Notation of General Relativity

In summary: But this is just a special coordinate system, not a special kind of coordinate system. In summary, The brackets denote that tensor is symmetric in the indices inside the brackets. The reason why the index of the \nabla can be included in the brackets is simple: the entire construction \nabla_a R_{bcd}{}^{e} is itself another tensor with five indices, under four of which it is symmetric. This is similar to a new tensor T_{abcd}{}^{e} = \nabla_a R_{bcd}{}^{e} where all of the lower indices of T are symmetric. The subscript on the \nabla indicates covariant differentiation with respect to the ath coordinate of some unspecified coordinate system,
  • #1
Mike2
1,313
0
I'm reading, General Relativity, by Robert M. Wald. On page 39 he has the following notation:

[tex]
\displaystyle{
{\rm R}_{{\rm [abc]}}^{\,\,\,\,\,\,\,\,\,\,\,\,{\rm d}} = 0
}

[/tex]

and

[tex]
\displaystyle{
\nabla _{{\rm [a}} {\rm R}_{{\rm bc]d}}^{\,\,\,\,\,\,\,\,\,\,{\rm e}} = 0
}
[/tex]

What do the square brakets [ ] mean? How can the subscript of the [tex]\nabla[/tex] be included in these brakets? I've not seen this notation in the other books I have, so I could use a little help.

Thanks.
 
Last edited:
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  • #2
The brackets denote that tensor is symmetric in the indices inside the brackets.

The reason why the index of the [itex]\nabla[/itex] can be included in the brackets is simple: the entire construction [itex]\nabla_a R_{bcd}{}^{e}[/itex] is itself another tensor. It has five indices, under four of which it is symmetric.

You could think of it as a new tensor T:

[tex]T_{abcd}{}^{e} = \nabla_a R_{bcd}{}^{e}[/tex]

where all of the lower indices of T are symmetric:

[tex]T_{[abcd]}{}^{e} = 0[/tex]

- Warren
 
Last edited:
  • #3
Originally posted by chroot
The brackets denote that tensor is symmetric in the indices inside the brackets.

The reason why the index of the [itex]\nabla[/itex] can be included in the brackets is simple: the entire construction [itex]\nabla_a R_{bcd}{}^{e}[/itex] is itself another tensor. It has five indices, under four of which it is symmetric.

You could think of it as a new tensor T:

[tex]T_{abcd}{}^{e} = \nabla_a R_{bcd}{}^{e}[/tex]

where all of the lower indices of T are symmetric:

[tex]T_{[abcd]}{}^{e} = 0[/tex]

- Warren

So this bracket notation is the same as the sum of all permutations with a coefficient of +1 for even permutations and -1 for odd permutations?

The subscript on the [itex]\nabla[/itex] confuses me because on page 31, Wald writes, "It is often notationally convenient to attach an index directly to the derivative operator and write it as [itex]\nabla[/itex]a, although this is to some extent an abuse of the index notation since [itex]\nabla[/itex]a is not a dual vector."

So what is he doing using it like this?
 
  • #4
Originally posted by Mike2
What do the square brakets [ ] mean?

This notation is defined on p26.

Originally posted by chroot
The brackets denote that tensor is symmetric in the indices inside the brackets.

It wasn't stated explicitly by either of you so, just in case, I'll point out that it's the vanishing of the antisymmetric part that means it's completely symmetric in those indices. In other words

[itex]R_{[abc]}{}^d=0 \Leftrightarrow
R_{abc}{}^d=R_{(abc)}{}^d[/itex].

Originally posted by chroot
The reason why the index of the [itex]\nabla[/itex] can be included in the brackets is simple: the entire construction [itex]\nabla_a R_{bcd}{}^{e}[/itex] is itself another tensor.

It doesn't matter that it's a tensor.

Originally posted by Mike2
...since [itex]\nabla[/itex]a is not a dual vector."

So what is he doing using it like this?

To make operations on the components of [itex]\nabla[/itex] explicit.
 
  • #5
Originally posted by jeff

To make operations on the components of [itex]\nabla[/itex] explicit.

Does the subscript on [itex]\nabla[/itex] stand for differentiation with respect to the a'th coordinate curve of some as yet unspecified system?

Is that system always orthogonal?

Thanks.
 
  • #6
Originally posted by Mike2
Does the subscript on [itex]\nabla[/itex] stand for differentiation with respect to the a'th coordinate curve of some as yet unspecified system?

If you mean covariant differentiation with respect to the ath coordinate of some as yet unspecified coordinate system, then yes.

Originally posted by Mike2
Is that system always orthogonal?

Again, by "system" I'll assume you mean coordinate system. A coordinate system S is orthogonal with respect to some inner product ( , ) if S's basis vectors are mutually orthogonal with respect to ( , ). In GR, ( , ) is the spacetime metric, and S needn't be chosen orthogonal with respect to it. On the other hand, given any point p on a manifold, one may always choose a coordinate system which is orthogonal at p.
 

1. What is the notation used in General Relativity?

The most commonly used notation in General Relativity is the tensor notation, which uses indices to represent the components of a tensor. Greek letters are often used for spacetime indices while Latin letters are used for spatial indices.

2. How is the metric tensor represented in General Relativity?

The metric tensor in General Relativity is represented by gμν, where μ and ν are indices that represent the components of the tensor. It is used to describe the curvature of spacetime.

3. What is the Einstein summation convention in General Relativity?

The Einstein summation convention is a shorthand notation used in General Relativity to simplify equations involving tensors. It states that when an index appears twice in a term, it is implicitly being summed over all possible values.

4. How is the Christoffel symbol represented in General Relativity?

The Christoffel symbol in General Relativity is represented by Γρμν, where ρ is the index for the coordinate system and μ and ν represent the components of the metric tensor. It is used to describe the connection between the metric tensor and the curvature of spacetime.

5. What is the difference between covariant and contravariant indices in General Relativity?

In General Relativity, covariant indices are used to represent the components of a tensor in a particular coordinate system, while contravariant indices are used to represent the components of the same tensor in a different coordinate system. The transformation between the two types of indices is described by the metric tensor.

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