Index Notation Confusion in GR - Help Needed!

In summary, index notation in GR can be confusing, particularly when it comes to the order of index placement. The best explanation of tensors can be found in the classic GR textbook by Misner, Thorne, & Wheeler. The order of the slots in a tensor matters, with upper indexes corresponding to vectors and lower indexes corresponding to covectors. The tensors Λab and Λba are not the same, but can be considered equivalent if their outputs are equal. It is important to pay attention to the placement of indexes, as some tensors are symmetric while others are anti-symmetric.
  • #1
dyn
773
61
Index notation in GR is really confusing ! I'm confused about many things but one thing is the order of index placement , ie. is Λa b the same as Λba ? And if not what is the difference ? Thanks
If anyone knows of any books or lecture notes that explain index gymnastics step by step that would be great.
 
Physics news on Phys.org
  • #2
dyn said:
is Λab the same as Λba ?

Strictly speaking, no. See below.

dyn said:
if not what is the difference ?

The best simple explanation of how a tensor works that I've seen is in Misner, Thorne, & Wheeler, the classic GR textbook. Basically, a tensor is a linear machine with some number of slots, that takes geometric objects as input into the slots and outputs numbers; each slot corresponds to an index. If the index is an upper index, the slot takes a vector as input; if the index is a lower index, the slot takes a covector (or 1-form) as input. The order of the slots matters, so Λab, which takes a vector in the first slot and a 1-form in the second, is not the same as Λba, which takes a 1-form in the first slot and a vector in the second.

In a manifold with metric (which is all we work with in GR), you can always use the metric to convert vectors to 1-forms or vice versa. So you could take a vector and a 1-form that you inserted into the slots of Λab, and insert them into the slots of Λba, by converting the vector to a 1-form (so it will go in the first slot of Λba) and the 1-form to a vector (so it will go in the second slot of Λba). If these two operations both give the same number as output, then the two tensors Λab and Λba can be considered "the same"; in this case, we say the second is just the first with one index lowered and one index raised, using the metric.
 
  • Like
Likes cianfa72 and HallsofIvy
  • #3
Thanks for your answer. Does that mean indices should never be directly in a vertical line as in that case we wouldn't know the order of the "slots" ?
 
  • #4
dyn said:
Does that mean indices should never be directly in a vertical line as in that case we wouldn't know the order of the "slots" ?

Yes, although some sources are sloppy about this, probably because in some cases it doesn't actually matter. For example, if a two-index tensor is symmetric, its indexes can be exchanged (i.e., slots swapped) without changing its output. Many key tensors that appear in GR are symmetric (e.g., the metric and the stress-energy tensor).
 
  • Like
Likes dyn
  • #5
dyn said:
Thanks for your answer. Does that mean indices should never be directly in a vertical line as in that case we wouldn't know the order of the "slots" ?
Just to add (pedantically) that some tensors are anti-symmetric so ##T_{ab}=-T_{ba}##.
 

1. What is index notation in general relativity (GR)?

Index notation is a mathematical notation commonly used in the field of general relativity to represent tensors and operations on them. It involves using indices as subscripts and superscripts to denote the components of a tensor and the coordinate system in which it is defined.

2. Why is there confusion surrounding index notation in GR?

Index notation in GR can be confusing because it involves multiple indices, each representing different quantities, and the order of indices can also affect the meaning of the expression. Additionally, there are different conventions and notations used by different authors, which can further add to the confusion.

3. How can I overcome confusion with index notation in GR?

One way to overcome confusion with index notation in GR is to familiarize yourself with the basic concepts and rules of index notation. This includes understanding the meaning of indices, the Einstein summation convention, and the properties of tensors. It is also helpful to practice working with index notation and to consult reliable sources for guidance.

4. Are there any common mistakes to watch out for when using index notation in GR?

Yes, there are a few common mistakes that can occur when using index notation in GR. These include using the wrong index order, forgetting to apply the summation convention, and mixing up upper and lower indices. It is important to double-check your notation and calculations to avoid these mistakes.

5. Can you provide an example of index notation in GR?

Sure, an example of index notation in GR is the Einstein field equation:

Gμν + Λgμν = (8πG)/c4 Tμν

Here, the indices μ and ν represent the components of the metric tensor gμν and the energy-momentum tensor Tμν. The index Λ represents the cosmological constant, and the index G represents the gravitational constant. The Einstein summation convention is applied, where repeated indices are summed over.

Similar threads

  • Special and General Relativity
Replies
3
Views
695
  • Special and General Relativity
Replies
12
Views
2K
  • Special and General Relativity
Replies
10
Views
1K
Replies
2
Views
1K
  • Special and General Relativity
Replies
1
Views
619
  • Special and General Relativity
Replies
8
Views
2K
  • Programming and Computer Science
Replies
1
Views
946
Replies
9
Views
1K
  • Special and General Relativity
Replies
4
Views
2K
  • Special and General Relativity
Replies
6
Views
1K
Back
Top