j - \frac{1}{2} \partial_a g_{ij} u^i u^j = \frac{1}{2} u^i u^j \partial_a g_{ij}[/tex]

This equation doesn't appear anywhere in the link you gave. The closest thing I can find at that link is:

$$
\partial_i g_{aj} u^i u^j + g_{aj} \frac{d u^j}{d \tau} - \frac{1}{2} \partial_a g_{ij} u^i u^j = 0
$$

which gives, expanding out the second term and moving it to the RHS,

$$
\partial_i g_{aj} u^i u^j - \frac{1}{2} \partial_a g_{ij} u^i u^j = - g_{aj} u^i \partial_i u^j
$$

So where does the equation you wrote come from?​
 
  • #3
PeterDonis said:

[tex]\partial_i g_{aj} u^i u^j - \frac{1}{2} \partial_a g_{ij} u^i u^j = \frac{1}{2} u^i u^j \partial_a g_{ij} [/tex]

So where does the equation you wrote come from?​
Starting from the geodesic equation:

[tex] \partial_i g_{aj} u^i u^j + g_{aj} \frac{d u^j}{d \tau} - \frac{1}{2} \partial_a g_{ij} u^i u^j = 0 [/tex]

I'm confused by this step when 'summation' was mentioned. Without multiplying ##u^a## we get:

[tex] LHS = g_{aj} \frac{d u^j}{d \tau} + \partial_i g_{aj} u^i u^j - \frac{1}{2} \partial_a g_{ij} u^i u^j [/tex]

[tex] = g_{aj} \frac{d u^j}{d \tau} + \frac{1}{2} u^i u^j \partial_a g_{ij} [/tex]Comparing 2nd and 3rd terms, that's where I couldn't get my head around it.
 
  • #4
unscientific said:
Without multiplying ##u^a##

If you don't multiply by ##u^a## then the derivation doesn't work; that multiplication is crucial because it means there are no free indexes in any of the terms, so you can swap dummy indexes at will. That is what allows the ##a## and ##i## indexes to be swapped so that you can combine terms.

That said, I'm not entirely sure the article you linked to has everything correct. It would be easier if it stuck to index notation everywhere. I don't have time now to dig into it in more detail, unfortunately.
 
  • #5
PeterDonis said:
If you don't multiply by ##u^a## then the derivation doesn't work; that multiplication is crucial because it means there are no free indexes in any of the terms, so you can swap dummy indexes at will. That is what allows the ##a## and ##i## indexes to be swapped so that you can combine terms.

That said, I'm not entirely sure the article you linked to has everything correct. It would be easier if it stuck to index notation everywhere. I don't have time now to dig into it in more detail, unfortunately.

How does the multiplication of ##u^a## change anything at all? Also, what do you mean by 'it would be easier if it stuck to index notation everywhere. ' ? I thought this was all in index notation.
 
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  • #6
I don't understand the problem.

[itex] u^a g_{aj,i} u^i u^j + g_{aj} u^a d_\tau u^j - \frac{1}{2} u^a g_{ij,a} u^i u^j [/itex]

The middle (2nd term) remains the same, but they rewrite the 1st and 3rd term as:

[itex] u^a g_{aj,i} u^i u^j -\frac{1}{2} u^a g_{ij,a} u^i u^j = \frac{1}{2} g_{ij,a} u^i u^j u^a [/itex]

Why is this done? because of symmetry reasons he rewrites the 1st term above by interchanging i with a [or if you like the other phrase better "by renaming them as null-indices"]...

Also you sum when you write the same indices... that's what the writter means by summing up ...in this case summing the a-index,
 
  • #7
ChrisVer said:
I don't understand the problem.

[itex] u^a g_{aj,i} u^i u^j + g_{aj} u^a d_\tau u^j - \frac{1}{2} u^a g_{ij,a} u^i u^j [/itex]

The middle (2nd term) remains the same, but they rewrite the 1st and 3rd term as:

[itex] u^a g_{aj,i} u^i u^j -\frac{1}{2} u^a g_{ij,a} u^i u^j = \frac{1}{2} g_{ij,a} u^i u^j u^a [/itex]

Why is this done? because of symmetry reasons he rewrites the 1st term above by interchanging i with a [or if you like the other phrase better "by renaming them as null-indices"]...

Also you sum when you write the same indices... that's what the writter means by summing up ...in this case summing the a-index,

I'm still not convinced that you can simply swap ##i## with ##a##..
 
  • #8
do you know that you can write:
[itex] u^i u^j x_{ij} = u^j u^i x_{ji} [/itex]
simply by renaming indices?
If not, let's say that i,j run from 1 to 2...

[itex] u^i u^j x_{ij} = u^1 u^1 x_{11} + u^1 u^2 x_{12} + u^2 u^1 x_{21} + u^2 u^2 x_{22} [/itex]
renaming the null indices i->j and j->i you get the same result:
[itex] u^j u^i x_{ji} =u^1 u^1 x_{11} + u^1 u^2 x_{12} + u^2 u^1 x_{21} + u^2 u^2 x_{22} [/itex]

As I said, you just have to rename null [summed up]-indices...

Otherwise you can use the symmetry, to say that since [itex]u^i u^j[/itex] is symmetric under the interchange of [itex]i \leftrightarrow j[/itex] then [itex]x_{ij}[/itex] is also symmetric [or better put, only its symmetric part contributes].

In that case you would have to symmetrize the first term in (i,a) and write:
[itex]g_{aj,i} u^i u^a u^j - \frac{1}{2} g_{ij,a} u^a u^i u^j= \frac{1}{2} ( g_{aj,i} + g_{ij,a}) u^i u^a u^j - \frac{1}{2} g_{ij,a} u^a u^i u^j= \frac{1}{2} g_{aj,i} u^a u^i u^j[/itex]

and rename the indices again [although you don't have to, because both expressions with renaming or not are equivalent*]...Of course that would be a very lame thing to do... since you could have done the renaming from the begining.

* the author could as well write as a result:
[itex]g_{aj,i} u^i u^a u^j - \frac{1}{2} g_{ij,a} u^a u^i u^j= \frac{1}{2} g_{mn,r} u^m u^n u^r[/itex]
or
[itex]g_{aj,i} u^i u^a u^j - \frac{1}{2} g_{ij,a} u^a u^i u^j= g_{mn,r} u^r u^m u^n - \frac{1}{2} g_{mn,r} u^r u^m u^n =\frac{1}{2} g_{mn,r} u^m u^n u^r[/itex]
I'm sorry, but it can't get more basic... The next thing one would have to do for illustrating what is going on, is to expand the summation ... but that would be really, really awful to read, so it's better to do that yourself on a scrap of paper.
 
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  • #9
Got it, thank you!
 

1. What is index notation?

Index notation is a mathematical convention used to simplify and generalize equations involving variables with multiple dimensions or components. It involves representing each component of a variable with a subscript index, allowing for concise and consistent notation in complex expressions.

2. What does LHS and RHS stand for in index notation?

In index notation, LHS stands for "left-hand side" and RHS stands for "right-hand side". These refer to the two sides of an equation, with the LHS being the expression on the left and the RHS being the expression on the right.

3. How do I read index notation?

To read index notation, start by identifying the variable being represented by the index, and then read the expression as "the component of (variable) with index (index number)". For example, Ai would be read as "the i-th component of A".

4. What is the purpose of using index notation?

The purpose of index notation is to simplify and generalize equations involving variables with multiple dimensions or components. It allows for concise and consistent notation, making it easier to express and manipulate complex mathematical expressions.

5. Are there any rules for using index notation?

Yes, there are a few rules to keep in mind when using index notation. Firstly, the index must be a positive integer. Additionally, repeated indices in an expression indicate a summation over those indices. Lastly, the order of indices in an expression does not affect the result, as long as they are consistently applied throughout the expression.

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