Ch 7 reorder v1

Chapter 7 pick and place Sections 1,2,3 as is. Below are Sections 7.4 through 7.9, most of it completely reordered and rewritten and renumbered. 7.4 Some Preliminary Translations; raising and lowering tensor indices 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". 1. Basic translations. 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. In this same spirit, we have the following translation for a bolded vector: V → V → V (7.4.2) 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. (7.4.3) 2. Raising and Lower Tensor Indices with g 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 (7.4.4) In the Standard Notation, the nature of a vector (contravariant or covariant) is determined by the up or down position of the index. Looking at gabVb = Va, we see that " gab lowers the index on Vb to give Va " which of course just means that abVb = a in the developmental notation. It means nothing more, and nothing less. Similarly, gab Vb = Va says that " gab raises the index on Vb to give Va " and this is just the Standard Notation way of saying gab b = Va. The key idea here is that gab lowers a tensor index of a tensor, and gab raises a tensor index (with an implied sum as above). We know from the outer product idea shown in ** that we can construct a covariant rank-2 tensor from two covariant rank-1 tensors in this way. Tab ≡ UaVb (7.1.1) We can then use gca to raise the a index on both sides of this equation, gcaTab = Tcb gca(UaVb) = ( gcaUa)Vb = UcVb => Tcb = UcVb (7.4.5) Here, Tca is a "mixed" rank-2 tensor, the first index is contravariant, the second index is covariant. This fact matches the nature of the right side of the equation UcVb. We could have but did not deal with mixed tensors in the developmental notation because the notation could not really handle it well, as shown in (7.2.4). So in developmental notation, we dealt only with "pure" rank-2 tensors like Mab and ab. As a next step, we can apply gdb to both sides of (*), gdb Tcb = Tcd gdb (UcVb) = Uc(gdbVb) = UcVd => Tcd = UcVd (7.4.6) We end up with a contravariant rank-2 tensor that is the outer product of two contravariant rank-1 tensors. Now let's do both index raising operations at the same time: Tab ≡ UaVb gca gdb Tab = Tcd gca gdb UaVb = ( gcaUa)(gdbVb) = UcVd => Tcd = UcVd (7.4.7) This shows that we are free to raise or lower corresponding indices on both sides of a tensor equation at will, and we don't have to write out the details. Thus if the first of these is true, then the other three forms are also valid: Tab ≡ UaVb => Tab = UaVb Tab = UaVb Tab = UaVb (7.4.8) This idea works with any rank-2 tensor because any rank-2 tensor transforms under x' = F(x) in the same way that the outer product of two vectors transforms. If M is some arbitrary contravariant rank-2 tensor, then gcaMab = Mcb and so on, repeating all of the above. The idea applies as well to rank-3 tensors and rank-n tensors as we shall describe below in ***. For example, if Tabc = UaVbWc, or if M is some arbitrary rank-3 tensor Mabc, then Tabc ≡ UaVbWc Mabc gdaTabc = Tdbc gdaMabc = Mdbc (gdaUa)VbWc = UdVbWc => Tdbc = UdVbWc (7.4.9) In this case, the validity of the first equation implies the validity of the 7 others: Tabc ≡ UaVbWc => Tabc ≡ UaVbWc Tabc ≡ UaVbWc Tabc ≡ UaVbWc Tabc ≡ UaVbWc Tabc ≡ UaVbWc Tabc ≡ UaVbWc and finally Tabc ≡ UaVbWc (7.4.10) To summarize, in this new Standard 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. The g raising and lower rules may be represented generically in this manner gaa' [----a'---] = [----a---] gaa' [----a'---] = [----a---] (7.4.11) where [----a'---] represents an arbitrary tensor expression (perhaps just one tensor) with lots of tensor indices indicated by dashes. Each of these dash indices could be up or down, it does not matter. An example of the second line is gaa'Ma'bc = Mabc . 3. The invariant distance and covariant dot product:   Based on the discussion so far, we can write the following translations: dxi → dxi (ds)2 = ab dxa dxb → gabdxadxb AB = ab Aa Bb → AB = gab Aa Bb . (7.4.12) Using the raising and lowering idea above, we can write the dot product in other ways: AB = gab Aa Bb = Aa( gabBb) = AaBa AB = gab Aa Bb = gba Aa Bb = (gba Aa)Bb = AbBb AB = AbBb = Ab (gbcBc) = gbcAbBc (7.4.13) To summarize: AB = gab Aa Bb = AaBa = AaBa = gabAaBb (7.4.14) 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. 4. Contraction Rules When an index is summed with one index down and the other up, one says that the two indices are contracted. For example, in the expressions AaBa or AaBa, index a is contracted. In gab Aa Bb both indices a and b are contracted. Tilt Reversal Theorem. The tilt of any pair of contracted indices can be reversed without affecting anything. (7.4.15) A proof is given below in ***. An example is AaBa =AaBa as shown above. Contraction Theorem. In analyzing the transformation nature of a tensor expression, one can simply ignore all contracted indices and look only at remaining indices to determine the tensor nature of the object. (7.4.16) A proof is given below in ***. As examples, in the cases AaBa or gab Aa Bb, if we ignore contracted indices, there are no indices left, so these objects must transform as a tensor with no indices, which is a tensorial scalar. Thus AB above is a scalar, something we already knew. The object AaMab transforms as a covariant vector, and the object Maa transforms as a scalar. 5. What about having g raise and lower indices on itself? After all, g is a valid tensor. We find gabgbc = gac // lower the first index of rank-2 tensor gbc gabgbc = gac // lower the second index gba gac = gbc // lower both indices (lower second, then lower first) (7.4.17) We know from ** above that gab and gab are inverses. Thus, it must be true that gabgbc = (1)ac = δac = δa,c gabgbc = (1)ac = δac = δa,c (7.4.18) Here δac and δac are just cosmetically nice ways to write the Kronecker delta δa,c . Comparing ** and ** we learn that gac = δac = δa,c gac = δac = δa,c (7.4.19) Thus, the mixed form of the metric tensor is trivial and is not a function of the nature of x-space. 7.5 Dealing with the matrices R and S ; various Rules and Theorems 1. Translations of R and S 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) . (7.5.1) 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) . (7.5.2) 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 (7.5.3) 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. However, it is a contravariant vector in x'-space, which will be further discussed below. 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 component in x'-space) 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 x-space, and V' and the other half of R are in a x'-space. There is no object called R', as if R were in x-space and R' were in x'-space. We can repeat the above discussion for S instead of R. 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 (7.5.4) 2, Transformation rule for rank-2 tensors For a contravariant rank-2 tensor, the translation of the transformation rule *** is this: M'ab = Raa'Rbb'Ma'b' → M'ab = Raa'Rbb'Ma'b' contravariant rank-2 tensor (7.5.5) 'ab = Sa'aSb'ba'b' → M'ab = Sa'aSb'bMa'b' . covariant rank-2 tensor Applying this to the metric tensor M = g we find these translations of (5.7.6), g' = R g RT => g'ab= Raa'Rbb'ga'b' → g'ab = Raa'Rbb'ga'b' (7.5.6) ' = ST S => 'ab = Sa'aSb'b a'b' → g'ab = Sa'aSb'b ga'b' . The inverses of the above equations appear in (5.7.7) and translate this way , g = S g' ST gab = Saa'Sbb'g'a'b → gab = Saa'Sbb'g'a'b (7.5.7) = RT ' R ab = Ra'aRb'b'a'b' → gab = Ra'aRb'b g'a'b' By raising and lowering indices on the first result in (7.5.5) we obtain these four transformation rules 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' . (7.5.8) 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. ] 3. Raising and lowering indices on R and S Although R and S are not tensors, one can still raise and lower their two indices using metric tensors, but things are a little different from the tensor situation, the reason being that R and S each have one foot in x-space and the other foot in x'-space. 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 (7.5.9) 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. In the same manner, we arrive at these index configurations for Sab : 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 (7.5.10) For object S, the metric tensor g raises or lowers the first index of S, while g' raises or lowers the second index of S. This is just the reverse of what happens for object R as reported above. This is not surprising since one gets S↔R when one swaps x-space ↔ x'-space. 4. Inverse of R and S We have showed the translation 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 , (∂x'c/∂xa) (∂xa/∂x'b) = δcb or Rca Sab = δcb , (7.5.11) and therefore from the first line it must be that (R-1)ca = Sca. The second line shows that (S-1)ca = Rca. Using the above rules for raising and lowering indices on both sides of an equation, we obtain these four versions of the developmental notation fact that R-1 = S and S-1 = R : (R-1)ik = Sik (S-1)ik = Rik (R-1)ik = Sik (S-1)ik = Rik (R-1)ik = Sik (S-1)ik = Rik (R-1)ik = Sik (S-1)ik = Rik (7.5.12) In (7.5.11) we see a certain "down tilt" matrix multiplication which will be explained later in ***. 5. The R-S Tilt Theorem: Sab = Rba which is the same as: (∂xa/∂x'b) = (∂x'b/∂xa) (7.5.13) Notice in this rule that the indices are reflected in a vertical line running between the indices. Proof: 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, see (7.4.18) 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 // δb'b" = (Rb'a' Sa'b") from (7.5.11) g' bb' Rb'a' (Saa" Sa'b" g'a"b") = Sab // regroup (7.5.14) g' bb' Rb'a' (gaa') = Sab // gaa' = Saa" Sa'b" g'a"b" from (7.5.7) (g' bb' Rb'a' gaa') = Sab // regroup Rba = Sab // g' lowers left index of Rij , g raises right index, as in (7.5.9) Notice that the above theorem says Sab = (∂xa/∂x'b) = (∂x'b/∂xa) = Rba (7.5.15) 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) (7.5.16) Here index a is always in x-space, while index b is in x'-space. 6. Determinants of R and S. We showed earlier that Rij→ Rij and Sij→ Sij . The determinants det(R) and det(S) translate as follows: 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 (7.5.17) where ε is the bookkeeping permutation tensor discussed in *** below. 7.6 Orthogonality Rules, Inversion Rule, Cancellation Rule 1. Orthogonality Rules: The above theorem Sab = Rba can be used to eliminate S in various forms of SR = 1: SR = 1 → Sab Rbc = δac => Rba Rbc = δac => Rba Rbc = gac from (7.4.19) (7.6.1) Although R is not a tensor, the object Rba Rbc is a tensor of the mixed type Mac. Thus, both sides of this last equation transform as this type of mixed rank-2 tensor. In this last equation, we can lower index a on both sides, raise index c on both sides, and reverse the tilt of the b contraction to get this result, again using (7.4.19), Rba Rbc = gac = δac. (7.6.2) We now repeat this process starting instead with RS = 1: RS = 1 → Rab Sbc = δac => Rab Rcb = δac => Rab Rcb = gac Rab Rcb = gac = δac (7.6.3) Collecting the four forms written with δ we obtain the four orthogonality rules for R: Rba Rbc = δac Rba Rbc = δac // b sum is on first indices Rab Rcb = δac Rab Rcb = δac // b sum is on second indices (7.6.4) 2. Inversion Rule. Consider now an equation which one wants to invert for the object on the right, [----a-----] = Rab [--------b------] // before (7.6.5a) The bracketed objects are two arbitrary tensors which might have many up-down indices which are indicated by dashes. The inversion rule for moving R to the other side of the equation is to reflect R's two indices in the horizontal line between the indices: Rab [----a-----] = [--------b------] // after (7.6.5b) Proof: Rename b→b' in (7.6.5a), 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 application, Sab = Rba (7.5.13) 3. Cancellation Rule. Next, consider a different generic equation ( bracketed objects are again arbitrary tensors) Rab [----b-----] = Rab [--------b------] // before (7.6.6a) 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 (7.6.6b) Proof: Rename b→b' in (7.6.6a), 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 7.7 About δ and ε The Kronecker δ is sometimes written in different ways to make things "look nice", δab = δab = δba = δba = δa,b . (7.7.1) Eq (7.4.19) showed that that one can regard the above sequence of equalities as saying gab = gab = gba = gba = δa,b (7.7.2) 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 (7.7.3) or in an ordinary cross product Aa = εabcBbCc . (7.7.4) 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 in Section 7.4 item 2, 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 (7.7.5) 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. 7.8 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' (7.8.1) 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. (7.8.2) 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' (7.8.3) 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 (7.8.4) 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. (7.8.5) 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 (7.8.6) 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 (7.8.7) 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 (7.8.8) 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 (7.8.9) 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 ) 7.9 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 (7.9.1) 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 (7.9.2) 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 (7.9.3) Then we make the conversion using (7.6.2) M'ad = RabRdcMbc → (M')ad = RabRdcMbc (Standard Notation) (7.9.4) 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..... (7.9.5) 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..... (7.9.6) det(Aut) = det(Aij) = εabc... A1a A2bA3c..... = εabc... Aa1 Ab2Ac3..... (7.9.7) 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 ****.