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.
I apologise to the author and other readers for this, and for the ensuing
loss of threading in many newsreaders.
-- jt]]

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]

Bossavit schrieb:
> 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.

Right, that's the better choice because it also distinguishes basis
vectors from their reciprocal duals when we omit the abstract index "a".
After choosing a metric we recognize this notation to be superior to
others like (e_mu)* for e^mu (Wolfgang Pauli, Theory of Relativity),
because the index is formally moved up and down by the components of the
metric in the usual way.

> 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.)

I think in the last comment you describe the kinship that relates a
basis vector e_mu to e^mu (also if there is no metric) because a basis
vector e_mu should alway be considered as a member of a family (the
basis). This kinship is different from the one induces by the metric, of
course. The difference is expressed by Greek versus Latin indices.

Compare this to the following usage of Greek indices after choosing a
metric. Let v^a be a vector. Then with respect to a basis, v^mu as the
component with number mu is not determined by v_mu for a fixed value mu.
Instead we have to use all components v_nu when we calculate:
v^mu = g^{mu nu}*v_nu (summation convention!)

> 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.

Which one seems to be the received convention, the former or the latter?
I consider "e_mu" as the "the mu-th basis vector". Greek indices have
the traditional meaning of identifying an element of a family (for
instance a family of basis vectors or the components of a vector).
Abstract indices as being like placeholders for arguments of functions
have a different meaning.

> 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).

I think this is not the way we are supposed to think about it when "i"
is considered to be an abstract index. v^i is indeed "a vector". But the
index "i" identifies the slot of the function v:V*->R. Compare this to
the notation, where a dot is used to denote the slot of a function f
with a single argument. One writes f=f(.) and f(x)=f(.)(x). Now in
multilinear algebra we have more the one argument, therefore a dot is
not enough. We use Latin indices instead (Mathematica uses #1,#2,...).
Furthermore we want to be independent of the order of the arguments, so
we tag arguments with Latin indices as well. For example we can write:

g(v,w) = g_ab v^a w^b = g_ab w^b v^a

or in Mathematica notation, where the order of arguments is significant:

g[v,w] = g[#1,#2]&[v,w] = g[#2,#1]&[w,v]

Notice that by no means the indices should be interpreted as labeling
components and therefore no summation convention is implied.

If we want to describe components of the vector v^i = v^mu (e_mu)^i
(summation over Greek indices!), then we use Greek indices: v^mu would
be the mu-th member of a dim(V)-tuple of scalars.

> 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.

As described above, in abstract index notation the index "i" is not an
element of an index set. So from what you write I can't see an
inconsistency. (Also notice that you are not using abstract index
notation rigorously, when you write "e_mu". Since e_mu is an vector it
should have an abstract index: (e_mu)^i.)

> 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.

I agree on this observation.

> 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.

Right, and I think it is successful in trying so. Maybe an equation says
more than a thousand words, so I write down how the translation between
both traditions works in the example I used above:

g(v,w) = g_ab v^a w^b = g_ab v^mu (e_mu)^a w^nu (e_nu)^b
= g_ab (e_mu)^a (e_nu)^b v^mu w^nu
= g_{mu nu} v^mu w^nu

I started with the abstract notation in mathematics and ended with the
component notation in physics. Notice that the component version looks
exactly like the abstract index version with Latin indices replaced by
Greek indices.

> The background of our discussion is, is this desirable, is this
> feasible? I'm skeptical on both points, yet ready to taste the
> pudding.

>From what I wrote above you can see that I believe it is feasible. I
also believe it is desirable, for the following reasons. Both notations
have advantages that makes it desirable to reconcile them. The abstract
mathematical notation doesn't depend on some arbitrary choice of
coordinates, which is an "act of violence" according Hermann Weyl. I
think a coordinate free description gives a better understanding of the
described physics because the real world doesn't provide any coordinate
systems. Coordinates are extremely useful to do concrete calculations,
that's why physicist learn to use them. But sometimes they tend to
forget the meaning of what they are calculating. "People like to
calculate before they think", as Juergen Ehlers once remarked in some
lecture I attended. Since the formalisms used in physics tend to be
useful and fool-prove, also mathematicians that have to do a lot of
calculation can benefit from a translation. Just compare how tensor
contraction is described in both notations.

[Bossavit]

>Then with respect to a basis, v^mu as the
>component with number mu is not determined by v_mu
>for a fixed value mu. Instead we have to use all
>components v_nu when we calculate:


Sorry, I forgot to mention the all important clause: We
start (because we can, since there is a metric), from an
*orthonormal* set of basis vectors {e_mu}. Now pick *one*
of them, e_mu, and obtain the covector e^mu defined by

<v ; e^mu> = v . e_mu,

where . is the dot product and < ; > the duality pairing.
(The r.h.s. is a linear function of v, so we do have an element
of the dual there, namely the map v --> v . e_mu.) We note
that <e_nu ; e^mu> = 0 if nu =/= mu. So we have been able to
find e^mu, using only e_mu, whatever the other basis vectors,
provided they (including e_mu) make an orthonormal family. This
is why giving e_mu and e^mu the same "family name" (e)
is all right, then, and why the pair {e_mu, e^mu} is considered
as a single entity, called "Euclidean tensor" (here, vector), as are
also called, of course, linear combinations of such pairs.
Euclidean tensors require a Euclidean structure (the dot product)
to thrive in, so when such a structure does not impose itself
naturally, they are rather a hindrance.

>I consider "e_mu" as the "the mu-th basis vector".

We agree. Yet,

>v^i is indeed "a vector". But the
>index "i" identifies the slot of the function v:V*->R.

... can't parse this. Indeed, v maps V* to R, since it's an
element of V**. But there is no "i" in the description of this
function you point at, which makes your sentence incomplete, at best.
Try, perhaps, to define "slot". But be careful: Usage is that
"slot" serves when one has functions of *several* variables, as
e.g. in f(x, y, z) = ..., we have three slots, to be filled by
(say) real values. In "the function v:V*->R", above, you have *one*
slot, to be filled out by a covector. Unless you (unconsciously?)
think of this covector as the collection of its components (which
is the "old tensor", nowadays discredited, way). What you write
later, "v^mu would be the mu-th member of a dim(V)-tuple of scalars"
smacks of this. At least, it goes against the grain of the
"coordinate-free decription" spirit, which I think we would agree to share.

I see your point in the derivation,

g(v,w) = g_ab v^a w^b = g_ab v^mu (e_mu)^a w^nu (e_nu)^b
= g_ab (e_mu)^a (e_nu)^b v^mu w^nu
= g_{mu nu} v^mu w^nu

which is all right. But see how, with only some very little grain of
bad faith, it can be subverted, by just omitting intermediate steps:

g(v,w) = g_ab v^a w^b = (...) = g_{mu nu} v^mu w^nu,

showing that only a change of names of dummy variables has been done
here, i.e., a blank operation. So why bother with the Latin-Greek
double decker?

[I remember a seminar, back in the 70s, when a respected specialist
in programming languages said, "There are two circumstances when people
behave like animals: when driving, and when discussing concrete
syntax." Let's not dwell on concrete syntax too long.]

[Theo Wollenleben]

Bossavit schrieb:
> Sorry, I forgot to mention the all important clause: We
> start (because we can, since there is a metric), from an
> *orthonormal* set of basis vectors {e_mu}. Now pick *one*
> of them, e_mu, and obtain the covector e^mu defined by
>
> <v ; e^mu> = v . e_mu,
>
> where . is the dot product and < ; > the duality pairing.
> (The r.h.s. is a linear function of v, so we do have an element
> of the dual there, namely the map v --> v . e_mu.)

I note that we make two steps here. First we assign to the basis vector
(e_mu)^a the dual (e_mu)_a by

(e_mu)_a v^a = g_ab (e_mu)^a v^b,

which is your equation in abstract index notation. To write this in your
notation it requires indeed another symbol for the dual basis vector,
because the Greek index mu is in the same position on both sides. The
assignment is applicable to every basis. If {(e_mu)^a} is orthonormal then

(e_mu)_a = (e^mu)_a,

where (e^mu)_a is the reciprocal basis vector corresponding to (e_mu)^a.

> We note
> that <e_nu ; e^mu> = 0 if nu =/= mu. So we have been able to
> find e^mu, using only e_mu, whatever the other basis vectors,
> provided they (including e_mu) make an orthonormal family. This
> is why giving e_mu and e^mu the same "family name" (e)
> is all right, then, and why the pair {e_mu, e^mu} is considered
> as a single entity, called "Euclidean tensor" (here, vector), as are
> also called, of course, linear combinations of such pairs.
> Euclidean tensors require a Euclidean structure (the dot product)
> to thrive in, so when such a structure does not impose itself
> naturally, they are rather a hindrance.

If I understand this correctly then we can say more generally: If there
is a metric then the pair {v^a,v_a} for every vector v^a is such an
(pseudo-)Euclidean vector. For basis vectors we have pairs
{(e_mu)^a,(e_mu)_a}, which equal {(e_mu)^a,(e^mu)_a} for an orthonormal
basis. So far, so good, but I'm afraid I'm missing the point you were
trying to make...

>> v^i is indeed "a vector". But the
>> index "i" identifies the slot of the function v:V*->R.
>
> ... can't parse this. Indeed, v maps V* to R, since it's an
> element of V**. But there is no "i" in the description of this
> function you point at, which makes your sentence incomplete, at best.
> Try, perhaps, to define "slot". But be careful: Usage is that
> "slot" serves when one has functions of *several* variables, as
> e.g. in f(x, y, z) = ..., we have three slots, to be filled by
> (say) real values. In "the function v:V*->R", above, you have *one*
> slot, to be filled out by a covector.

Exactly.

> Unless you (unconsciously?)
> think of this covector as the collection of its components (which
> is the "old tensor", nowadays discredited, way). What you write
> later, "v^mu would be the mu-th member of a dim(V)-tuple of scalars"
> smacks of this. At least, it goes against the grain of the
> "coordinate-free decription" spirit, which I think we would agree to share.

Indeed, we share this spirit. And right, mu is an element of some index
set. But no, v^i stands neither for the i-th component of a vector nor
for a collection of its components. Maybe I failed in trying to explain
my understanding of abstract index notation in my last post, so I'd like
to remind us again of the dot-notation used to label the slot of a
function. For instance using standard mathematical notation we could
write g(v,.):V->R for the dual vector associated with the vector v. The
dot labels the slot and we can then write

g(v,u)=g(v,.)(u)

Now I rename the slot, calling it "i" instead "." and write it as an
lower index to the function. I also rename the argument u by labeling it
with the same letter as the slot in which this argument is plugged in
and omit the parentheses of the argument:

g(v,.)(u)=g(v)_i u^i

which is an intermediate expression between the traditional mathematical
and the abstract index notation. Now we allow ourselves to dub different
argument of a function with different Latin indices. For the metric
g:VxV->R we can then write

g(v,u) = g_ji v^j u^i = g_ji u^i v^j

where we have the freedom to change the order of the arguments u^i and
v^j because their labels tell us which slot to plug them in. This mimics
the commutativity of multiplication in component notation.


We note that it is a matter of interpretation whether an abstract index
is labeling a slot or an argument. For instance the canonical
identification F:V->V**, which is given by

F(v)(w) = w(v)

reads in abstract index notation

v^i w_i = w_i v^i

which mimics again the commutativity of multiplication in component
notation. Another example is the metric induced isomorphism G:V->V* given by

G(v) = g(v,.)

which gives another interpretation to g_ji as this isomorphism V->V*:

g_ji v^j = v_i

(above we introduced g as a map VxV->R). These ambiguities are a benefit
because we needn't introduce symbols for the maps F,G. These
identifications are inherent in abstract index notation. We needn't care
about them and we are released from thinking about what we are doing
when we are calculating (of course we should nevertheless be able to
explain it if someone asks).


> I see your point in the derivation,
>
> g(v,w) = g_ab v^a w^b = g_ab v^mu (e_mu)^a w^nu (e_nu)^b
> = g_ab (e_mu)^a (e_nu)^b v^mu w^nu
> = g_{mu nu} v^mu w^nu
>
> which is all right. But see how, with only some very little grain of
> bad faith, it can be subverted, by just omitting intermediate steps:
>
> g(v,w) = g_ab v^a w^b = (...) = g_{mu nu} v^mu w^nu,
>
> showing that only a change of names of dummy variables has been done
> here, i.e., a blank operation.

Actually, I was trying to make the point that this is the essential
advantage of abstract index notation (the absence of bad faith assumed).
We get a formalism that is coordinate-free and as practical as the
classical component notation, the transition to coordinates being a
no-brainer.

If possible we use Latin indices, indicating that we need no coordinate
system when we are formulating physical laws.

> So why bother with the Latin-Greek double decker?

Sometimes we want to do computations in some coordinates that fit our
physical problem. Then some equations will hold only with respect to the
chosen coordinates. We can't use Latin indices then and need the Greek
indices. Another example are the Christoffel symbols, which can't be
expressed as abstract index objects because they do not transform like
tensors.

> [I remember a seminar, back in the 70s, when a respected specialist
> in programming languages said, "There are two circumstances when people
> behave like animals: when driving, and when discussing concrete
> syntax." Let's not dwell on concrete syntax too long.]

I apologize for ignoring your suggestion and writing a long answer
instead. If no one else reads through it then at least it helps me to
disentangle my own thoughts.

[Bossavit]

Wollenleben:

>If I understand this correctly then we can say more generally: If there
>is a metric then the pair {v^a,v_a} for every vector v^a is such an
>(pseudo-)Euclidean vector.

Yes, I think we agree there.


> g(v,u) = g_ji v^j u^i = g_ji u^i v^j
>
>where we have the freedom to change the order of the arguments (...)
>This mimics the commutativity of multiplication in component
>notation.

"Mimics"? I don't follow you here. This seems to address symmetry of
the metric tensor, actually, so "commutativity" may be misleading.

>I apologize for ignoring your suggestion