How to denote tetrad in Abstract Index Notation ?

[yicong2011]

I like Penrose's Abstract Index Notation very much. I am familiar with using Abstract Index Notation to denote Coordinate Basis.

But when I try to denote tetrad with Abstract Index Notation, I meet problems.

How to denote tetrad in Abstract Index Notation?

 

[yicong2011]

In Landau's Book, (Page 313, The Classical Theory of Fields, Fourth edition, Elsevier),

tetrads are expressed

eⁱ_(a), e_(b)i, e_i^(a), (*)

Of course, I cannot denote tetrads as

(eⁱ)_a, (e_i)_b, (e_i)^a

 

[Mentz114]

I've only ever seen them written like this

$$\Lambda^A_a$$

where A is the frame basis index and a is a holonomic frame index. So to change basis from holonomic to frame is

$$V^A = \Lambda^A_a\ V^a$$

The inverse tetrad is denoted

$$\Lambda^a_A$$

 

[yicong2011]

I've only ever seen them written like this

$$\Lambda^A_a$$

where A is the frame basis index and a is a holonomic frame index. So to change basis from holonomic to frame is

$$V^A = \Lambda^A_a\ V^a$$

The inverse tetrad is denoted

$$\Lambda^a_A$$

Can we say tetrad is rank 2 tensor, since two index can both rise and lower down?

 

[Mentz114]

Strictly it is not a tensor, so there is always one lower and one upper index. The tetrad performs a basis transformation on a tensor without altering its rank.

$$T^{AB}\ =\ \Lambda^A_a \ \Lambda^B_b\ T^{ab}$$

Because the indexes refer to two bases, there is no metric to raise or lower them. It is never necessary to raise or lower a tetrad index, because the operand tensor or the result can be adjusted.

The inverse is defined thus

$$\begin{align*} &V^A\ = \ \Lambda^A_a \ V^a\\ \Rightarrow\ &\Lambda^a_A \ V^A\ = \Lambda^a_A \ \Lambda^A_a \ V^a = \ V^a \end{align*}$$

Recently in a paper on TP gravity ( where the tetrad is a gauge potential) I saw a tetrad written with 2 lower indexes which could be raised. But I have not worked out what it means.

 

[yicong2011]

Strictly it is not a tensor, so there is always one lower and one upper index. The tetrad performs a basis transformation on a tensor without altering its rank.

$$T^{AB}\ =\ \Lambda^A_a \ \Lambda^B_b\ T^{ab}$$

Because the indexes refer to two bases, there is no metric to raise or lower them. It is never necessary to raise or lower a tetrad index, because the operand tensor or the result can be adjusted.

The inverse is defined thus

$$\begin{align*} &V^A\ = \ \Lambda^A_a \ V^a\\ \Rightarrow\ &\Lambda^a_A \ V^A\ = \Lambda^a_A \ \Lambda^A_a \ V^a = \ V^a \end{align*}$$

Recently in a paper on TP gravity ( where the tetrad is a gauge potential) I saw a tetrad written with 2 lower indexes which could be raised. But I have not worked out what it means.

But I have seen it in

"Metric Compatibility
Condition And Tetrad
Postulate"

(Myron W. Evans)

See just below the formula (8.1)




"q^a_μ is the tetrad [3]-[5], a mixed index rank two tensor"



However, in Landau's Book "The Classical Theory of Fields" (Fourth Edition, Elsevier) Page 313, it is definitely pointed out that "a set of four linearly independent coordinate four-vectors"

Thus, I am a bit confused...

 

[Mentz114]

Thus, I am a bit confused...

Me too, about the nomenclature. In practise it is usually clear what is meant, though.