# Summation Convention in Einstein Notation

• I
• olgerm
In summary, the conversation discusses Einstein notation and whether the summation should start from the first occurrence of an index or from the beginning of the equation. The conversation also includes an equation from Wikipedia and asks for clarification on its meaning. The expert summarizer explains that the equation means a summation of various terms and reminds the individual that using non-standard notation can make it difficult for others to check their work. The thread is then closed.
olgerm
Gold Member
I got another basic question: should the summation in einstein notation start from first occurance of index or in beginning of equation?
For eampledoes this equation ##R_{\alpha \beta }={R^{\rho }}_{\alpha \rho \beta }=\partial _{\rho }{\Gamma ^{\rho }}_{\beta\alpha }-\partial _{\beta }{\Gamma ^{\rho }}_{\rho \alpha }+{\Gamma ^{\rho }}_{\rho \lambda }{\Gamma ^{\lambda }}_{\beta \alpha }-{\Gamma ^{\rho }}_{\beta \lambda }{\Gamma ^{\lambda }}_{\rho \alpha }## from wikipedia mean:
## \sum_{j_1=0}^D(\sum_{j_2=0}^D(\frac{\partial{\Gamma ^{j_1}}_{\beta \alpha }}{\partial x^{j_1}}-\frac{{\Gamma^{j_1}}_{j_1 \alpha}}{\partial x^{\beta}}+{\Gamma^{j_1}}_{j_1 j_2}{\Gamma ^{j_2}}_{\beta \alpha }-{\Gamma ^{j_1}}_{\beta j_2 }{\Gamma^{j_2}}_{j_1 \alpha }))##
or
##\sum_{j_1=0}^D(\frac{\partial{\Gamma ^{j_1}}_{\beta \alpha }}{\partial x^{j_1}}-\frac{{\Gamma^{j_1}}_{j_1 \alpha}}{\partial x^{\beta}}+\sum_{j_2=0}^D({\Gamma^{j_1}}_{j_1 j_2}{\Gamma ^{j_2}}_{\beta \alpha }-{\Gamma ^{j_1}}_{\beta j_2 }{\Gamma^{j_2}}_{j_1 \alpha }))##?

Last edited:
olgerm said:
I got another basic question: should the summation in einstein notation start from first occurance of index or in beginning of equation?
Neither. Each individual term is summed by itself: ##A^iB_i+P^kQ_k## is to be read as##\sum_iA^iB_i+\sum_kP^kQ_k##

Nugatory said:
Neither. Each individual term is summed by itself: ##A^iB_i+P^kQ_k## is to be read as##\sum_iA^iB_i+\sum_kP^kQ_k##
so it is same as last one beacuse ##\sum_i(A^iB_i)+\sum_k(P^kQ_k)=\sum_i(A^iB_i+P^iQ_i)##?

olgerm said:
so it is same as last one beacuse ##\sum_i(A^iB_i)+\sum_k(P^kQ_k)=\sum_i(A^iB_i+P^iQ_i)##?
In this particular case, yes, that just happens to work. In more complex expressions you won't be able to combine terms under one summation the way you're trying to do. Consider, for example, ##A^iB_i+P^{ij}Q_{ij}## - the first term is a four-element summation and the second is a sixteen-element summation.

Nugatory said:
Consider, for example, ##A^iB_i+P^{ij}Q_{ij}##
if every term has its own summation it is ##\sum_{i=0}^D(A^iB_i)+\sum_{i=0}^D(\sum_{j=0}^D(P^{ij}Q_{ij}))=\sum_{i=0}^D(A^iB_i+\sum_{j=0}^D(P^{ij}Q_{ij}))##?

Can you say clearly whether it is true that the equation ##R_{\alpha \beta }={R^{\rho }}_{\alpha \rho \beta }=\partial _{\rho }{\Gamma ^{\rho }}_{\beta\alpha }-\partial _{\beta }{\Gamma ^{\rho }}_{\rho \alpha }+{\Gamma ^{\rho }}_{\rho \lambda }{\Gamma ^{\lambda }}_{\beta \alpha }-{\Gamma ^{\rho }}_{\beta \lambda }{\Gamma ^{\lambda }}_{\rho \alpha }## from wikipedia means:
##\sum_{j_1=0}^D(\frac{\partial{\Gamma ^{j_1}}_{\beta \alpha }}{\partial x^{j_1}}-\frac{{\Gamma^{j_1}}_{j_1 \alpha}}{\partial x^{\beta}}+\sum_{j_2=0}^D({\Gamma^{j_1}}_{j_1 j_2}{\Gamma ^{j_2}}_{\beta \alpha }-{\Gamma ^{j_1}}_{\beta j_2 }{\Gamma^{j_2}}_{j_1 \alpha }))##?

Last edited:
olgerm said:
Can you say clearly whether it is true that the equation ##R_{\alpha \beta }={R^{\rho }}_{\alpha \rho \beta }=\partial _{\rho }{\Gamma ^{\rho }}_{\beta\alpha }-\partial _{\beta }{\Gamma ^{\rho }}_{\rho \alpha }+{\Gamma ^{\rho }}_{\rho \lambda }{\Gamma ^{\lambda }}_{\beta \alpha }-{\Gamma ^{\rho }}_{\beta \lambda }{\Gamma ^{\lambda }}_{\rho \alpha }## from wikipedia means:
##\sum_{j_1=0}^D(\frac{\partial{\Gamma ^{j_1}}_{\beta \alpha }}{\partial x^{j_1}}-\frac{{\Gamma^{j_1}}_{j_1 \alpha}}{\partial x^{\beta}}+\sum_{j_2=0}^D({\Gamma^{j_1}}_{j_1 j_2}{\Gamma ^{j_2}}_{\beta \alpha }-{\Gamma ^{j_1}}_{\beta j_2 }{\Gamma^{j_2}}_{j_1 \alpha }))##?
I can say clearly that it means$$R_{\alpha \beta }={R^{\rho }}_{\alpha \rho \beta }=\sum_\rho\partial _{\rho }{\Gamma ^{\rho }}_{\beta\alpha }-\sum_\rho\partial _{\beta }{\Gamma ^{\rho }}_{\rho \alpha }+\sum_{\rho,\lambda}{\Gamma ^{\rho }}_{\rho \lambda }{\Gamma ^{\lambda }}_{\beta \alpha }-\sum_{\lambda,\rho}{\Gamma ^{\rho }}_{\beta \lambda }{\Gamma ^{\lambda }}_{\rho \alpha }$$I haven't checked your algebra, but it is plausible that you've found a valid way of obscuring manipulating the formula.

Nugatory said:
obscuring
more easily readable for me. waiting foranswer whether it's correct or not.

weirdoguy
olgerm said:
more easily readable for me. waiting foranswer whether it's correct or not.

If you insist on writing things differently from the way everybody else writes them, you can't expect everybody else to check your work. We have standard ways of writing things in physics for a reason.

vanhees71 and weirdoguy

## 1. What is the summation convention in Einstein notation?

The summation convention in Einstein notation is a mathematical convention used to simplify the writing of equations involving repeated indices. It states that when an index appears twice in a single term of an equation, it is implicitly summed over all possible values.

## 2. How is summation convention denoted in Einstein notation?

In Einstein notation, the summation convention is denoted by placing the repeated index in both the upper and lower positions of a tensor or vector. This indicates that the index is being summed over.

## 3. What are the benefits of using summation convention in Einstein notation?

The main benefit of using summation convention in Einstein notation is that it simplifies the writing of equations, making them more concise and easier to understand. It also helps to avoid errors that may occur when manually summing over repeated indices.

## 4. Can summation convention be applied to any type of tensor or vector?

Yes, summation convention can be applied to any type of tensor or vector, as long as the indices are repeated within a single term of an equation. It is commonly used in equations involving tensors and vectors in physics and mathematics.

## 5. How does summation convention relate to the Einstein summation convention?

The summation convention in Einstein notation is a specific instance of the more general Einstein summation convention. The Einstein summation convention states that when an index appears twice in a single term of an equation, it is summed over all possible values, regardless of its position in the term.

• Special and General Relativity
Replies
17
Views
2K
• Special and General Relativity
Replies
2
Views
1K
• Special and General Relativity
Replies
44
Views
2K
• Special and General Relativity
Replies
10
Views
1K
• Special and General Relativity
Replies
1
Views
1K
• Special and General Relativity
Replies
9
Views
2K
• Special and General Relativity
Replies
24
Views
2K
• Special and General Relativity
Replies
2
Views
1K
• Special and General Relativity
Replies
5
Views
1K
• Special and General Relativity
Replies
1
Views
825