entire original Chapter 7

Entire Original Chapter 7 PhL 4.6.15 7. Translation to the Standard Notation 1 7.1 Outer Products 2 7.2 Mixed Tensors and Notation Issues 2 7.3 The up/down bell goes off 3 7.4 Some Preliminary Translations: raising and lowering indices on a vector with g 4 7.5 Contraction of a Pair of Indices 5 7.6 Dealing with the matrix R 6 7.7 Repeat the above section for S 7 7.8 About ε and δ 7 7.9 More translations from developmental to standard notation 8 (a) Determinants of R and S 8 (b) Inverse of R and S. 8 (c) Tensor g raises and lowers any index. 9 (d) Raising Lowering Rule: 10 (e) Contraction Tilt Reversal Rule: 11 (f) The Diagonal g Rule: 11 (g) Covariance and Matrix Multiplication 11 (h) Matrix Inverse, Transpose and Determinant 14 (i) Orthogonality rules. 16 (j) Rotation matrices. 16 (k) Variations on the relation between g and g'. 18 7.10 Tensors of Rank n, direct products, Lie groups, symmetry and Ricci-Levi-Civita 19 7.11 The Contraction Tilt-Reversal Rule 22 7.12 The Contraction Neutralization Rule 24 7.13 Raising and lowering indices on g 25 7.14 Other forms of R 26 7.15 Summary of facts about R 26 7.16 Repeat all the above for S 27 7.17 Theorem: Sab = Rba and Sab = Rba ( reflect indices in vertical line between them) 27 7.18 Orthogonality Rules, the Inversion Rule, and the Cancellation Rule 29 7.19 The tangent and reciprocal base vectors and expansions on same 30 7.20 Comment on Covariant versus Contravariant 33 7.21 The Significance of Tensor Analysis 34 7.22 The Christoffel Business: covariant derivatives 38 7.23 Expansions of higher order tensors 39 7. Translation to the Standard Notation In this Chapter we discuss the "translation" from our developmental notation (all lower indices; overbars for covariant objects) to the Standard Notation used in tensor analysis. The developmental notation has served well in the discussion of scalars and vectors, tensors of rank-0 and rank-1. For pure (unmixed) tensors of rank-2 it does especially well, allowing the use of matrix algebra to leverage the use of familiar matrix theorems such as det(ABC) = det(A)det(B)det(C) and A-1 = cof(AT)/det(A). The transformation of the contravariant metric tensor is cleanly expressed as g' = R g RT, and so on. The notation in fact works fine for unmixed tensors of any rank, but runs into big trouble with "mixed" tensors as shown in the next sections. 7.1 Outer Products It is possible to form larger tensors from smaller ones using the "outer product" method. For example, consider, Tab ≡ UaVb (7.1.1) where U and V are assumed to be contravariant vectors. One then has T 'ab = U'aV'b = (Raa'Ua') (Rbb'Vb') = Raa' Rbb' Ua'Vb' = Raa' Rbb' Ta'b' (7.1.2) so in this way a contravariant rank-2 tensor like M in (5.6.3) has been successfully constructed from two contravariant vectors. Similarly, ab ≡ ab => 'ab = STaa' STbb' a'b' (7.1.3) so the outer product of two covariant vectors transforms as a covariant rank-2 tensor, again as in (5.6.3). 7.2 Mixed Tensors and Notation Issues Suppose we take the "outer product" of a contravariant vector with a covariant vector, [ ... ]ab ≡ Uab (7.2.1) where we are not sure what to call this thing, so we just call it [...]. Here is how this new object transforms (always: with respect to the underlying transformation x' = F(x) ) [ ... ]'ab = U'a'b = (Raa'Ua') (STbb'b') = Raa' STbb' Ua'b' = Raa' STbb' [...]ab . (7.2.2) This object transforms as a contravariant vector on the first index (ignoring the second), and as a covariant vector on the second index (ignoring the first). This is an example of a "mixed" rank-2 tensor. Extending this outer product idea, one can make elaborate tensor objects with an arbitrary mixture of "contravariant indices" and "covariant indices". For example [.....]abcd = Uab Xcd . (7.2.3) To write down the transformation rule for such an object, one must know which indices are contravariant and which are covariant. It is totally clear how the object transforms, looking at the right side of (7.2.3), but somehow this information has to be embedded in the notation [.....]abcd because once this object is defined, the right hand side might not be immediately available for inspection. Worse, there may be no right hand side for a mixed tensor, because not all mixed tensors are outer products of vectors (they just transform as if they were). Just as we can use the idea ≡ V to convert a contravariant vector to its covariant partner, we can similarly use to convert the 1st or 3rd index on [.....]abcd from contravariant to covariant. We could apply two 's with the proper linkage of indices to convert them both at once. So given the ability of to change any index one way, and g to change it the other way, one can think of the 4-index object [.....]abcd as a family of 16 different 4-index objects, each corresponding to a certain choice for the indices being one type or the other. We know how to interconvert between these 16 objects just applying g or factors. So how does one annotate which of the 16 objects [.....] one is staring at for some choice of index types? Here is a somewhat facetious possibility, the Morse Code method ab ≡ Uab abcd = Uab Xcd . (7.2.4) Instead of having a bar over the entire object, in the first case the bar it is placed just over the right side of the W to indicate that b is a covariant index, while no bar means the first index is contravariant. The second example shows how horrible such a notation would be. We really want to put some kind of notation on the individual indices, not on the object! Here is a notation that is slightly better than the Morse code option, though similar to it, Wac = UaV XcY (7.2.5) Here overbars on covariant indices distinguish them. Now one can dispense with the overbars on covariant vectors as well, putting the overbar on the index, for example a = abVb → V = g Vb . There are several problems with this scheme. One is that in the spinor application of tensor analysis used in special relativity, dots are placed on certain indices, such as in (5.14.8), and these would conflict with the proposed overbars. A more substantial reason is that this last notation is hard to type (or typeset, as one used to say), it looks cluttered with all the overbars, and the subscripts are already hard to read without extra decorations since they are in a smaller font than the main text. 7.3 The up/down bell goes off This is where a bell went off somewhere, perhaps in the mind of Gregorio Ricci in the 1880-1900 time frame (1900 snippet quoted in section (j) below). Someone might have said: suppose, instead of using overbars on indices or some other decoration, we distinguish covariant indices by making them be superscripts instead of subscripts. Superscripts are as easy to type as subscripts, and the result is fairly easy to read and totally unambiguous. We would then have for our ongoing example of (7.2.4 and 5), Wabcd = UaVbXcYd // a path not taken This is almost what happened, but the up/down decision went the other way and we now have: superscripts = contravariant = up subscript = covariant = down (7.3.1) and then we get this translation Wac = UaV XcY → Wabcd = UaVbXcYd // the path taken (7.3.2) and this has become The Standard Notation. Perhaps the reason for this choice was that the covariant gradient ∂n object appeared more commonly in equations than idealized objects such as dx, and ∂n already used a lower index. A downside of this particular up/down decision is that every student has be be confused by the fact that his or her familiar position, velocity and momentum vectors that always had subscripts suddenly have superscripts in the Standard Notation. The silver lining is that this shocking change alerts the student to the fact that whatever subject is being studied is going to have two kinds of vectors. Despite appearances, it is not completely obvious how one should translate the whole world as presented in the previous six Chapters into this new notation. There are quite a few subtle details that will be discussed in the following sections. 7.4 Some Preliminary Translations: raising and lowering indices on a vector with g In the rest of this entire Chapter, anything to the left of a → arrow is in "developmental notation", while anything to the right of → is in "Standard Notation". So we start translating some of the results above: s → s // a scalar Va → Va // a contravariant rank-1 tensor (vector) a → Va // a covariant rank-1 tensor (vector) Mab → Mab // a contravariant rank-2 tensor ab → Mab // a covariant rank-2 tensor gab → gab // the contravariant rank-2 metric tensor ab → gab // the covariant rank-2 metric tensor g is inverse of → gab is inverse of gab . (7.4.1) As noted earlier, one "feature" of the Standard Notation is that it is no longer sufficient to specify an object by a single letter. One has to somehow indicate the index nature by showing index positions. Thus, "g" stands for all four metric tensors gab , gab, gab and gab. The pure covariant metric tensor is gab or perhaps g** . At first this seems a disadvantage of the notation, but one then realizes that the true object really is "g", and it has four different "representations" and the notation makes this very clear. Still, one cannot just write det(g) because det(g) is representation dependent, so one must say something like det(gab) or det(g**) to denote a particular determinant. As for converting a vector from one type to the other, a = abVb → Va = gabVb // gab "lowers" a contravariant index Va = gab b → Va = gab Vb // gab "raises" a covariant index , (7.4.2) and so in this new notation, the covariant metric tensor gab becomes an "index lowering operator" and the contravariant metric tensor gab becomes an "index raising operator". This is a huge advantage of the Standard Notation. It pretty much eliminates the need to think, something universally appreciated. In a certain obscure sense, it is like double entry accounting (credits and debits), where the notation itself serves as a check on the accuracy of bookkeeping entries, as will be seen below. As for bolded vectors, the translation rule is, V → V → V (7.4.3) The reason is that the overbar is no longer used to denote covariancy. The above lines show a subtle change in the interpretation of the bolded symbol V in the standard notation: the single symbol V stands for both the developmental vector V and for its developmental covariant partner vector . The new symbol V is both contravariant with components Vn and it is covariant with components Vn. The invariant distance and covariant dot products: dxi → dxi (ds)2 = ab dxa dxb → gabdxadxb AB = ab Aa Bb → AB = gab Aa Bb = AbBb = AaBa = gabAaBb . (7.4.4) The general idea is this: any tensor index on any tensor object can be raised by gab and can be lowered by gab. Remember that a tensor object lives in some space like x-space, so we shall have to ponder what to do for our matrices Sab and Rab which live half in x-space and half in x'-space, a subject we defer for a short while. 7.5 Contraction of a Pair of Indices When two indices are summed together in a tensor expression and one is up and the other down, one says that the two indices are contracted. Here is an example, where the index b is contracted (as usual, summation is implied for repeated indices), Va = gabVb . (7.5.1) It will be shown below that contracted indices neutralize each other in terms of how an object transforms. Thus, for example, the RHS above gabVb transforms as a covariant vector, since only a lower index a survives the contraction, which conveniently matches the LHS. Similarly, AaBa transforms as a scalar. 7.6 Dealing with the matrix R Consider the translation of this partial derivative into the new up/down notation. Since the differential dx element is contravariant and is now written dxi , (∂x'i/∂xk) → (∂x'i/∂xk) . (7.6.1) In terms of "existence", this object has one leg in each space of Picture A. The gradient operator ∂/∂xk is an x-space thing, while x'i is an x'-space thing. Since this object does not live in x-space or in x'-space exclusively, but straddles the two spaces, it cannot possibly be a tensor of any kind. Recall that a tensor object must be entirely within a space, it cannot have body parts hanging out into other spaces. Nevertheless, it seems clear that each of the two indices has a well-defined nature. We showed that the gradient is a covariant vector, so we regard k as a covariant index. And of course dx'i is a contravariant vector, so i is a contravariant index. Here then is the proper translation starting with (2.5.1), Rik ≡ (∂x'i/∂xk) → Rik ≡ (∂x'i/∂xk) . (7.6.2) To summarize, Rik is not a mixed rank-2 tensor, though it looks just like one. Therefore , Rik can never appear in a tensor equation -- it just appears in the equations that show how tensors transform. However, each of the two indices of R has a well-defined transformational nature, and we place them up and down in the proper manner. It is very typical for an object to have up and down indices but the object is not a tensor. As was noted below (2.3.2), the canonical example is that for a non-linear transformation x' = F(x), xi has a contravariant index but is not a contravariant vector. Consider now the translation of the transformation rule for a contravariant vector from (2.5.1), V'a = RabVb → V'a = RabVb . // contravariant (7.6.3) Even though R is not a tensor, we see that index b is contracted and is thus neutralized from the evaluation of the tensor nature of the RHS. This leaves upper index a as the only free index, indicating that the RHS is a contravariant vector, and this of course then matches the LHS. So we can deal with the indices on R just as we deal with indices on true tensors. Notice that, even though both sides of V'a = RabVb have the same "tensor nature" (both sides are a contravariant vector) one cannot ask how the equation V'a = RabVb "transforms" under a transformation. That question can only be asked about equations constructed of objects all of which are tensors in the same space. Here V and half of R are in one space, and V' and the other half of R are in a different space. There is no object called R', as if R were in x-space and R' were in x'-space. 7.7 Repeat the above section for S We omit the words and just show the translations, quoting expressions from (2.1.6) and (2.5.1) : (∂xi/∂x'k) → (∂xi/∂x'k) Sik ≡ (∂xi/∂x'k) → Sik ≡ (∂xi/∂x'k) 'a = STabb = Sba b → V'a = SbaVb // covariant (7.7.1) 7.8 About ε and δ The Kronecker δ is sometimes written in different ways to make things "look nice", δab = δab = δba = δba = δa,b . (7.8.1) Section 7.x will show that one can regard the above sequence of equalities as saying gab = gab = gba = gba = δa,b (7.8.2) where these g objects are mixed versions of the symmetric rank-2 metric tensor gab. There is no "δ tensor", it is the g tensor, but tradition is to write the diagonal objects using the δ symbol. The object εabc... is a bit more complicated. It can at first be regarded as a mere bookkeeping device, in which context it is usually called "the permutation tensor". It appears for example in the expansion of a determinant det(M) = εabc...xM1aM2b.....MNx = εabc...xMa1Mb2.....MxN (7.8.3) or in an ordinary cross product Aa = εabcBbCc . (7.8.4) This permutation tensor has the usual properties that ε123...N = +1 , that ε changes sign when any two indices are swapped, and that ε vanishes if two or more indices are the same. This permutation "tensor" is not really a tensor since one would regard it as being the same in x-space or x'-space. Whether indices are written up or down on this ε is immaterial. At another level, however, εabc...x with N indices (the same ε symbol is used) is a covariant rank-N tensor density of weight -1 known as the Levi-Civita tensor. This subject is addressed in Appendix D in much detail. In what we call the Weinberg convention, individual indices of ε can be raised and lowered by g as discussed in Section 7.4 just as with any tensor. Therefore, in Cartesian space with g = 1, indices on ε are raised and lowered with no consequence [ see (5.9.1) or (7.4.2) ], and then one can identify any form of ε as being the permutation tensor. For example, εabc = εabc = εabc and so on. In a non-Cartesian x-space, however, one would say that εabc = gbb'εab'c ≠ εabc. In the Weinberg convention, one sets ε123..N = ε'123..N = 1 and εabc..x = ε'abc..x has the properties of the permutation tensor described above and these properties are the same in x-space as in x'-space. Then for general g≠1, εabc..x (lower indices) is NOT the permutation tensor. The bottom line is that one must be aware of the space in which one is working (the Picture). The ε appearing above in the determinant expansion (7.8.3) is always just the permutation tensor, but in the cross product that is not the case, and one would properly write Aa = εabcBbCc (7.8.5) and conclude that the cross product of two ordinary contravariant vectors is a covariant vector density (Appendix D (g)). Again, in Cartesian space where one often works, this would be the same as Aa = εabcBbCc = εabcBbCc , but the "properly tilted form" Aa = εabcBbCc reveals the tensor nature of the object Aa. As mentioned below in section (u), this "covariant" equation would appear as A'a = ε'abcB'bC'c in x'-space, but since A'a is a covariant vector density, A'a ≠ RabAb, and in fact A'a = J RabAb. The permutation tensor εabc... and the contravariant Levi-Civita tensor εabc...x are both "totally antisymmetric" which just means ε changes sign if any pair of indices is swapped. In fact, as discussed in Appendix D (c), there IS only one antisymmetric tensor of rank N apart from a multiplicative scalar factor, and εabc...x is it. This fact simplifies various calculations. Technically, εabc...x is a totally antisymmetric tensor density, but normally it is just called "the totally antisymmetric tensor". As shown in Appendix D, the covariant Levi-Civita tensor εabc...x is also totally antisymmetric and is therefore a multiple of εabc...x. The reader is invited to peruse Appendix D at some appropriate time for more about tensor densities and the ε tensor. 7.9 More translations from developmental to standard notation After reading this section, one will understand why matrix multiplication notation is often avoided in the standard notation. Since both indices of rank-2 tensors and tensor-like objects can be up or down, the meaning of matrix multiplication requires some clarification. Part of the complication is defining a reasonable transpose for the various up and down matrix forms. The smaller subsections below are presented here in a "short form" to allow for a reasonable logic flow since certain "facts" have not yet been introduced. The "long form" presentations of these facts occur in later sections of Chapter 7. (a) Determinants of R and S Chapter (f) showed that Rab → Rab, so one translates from old to new notation, det(R) = εabc...R1aR2b....RNx → det(Rij) = εabc...R1aR2b....RNx det(S) = εabc...S1aS2b.....SNx → det(Sij) = εabc...S1aS2b....SNx det(R) = εabc...Ra1Rb2....RxN → det(Rij) = εabc...Ra1Rb2....RxN det(S) = εabc...Sa1Sb2.....SxN → det(Sij) = εabc...Sa1Sb2....SxN (7.9.a.1) where ε is the bookkeeping permutation tensor discussed in section (h). (b) Inverse of R and S. Again, section (f) showed that Rab → Rab. In the standard notation, imagine that there is some inverse R-1 defined by (R-1)caRab = δcb. The chain rule says that (∂xc/∂x'a) (∂x'a/∂xb) = δcb or Sca Rab = δcb , (7.9.b.1) and therefore it must be that (R-1)ca = Sca. A similar argument shows that (S-1)ca = Rca. Using the results of the next section which allow us to raise and lower indices on both sides of an equation, this relationships R-1 = S is valid for all four matrix position possibilities, (R-1)ik = Sik (R-1)ik = Sik (R-1)ik = Sik (R-1)ik = Sik (7.9.b.2) and of course the same is true for S-1 = R. Thus arise these translations from old to new notation: R-1 = S → (R-1)ik = Sik and all other index combinations S-1 = R → (S-1)ik = Rik and all other index combinations RR-1 = RS = 1 etc → Rik(R-1)ka = RikSka = δia etc (7.9.b.3) We have silently used a certain "down tilt" matrix multiplication here which will be explained later. (c) Tensor g raises and lowers any index. So far the following translation rules have been established: gab → gab ab → gab Rik → Rik Sik → Sik . (7.9.c.1) It was shown in developmental notation (5.6.3) how rank-2 contravariant and covariant tensors transform. Here then is how those statements translate to the new notation : M'ab = Raa'Rbb'Ma'b' → M'ab = Raa'Rbb'Ma'b' contravariant rank-2 tensor (7.9.c.2) 'ab = Sa'aSb'ba'b' → M'ab = Sa'aSb'bMa'b' . covariant rank-2 tensor Since g itself is such a rank-2 tensor, replace M by g to get g'ab = Raa'Rbb'ga'b' → g'ab = Raa'Rbb'ga'b' (7.9.c.3) 'ab = Sa'aSb'b a'b' → g'ab = Sa'aSb'b ga'b' . It was shown in section (d) that Va = gaa'Va' and Va = gaa' Va' so that gaa' lowers a vector index and gaa' raises a vector index. That is to say, gaa' converts a contravariant vector index into a covariant one, and gaa' does the reverse. What does gaa' do to the index of a rank-2 tensor? Consider the following definition: Mab ≡ gaa'Ma'b . (7.9.c.4) Since gab and gab are inverses, it follows that Mab = gaa' Ma'b . (7.9.c.5) How does this new object Mab transform? The claim is that it transforms as a mixed rank-2 tensor, which would mean that M'ab = Raa' Sb'b Ma'b' . (7.9.c.6) The upper index gets a factor Raa' and the lower index gets a factor Sb'b , consistent with (*) above. It is not hard to prove this claim: (7.32) (7.31) (7.30) (7.33) M'ab = g'aa'M'a'b = ( RacRa'd gcd ) ( Sea'SfbMef) = ( RacRa'd gcd ) ( Sea'Sfb gei Mif ) = [RacRa'd gcd Sea'Sfb gei] Mif = [Rac (Sea'Ra'd) gcd Sfb gei] Mif = [Rac (SR)ed gcd Sfb gei] Mif = [Rac δed gcd Sfb gei] Mif = [Rac gcd Sfb gdi] Mif (7.28) = [Rac (gcd gdi) Sfb] Mif = [Rac δci Sfb] Mif = [Rai Sfb] Mif inverses = Raa' Sb'b Ma'b' QED (7.9.c.7) Similarly one could define Mab ≡ gaa'Ma'b and one would find that M'ab = Sa'a Rbb' Ma'b' (7.9.c.8) so Mab is then another member of the family of rank-2 tensors. Finally were one to define Mab ≡ gbb'Mab' one would find that Mab transforms as in (7.30). To summarize the four transformation results M'ab = Raa' Rbb' Ma'b' M'ab = Raa' Sb'b Ma'b' M'ab = Sa'a Rbb' Ma'b' M'ab = Sa'a Sb'b Ma'b' . (7.9.c.9) One sees then a family of four tensors associated with M. One is contravariant, one is covariant, and the other two are mixed. [Later we will show that Sij = Rji and this allows one to write the above equations in a manner that is easier to remember. ] (d) Raising Lowering Rule: gaa' [----a'---] = [----a---] and gaa' [----a'---] = [----a---] (7.9.d.1) Here [----i---] represent a tensor with a certain contravariant index i and dashes indicate other indices which each could be up or down. Similarly [----i---] is another tensor in the same family where the index i that was up is now down. The notion of higher rank tensors is coming soon, but we just want to establish the general idea that ANY index on ANY tensor can be raised or lowered by an appropriate g tensor. For the rank-2 tensors this was demonstrated explicitly above, and section (d) showed it was valid for rank-1 tensors (vectors), gaa'Va' = Va gaa' Va' = Va . (7.9.d.2) Comment: Notice that in every equation shown above, the summed indices always occur in the contracted form discussed in section (e) above, which is to say, one index is up and the other is down. (e) Contraction Tilt Reversal Rule: [-----a---------a----] = [-----a---------a----] (7.9.e.1) This is proved in section (k) below, but since we are going to need it right now, here is a preview of that proof: ( note that gab gac = gba gac = δbc ) [-----a---------a----] = gab gac [-----b---------c----] = δbc [-----b---------c----] = [-----b---------b----] The upshot is that one can always "reverse the tilt" on any pair of contracted indices. (f) The Diagonal g Rule: gab = δab and gab = δab // and same for g' (7.9.f.1) This is proved in section (m) below, and here a preview: gab = gaa' ga'b // gaa'raises the first index of tensor ga'b = δab // because gij and gij are inverses of each other (g) Covariance and Matrix Multiplication Before continuing the process of translation from developmental to standard notation, we digress momentarily to consider the notion of covariance in developmental notation. As we shall discuss in more detail below in Section **, an equation is said to be covariant under the transformation x' = F(x) if it has "the same form" in both x-space and x'-space. The "same form" means that the equation looks the same but everything is primed in x'-space. Example 1: Newton's Law F = ma is covariant under rotations (x' = F(x) = Rx), and in x'-space this law takes the form F' = m'a' which has the same form as the equation in x-space F = ma. Once we know that F and a are contravariant vectors and m is a scalar, this conclusion is automatic from (2.3.2), F = ma F' = m'a' proof: F' = RF = R(ma) = m Ra = m a' = m' a' (7.9.g.1) where m = m' follows since mass is a scalar under rotation. Thus, Newton's Law has the same form when it is examined in two frames of reference related by a rotation. It is covariant. Example 2: Consider the equation A B = π where A and B are contravariant vectors and is the covariant dot product defined in (5.10.1), A B ≡ abAaBb . It was shown in (5.10.2) that the quantity A B transforms as a scalar under general transformation x' = F(x) so that A' B' = A B. Since the number π is also a scalar under any transformation (it is a constant), one could say that π' = π (it is the same number 3.14 in x'-space and x-space), so A B = π A' B' = π' , equation is covariant. (7.9.g.2) What we see here is that an equation is covariant IFF both sides of the equation transform as the same tensorial tensor type under the transformation of interest. In Example 1, both sides of F = ma transform as contravariant vectors under rotations, and in Example 2 both sides of A B = π transform as scalars under a general transformation. Example 3: Consider the outer product equation (7.1.1) Tab = UaVb where U and V are contravariant vectors. We show in (7.1.2) that Tab transforms as a contravariant rank-2 tensor. Both sides of this equation transform in this way, so in x'-space the equation becomes T'ab = U'aV'b. The equation is therefore covariant under the transformation x' = F(x) with dx' = Rdx. Approaching this example in a slightly different manner, suppose we define Tab ≡ UaVb where U and V are contravariant vectors. We then ask: Is Tab a contravariant rank-2 tensor? Line (7.1.2) shows that the answer is yes, T 'ab = U'aV'b = (Raa'Ua') (Rbb'Vb') = Raa' Rbb' Ua'Vb' = Raa' Rbb' Ta'b' (7.9.g.3) which matches the transformation rule as stated in (5.6.3). Example 4: Suppose A and B are tensorial contravariant rank-2 tensors. Is the equation AB = C covariant? If it were, we would have to show that in x'-space we have A'B' = C' where C is a contravariant rank-2 tensor. To investigate, we use the rule (5.7.1) which states how a contravariant rank-2 tensor transforms in terms of Picture A shown in (5.7.2) : A'B' = (RART)(RBRT) = RA(RTR)BRT // since A and B are contra rank-2 tensors (7.9.g.4) C' = RCRT = RABRT // assuming C is also a contra rank-2 tensor and AB = C If it were true that RTR = 1, one would find from the first line above that A'B' = RABRT = RCRT = C' and the answer would be yes, the equation AB = C is covariant. However, for a general Picture A transformation with metric tensor g in x-space and g' in x'-space, what we know about R comes from (5.7.6) : g' = R g RT . Even if g = 1 so x-space is Cartesian, this says g' = RRT, but this tells us nothing about RTR. So for a general transformation, we have RTR ≠ 1 and so the equation AB = C is NOT covariant. [In the special case that R is a rotation, so RT = R-1 (real orthogonal), then RTR= R-1R = 1.] As with Example 3, we can reformulate the current example in a different manner. Suppose we define C ≡ AB and specify that both A and B are contravariant rank-2 tensors. In this case, is C a contravariant rank-2 tensor? If it were, we would have to have (A'B') = R(AB)RT from (5.7.1). But we showed above that, since RTR ≠ 1, we end up with (A'B') ≠ R(AB)RT . Therefore, C ≡ AB is not a contravariant rank-2 tensor. Could C be a covariant rank-2 tensor? If it were, we would need to have (A'B') = ST(AB)S from (5.7.1). But above we show that A'B' = RA(RTR)BRT and this is completely different from (A'B') = ST(AB)S. Thus, C is not a covariant rank-2 tensor. Since C has two indices, the only way it could be a tensorial tensor is if it is either a contravariant or a covariant rank-2 tensor, but we have just ruled out both these possibilities. Therefore C ≡ AB is not a tensorial tensor of any kind whatsoever, even though A and B are tensorial tensors. Matrix Rule #1. In developmental notation, if A and B are contravariant rank-2 tensors, the matrix product AB is (in general) not a rank-2 tensor and is in fact not any kind of tensor. The equation C = AB is not covariant. Mimicking the above discussion, the reader can show that the same conclusion applies to C = B, C = A and C = : in none of these cases is C a tensor of any kind, and all these equations are non-covariant. Similarly, the Rule applies to X = ABC or X = ABCD and so on. (7.9.g.5) For this reason, we shall never ask how to transform an equation like X = ABC... from developmental to standard notation. Equations which are non-covariant are simply of no interest, and can never describe a physical relationship as we explain below in Section ***. The attentive reader might ask: What about the equation g' = R g RT which has the form X = ABC. And if g = 1, what about g' = RRT whose form is X = AB? In both these cases, the left hand side is a contravariant rank-2 tensor. These equations do not violate the Matrix Rule #1 above because the matrices R and RT are not tensors of any kind, as noted above in ***. Furthermore, one does not ask whether g' = R g RT is covariant or not because it is an equation relating objects in different spaces and not all objects in the equation are tensors. We now consider the notion of matrix multiplication using mixed rank-2 tensors. Since we never introduced such mixed tensors in our developmental notation, we have this discussion entirely in the Standard Notation. Consider Cij = AikBkj . // implied sum on k (7.9.g.6) The indices k have the right adjacency so one could think of this as being a matrix equation C = AB where all three objects are "down-tilt rank-2 mixed tensors". Down-tilt just means the two indices are tilting down like ij. We can ask again our questions of Example 4. If A and B are rank-2 tensors, is C = AB covariant? And if we define C ≡ AB, is C a rank-2 tensor? The answer to both questions is yes. To show that AB = C is covariant, we start with (7.9.c.9) applied to A and B: M'ab = Raa' Sb'b Ma'b' (7.9.c.9) so A'ikB'kj = (Ria' Sb'k Aa'b') (Rka" Sb"j Ba"b") = Ria' Sb'k Rka" Sb"j Aa'b'Ba"b" = Ria' (Sb'k Rka") Sb"j Aa'b'Ba"b" = Ria' (SR)b'a" Sb"j Aa'b'Ba"b" = Ria' δ b'a" Sb"j Aa'b'Ba"b" = Ria' Sb"j Aa'b'Bb'b" = Ria' Sb"j (AB)a'b" = Ria' Sb"j Ca'b" = C'ij . // using (7.9.c.9) a third time with M = C (7.9.g.7) Thus we have shown that AB = C A'B' = C' so our down-tilt matrix equation is covariant. If we define C ≡ AB where A and B are down-tilt mixed rank-2 tensors, then C will be a rank-2 down-tilt tensor providing we can show that (A'B')'ab = Raa' Sb'b (AB)a'b = Raa' Sb'b Ca'b . But this is just what was shown above (albeit with different indices), so yes, C is also a down-tilt mixed rank-2 tensor. One way to clarify the intention of C = AB is to write the matrix equation as Cdt = AdtBdt where the notation Adt means the down-tilt mixed rank-2 tensor having components Aij. It is easy to show that the conclusions reached above apply similarly to an all up-tilt matrix equation Cij = Aik Bkj . // implied sum on k (7.9.g.8) We thus arrive at: Matrix Rule #2. In Standard Notation, it is reasonable to use matrix notation in the following two situations involving mixed rank-2 tensors: Cij = AikBkj Cdt = AdtBdt dt = down-tilt Cij = AikBkj Cut = AutBut ut = up-tilt (7.9.g.9) In special relativity the down-tilt matrix form is most often used, and one just writes C = AB without bothering with the dt clarifying subscripts. This is consistent with the usual statement x'μ = Λμνxν to describe a Lorentz transformation acting on the contravariant vector xν (where Λ = our R ) (h) Matrix Inverse, Transpose and Determinant Matrix Inverses. Consider the matrix equation AB=1 where (assuming det(A) ≠ 1) we can write B = A-1. As demonstrated above, AB = 1 can only be a covariant equation if A and B are both down-tilt or both up-tilt mixed rank-2 tensors. Then we are talking about either AdtBdt = 1dt or AutBut = 1ut, and the corresponding Bdt = (A-1)dt and But = (A-1)ut, all these being matrix equations . Thus AdtBdt = 1dt Bdt = (A-1)dt AikBkj = δij Bij = (A-1)ij AutBut = 1ut But = (A-1)ut AikBkj = δij Bij = (A-1)ij (7.9.h.1) In this context, we have shown that if A is a mixed rank-2 tensor, then (A-1) is a mixed rank-2 tensor as well, assuming it exists. In Section (b) above we considered SR = 1 and reached the conclusions shown just above for the cases A = S and R = B. It happens that in this special case, S and R are not tensors, but the results are still valid. We shall see below that the matrix 1dt is really gdt, the down-tilt form of the metric tensor g, and similarly for 1ut : 1dt = gdt (1)ij = gij = δij = δi,j 1ut = gut (1)ij = gij = δij = δi,j (7.9.h.2) Thus first equation above can be written AdtBdt = gdt where all three matrices are down-tilt mixed rank-2 tensors. And this is also true for AutBut = gut . Transpose Matrices. In the developmental notation we have equations like (5.7.1) M' = R M RT and = RT ' R which involve transposes of matrices. Although it is possible to define a notion of "matrix transpose" in the Standard Notation, we have found that this leads to much confusion, and it is best to simply not allow such a notation. In converting an equation to standard notation, one should remove the transpose notation right at the start within the developmental notation, and then convert the equation to standard notation. For example we start in developmental notation, M' = R M RT => M'ad = RabMbc(RT)cd = RabMbcRdc = RabRdcMbc (7.9.h.3) Then we make the conversion using (7.6.2) M'ad = RabRdcMbc → (M')ad = RabRdcMbc (Standard Notation) (7.9.h.4) and this then is the Standard Notation rule for the way a contravariant rank-2 tensor transforms, as was shown in (7.9.c.9). Determinant of a Matrix. . In developmental notation one writes det(A) = εabc... A1aA2bA3c..... = εabc... Aa1 Ab2Ac3..... (7.9.h.5) where the Aij are components of the contravariant rank-2 tensor A and where ε is the permutation tensor discussed above in **. We have argued above that the notion of a rank-2 tensor being a matrix in Standard Notation is only viable for mixed rank-2 tensor of either the down-tilt or up-tilt variety. Thus, the matrix determinants of interest in Standard Notation would be these: det(Adt) = det(Aij) = εabc... A1a A2bA3c..... = εabc... Aa1 Ab2Ac3..... (7.9.h.6) det(Aut) = det(Aij) = εabc... A1a A2bA3c..... = εabc... Aa1 Ab2Ac3..... (7.9.h.7) These determinants for a rank-2 tensor A will never come up in this document, but they have come up for the non-tensor objects R and S as shown in ****. (i) Orthogonality rules. The rules are easy to show: [RTR = 1]SN ,dt => (RT)abRbc = δac => RbaRbc = δac [RRT = 1]SN ,dt => Rab(RT)bc = δac => RabRcb = δac [RTR = 1]SN ,ut => (RT)abRbc = δac => RbaRbc = δac [RRT = 1]SN ,ut => Rab(RT)bc = δac => RabRcb = δac (7.9.i.1) These orthogonality rules are derived in a slightly different manner and order in section (r) below. (j) Rotation matrices. In developmental notation, a real orthogonal matrix R has the property that RRT = 1 R-1 = RT RTR = 1 (7.9.j.1) The determinant of R must be ± 1 since RRT = 1 => (detR)2 = det(1) = 1 => detR = ± 1 (7.9.j.2) If detR = +1, then R can be a rotation matrix (in any number of dimensions N). If detR = -1, then R can be that same rotation matrix combined in any manner with an odd number of axis reflection matrices, each represented by a unit matrix in which one of the diagonal 1's is replaced by a -1. For example, for N = 2 one can write, R = R = = rotation matrix detR = +1 rx = = reflect the x-axis ry = = reflect the y-axis R = = = R rx detR = -1 (7.9.j.3) For N = 3 one might write R = R = rotation matrix (Euler angles) detR = +1 rx = ry = rz = P = rxryrz = A = R rx or R ry or rxR or RP etc all have: detA = -1 (7.9.j.4) The parity P operation is included only when N = odd. In general, then, a real orthogonal matrix R satisfying RRT = 1 is a rotation matrix combined with any even number (including 0) of axis reflection matrices (detR = +1), or it is a rotation matrix combined with any odd number of axis reflections (detR = -1). If one is interested in real-orthogonal transformations that can be arrived at in a continuous morphing manner from the identity transformation, then one would include only the rotation matrices in the set of real orthogonal matrices. So, our developmental notation characterization of a rotation matrix is this: [ RRT = 1 ]DN (7.9.j.5) where we now use the notation described in item 7 above. Up to this point R has stood for a "rotation matrix", but now we shall assume that this R is also the matrix which represents the linearized F transformation as in Chapter 2. In standard notation one finds RRT = 1 RabRcb = δa,c → RabRcb = δa,c . (7.9.j.6) The equation RabRcb = δa,c cannot be written in any of the four standard notation matrix multiplication forms discussed in item 7, so we just leave it "as is". There are several other characterizations of R being rotation matrix. First consider this sequence: RabRcb = δa,c // the above RabSbc = δa,c // Rcb = Sbc as shown above in Theorem 3 RabSbc Rcd = δa,c Rcd // apply Rcd to both sides and sum on c Rab (SR)bd = Rad // SR = 1 Rab δbd= Rad Rad = Rad Sda = Sda (7.9.j.7) where on each of the last two equations the two sides are related by reflection of the indices in a horizontal line running through the indices. Since it has now been shown that for a rotation Rab = Rab, it follows that RabRcb = δa,c => RabRcb = δa,c (7.9.j.8) The following theorem has now been proven: Theorem 5 (Rotations): A general transformation F can be represented at a point x by the linearized matrices R and S as discussed in Chapter 2. In developmental notation if R is a rotation matrix then it can be characterized by any of these equivalent statements in developmental notation (DN) : [RRT = 1]DN [RTR = 1]DN [R-1 = RT]DN . (7.9.j.9) In standard notation, the fact that R is a rotation matrix can be characterized by any of the following equivalent statements: RabRcb = δa,c RabRcb = δa,c Rab = Rab . (7.9.j.10) Of course if R is a rotation, so is S = R-1. Since Rij = Sji, the above are the same as SbaSbc = δa,c SbaSbc = δa,c Sba = Sba . (7.9.j.11) In principle, R = R(x) might be a different rotation matrix at different points x in x-space, so that F in that case would be a non-linear transformation. This would happen in 3D space if the Euler angles which parameterize a general rotation were arbitrary (but reasonable) functions of position x. What can be said about a rotation matrix Q which is not the R matrix which linearizes F? The translation from developmental to standard notation could be defined this way (same as for R and S) QQT = 1 QabQcb = δa,c → QabQcb = δa,c QTQ = 1 QbaQbc = δa,c → QbaQbc = δa,c (7.9.j.12) Since Qab(Q-1)bc = δa,c comparison shows that one must have (Q-1)bc = Qcb which is a straight swap of the indices. Since we have no information regarding the possible tensor or tensor-like nature of Q, we don't know how to raise or lower its indices, and nothing much more can be said. (k) Variations on the relation between g and g'. The third subsection above gave the basic statement of the transformation properties of tensor g. These can be inverted as follows, g'ab = Raa'Rbb'ga'b' => gab = (R-1)aa'(R-1)bb'g' a'b' = Saa' Sbb' g' a'b' g'ab = Sa'a Sb'b ga'b' => gab = (S-1)a'a (S-1)b'b g'a'b' = Ra'a Rb'b g'a'b' (7.9.k.1) and here then is a summary, g'ab = Raa'Rbb' ga'b' gab = Saa'Sbb' g'a'b' g'ab = Sa'a Sb'b ga'b' gab = Ra'a Rb'b g'a'b' (7.9.k.2) If x-space is Cartesian with g = 1, the first column above simplifies to g'ab = RacRbc // g = 1 g'ab = Sca Scb = Rac Rbc // g = 1 (7.9.k.3) Notice that the summation index c is not a contraction here. Also, although R and S are not tensors, the sums shown produce the true tensors g'ab and g'ab. In the previous section is was shown that if R is a rotation matrix, then [RRT = 1]DN and in standard notation one gets RacRbc = δa,b . where was this shown? (7.9.k.4) If x-space is Cartesian so metric tensor g = 1, then it was just shown above that for any R, g'ab = RacRbc . (7.9.k.5) Therefore, when R is a rotation, g' = 1 as well as g = 1, just as one would expect. 7.10 Tensors of Rank n, direct products, Lie groups, symmetry and Ricci-Levi-Civita The most general tensor of rank n (aka order n) will have some number s of contravariant indices and then some number n-s of covariant indices. If s = n, the tensor is pure contravariant, and if s = 0, it is pure covariant, otherwise it is "mixed" (as opposed to "pure"). The transformation of the tensor under F will show a factor Raa' for each contravariant index, and a factor Sa'a ( = Raa' as shown below in section (q)) for each covariant index, as illustrated by this example : T 'abcde = Raa' Rbb' Rcc' Sd'd Se'e Ta'b'c'd'e' T 'abcde = Raa' Rbb' Rcc' Rdd' Ree' Ta'b'c'd'e' (7.10.1) Note that for Raa' and Raa' the second index is the summation index, but for Sa'a it is the first index. A rank-n tensor always transforms the way an outer product of n vectors transforms if those vectors have indices which type-match those of the tensor. In the above case, an object that would transform the same as Tabcde would be AaBbCcDdEe (7.10.2) A tensor of rank-n has 2n tensor objects in its family since each index can be up or down. For example, the tensor T above is one of 25 = 32 tensors one can form. Of these, one is pure covariant and one is pure contravariant and 30 are mixed. If any of these tensors is a tensor field, such as Tabcde(x), then of course all family members are tensor fields. Direct Products. Consider again the outer product of vectors AaBbCcDdEe. The transformation A'a = Raa'Aa' occurs in an N-dimensional contravariant vector space we shall call R . In this space one could establish a set of basis vectors, and of course there are rules for adding vectors and so on. Transformation B'a = RacBc occurs in an identical copy of the space R, but transformation D'd = Sd'dDd' = Rdd'Dd' occurs in a covariant version of R we call . Since dot products (inner products) have been established for vectors in these spaces, they can be regarded as full blown Hilbert Spaces with the caveats of Chapter 5 (i). The transformation of the outer product object, as already noted, is given by A'aB'bC'cD'dE'e = Raa' Rbb' Rcc' Rdd' Ree' Aa'Bb'Cc'Dd'Ee' (7.10.3) and one can consider the operator Raa' Rbb' Rcc' Rdd' Ree' as a transformation element in a so-called direct product space which in this case would be written Rdp = R R R (7.10.4) One could then define (Rdp)abcde ; a'b'c'd'e' ≡ Raa' Rbb' Rcc' Rdd' Ree' (7.10.5) so that A'aB'bC'cD'dE'e = (Rdp)abcde ; a'b'c'd'e' Aa'Bb'Cc'Dd'Ee' (7.10.6) and of course this would apply to any tensor of the same index configuration, such as T 'abcde = Rdpabcde ; a'b'c'd'e' Ta'b'c'd'e' (7.10.7) This suggests a definition of "tensor" as follows" : tensors are those objects that are transformed by all possible direct product representations formable from the two fundamental vector representations R and . To this set of spaces one would add the identity space 1 to handle tensorial scalars. Appendix E continues this direct product discussion in terms of the basis vectors that form a complete set for a direct product space such as Rdp and shows how to expand tensors on such bases. Lie Groups. The direct product notion is just a formalism, but the formalism has some implications when the space R is associated with a "representation" of a Lie group. In this case, a direct product Rdp = R R can be written as a sum of "irreducible" representations of that group. What this means is that the transformation elements of Rdp and the objects Tab can be shuffled around with linear combinations so that (Rdp)aba'b', when thought of as a matrix with columns labeled by N2 ab possibilities and rows labeled by the N2 a'b' possibilities, appears in "block diagonal form" with all zeros outside the blocks. In this case, the shuffled components of tensor Tab can be regarded as a non-interacting assembly of pieces each of which transforms according to one of those blocks of the shuffled (Rdp)aba'b'. The most famous example occurs with N=3 and the rotation group SU(2) in which case R(1) ≡ R can be decomposed according to R(1) R(1) = R(2) R(1) R(0) where the symbols indicate this block diagonal form. In this case the blocks are 5x5, 3x3 and 1x1, fitting onto the diagonal of the 9x9 matrix area. The numbers L = 0,1,2 here label the rotation group representations and that label is associated with angular momentum. The elements of the 5x5 block are called D(2)M,M'(φ,θ,ψ) where M,M' = 2,1,0,-1.-2, and where φ,θ,ψ are the "Euler angles" which serve to label a particular rotation. This D(2)object is the L=2 matrix representation of the rotation group. Taking two vectors A and B, one can identify AB as the combination transforming according to R(0) ("scalar") and AxB ( linearly combined) as that transforming as R(1) ("vector"). The traceless symmetric matrix AiBj - δi,jAB has 5 independent elements associated with R(2) ( "quadrupole"). This whole reduction idea can be applied to larger direct products such as R(1) R(1) R(1) and tensor components Tabc. The Standard Model of elementary particle physics is chock full of direct products of this nature, where the idea of rotational symmetry is extended to other kinds of "internal" symmetry, spin and isospin being two examples. Representations of the Lie symmetry group SU(3) are associated with quarks which are among of the fundamental building blocks of the Standard Model. The group discussion above can be applied generally to quantum physics. The basic idea is that if "the physics" (the Hamiltonian or Lagrangian) describing some quantum object is invariant under a certain symmetry group (such as rotational symmetry or perhaps some discrete crystal symmetry), then the quantum states of that object can be classified according to the representations of that group. The Bohr hydrogen atom "physics" H ~ 2-1/|r| has perfect rotation group symmetry and is also symmetric about the axis (angle ψ) from center to electron (no spin). The representation functions then must have M' = 0, and then D(L)M,0(φ,θ,ψ) ~ YLM (θ,φ), the famous spherical harmonics that describe the "orbitals" which have mystified first-year chemistry students for the last 100 years. Historical Note: Ricci and Levi-Civita (see Refs) referred to rank-n tensors as "systems of order n" and did not include mixed tensors in their 1900 paper. Nor did they use the Einstein summation convention, since Einstein thought of that later on. They did use the up and down index notation pretty much as it is used today, though the up indices are enclosed in parenthesis. Here is a direct quote from the paper where the nature of the contravariant and covariant tensors is described (with crude translation below for non-French readers). Equation (6) had typos which some thoughtful reader corrected: the y subscripts should be r's and the x subscripts should be s's. In our notation ∂xs/∂yr → ∂xs/∂x'r = Ssr = Rrs. (7.10.8) We will say that a system of order m is covariant (and in this case we will designate its elements by the symbol Xr1,r2....) (r1,r2.... can each take all the values 1...n), if the elements Yr1,r2.... of the transformed system are given by the formulas (6) . We will designate on the contrary by the symbols X(r1,r2....) the elements of a contravariant system, which is to say of a system where the transformation is represented by the formulas (7) , the elements X and Y being related respectively to (presumably "are functions of") the variables x and y. Their y is our x', and their n is our N. Notice that the indices on the coordinates themselves are taken down, contrary to current usage. They do not explain why the words contravariant and covariant are used. In our notation, the two equations above would be written A'abc... = Raa'Rbb'....... Aa'b'c'... = Σprimed Aa'b'c'... Raa'Rbb'....... (6) A'abc... = Raa' Rbb'....... Aa'b'c'... = Σprimed Aa'b'c'... Raa' Rbb''....... (7) (7.10.9) 7.11 The Contraction Tilt-Reversal Rule In some complicated combination of multiple tensors, imagine there is somewhere a pair of summed indices where one is up and the other is down. As noted above, such a sum is called a contraction. The contracted indices could be on the same object or they could be on different objects. We depict this situation with the following symbolic notation, [-----a---------a----] (7.11.1) where the dashes indicate indices that we don't care about and which won't change -- each one could be up or down. We know we can reverse the tilt this way, [-----a---------a----] = gab gac [-----b---------c----] (7.11.2) where the first g raises the index b to a, and the second g lowers the index c to a. But the two g's are inverses, gab gac = gba gac = δa,c, which at once gives the desired result [-----a---------a----] = [-----a---------a----] // the Contraction Tilt-Reversal Rule (7.11.3) A notable example of course is this: AaYa = AaYa = " A Y " // or perhaps " A.Y " as noted in Chapter 5 (i) (7.11.4) When is index tilt-reversal allowed and when is it not allowed? It is always allowed when both indices of the tilted contraction are valid tensor indices. Consider these four examples to be discussed below: RabAb = RabAb Proof: RabAb = Racgcb gbdAd = gcbgbdRacAd = δcdRacAd = RacAc AaRab ≠ AaRab Proof: AaRab = gacAc Rdb g'da = gac g'ad Ac Rdb ≠ AaRab Aa∂a = Aa∂a Proof: Aa∂af = gacAc gad∂df = gacgadAc∂df = δcdAc∂df = Ac∂cf (7.11.5) ∂aAa ≠ ∂aAa Proof: ∂aAa = (gac∂c)(gadAd) = gacgad(∂cAd) + gac(∂cgad)Ad = δcd(∂cAd) + gac(∂cgad)Ad = ∂cAc + (∂agad)Ad ≠ ∂cAc In the first example, since Rab is not a tensor, one is on dangerous ground doing the tilt reversal, but it happens to work because the second index is associated with metric tensor gij which is the same metric tensor that raises and lowers indices of Ab. In the second example, the tilt-reversal fails because the first index of Rab is associated with the x'-space metric tensor g'ij. In the third example, both indices are valid tensor indices (with the same metric tensor). The fourth example shows a failure of the tilt-reversal rule and this example is very important. The inequality becomes an equality only if the underlying transformation F is linear so that R and S and g are then constants independent of position. For general F, such as the F involved in curvilinear coordinate transformations, the object ∂aAb is not a rank-2 tensor and so the object ∂aAa does not represent contraction of two true tensor indices and therefore the "contraction tilt-reversal rule" does not apply. The rule of the next section also does not apply for this same reason, so ∂aAa does not transform as a scalar under general F. Chapter (v) below continues this topic. Here is one more example along the lines of the fourth example above that shows the potential danger of reversing a tilt when it is not justified. Consider the equation, Va = εabcBb;c (1) // valid where Bb = a tensorial vector Bb;c = ∂cBb – ΓnbcBn = the covariant derivative of vector Bb = a rank-2 tensor εabc = a rank-3 tensor density (weight -1) (the Levi-Civita tensor) Va = a vector density (weight -1) (7.11.6) With regard to Bb;c : (a) in comma notation one writes ∂cBb = Bb,c ; (b) Γnbc = Γncb . Since all indices on equation (1) are tensor indices, one can lower index a and reverse the b and c tilts to get Va = εabcBb;c (2) // valid (7.11.7) Now go back to equation (1). Because εabc is antisymmetric on b and c, whereas Γnbc is symmetric on b and c, one can write εabcBb;c = εabcBb,c since the Γ term vanishes by symmetry. Thus one gets Va = εabcBb,c (3) // valid (7.11.8) Were one to blindly lower a and reverse the b and c tilts, one would get Va = εabcBb,c (4) // NOT valid (7.11.9) The reason for "not valid" is that the reversal of the b tilt is not justified. That is, Va = εabcBb,c = εabc∂cBb = gbb' εab'c ∂c(gbb"Bb") => Va = gbb' εab'c ∂c(gbb"Bb") = gbb' εab'c ∂c(gbb"Bb") // c tilt reversal is OK = gbb'gbb" εab'c(∂c Bb") + gbb' εab'c (∂c gbb") Bb" = δb'b" εab'c(∂c Bb") + gbb' εab'c (∂c gbb") Bb" = εabc(∂cBb) + gbb' εab'c (∂c gbb") Bb" = εabcBb,c + gbb' εab'c (∂c gbb") Bb" = εabcBb,c + extra term! (7.11.10) It is due to this extra term that (4) is not valid. Basically this is the same as the fourth example above, but the situation is embedded in a more complicated environment (extra tensors, tensor densities, comma notation, covariant derivatives, other tilted indices, etc). One way to summarize the example is this: (Va = εabcBb,c) (Va = εabcBb;c) (Va = εabcBb;c) (Va = εabcBb,c) (7.11.11) 7.12 The Contraction Neutralization Rule A contracted index pair plays no role in how an object transforms, the two indices neutralize each other, as we now show. First, recall that the indices on a general rank-n tensor (perhaps formed from several tensors) transform the same way an outer product of n vectors transforms, where the vector index types match those of the tensor. The vectors transform this way: V'a = RabVb V'a = SbaVb (7.18) and (7.19) So, we take our same "big object" above and now ask how it transforms. In the following, the X's represent either R or S factors for the dash indices (each of which might be up or down): [-----a---------a----]' = XXXXX Rab XXXXXXXXX Sca XXXX [-----b---------c----] = Sca Rab XXXXX XXXXXXXXX XXXX [-----b---------c----] = δcb XXXXX XXXXXXXXX XXXX [-----b---------c----] = XXXXX XXXXXXXXX XXXX [-----a---------a----] (7.12.1) where now the only X's left are for the other indices. Again we look at our canonical example, AaYa = AaYa = AY as in (7.14) or (7.105) The contracted vector indices cancel each other out and the resulting object transforms as a scalar. Here are some examples of tensor transformations with 0,1 and 2 index pairs contracted: T 'abcde = Raa' Rbb' Rcc' Sd'd Se'e Ta'b'c'd'e' // no pairs contracted T 'abcae = Rbb' Rcc' Se'e Ta'b'c'a'e' // index a contracted T 'abcab = Rcc' Ta'b'c'a'e' // index a and index b contracted Q' = Q where Q = AaYa and Q' = A'aY'a // index a contracted (7.12.2) This shows the idea that one can take a larger tensor like Tabcde and form from it smaller (lower rank) tensors by contracting tilted pairs of indices. In the above example list we really have Dbce ≡ Tabcae = a mixed rank-3 tensor Ec = Tabcab = a contravariant vector (rank-1 tensor) Q = AaYa = a scalar (rank-0 tensor) (7.12.3) It is similarly possible to build larger tensors from smaller ones, for example Zabcde = Va We gab Lc (7.12.4) which goes under the same rubric "outer product" mentioned earlier. 7.13 Raising and lowering indices on g On the one hand, since gab and gab are inverses of each other (formerly and g) , one has gabgbc = δa,c = δac (7.13.1) where the above-mentioned "look-nice" form of δa,c makes indices match. On the other hand, gabgbc = gbc // left g lowers the left index of the right g, or the converse (7.13.2) Comparison shows that gbc = δbc (7.13.3) As a sanity check, consider Va = gabVb Applying our Contraction Tilt-reversal Rule, this can be written Va = gabVb but this is = δabVb = Va (7.13.4) There are many ways to write things, here is a collection (gab = gba !) gabgbc = δac = gac // start out gabgbc = δac = gac // tilt reversal of the line above; just says δabδbc = δac gabgbc = δac = gac gabgbc = δac = gac (7.13.5) 7.14 Other forms of R Object Rab = (∂x'a/∂xb) was considered above. One could lower the a index using g'** since x'a is in x'-space and is an up index. The index in ∂/∂xb = ∂b is really a lower index (gradient), so one could in effect raise it using g** (no prime) because ∂/∂xb is in x-space. So when raising and lowering indices on Rab one has the unusual situation that one must use g' when acting on the first index, and g when acting on the second. With this in mind, we can now write three other index configurations of Rab Rab = ( ∂x'a/∂xb) // original object (formerly Rab) Rab = Rab' gb'b = (∂x'a/∂xb) // g pulls up the second index of Rab Rab = g'aa'Ra'b = (∂x'a/∂xb) // g' pulls down the first index of Rab Rab = g'aa'Ra'b' gb'b = (∂x'a/∂xb) // both actions at once (7.14.1) Although the g and g' factors can be placed anywhere, we have put g' factors on the left of R, and g factors on the right, each next to its appropriate leg of R. In each case, examination of the corresponding partial derivative shows that that the index sense matches on both sides. For example, in Rab = (∂x'a/∂xb) = ∂bx'a, both indices are contravariant on both sides. Remember that Rab is not a contravariant rank-2 tensor due to its dual-space nature. 7.15 Summary of facts about R Rik ≡ (∂x'i/∂xk) → Rik ≡ (∂x'i/∂xk) V'a = RabVb → V'a = RabVb Va = SabV'b [= RbaV'b] Rab(x) = (∂x'a/∂xb) // original object (formerly Rab) Rab = Rab' gb'b = (∂x'a/∂xb) // g pulls up the second index Rab = g'aa'Ra'b = (∂x'a/∂xb) // g' pulls down the first index Rab = g'aa'Ra'b' gb'b = (∂x'a/∂xb) // both actions at once Rab = g'aa'Ra'b' gb'b // the inverse of the previous line (using gabgbc = δac twice) (7.15.1) 7.16 Repeat all the above for S Sik ≡ (∂xi/∂x'k) → Sik ≡ (∂xi/∂x'k) 'a = STab b = Sba b → V'a = SbaVb Va = RbaV'b [= SabV'b] Sab(x) = (∂xa/∂x'b) // original object (formerly Sab) Sab ≡ Sab' g'b'b = (∂xa/∂x'b) // g' pulls the second index up Sab ≡ gaa'Sa'b = (∂xa/∂x'b) // g pulls the first index down Sab ≡ gaa'Sa'b' g' b'b = (∂xa/∂x'b) // both actions at once Sab ≡ gaa'Sa'b' g' b'b // the inverse of the previous line (using gabgbc = δac twice) (7.16.1) 7.17 Theorem: Sab = Rba and Sab = Rba ( reflect indices in vertical line between them) This theorem has already been proven in section (i) Theorem 3 as part of the discussion there of the fact that, when tilted matrix forms are consider for R and S, one has all of the following matrix results: S-1 = ST SST = 1 STS= 1 S = RT S = R-1 R-1 = RT RRT = 1 RTR= 1 R = ST R = S-1 (7.17.1) Here two slightly different lower-level proofs of this theorem that Sab = Rba . Once this is established, one ran raise and lower indices on either side to get all of the following Sab = Rba Sab = Rba Sab = Rba Sab = Rba (7.17.2) As noted earlier, an implication is that one can completely eliminate references to matrix S in tensor analysis and that is what is usually done! Proof of Theorem: This proof is a bit long-winded, but brings in many earlier results: δba" Saa" = Sab // introduce a δ . Remember all g's are symmetric. (g' bb' g' a"b') Saa" = Sab // since g'ab and g'ab are inverses of each other. g' bb' δb'b" Saa" g' a"b"= Sab // reorder and introduce another δ g' bb' (Rb'a' Sa'b") Saa" g'a"b"= Sab // 1 = RS so δb'b" = (Rb'a' Sa'b") g' bb' Rb'a' (Saa"Sa'b" g'a"b") = Sab // regroup (7.17.3) g' bb' Rb'a' (gaa') = Sab // use gaa' = Saa"Sa'b" g'a"b", see end of (i) above (g' bb' Rb'a' gaa') = Sab // regroup Rba = Sab // g and g' raise and lower R's indices, see (n) above Notice that the above theorem says Sab = (∂xa/∂x'b) = (∂x'b/∂xa) = Rba (7.17.4) A faster way to derive this result is to differentiate dxcdxc = dx'cdx'c and use the chain rule: dx'b = ( ∂( dx'cdx'c)/∂x'b ) = (∂(dxcdxc)/∂xa) (∂xa/∂x'b) = dxa (∂xa/∂x'b) => (∂x'b/∂xa) = (∂xa/∂x'b) => Rba = Sab (7.17.5) Similar results can be derived for other index positions (or we can just raise and lower indices!) to get Sab = Rba = (∂xa/∂x'b) = (∂x'b/∂xa) Sab = Rba = (∂xa/∂x'b) = (∂x'b/∂xa) Sab = Rba = (∂xa/∂x'b) = (∂x'b/∂xa) Sab = Rba = (∂xa/∂x'b) = (∂x'b/∂xa) (7.17.6) Here index a is always in x-space, while index b is in x'-space. The two vector transformation rules V'a = RabVb V'a = SbaVb (7.18) and (7.19) can now be written V'a = RabVb V'a = RabVb (7.17.7) which has the advantage that the indices are properly arranged for matrix multiplication in both cases. Here then is a restatement of the transformation of the example given in section (l) , T 'abcde = Raa' Rbb' Rcc' Sd'd Se'e Ta'b'c'd'e' // no pairs contracted (7.114) becomes T 'abcde = Raa' Rbb' Rcc' Rdd' Ree' Ta'b'c'd'e' // no pairs contracted (7.17.8) It is easy to remember since the second index is always the summed index and the other index has to match (up or down) the left side of the equation. 7.18 Orthogonality Rules, the Inversion Rule, and the Cancellation Rule The above theorem Sab = Rba can be used to eliminate S in various forms of RS = 1: SR = 1 Sab Rbc = δac Rba Rbc = δac Rba Rbc = δac Σ 1st RTST = 1 Rba Scb = δac RbaRbc = δac Rba Rbc = δac Σ 1st RS = 1 Rab Sbc = δac Rab Rcb = δac Rcb Rab = δca Σ 2nd STRT = 1 Sba Rcb = δac Rab Rcb = δac Rcb Rab = δca Σ 2nd (7.18.1) The four results in the right column are called orthogonality rules for R. The first pair is summed on the first index, the second on the second. In section (i) it was shown that these rules are just statements of the fact that in up or down tilted standard notation R is a real-orthogonal matrix so RRT = RTR= 1. Inversion Rule. Consider now an equation which one wants to invert for the object on the right, [----a-----] = Rab [--------b------] (*) // before (7.18.2) The inversion rule for moving R to the other side of the equation is to reflect R's two indices in the horizontal index plane, so the result will be Rab [----a-----] = [--------b------] // after (7.18.3) Proof: Rename b→b' in (*), apply Rab to both sides and sum on a, then use an orthogonality rule: Rab [----a-----] = Rab Rab' [--------b'------] = δbb' [--------b'------] = [--------b------] QED Recall from the last section that reflection in the vertical index plane has a different implication, Sab = Rba or (R-1)ab = Rba or (RT)ab = Rba (7.124) and (7.125) Cancellation Rule. Next, consider a different generic equation ( bracketed objects are different) Rab [----b-----] = Rab [--------b------] (**) // before (7.18.4) The cancellation rule says the equation is still valid if identical contracted R factors are canceled on both sides such that the contraction index becomes a free index, [----b-----] = [--------b------] // after (7.18.5) Proof: Rename b→b' in (**), apply Rab to both sides and sum on a, then use an orthogonality rule, RabRab' [----b'-----] = RabRab' [--------b'------] δbb' [----b'-----] = δbb' [--------b'------] [----b-----] = [--------b------] QED (7.18.6) 7.19 The tangent and reciprocal base vectors and expansions on same Tangent and reciprocal base vectors Here are some basic translations: (en)i → (en)i // contravariant index i (n)i → (en)i // covariant index i (En)i → (en)i // contravariant index i (n)i → (en)i // covariant index i (en)i = Sin → (en)i = Sin = Rni // contravariant index i (n)i = Rni → (en)i = Rni // covariant index i (En)i = Rnkgki → (en)i = Rnkgki = Rni // contravariant index i (7.19.1) As noted earlier, writing a vector in bold such as en is not enough to say whether the vector is contravariant or covariant. If one form or the other is intended, one must show an index up or down, even if it is just a dummy placeholder index. As examples, S = [e1, e2, e3 .... eN ] → Sij = [(e1)k, (e2)k, (e3)k .... (eN)k] (7.19.2) R = [1, 2, 3 .... N ]T → Rij = [(e1)k, (e2)k, (e3)k .... (eN)k]T (7.19.3) The relationship between en and en is very simple, En ≡ g'ni ei → en = g'ni ei and en = g'ni ei (7.19.4) For either contravariant or covariant indices (indices are not shown), g'ni raises the label on ei , and inverting one finds that g'ni lowers the label on ei. This fact makes things easy to remember. Using the fact that ei = ∂'ix , one has g'ni ei = g'ni ∂'ix = ∂'nx so the above line can be expressed as En ≡ g'ni ei → en = ∂'nx and en = ∂'nx (7.19.5) The dot products are en em = 'nm → en em = g'nm = ∂'nx ∂'mx |en| = = h'n En em = δn,m → en em = δnm = ∂'nx ∂'mx En Em = g'nm → en em = g'nm = ∂'nx ∂'mx |en| = (7.19.6) The "labels" on the base vectors behave in this dot product structure the same way that up and down "indices" behave. This is the motivation for En → en . Thus, the three final equations can be regarded as the same equation en em = g'nm where we can raise either or both indices/labels to get the other equations. For example, en em = g'nm = δnm . The dot product at the end of Chapter 6 (a) becomes En um = Rnm → en um = Rnm // = <en | um> in bra-ket notation (App E (g)) (7.19.7) The matrix Rnm is a "basis change matrix" between basis uk and basis ek . Inverse tangent and reciprocal base vectors Using the rules given above, g'↔ g R ↔ S en → u'n e'n → un En → U'n E'n → Un (6.24) we can obtain the corresponding results for the inverse tangent and reciprocal base vectors: (u'n)i → (un)i // contravariant index i ('n)i → (un)i // covariant index i (U'n)i → (un)i // contravariant index i ('n)i → (un)i // covariant index i (u'n)i = Rin → (u'n)i = Rin = Sni // contravariant index i (')i = Sni → (un)i = Sni // covariant index i (U'n)i = Snkg'ki → (un)i = Snkg'ki = Sni // contravariant index i R = [u'1, u'2, u'3 .... u'N ] → Rij = [(u'1)k, (u'2)k, (u'3)k .... (u'N)k] S = ['1, '2, '3 .... 'N ]T → Sij = [('1)k, ('2)k, ('3)k .... ('N)k]T U'n ≡ gni u'n → u'n = gni u'i and u'n = gni u'i u'n u'm = nm → u'n u'm = gnm U'n u'm = δn,m → u'n u'm = δnm U'n U'm = gnm → u'n u'm = gnm (7.19.8) Summary table. The summary table given at the end of Chapter 6 (e) was this x'-space x-space (6.31) axis-aligned basis vectors e'n un (e'n)i= δn,i (un)i= δn,i dual partners to the above E'n Un (E'n)i = g'ni (Un)i = gni tangent base vectors u'n en (u'n)i= Rin (en)i = Sin reciprocal base vectors U'n En (U'n)i = g'ia Sna (En)i = gia Rna = gnaRia = g'naSia which translates into this → : x'-space x-space (7.19.9) axis-aligned basis vectors e'n un (e'n)i= δni (un)i= δni dual partners to the above e'n un (e'n)i = g'ni (un)i = gni tangent base vectors u'n en (u'n)i = Rin (en)i=Sin= Rni reciprocal base vectors u'n en (u'n)i = g'ia Sna (en)i = gia Rna (u'n)i = Sni (en)i = Rni x-space expansions The x-space expansions of Chapter 6 (f) were V = V1 u1 + V2 u2 +... = ΣnVn un where Un V = Vn Un = gni ui V = 1 U1 + 2 U2 +... = Σnn Un where un V = n V = V'1 e1 + V'2 e2 +... = Σn V'n en where En V = V'n En = g'ni ei V = '1 E1 + '2 E2 +... = Σn 'n En where en V = 'n (6.40) and they now become → : V = V1 u1 + V2 u2 +... = ΣnVn un where un V = Vn un = gni ui V = V1 u1 + V2 u2 +... = ΣnVn un where un V = Vn V = V'1e1 + V'2e2 +... = Σn V'n en where en V = V'n en = g'ni ei V = V'1e1 + V'2 e2 +... = Σn V'n en where en V = V'n (7.19.10) x'-space expansions Similarly, the x'-space expansions of Chapter 6 (f) were V' = V'1 e'1 + V'2 e'2 +... = ΣnV'n e'n where E'n V' = V'n E'n = g'ni e'i V' = '1 E'1 + '2 E'2 +... = Σn'n E'n where e'n V' = 'm V' = V1u'1 + V2u'2 +... = Σn Vn u'n where U'n V' = Vn U'n = gni u'i V' = 1U'1 + 2U'2 +... = Σn n U'n where u'n V' = n (6.46) and they now become → : V' = V'1 e'1 + V'2 e'2 +... = ΣnV'n e'n where e'n V' = V'n e'n = g'ni e'i V' = V'1 e'1 + V'2 e'2 +... = ΣnV'n e'n where e'n V' = V'm V' = V1u'1 + V2u'2 +... = Σn Vn u'n where u'n V' = Vn u'n = gni u'i V' = V1u'1 + V2u'2 +... = Σn Vn u'n where u'n V' = Vn (7.19.11) Summary of all expansions: Using implied sum notation, we can now summarize the eight expansions above, plus the unit vector expansion onto n, on just two lines : V = Vn un = Vn un = V'n en = V'n en = V'n n // x-space expansions, V'n = hnV'n V' = V'n e'n = V'n e'n = Vn u'n = Vn u'n // x'-space expansions (7.19.12) In all cases one sees a tilted index summation where one index is a vector index and the other is a basis vector label. Half the forms shown above can be obtained from the others by just "reversing the tilt". The power of the Standard Notation makes itself felt in relations like these. Due to this tilt situation, sometimes a basis like en appearing in V = V'n en is called a "covariant basis" while the basis en appearing in V = V'n en is called a "contravariant basis". Corresponding expansions of higher rank tensors are presented in section (w) below. If V is a tensor density of weight W (see Appendix D and E) the rule for adjusting the above expansions is to make the replacement V'n → JW V'n and V'n → JW V'n where J is the Jacobian of Chapter 5 (k). 7.20 Comment on Covariant versus Contravariant Consider this expansion for a vector V in x-space, V = Vnbn Vn = V bn (7.20.1) where bn is some basis having dual basis bn where as usual bn bm = δnm. Imagine taking Vn → V'n = Rnm Vm and bi → bi' = Qij bj. What Q would cause the following to be true? V = Vnbn = V'nb'n (7.20.2) In other words, how does one transform that basis bn such that the vector V remains unchanged if Vn is transformed contravariantly? The answer to this question is that Qij = Rij since then (using R orthogonality as in section (r)) V'nb'n = [Rnm Vm][ Rnj bj] = (Rnm Rnj) Vm bj = δmj Vm bj = Vj bj = Vnbn (7.20.3) Compare then the transformation of Vn with that of the basis bn: V'n = Rnm Vm bn' = Rnm bm (7.20.4) The Vm vector components transform with Rnm but the basis vectors have to transform with Rnm to maintain the invariance of the vector V. One varies with the down-tilt R, while the other varies with the up-tilt R, so the two objects are varying against each other in this tilt sense. They are "contra-varying", so one refers to the components Vm as contravariant components with respect to the basis bm . If one starts over with Vn components and the bn "dual" (reciprocal) expansion vectors and asks for a solution to this corresponding problem, V = Vnbn = V'nb'n (7.20.5) one finds not surprisingly that the dual basis must vary as bn' = Rnm bm and then one has V'n = Rnm Vm bn' = Rnm bm (7.20.6) which is the previous result with all indices up↔down. Comparing the tilts, one would say that the Vm again "contra vary" with the way the bm vary to maintain invariance of V. But one does not care about the dual basis, one cares about the basis, so relative to the basis bn one has V'n = Rnm Vm bn' = Rnm bm (7.20.7) If the basis bm is varied as shown here, then the dual basis bm varies as shown above and V remains invariant. Comparing now the way the Vn transform with the way the basis vectors bm transform, one sees that both equations have the same tilted Rnm. They are "co-varying", so one refers to the components Vm as covariant components with respect to the basis bm . 7.21 The Significance of Tensor Analysis "Why is tensor analysis important?", the reader might ask in the midst of this storm of index shuffling. Now is a good time to answer the question. Consider the following sample equation in x-space, where the fields Q, H, T and B may or may not be tensor fields: Qadc(x) = Hab(x)Tbc(x) Bd(x) (7.21.1) Notice that when contracted indices are ignored, the remaining indices have the same type on both sides. If the various objects really were tensors, one would say this was a "valid tensor equation" based on the index structure just described. One says that an equation is "covariant with respect to transformation x' = F(x)" if the equation has exactly the same form in x'-space that it has in x-space , which for our example would be Q'adc(x') = H'ab(x')T 'bc(x') B'd(x') (7.21.2) Here the word "covariant" has a new meaning, different from its being a type of vector or index. The meaning is related in the sense that, comparing the above two equations, everything has "moved" in the same manner ("co-varied") under the transformation. (Some authors think the word "invariant" is more appropriate; Ricci and Levi-Civita used the term "absolute"; continuum mechanics uses the term "frame-indifferent".) If the objects Q, H, T and B are tensors under F, then covariance of any valid tensor equation like the one shown above is guaranteed!! The reason is that, once the contracted indices on the two sides are ignored according to the "contraction neutralization rule", the objects on the two sides of the equation have the same indices which are of the same type, so both sides are tensors of the same type, and therefore both sides transform from x-space to x'-space in the same way. If one starts, for example, with the primed equation and installs the known transformations for all the pieces, one ends up with the unprimed equation. If this explanation is not convincing, a brute force demonstration can perhaps help out. The following is also a good exercise is using the two tilt forms of the R matrix. Recall from section (q) that Sba = Rab and that SR = 1 is replaced by the various orthogonality rules of section (r). We shall process the primed equation into the unprimed one : Q'adc(x') = H'ab(x')T 'bc(x') B'd(x') (*) // x'-space equation (7.21.3) [Raa'Rdd'Rcc'Qa'd'c'(x)] = [Raa'Rbb' Ha'b'(x)] [Rbb"Rcc' Tb"c'(x) ] [Rdd'Bd'(x)] = Raa' Rdd' Rcc'(Rbb' Rbb") Ha'b'(x) Tb"c'(x)Bd'(x) (7.21.4) Using one of the orthogonality rules of section (r), = Raa' Rdd' Rcc'(δb'b") Ha'b'(x) Tb"c'(x) Bd'(x) = Raa' Rdd' Rcc' Ha'b'(x) Tb'c'(x) Bd'(x) (7.21.5) so that, using the fact that Q is a tensor to replace Q' on the left side of (*), (Raa'Rdd'Rcc') Qa'd'c'(x) = (Raa' Rdd' Rcc') Ha'b'(x) Tb'c'(x) Bd'(x) (7.21.6) Now apply the Cancellation Rule of section (r) three times to conclude that Qa'd'c'(x) = Ha'b'(x) Tb'c'(x) Bd'(x) (7.21.7) and then remove all primes to get Qadc(x) = Hab(x)Tbc(x) Bd(x) // x-space equation (7.21.8) Thus it has been shown that, if all the objects transform as tensors, the equation is covariant. Tensor density equations are also covariant. As discussed in Appendix D, a tensor density of weight W is a generalization of a tensor which has the same transformation rule as a regular tensor, but there is an extra factor of J-W on the right hand side of the rule, where J is the Jacobian J = detS. For example, Q'adc(x') = J-WQ Raa'Rdd'Rcc'Qa'd'c'(x) (7.21.9) would indicate that Q was a tensor density of weight WQ. If WQ = 0, then Q is a regular tensor. With this definition in mind, it is easy to generalize the notion of a "covariant equation" to include tensor densities. Consider some arbitrary tensor equation which we represent by our example above, Qadc(x) = Hab(x)Tbc(x) Bd(x) (7.155) Suppose all four objects Q, H, T, B are tensor densities with weights WQ, WH, WT, WB. If the four objects Q, H, T, B are tensor densities, and if the up/down free indices match on both sides (the non-contracted indices), and if WQ = WH + WT + WB, then this is a "valid tensor density equation" and covariance is guaranteed, so it follows that Q'adc(x') = H'ab(x')T 'bc(x') B'd(x') . (7.156) It is trivial to edit the above proof by just adding weight factors in the right places and then of course they cancel out on the two sides. Examples of covariant tensor equations: In special relativity, which happens to involve linear Lorentz transformations, a fundamental principle is that any "equation of motion" describing anything at all (particles, EM fields, etc) must be covariant with respect to Lorentz transformations, or it cannot be a valid equation of motion (ignoring general relativity). An equation of motion must look the same in a reference frame which is rotated, boosted, or related by any combination of boosts and rotations to some original frame of reference (see Chapter 5 (m)). As was noted earlier, the tradition is to write 4-vector indices as Greek letters and 3-vector spatial indices as Latin letters. For example, we can define the "electromagnetic field-strength tensor" (rank-2) this way in terms of the 4-vector "vector potential" Aμ: Fμν ≡ ∂μAν - ∂νAμ (7.21.10) where ∂μ means gμα∂α, the contravariant form of the gradient operator. The components are then (7.21.11) where c is the speed of light and of course E and B are the electric and magnetic fields. Maxwell's two inhomogeneous equations (that is, the two with sources) are, in SI units where ε0μ0= 1/c2, ∂νFμν = μ0 Jμ with Jμ = (cρ,J) (7.21.12) while the two homogeneous equations become ∂αFμν + ∂μFνα + ∂νFαμ = 0 or ∂αFμν + cyclic = 0 . (7.21.13) One can see that each of these equations involves only tensors and we expect that in x'-space these equations will take the form ∂'νF 'μν = μ0 J'μ with J'μ = (cρ',J') ∂'αF 'μν + ∂'μF 'να + ∂'νF 'αμ = 0 or ∂'αF 'μν + cyclic = 0 (7.21.14) Objects like ∂νFμν and ∂αFμν are true rank-3 tensors because the transformation F is linear. Covariance of tensor equations involving derivatives with non-linear F. A tensor equation which involves derivatives of tensors is non-covariant under transformations F which are non-linear. The reason is that the derivative of a tensor is, in that case, not a tensor, as shown in the next section. Such tensor equations can be made covariant by replacement of all derivatives by covariant derivatives (which are indicated by a semicolon). In general relativity, this is known as the Principle of General Covariance (Weinberg p 106). A simple example is the tensor equation gab;c = 0 (Appendix F (i)). Examples relating to the transformation from Cartesian to curvilinear coordinates appear in Chapter 15. 7.22 The Christoffel Business: covariant derivatives This subject is treated in full detail in Appendix F, but here we provide some motivation. It should be noted that a normal derivative is sometimes written ∂aVb = Vb,a with a comma, whereas the covariant derivative discussed below is written Vb;a with a semicolon. When a transformation F is non-linear, the matrix Rab is a function of x. Thus one gets the following transformation for a lower index derivative of a covariant vector field component ∂aVb(x), where a "second term" quite logically appears, (∂'aV'b) = (Rad∂d) (RbcVc) = Rad Rbc (∂dVc) + Rad(∂d Rbc)Vc (7.22.1) This second term did not arise earlier when we looked at ∂a on a scalar field φ'(x') = φ(x) , (∂'aφ') = (Rad∂d) φ = Rad (∂d φ). (7.22.2) In special relativity, for example, where transformations are linear, ∂d Rbc = 0, there is no second term, and the object ∂aVb transforms as a covariant rank-2 tensor, (∂'aV'b) = Rad Rbc (∂dVc) , // F is a linear transformation (7.22.3) but in the general case the second term is present, so ∂dVc fails to transform as a rank-2 covariant tensor. In this case, one defines a certain "covariant derivative" which itself has an extra piece Vb;a ≡ ∂aVb – Γkab Vk => Vd;c ≡ ∂cVd – Γkcd Vk (7.22.4) where Γcab is a certain function of the metric tensor g. One then finds that V'b;a = Rad Rbc V d;c // the notation cVd ≡ V d;c is also commonly used or [∂'aV'b – Γ 'kab V'k] = Rad Rbc [∂dVc – Γkcd Vk] (*) (7.22.5) so that this covariant derivative of a covariant vector field Vc transforms as a covariant rank-2 tensor even with non-linear transformation F (see Christoffel Ref., 1869). This issue arises in general relativity and elsewhere. The object Γcab (sometimes called the "Christoffel connection") is given by Γcab ≡ {ab,c} ≡ ≡ gcd [ab,d] = ½ gcd( ∂agbd + ∂bgad – ∂dgab ) // Christoffel 2nd kind (7.22.6) Γdab ≡ [ab,d] ≡ ½ ( ∂agbd + ∂bgad – ∂dgab ) // Christoffel 1st kind and this is where the various "Christoffel symbols" come into play. In general relativity and elsewhere, Γcab is known as the "affine connection" which represents the effect of "curved space" appearing as a force which acts on a mass (that is to say, a gravitational force), see Chapter 5(n). Warning: There is a differently defined version of Γcab floating around in the literature. The version used above and everywhere in this document is that of Weinberg and is the most common form. The derivative of any tensor field other than a scalar field shows this same complication when the underlying transformation F is non-linear. For example, ∂agbd(x) does not transform as a rank-3 tensor, ∂'ag'bd(x') = (Rad∂d)(Rbb'Rdd'gb'd') = Rbb'Rdd'(∂d gb'd') + other terms (7.22.7) and therefore neither of the Christoffel symbols Γdab or Γcab transforms as a tensor in this case. See Appendix F for more detail. 7.23 Expansions of higher order tensors Appendix E clarifies the use of direct product and polyadic notations for describing the basis vector combinations onto which higher order tensors can be expanded in a simple generalization of the vector expansions presented in section (s) above. There it was shown that a vector A can be expanded in two interesting ways : A = Σn An un An are the contravariant components of A in x-space A = Σn A'n en A'n are the contravariant components of A in x'-space (7.23.1) In the first, un are axis-aligned basis vectors, and in the second en are the tangent base vectors. If A is instead a tensor of rank n, these expansions are replaced by A = Σijk... Aijk... (uiujuk...) Aijk... are the contravariant components of A in x-space A = Σijk... A'ijk... (eiejek...) A'ijk... are the contravariant components of A in x'-space (7.23.2) where there are n indices in each sum, n factors in the direct products, and n contravariant indices on the components of tensors A in x-space and in x'-space. In the polyadic notation the direct-product crosses are eliminated giving A = Σijk... Aijk... uiujuk... Aijk... are the contravariant components of A in x-space A = Σijk... A'ijk... eiejek... A'ijk... are the contravariant components of A in x'-space . (7.23.3) In the case of rank-2 tensors, a product like uiuj = uiuj is called a dyadic (see Appendix E). In this case (only) the product can be visualized as uiuTj which is a matrix constructed from a column vector to the left of a row vector. Thus one can write A = Σij Aij uiuTj Aij are the contravariant components of A in x-space A = Σij A'ij eieTj. A'ij are the contravariant components of A in x'-space . (7.23.4) Appendix E (g) promotes the interpretation of a rank-2 tensor A as an operator in a Hilbert space, where the matrices Aij and A'ij are matrices associated with the operator A in different bases, Anm = <un | A | um > = the x-space components of tensor A A'nm = <en | A | em > = the x'-space components of tensor A (7.23.5) As Appendix E shows, these two matrices are related to each other by a similarity transformation A' = R A R-1. These expansion methods are used in Appendices G and H to derive curvilinear expressions for two objects that play a role in continuum mechanics, (v) and div(T) (where T is a tensor).