tensor doc and component types

Tensor Doc and Component Types PhL 11.24/15 All this stuff is resolved, shows what a tiny misunderstanding can do. 11.25.15 [ But where is the resolution written up? 1.15.16 ] Resolution is explained below in this doc: the problem is that I was thinking there were two different kinds of component indices, maybe a u kind and an e kind. I thought I was labeling the index with a u or an e, for example, [V]i(e) where the component index i is of "e type". This was completely wrong, and what one really means is [V(e)]i where the (e) label goes with the object, not with the index i. The index i just serves to label the components from 1 to n and has no "type". There are not two kinds of indices. Motivation. In tensor/wedge doc I talk about vectors having more than one kind of component. But in tensor doc I never talk about "types" of vector components. I just write Vi and I don't wonder whether it is of some type like V(u)i. I find this disconcerting, to say the least. Is all of tensor doc "ambiguous" ? For me, this is a very serious question. Other authors don't talk about "component types", by which I mean whether a component is with respect to basis A or basis B. I really never even thought about this until recently writing Chapter 2 in tensor/wedge doc. This topic is a complete showstopper for me, and I cannot continue to develop tensor/wedge until this matter is fully resolved. This could turn out to be a 3 month long question, it has happened before! Background. OK, so why is it that tensor/wedge things we need to have "component types"? I make no mention of such things in tensor/wedge as I parrot off facts about covariant notation in Chapter 2. For example, one such equation is dx'a = Rabdxb where you clearly see components of vectors. I do see in tensor doc (2.1.8) this paragraph directly related to this equation: One might wonder at this point how the vector dx is related to its components dxi and the same question for dx'i and dx'i. As will be shown later in (6.6.9) and (6.6.15), dx = Σndxn un where the un are x-space axis-aligned basis vectors of the form u1 = (1,0,0,..0) dx' = Σndx'n e'n where the e'n are x'-space axis-aligned basis vectors of the form e'n = (1,0,0,..0) (2.1.8) If x-space and x'-space were Cartesian, one could write un = and e'n = ', but in general the un and e'n vectors do not have (covariant) unit length, as shown later in (6.5.3) and (6.4.1). This basically says that the meaning of dxn is dx(u)n because these are the coefficients of the expansion shown. That is dx = Σndxn(u) un Here I don't have to ask about the components of un , they are just full vectors. In the second equation I guess I would say dx' = Σndx'n(e') e'n Whatever the expansion vectors are, that is what you must put into the component type (..). I would also write dx = Σndxn(e) en But I know that later with all my expansions I will write this expansion as dx = Σndx'n en This is the expansion that always surprised me, the coefficients being those of dx in x'-space. Question: How do you put a component label on dx'n ? So it doesn't take long for me to get to a question I can't answer! So getting back now to tensor/wedge, I carry on without mentioning component types. On and on I go, talking about the basis vectors un and en which I am quite comfortable with. Now I get to (2.5.4) and if I ignore the right column, still no component types mentioned. Then in 2.6 I make the change from Picture A to Picture E. I am merely renaming things, I think. OK, now we have the first sighting of type superscripts! I write g→ g(u) dx'a = Rabdxb → dx(e)a = Rabdx(u)b g' →g(e) V'a = RabVb → V(e)a = RabV(u)b OK, what is this saying. I am doing a blind replacement no prime → (u) superscript prime → (e) superscript So prior to drawing Picture E, I have no mention of such superscripts. I guess I had forgotten about this simple idea. Example: My question above asks should you say dx'(u)n or dx'(e)n . But this question makes no sense because you are just supposed to replace the prime with (e), but here I am showing in effect two superscripts. A total screw up question! The superscripts mean prime or no prime, nothing more and nothing less. Question: does this rule apply only to components, or also to bolded vectors? My "instructions" below (2.6.1) say to do this X → X(u) where X is any x-space object X' → X(e) where X' is any x'-space object . So does that mean V → V(u) = a vector in u-space V' → V(e) = a vector in e-space yes! I did not do that in what follows. Consider V = Vn un Am I supposed to write that as V(u) = V(u)n un(u) ?? yes! And how about V = V'n en That should really say V(u) = V(e)n en(u) yes! Why did I not do this there in Sec 2.6 of tensor/wedge? Suddenly my whole idea of Section 2.6 seems wobbly. I am labeling only components, not vectors. The equations (2.6) should really read V(u) = Σn V(u)n un(u) where un(u) V(u) = V(u)n V(u) = Σn V(u)n un(u) where un(u) V(u) = V(u)n V(u) = Σn V(e)n en(u) where en(u) V(u) = V(e)n V(u) = Σn V(e)n en(u) where en(u) V(u) = V(e)n from (2.5.1) yes! (2.6.2) Suppose I rigorously do this as shown above. Then what happens to components? Let's try out the above equations on some special vectors Try first equation on um . Then um(u) = Σn um(u)n un(u) where un(u) um(u) = um(u)n um(u) = Σn um(u)n un(u) where un(u) um(u) = um(u)n um(u) = Σn um(e)n en(u) where en(u) um(u) = um(e)n um(u) = Σn um(e)n en(u) where en(u) um(u) = um(e)n from (2.5.1) yes! (2.6.2) I just replaced the letter V by um everywhere, don't worry about the bolding of the u. Now what comes next? Now in early tensor doc I say that umi = δmi so in my new notation this equation becomes u(u)mi = δmi yes! OK, this is way too messy. So let's try out a new rule set: X → X where X is any x-space object X' → X(e) where X' is any x'-space object . good idea So using this rule, only something with a prime gets a superscript! I could then redo Picture E this way, Since I am mostly interested in non-primed objects, this is a much more efficient change to make. But I guess I could always add a (u) for emphasis. So now the default no-label means u-space and is as if there were a (u) label. Then our equations are Examples: Rab → Rab g→ g dx'a = Rabdxb → dx(e)a = Rabdxb g' →g(e) V'a = RabVb → V(e)a = RabVb un → un en → en same for n index being up. V → V V = Vn un → V = Vn un V = V'n en → V = V(e)n en . The four expansions of (2.5.1) are now (showing the n sum explicitly) V = Σn Vn un where un V = Vn V = Σn Vn un where un V = Vn V = Σn V(e)n en where en V = V(e)n V = Σn V(e)n en where en V = V(e)n from (2.5.1) (2.6.2) So cosmetically this is a lot nicer convention to use. Now apply these equations to the four basis vectors. Notice also the clear idea: If you expand on en you get an (e) in your coefficient! Things are much simpler to keep track of I think in this notation. Those are shown in (2.5.4), all 16 of them! But this is in the old notation . I would rewrite this as um = (un um) un = δnm un = um // not very interesting (um)n = δnm um = (un um) un = gnm un // a fact already known in (2.4.3) (um)n = gnm um = (en um) en = Rnm en (um(e))n = Rnm um = (en um) en = Rnm en (um(e))n = Rnm um = (un um) un = gnm un // a fact already known in (2.4.3) (um)n = gnm um = (un um) un = δnm un = um // not very interesting (um)n = δnm um = (en um) en = Rnm en (um(e))n = Rnm um = (en um) en = Rnm en (um(e))n = Rnm em = (un em) un = Rmn un (em)n = Rmn em = (un em) un = Rmn un (em)n = Rmn em = (en em) en = δnm en = em // not very interesting (em(e))n = δnm em = (en em) en = g(e)nm en // a fact already known in (2.3.3) (em(e))n = g(e)nm em = (un em) un = Rmn un (em)n = Rmn em = (un em) un = Rmn un (em)n = Rmn em = (en em) en = g(e)nm en // a fact already known in (2.3.3) (em(e))n = g(e)nm em = (en em) en = δnm en = em // not very interesting (em(e))n = δnm There were very few primes here, they are shown in red. For example, look at um = (en um) en = Rnm en (um(e))n = Rnm The coefficient here should be written (um(e))n because we are expanding um and we know that the coefficient is the V'a type object when you expand on en. Good. Now (2.5.5) is unchanged because there is not a single prime! So here is the new version of (2.6.4) (um)n = δmn (um)n = gmn (em)n = Rmn (em)n = Rmn (um)n = gmn (um)n = δmn (em)n = Rmn (em)n = Rmn (um(e))n = Rnm (um(e))n = Rnm (em(e))n = δmn (em(e))n = g(e)mn (um(e))n = Rnm (um(e))n = Rnm (em(e))n = g(e)mn (em(e))n = δmn . The only (e) appear in the bottom two rows. I can take (e)→' and check all these against tensor doc. I will do that right now just to be stubborn. Checked ones in red. All are OK, I avoided tilted R things in my tensor doc displays. Now look at my paradox resolution again. Paradox? Let v = Σnv(e)nen be the expansion of a vector v onto a basis en. Since em is a vector, use this same expansion to expand v = em to get em = Σn(em(e)n) en. This implies that (em(e))n = δmn. This agrees with (2.6.4) column 3 row 3. There is no paradox. Without clear labeling, one assumes that the coefficient should just be (em)n = Rmn which gives the garbage equation em = Σn(em)n en = Σn Rmn en which is not self-consistent since Rmn ≠ δmn. OK, I am still alive here. Comment: You should write things as (em(e))n to stress that it is e'm which is in there. I was writing things in the manner (em)(e)n as if (e)n specified a component of the (e) type. Question. This is all fine. But what exactly do I then mean by a "component index"? Go back to V = Σn Vn un where un V = Vn V = Σn Vn un where un V = Vn V = Σn (V(e))n en where en V = (V(e))n V = Σn (V(e))n en where en V = (V(e))n from (2.5.1) (2.6.2) You see the component index here, it is always n. Pause. What about those dot products? I developed them earlier. Recall en um = (en)i(um)i = Rni δmi = Rnm . (2.4.4) I don't write out the metric tensor, I just use the tilts. Then I look things up. I think my development is all. So again, what is the meaning of a component index? I can just define them using the four dot products on the right there in (2.6.2). Everything is unambiguous. There is only one "kind" of component index, it has no (u) or (e) label. Conclusion #1: There is only one kind of component index, there are not two kinds as I was somehow thinking. yes! Conclusion #2: Any equation from tensor doc you translate by replacing a prime with an (e). yes! Conclusion #3: Most tensor doc equations I use have no primes on things, so in tensor/wedge they would look exactly the same. yes! Conclusion #4: There is no "ambiguity" in tensor doc "components". yes! So I can rest easy again. Go back to our original equations of interest directly from tensor doc dx = Σndxn un where the un are x-space axis-aligned basis vectors of the form u1 = (1,0,0,..0) dx' = Σndx'n e'n where the e'n are x'-space axis-aligned basis vectors of the form e'n = (1,0,0,..0) I would now write these as dx = Σndxn un dx(e) = Σn(dx(e))n (en(e)) I guess I now have to back and review my various confusing questions.