Questions on Contracting (1,1) Tensors & (2,0) Tensors

[Ratzinger]

1. When can I contract a tensor? Is that an option to contract for example a (1, 1) tensor to a real number or has that to happen for some tensors and for some don’t? Or put differently, when am I allowed to sum over one upper and one lower index?


2. I read in Schutz “Geometrical methods..” that a general (2, 0) tensor cannot be expressed as a simple outer product of two vectors. He says that’s because in n dimensions a (2, 0) tensor has n^2 independent components, while two vectors have between them only 2n components and in general this is inadequate.
I don’t quite understand. Is the independence of components here crucial? And does that really mean that tensors in general can not be constructed by forming tensor products of vectors and covectors? I thought that was kind of the definition of tensors?

As always I hope my questions make sense and thanks for replies.

 

[matt grime]

Let me do the algebraic bit.

There is no way to write

x\otimes y + y \otimes x

as an 'elementary tensor', where elementary is 'of the form

u\otimes v

that's all it's saying, ie a linear combination of elementary tensors is not necessarily an elementary tensor. It was explaining it in a sort of information theoretic sense: if we just look at the elementary tensors then they convey 2n units of information, but a general tensor needs n^2 bits of information to be specified.

 

[robphy]

You can contract a tensor when your tensor has at least an upper-index and at least a lower-index. Contraction yields a tensor with one fewer upper-index and one fewer lower-index. Contracting over different pairs of upper-and-lower indices yields different tensors: Given the (2,3)-tensor T^ab_def, the contraction T^ab_aef is a different (1,2)-tensor from T^ab_dea.

 

[Ratzinger]

So does that mean that I can contract any (5, 4) tensor to a vector if I like.

 

[mathwonk]

i will try to control myself slightly, but this kind of thing drives me absolutely bats, as it has done on this forum for over two years now.

this is a wonderful illustration of the stupidity of the upper lower tensor notations as opposed to the intrinsic conceptual version.

tensors are combinations of vectors and covectors, where covectors are linear functionals on vectors.

quite obviously, if you have both a vector and a covector, i.e. a vectior and a function on vectors, then you can evaluate the covector on the vector.
this is called "contraction".

i.e. let v be a vector and f be a linear function from vectors to numbers, then obviously you can evaluate f on v, hence you can "contract" the pair (f,v) to the number f(v).

similarly, given two vectors (v,w) and one function f, you can contract the triple (f,v,w) to either the vector f(v)w or the vector f(w)v.

this kind of thing is utterly trivial, and only seems confusing when people insist on ignoring the meanings of vectors and tensors and referring to them by the stupid notation of their upper and lower indices.


forgive me, i am going to go get drunk.

best regards,

roy

 

[robphy]

this is a wonderful illustration of the stupidity of the upper lower tensor notations as opposed to the intrinsic conceptual version.

Please enlighten me on the stupidity.
I am using Penrose's abstract index notation... and am not making any reference to any coordinates or choice of basis vectors. Such notation is meant to encode the conceptual meaning while facilitating computations.

 

[matt grime]

Leaping to mathwonk's defence, the point of a lot of differential geometry is to come up with compact notation that hides the wood behind the trees. Since a covector is simply an element in the dual space to a vector of course one can contract a vector and a covector by the standard properties of linear algebra that any undergraduate is taught in their first course. But in needing complicated notation differential geometry hides this obvious fact.

If you like, the reason you can contract covector with vector is because of their nature; the rule you're told is that upper indices can be contracted with lower indices to form something without explanation of the whys and wherefores. Admittedly any course that doesn't explain the nature of covectors and vectors is remiss, but I strongly suspect that plenty of courses will omit this material for one of two reasons: it's deemed obvious; it's deemed unimportant.

 

[robphy]

Admittedly, my description of contraction was meant to give a rule when contraction is possible [assuming the OP is aware of the other rules of tensor algebra and their interpretation]... but not necessarily to give an interpretation or justification for it.

In my initial reading of the OP's question, it seems the OP was asking about contracting a single tensor to obtain one of lower valence, specifically, obtaining, from a certain multilinear mapping of k-vectors and l-covectors, a multilinear mapping of (k-1)-vectors and (l-1)-covectors.

I can sympathize with complaints about the old-style use of index-notation as "components". Doesn't the new-style "abstract index notation" (by Penrose) approach the conceptual version presented by geometers? (See the discussions in, say, Wald's General Relativity.)

 

[mathwonk]

anyone who knows that an upper indexed gadget represents a vector and a lower indexed gadget represents a covector (if this is your convention) KNOWS you can contract them, because that is the definition of vector and covector.
he also knows that if you have 2 different covectors and one vector, then evaluating one of the covectors yields a different result from evaluating the other one.
by contraposition, anyone who has to ask whether you can contract one with the other does not know what they represent.
[these obvious facts are equivalent to the answers provided in post 3 but where they are given without any justification.]
this to me is absurd, i.e. employing notation that one does not understand, or memorizing only the rules for manipulating it instead of just learning what it means.
Penrose is not a God. Saying that his notation is standard means the interpretation of tensors is dependent on familiarity with penrose's use of notation.
it is just as feasible to make lower indices represent vectors and upper indices represent covectors, for example, as the opposite.
i realize that robphy was answering the question actually posed by the OP. I just hate to see one more person being encouraged to stagger along as a blind person using a mental cane, rather than being shown how to open his eyes and see.
i am not aware of penrose abstract index notation, but as i said it still seems arbitrary as to which altitude represents vectors and which represent covectors.
perhaps you could enlighten me on the use of penroses notation, and thus blunt my distaste for such things.
it seems quite persuasive to me however from the OP's original question concerning summing over the indices, that there are coordinates in use or else summing would be unnecessary.
i.e,. if f is an abstract covector and v is an abstract vector then f(tens)v is a (1,1) tensor whose contraction is merely f(v), no summing (and no indices) involved.

what i was set off by is that the original question, translated into plain english, becomes essentially: "when can I evaluate a function on one of its arguments?" a patently ridiculous question, whose answer is obviously "whenever you darned well wish to".

on the other hand, "when can i evaluate a point of the domain of a function on another point of that domain", has as answer: "huh? what are you talking about? that does not make sense!"

(in the absence of an isomorphism between two of these dual spaces, i.e. an inner product.)


what am I missing??

 

[mathwonk]

Let me apologize. The problem I have is unrelated to the present question or its answers. I am just geting old and tired.
I have posted literally hundreds and thousands of words explaining this, or trying to, over two years, and I have made perhaps no headway at stemming the tide of new physicists insisting on not learning what tensors mean.
It is a bit like the calculus prof who goes in the class the first day of fall and says, "what, you students till do not know what derivatives are? but I have been telling you for 40 years!"

my point is merely that tensors are not symbols, not indexed letters, a tensor is a sort of multiheaded hydra, some heads being functions, some heads being arguments for those functions, and sometimes some of the heads devour other heads.
peace

 

[robphy]

Abstract-index notation

Here's some motivation on the "abstract-index notation" from Roger Penrose, The Road to Reality (Ch 12.8)
"
There is an issue that arises here which is sometimes seen as a conflict between the notations of the mathematician and the physicist. The two notations are exemplified by the two sides of the above equation, \mbox{\boldmath$ \beta\cdot \xi$}=\beta_r\xi^r. The mathematician's notation is manifestly independent of coordinates, and we see that the expression \mbox{\boldmath$ \beta\cdot \xi$}} (for which a notation such as \mbox{\boldmath $ (\beta, \xi)$} or \mbox{\boldmath $<\beta, \xi>$} might be more common in the mathematical literature) makes no reference to any coordinate system, the scalar product operation being defined in entirely geometric/algebraic terms. The physicist's expression \beta_r \xi^r}, on the other hand, refers explicitly to components in some coordinate system. These components would change when we move from coordinate patch to coordinate patch; moreover, the notation depends upon the `objectionable' summation convention (which is in conflict with much standard mathematical usage). Yet, there is a great flexibility in the physicist's notation, particularly in the facility with which it can be used to construct new operations that do not come readily withing the scope of the mathematician's specified operations. Somewhat complicated calculations (such as those that the relate the last couple of displayed formulae above [the previous section had expressions involving symmetrized and antisymmetrized tensors and tensors with an arbitrary but finite number of indices]) are often unmanageable if one insists upon sticking to index-free notations. Pure mathematicians often find themselves resorting to `coordinate-patch' calculations (with some embarrassment!)--when some essential caculational ingredient is needed in an argument--and they rarely use the summation convention.

To me, this conflict is a largely artificial one, and it can be effectively circumvented by a shift in attitude. When a physicist employs a quantity `\xi^a, she or he would normally have in mind the actual vector quantity that I have been denoting by \mbox{\boldmath$ \xi$}, rather than its set of components in some arbitrarily chosen coordinate system. The same would apply to a quantity `\alpha_a', which would be thought of as an actual 1-form. In fact, this notion can be made completely rigorous within the framework of what has been referred to as the abstract-index notation.¹⁶ In this scheme, the indices do not stand for one of 1,2, ..., n, referring to some coordinate system; instead they are just abstract markers in terms of which the algebra is formulated. This allows us to retain the practical advantages of the index notation without the conceptual drawback of having to refer, whether explictly or not, to a coordinate system. Moreover, the abstract-index notation turns out to have numerous additional practical advantages, particularly in relation to spinor based formalisms. ¹⁷
...

First, we should know what a tensor actually is. In the index notation, a tensor is denoted by a quantity such as Q^{f\ldots h}_{a\ldots c}, which have q lower and p upper indices for any p, q \geq 0, and need have no special symmetries. We call this a tensor of valence¹⁸ \left[ \begin{array}{cc} p\\ q \end{array} \right] (or a \left[ \begin{array}{cc} p\\ q \end{array} \right]-valent tensor or just a \left[ \begin{array}{cc} p\\ q \end{array} \right]-tensor). Algebraically, this would represent a quantity \mbox{\boldmath $ Q$} which can be thought of as a function (of a particular kind known as multilinear¹⁹) of q vectors \mbox{\boldmath $ A, \ldots, C$} and p covectors \mbox{\boldmath $ F, \dots, H$}, where \mbox{\boldmath $ Q(A,\ldots,C; F, \ldots, H)$}= A^a\ldots C^c Q^{f \ldots h}_{a\ldots c}F_f \ldots H_h}.

[16] Penrose (1968a), p.135-41 ; Penrose and Rindler (1984), pp. 68-103 ; Penrose (1971)
[17] Penrose (1968a); Penrose and Rindler (1984, 1986) ; Penrose (1971) and O'Donnell (2003)
[18] Sometimes the term rank is used for the value of p+q, but this is confusing because of a separate meaning for `rank' in connection with matrices.
[19] This means separately linear in each of \mbox{\boldmath $ A, \ldots, C$} ; \mbox{\boldmath $ F, \dots, H$}

Penrose (1968a) is "Structure of Space-time". In Battelle Rencontres, 1967 (ed. C.M. DeWitt and J.A. Wheeler).
Penrose and Rindler (1984, 1986) are "Spinors and Space-Time, I and II"

"

Here are some web-based references on the "abstract index notation":

http://en.wikipedia.org/wiki/Abstract_index_notation

Wald, "Teaching General Relativity"
http://arxiv.org/abs/gr-qc/0511073
start at p.4, the abstract-index notation is discussed on p.7

(from http://www.lps.uci.edu/home/fac-staff/faculty/malament/FndsofGR.html )
http://www.lps.uci.edu/home/fac-staff/faculty/malament/FndsofGR/GRNotes.Chapter1.pdf
start at page 24

(using the Wayback machine:
http://web.archive.org/web/20030203194024/math.harvard.edu/~allcock/expos.html )
http://web.archive.org/web/20030203194024/http://math.harvard.edu/~allcock/expos/notation.ps
(not on his current webpage http://www.ma.utexas.edu/~allcock/#expos )

 

[mathwonk]

well this may be heresy, but having read it, even if it was written by penrose, it still seems like utter imprecise nonsense to me.

i will reread it to tomorrow after sleeping. thanks anyway. no wonder you guys are led astray if you take this stuff as gospel.

 

[George Jones]

.. it still seems like utter imprecise nonsense to me.

But robphy said

Here's some motivation on the "abstract-index notation" from Roger Penrose

In my experience, even pure mathematicians use "utter imprecise nonsense" for motivational purposes, hoping that rigor can be supplied latert. Don't you?

As Penrose said in the passage (from a book written ostensibly for the educated layperson) robphy quoted

In fact, this notion can be made completely rigorous within the framework of what has been referred to as the abstract-index notation.16

As promised, Penrose does have a go in [16] at supplying the rigor. In particular, see Penrose and Rindler (1984), pp. 68-103, where Penrose talks about totally refexive modules and the like. You might find it to be a total waste of time, but I don't think you'll quibble over the rigor. I could be wrong, though.

The abstract-index notation is notational convenience for telling which types of hydra one is tryin to tame. One should, however, always be aware that the beasts are hydra, and of what a hydra is.

Don't get me wrong mathwonk, I agree with some of your points, but there are other points of view.

By the way, Penrose, when sitting at his desk doing complicated tensorial calculations often uses neither the standard math notation, nor the index notation.

Again, I might make more comments later about the uneasy relationship (with respect to mathematics) between physicists and mathematicians.

Regards,
George

 

[robphy]

Here is an attached scan of the section from Penrose 1968a. (Scanning-quality was degraded to fit within the filesize limits.)

 

[matt grime]

Mathematicians always use summation convention when it is convenient and unambiguous to do so, and they explicitly say they are using it, contrasted to a physicist who can write t_iv_i and mean two totally different things entirely, especially in algebraic terms: who doesn't want to say that t_i, is an eigenvalue and v_i an eigenvector... Summation convention though useful in many ways is equally useless in many other ways (ie triply repeated indices). I would never attempt to prove vector identities without summation convention. Sadly summation convention is not taught enough in my experience.

And don't even get me started on the entire bra and ket rubbish.

 

[mathwonk]

i wish to remind everyone that, whereas the OP has here asked whether he can or cannot contract various upper and lower indices of tensors, no one could possibly question whether he can or cannot evaluate a function on one of its arguments.
Even if the respondent was using a justifiable abstract index notation, he still gave an answer which gave absolutely no insight into this simple point.
This is my objection to the index point of view, it ignores the meaning in favor of a mechanical form of prescribed behavior. it makes automatons out of humans. The pod people are coming!
This simple point is being ignored by appeals to the authority of various deities of physics writing.
In regard to clarity and precision, how much more precise can one be about tensors other than by saying plainly that they are multiheaded hydras which occasionally devour their own heads.?
This gives a very precise and verifiable algorithm for recognizing tensors: i.e. if it is not a hydra, or even if it is a hydra, but if it never devours any of its own heads, then it is not a [mixed] tensor.
I rest my case (temporarily).

 

[mathwonk]

to re - iterate (understatement):
instead of telling peopel what to do with tensors, we should also say why.
e.g. can i contract an upper with a lower index? answer: yes, because each lower index denotes a function, and each upper index denotes an argument of that function, so "contracting" is merely evaluating that function on that argument.
This is along the principle of "teach a man to fish" and he won't come asking for your fish again.
E.g. when my doctor tells me to take 2 lexapro each day, I ask why? When he says: it will help control your persistent paranoia that indices are destroying the world, then I may take them... but not always.