# Wald's Abstract Index Notation: Explaining T^{acde}_b

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In summary, Wald's General Relativity rewrites T^{acde}_b as g_{bf}g^{dh} g^{ej}T^{afc}_{hj} . Can anyone explain this? I am confused by the explantion given in the book. Especially puzzling is that the inverse of g seems to be applied twice, which I can't make sense of.
TL;DR Summary
Confusion by the abstract index notation introduced in Wald's General Relativity.
In the second paragraph on page 25 of Wald's General Relativity he rewrites T^{acde}_b as g_{bf}g^{dh} g^{ej}T^{afc}_{hj} . Can anyone explain this? I am confused by the explantion given in the book. Especially puzzling is that the inverse of g seems to be applied twice, which I can't make sese of.

The inverse metric is applied twice to raise the last two indices. The metric is used once to lower the second index.

Also note that the index horizontal placement is important.

This isn't particularly to do with abstract index notation - it applies in all tensor index notations. You do need to pay attention to which index is being contracted over and the order of indices is important. But all that's happening here is that the metric is being used to lower two of the indices and the inverse metric is being used to raise one. That's just what the metric does. Just as the metric in ##g_{ab}v^b## lowers the ##b## to give you the one form ##v_a##, the metric applied to any tensor lowers the repeated index - so ##g_{ad}T^{bcdefg}## lowers the ##d## to give you ##T^{bc}{}_a{}^{efg}##. Note that the repeated index ##d## was replaced with the other index on the metric because we summed over the dummy index.

In the cited section Wald is just randomly lowering a couple of indices and raising one to show you can do it to multiple indices at once.

Quote my post to see a way to do index notation with correct positioning in ##\LaTeX##.

Ibix said:
In the cited section Wald is just randomly lowering a couple of indices and raising one to show you can do it to multiple indices at once.
Lowering one and raising two.

cianfa72 and Ibix
Orodruin said:
Lowering one and raising two.

## 1. What is Wald's Abstract Index Notation?

Wald's Abstract Index Notation is a mathematical notation system used to represent tensors, which are multidimensional arrays of numbers. It was developed by physicist Robert Wald and is commonly used in the field of general relativity.

## 2. How is Wald's Abstract Index Notation different from other notation systems?

Unlike other notation systems, Wald's notation does not use specific coordinate systems or basis vectors. Instead, it uses abstract indices to represent the components of a tensor, making it more general and applicable to various coordinate systems.

## 3. What does the "T^{acde}_b" notation mean?

The "T^{acde}_b" notation represents a tensor with four upper indices (a, c, d, e) and one lower index (b). The upper indices indicate the contravariant components of the tensor, while the lower index represents the covariant component.

## 4. How do you perform operations with tensors using Wald's notation?

To perform operations with tensors using Wald's notation, you can use the Einstein summation convention, which states that when an index appears both as an upper and lower index in a term, it is automatically summed over all possible values. Additionally, there are specific rules for manipulating tensors with repeated indices.

## 5. How is Wald's Abstract Index Notation used in physics?

Wald's notation is commonly used in the field of general relativity to represent the equations and concepts involved in the theory. It allows for a more concise and general representation of tensors, making it easier to work with complex equations and calculations.