Re: Wald's Isomorphism between V and V*

[Mike_Fontenot]

[[Mod. note -- I mistakenly deleted this article's References: headers
when saving it from my inbox into my s.p.r articles-to-be-moderated queue.
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I (Mike_Fontenot) wrote:

> I should have added that Wald also uses
> Greek indices for basis vectors and basis duals.
> So he uses a DIFFERENT notation for basis
> vectors than for other vectors (and likewise
> for basis duals versus other duals).

In spite of what I wrote above, after thinking about it a bit more, I
now suspect that Wald is NOT using different indexing rules for basis
vectors as for other vectors. I think his rules are:

1) Use a Latin superscript on a vector when you want the index to
identify the ARGUMENT of the vector, as opposed to identifying the
vector itself. (And I use the same rule in my notation, except that I
can use Latin, Greek, or numerals for the index. And I always use an
upper-case (Latin) letter for the root-name of a vector, versus a
lower-case (Latin) letter for the root-name of the coefficient of the
vector, to clearly distinguish between coefficients versus the entities
themselves; Wald uses both lower-case and upper-case, for both
coefficients and for the entities themselves).

2) Use a Greek or numeral subscript on a vector when you want the index
to identify the vector itself. (And I use the same rule in my notation,
except that I use a "prescript" index whenever it identifies the vector
itself, versus a "postscript" index whenever it identifies the argument
of the vector. And again, I can use Latin, Greek, or numerals for any
type of index).

3) The rules for duals are the opposite of the rules for vectors.

But this means that Wald DOES have to have DIFFERENT rules for
higher-order tensors than for vectors and duals. Tensors that have both
vector AND dual arguments need both superscripts and subscripts for
representing arguments. So there is no way in Wald's notation to use
an index to identify the tensor itself, as opposed to identifying the
arguments of the tensor. Since I reserve the "prescripts" for
identifying the entity itself (and postscripts for identifying the
arguments), I can use the same rules for ALL types of tensors, including
vectors and duals.

And I still think Wald's use of the same root-name for basis duals and
basis vectors (the letter "e"), is ill-advised. The same root-name for
vectors and duals should be reserved for the case where the vector and
dual are "metric-isomorphic". That is the basis for using the metric to
"raise" and "lower" indices on vectors and duals (and on tensors in
general). The basis duals are defined based on their "delta"
relationship with the basis vectors...they are NOT "metric-isomorphic":
you DON'T get a basis dual by using the metric to lower the index on a
basis vector. To make this clear, I use upper-case "E" for basis
vectors, and upper-case "D" for basis duals (following the lead of
Bossavit in a previous post).

[Theo Wollenleben]

Mike_Fontenot schrieb:
> In spite of what I wrote above, after thinking about it a bit more, I
> now suspect that Wald is NOT using different indexing rules for
> basis vectors as for other vectors. I think his rules are:
>
> 1) Use a Latin superscript on a vector when you want the index to
> identify the ARGUMENT of the vector, as opposed to identifying the
> vector itself. (And I use the same rule in my notation, except that I
> can use Latin, Greek, or numerals for the index. And I always use an
> upper-case (Latin) letter for the root-name of a vector, versus a
> lower-case (Latin) letter for the root-name of the coefficient of the
> vector, to clearly distinguish between coefficients versus the
> entities themselves; Wald uses both lower-case and upper-case, for
> both coefficients and for the entities themselves).

Right, upper and lower case have no special meaning, when Wald names
tensors. Wald uses Latin versus Greek indices to distinguish tensor
arguments from tensor components with respect to some basis. You use
upper-case versus lower-case to do the same job. So when expressing in
abstract equation in components we have to do the replacement Latin ->
Greek in Wald's notation or upper-case -> lower-case in your notation.
Seems to be equally efficient to me. I see no danger of confusion in
both notations. (Wald's notation has the advantage of being standard, of
course.)

> 2) Use a Greek or numeral subscript on a vector when you want the
> index to identify the vector itself. (And I use the same rule in my
> notation, except that I use a "prescript" index whenever it
> identifies the vector itself, versus a "postscript" index whenever it
> identifies the argument of the vector. And again, I can use Latin,
> Greek, or numerals for any type of index).

I haven't worked through Wald's book completely, but from what I have
read he needs a Greek index to identify a vector itself only for basis
vectors. Such Greek index always runs from 1 to the dimension of the
space and is used to expand tensors with respect to the basis. Therefore
Greek indices have a very special meaning.

If I understand you correctly, you want to identify some arbitrary
vector itself with an index from some index set having no special
meaning. So your prescript notation has a different (far more general)
scope of application than Wald's Greek indices. Your prescript notation
used in the special case of labeling basis vectors indicates this
special meaning by resulting in lower-case letters for tensor components
when a tensor is applied to the basis vectors.

> 3) The rules for duals are the opposite of the rules for vectors.
>
> But this means that Wald DOES have to have DIFFERENT rules for
> higher-order tensors than for vectors and duals. Tensors that have
> both vector AND dual arguments need both superscripts and subscripts
> for representing arguments. So there is no way in Wald's notation to
> use an index to identify the tensor itself, as opposed to identifying
> the arguments of the tensor.

When Wald uses abstract index notation, does he ever need to identify
tensors other than basis vectors? I don't think Wald's notation is
unclear or confusing. He has just fewer rules, because he doesn't need more.

> Since I reserve the "prescripts" for identifying the entity itself
> (and postscripts for identifying the arguments), I can use the same
> rules for ALL types of tensors, including vectors and duals.

I hope, it is clear from what I said above that Wald uses the same rules
for the usage of Latin indices for all sorts of tensors. Only in the
special situation of naming basis vectors he needs the Greek indices to
identify (dual) vectors itself.

> And I still think Wald's use of the same root-name for basis duals
> and basis vectors (the letter "e"), is ill-advised. The same
> root-name for vectors and duals should be reserved for the case where
> the vector and dual are "metric-isomorphic". That is the basis for
> using the metric to "raise" and "lower" indices on vectors and duals
> (and on tensors in general). The basis duals are defined based on
> their "delta" relationship with the basis vectors...they are NOT
> "metric-isomorphic": you DON'T get a basis dual by using the metric
> to lower the index on a basis vector. To make this clear, I use
> upper-case "E" for basis vectors, and upper-case "D" for basis duals
> (following the lead of Bossavit in a previous post).

Well, I think you are plain wrong on this. A. Bossavit may correct me,
but he just adopted the notation you introduced and he used notation not
consistent with Wald's notation. A. Bossavit wrote:
"The E-frame spawns the D-frame, and vice versa, but an *individual*
basis vector E_a does not point to a mate D^a on the dual side,
unless there is a metric structure."

According to Wald we must use a Greek index instead of "a". Furthermore
it is not abstract index notation because the abstract index is omitted.
The same statement in abstract index notation would read:
"The basis of V spawns its dual basis, and vice versa, but an
*individual* basis vector (e_mu)^a does not point to a mate (e_mu)_a (or
(e^mu)_a) on the dual side, unless there is a metric structure."

Firstly, I can't see any source of confusion, if we use abstract index
notation rigorously. We have four different kind of basis vectors:
(e_mu)^a and (e^mu)_a (the basis and its reciprocal dual basis) and via
the metric isomorphism (e_mu)_a and (e^mu)^a. The placement of the two
indices indicates the different meaning clearly. As far as I know this
notation is widely used and people are pleased with it.

Secondly, your naming convention has a disadvantage, which I explained
in an previous message. I wrote (with minor correction): "... in an
identity like

g_{mu nu} (e^nu)_a = g_ab (e_mu)^b

we needn't care about what we call "e" or "d". It's just a matter of
moving the indices of "e" in a very simple way."
Now with your convention it isn't that simple, because you must keep
track of where to put the "E" and the "D". This makes thing more
complicated, which is unnecessary because the different meaning of the
basis vectors is already encoded in the position of the indices.


Summary: I think your notation is okay, except for the introduction of
the unnecessary "D". Your dismissal of abstract index notation seems
unjustified, the main reason for your confusion probably being a
misnomer in the introduction of Wald's book. He shouldn't have called
basis vectors v_mu (not abstract index notation!), because this can
easily be confused with the component of an arbitrary vector v_a (in
abstract index notation!). He should have used e_mu instead as he does
later ((e_mu)^a in abstract index notation).

[Bossavit]

Still following this discussion with interest, I'd wish to address
the following point, a small one, but with some potential for confusion.

Wollenleben wrote:

>A. Bossavit wrote:
>
>"The E-frame spawns the D-frame, and vice versa, but an *individual*
>basis vector E_a does not point to a mate D^a on the dual side,
>unless there is a metric structure."
>
>(...) The same statement in abstract index notation would read:
>
>"The basis of V spawns its dual basis, and vice versa, but an
>*individual* basis vector (e_mu)^a does not point to a mate (e_mu)_a
>(or (e^mu)_a) on the dual side, unless there is a metric structure."

This leaves open the alternative (e_mu)_a versus (e^mu)_a on the
dual side. I'd definitely use the latter, (e^mu)_a.

But now -- the weakness of this notation -- we have (for some definite
value of mu) two objects, (e_mu)^a and (e^mu)_a, with the same last
name (the "e" part on their calling cards), in spite of lacking kinship,
*unless* there is a metric to make them related. In the absence of
metric structure, e^mu is not determined by e_mu (for some definite
value of mu), so their respective names suggest a kinship that doesn't
exist. (On the other hand, of course, if we know the whole e_mu
family, mu = 1 to n, we know the whole e^mu family on the dual side.)

A correlated weakness is the different roles attributed to latin and
greek diacritics. Let's agree that "v^i" doesn't mean "the i-th
component of vector v, a real (or complex) number", but just "the
vector v". Then we should, consistently, consider "e_mu" as meaning
"the (set comprised of the) n basis vectors of the n-dimensional
space V", and not as "the mu-th basis vector", which seems to be the
received convention. Thus, v^i would be "a vector" (whose components,
accessible one by one by assigning to i some definite value, are
scalars), whereas e_mu would be "a basis" (whose components,
accessible one by one by assigning to mu some definite value, are
vectors).

But if the intent is for both "v^i" and "e_mu" to be recognized, by just
looking at their labels, as vectors (a single vector, named v^i, and
another single vector, named e_mu), the inconsistency is hard to bear,
since this interpretation implies a *set* of i's (from 1 to n) but a
*unique* mu. Greek magic.

What's wrong with all that is the attempt to name mathematical objects
in such a way that their *type* be written on their face, ready to be
read off. That's what good old Fortran did: Names beginning with I,
J, K, etc., belonged to INTEGERS, unless "declared" otherwise. Then, in
(e.g.) a finite element code, a generic node, or element, or etc., would
be labelled INODE, JELEMENT, KWHATNOT, etc. Contrast that with modern
coding practice, which wants us to declare NODE, or ELEMENT, or etc., as
INTEGERS at the beginning of the subroutine, and thus be free to use
natural labels, with mnemonic value, in instructions that follow. (We
all do so, don't we?...)

The mainstream attitude in mathematics, I believe, is the same: We
carefully state that "labels such as v, w, etc., will apply to vectors,
labels such as omega, eta, etc., to covectors, labels such as e, or f,
will point to bases, (...)", and so forth, at the beginning of the
paper (or chapter, etc.), which then allows us to create new labels for
related subentities such as v^i (i-th component of v), e_mu
(mu-th basis vector), etc., in a hopefully consistent way, without
forfeiting useful tricks such as Einstein's convention or Kronecker deltas.

The "abstract index" notation departs from this tradition, presumably
because it tries to reconcile it with a *different* tradition (the "old
tensor" notation), prevalent in physics. The background of our
discussion is, is this desirable, is this feasible? I'm skeptical on
both points, yet ready to taste the pudding.