tensor doc 1-7

Tensor Analysis and Curvilinear Coordinates Phil Lucht Rimrock Digital Technology, Salt Lake City, Utah 84103 last update: March 8, 2012 Note: The Appendices for this document are located in a separate document. Overview and Summary 5 1. The Transformation F: invertibility, coordinate lines, and level surfaces 8 Example 1: Polar coordinates (N=2) 8 Example 2: Spherical coordinates (N=3) 9 Cartesian Space and Quasi-Cartesian Space 11 Pictures A,B,C and D 11 Coordinate Lines 12 Example 1: Polar coordinates, coordinate lines 13 Example 2: Spherical coordinates, coordinate lines 13 Level Surfaces 14 2. Linear Local Transformations associated with F : scalars and two kinds of vectors 16 (a) Scalars 17 (b) Contravariant vectors 18 (c) Covariant vectors 18 (d) Bar notation 19 (e) Origin of the names contravariant and covariant 19 (f) Other vector types? 20 (g) Linear transformations 20 (h) Vectors that are contravariant by definition 21 (i) Vector Fields 21 (j) Names and symbols 22 (k) Definition of the words "scalar" and "vector". 23 3. Tangent Base Vectors en and Inverse Tangent Base Vectors u'n 24 (a) Definition of the en ; the en are the columns of S 25 (b) en as a contravariant vector 26 (c) a semantic question: unit vectors 27 Example 1: Polar coordinates, tangent base vectors 27 Example 2: Spherical Coordinates, tangent base vectors 29 (d) The inverse tangent base vectors u'n and inverse coordinate lines 30 Example 1: Polar coordinates: inverse tangent base vectors and inverse coordinate lines 31 4. Notions of length, distance and scalar product in Cartesian Space 33 5. The Metric Tensor 35 (a) Definition of the metric tensor 35 (b) Inverse of the metric tensor 37 (c) A metric tensor is symmetric 38 (d) det(g) and gnn of a Cartesian-generated metric tensor are non-negative 38 (e) Definition of two kinds of rank-2 tensors 38 (f) Proof that the metric tensor and its inverse are both rank-2 tensors 39 (g) Metric tensor converts vector types 41 (h) Vectors in Cartesian space 42 (i) Metric tensor: covariant scalar product and norm 42 (j) Metric tensor and tangent base vectors 44 (k) The Jacobian J 45 (l) Some relations between g, R and S in Picture C 48 Example 1: Polar coordinates: metric tensor and Jacobian 49 Example 2: Spherical coordinates: metric tensor and Jacobian 50 (m) Special Relativity and its Metric Tensor: vectors and spinors 50 (n) General Relativity and its Metric Tensor 54 (o) Continuum Mechanics and its Metric Tensors 55 6. Reciprocal Base Vectors En and Inverse Reciprocal Base Vectors U'n 58 (a) Definition of the En 58 (b) The Dot Products and Reciprocity (Duality) 59 (c) Covariant partner for En 60 (d) Summary of the basic facts: 61 (e) Repeat the above for the inverse transformation: definition of the U'n 61 (f) Expanding vectors on different sets of basis vectors 62 (g) Another way to write the En 65 (h) Comparison of n and En 66 (i) Handedness of coordinate systems: the en , the sign of det(S), and Parity 66 7. Translation to the Standard Notation 70 (a) Outer Products 70 (b) Mixed Tensors and Notation Issues 70 (c) The up/down bell goes off 71 (d) Some Preliminary Translations: raising and lowering indices on a vector with g 72 (e) Contraction of a Pair of Indices 73 (f) Dealing with the matrix R 74 (g) Repeat the above section for S 75 (h) About ε and δ 75 (i) Further development of the Standard Notation 76 (j) Tensors of Rank n, direct products, Lie groups, symmetry and Ricci-Levi-Civita 83 (k) The Contraction Tilt-Reversal Rule 86 (l) The Contraction Neutralization Rule 87 (m) Raising and lowering indices on g 88 (n) Other forms of R 89 (o) Summary of facts about R 89 (p) Repeat all the above for S 89 (q) Theorem: Sab = Rba and Sab = Rba ( reflect indices in vertical line between them) 90 (r) Orthogonality Rules 91 (s) The tangent and reciprocal base vectors and expansions on same 92 (t) Comment on Covariant versus Contravariant 95 (u) The Significance of Tensor Analysis 96 (v) The Christoffel Business: covariant derivatives 99 (w) Expansions of higher order tensors 100 8. Transformation of Differential Length, Area and Volume 102 Overview 102 (a) The differential N-piped mapping 103 (b) Properties of the finite N-piped spanned by the en in x-space 104 (c) Back to the differential N-piped mapping: how edges, areas and volume transform 107 Examples of area magnitude transformation for N = 2,3,4 117 Example 2: Spherical Coordinates: area patches 117 (d) Transformation of Differential Volume applied to Integration 118 (e) Interpretations of the Jacobian 120 9. The Divergence in curvilinear coordinates 121 (a) Geometric Derivation of the Curvilinear Divergence Formula 121 (b) Various expressions for div B 124 (c) Translation from Picture B to Picture M&S 126 (d) Comparison of various authors' notations 127 10. The Gradient in curvilinear coordinates 129 (a) Expressions for grad f 129 (b) Expressions for grad f B 131 11. The Laplacian in curvilinear coordinates 133 12. The Curl in curvilinear coordinates 135 (a) Definition of curl B 135 (b) Computation of the line integral 137 (c) Solving for the curl 139 (d) Various forms of the curl 140 (e) The curl in orthogonal coordinate systems 141 (f) The curl in N > 3 dimensions 142 13. The Vector Laplacian in curvilinear coordinates 144 (a) Derivation of the Vector Laplacian in general curvilinear coordinates 144 (b) The Vector Laplacian in orthogonal curvilinear coordinates 148 (c) The Vector Laplacian in Cartesian coordinates 150 14. Summary of Differential Operators in curvilinear coordinates 152 (a) divergence 153 (b) gradient and gradient dot vector 153 (c) Laplacian 154 (d) curl 154 (e) vector Laplacian 155 Example 1: Polar coordinates: a practical curvilinear notation 157 15. Covariant derivation of all curvilinear differential operator expressions 159 (a) Review of Sections 9 through 13 159 (b) The Covariant Method 160 (c) divergence (Section 9) 161 (d) gradient and gradient dot vector (Section 10) 161 (e) Laplacian (Section 11) 162 (f) curl (Section 12) 162 (g) vector Laplacian (Section 13) 162 References 166 Overview and Summary This paper develops elementary tensor analysis (also known as tensor algebra or tensor calculus) starting from Square Zero which is an arbitrary invertible continuous transformation x' = F(x) in N dimensions. The subject was "exposed" by Gregorio Ricci in the late 1800's under the name "absolute differential calculus". He and his student Tullio Levi-Civita published a masterwork on the subject in 1900 (see References). Christoffel and others had laid the groundwork a few decades earlier. The general mathematical classification of this subject is now called differential geometry. Two somewhat different applications of tensor analysis are treated concurrently. One is the subject of curvilinear coordinates in N dimensions, while the other involves transformations connecting "frames of reference". These transformations could be spatial rotations, the Lorentz transformations of special relativity, the transformations involving the effects of gravity in general relativity, or the deformation transformations associated with the flow of continuous matter. Beyond establishing the tensor analysis formalism, not much is said about this second set of applications. On the other hand, all the basic expressions for the standard differential operators in general curvilinear coordinates are derived from scratch. These results are often stated but not so often derived. The first six sections develop the theory of tensor analysis in a simple developmental notation where all indices are subscripts, just as in normal college physics. After providing motivation, the seventh section translates this developmental notation to the Standard Notation in use today. The eighth section treats transformations of length, area and volume and then the curvilinear differential operator expressions are derived, one per section, with a summary in the penultimate section. The final section rederives all the same results using the notion of covariance and associated covariant derivatives. The information is presented informally as if it were a set of lectures. Little attention is paid to mathematical rigor. There is no attempt to be concise: examples are given, tangential remarks are inserted, almost all claims are derived in line, and there is a certain amount of repetition. The material is presented in a planned sequence to minimize the need for forward references, but the sequence is not perfect. The interlocking pieces of tensor analysis do seem to exhibit a certain logical circularity. Section 1 introduces the notion of the general invertible transformation x' = F(x) as a mapping between x-space and x'-space. The range and domain of this mapping are considered in the familiar examples of polar and spherical coordinates. These same examples are used to illustrate the general ideas of coordinate lines and level surfaces. Certain Pictures are introduced to allow different names for the two inter-mapped spaces, for the function F, and for its associated objects. Section 2 introduces the linear transformations R and S=R-1 which approximate the (generally non-linear) x' = F(x) in the local neighborhood of a point x. It is shown that two types of vectors naturally arise in the context of this linearization, called contravariant and covariant, and an overbar is used to distinguish a covariant vector. Vector fields are defined and their transformations stated. The idea of scalars and vectors as tensors of rank 0 and rank 1 is presented. Section 3 defines the tangent base vectors en(x) which are tangent to the x'-coordinate lines in x-space. In the example of polar coordinates it is shown that er = and eθ = r . The vectors en exist in x-space and form there a complete basis which in general is non-orthogonal. The tangent base vectors u'n(x') of the inverse transformation x = F-1(x') are also defined. Section 4 is brief review of the notions of norm, metric and scalar product in Cartesian Space. Section 5 addresses the metric tensor, called in x-space and ' in x'-space. The metric tensor is first defined as a matrix object , and then g ≡ -1. A definition is given for two kinds of (pure) rank-2 tensors (both matrices), and it is then shown that transforms as a covariant rank-2 tensor while g is a contravariant rank-2 tensor. It is demonstrated how applied to a contravariant vector V produces a vector that is covariant = V, and conversely g = V. In Cartesian space g = 1, so the two types of vectors coincide. The role of the metric tensor in the covariant vector dot product is stated, and the metric tensor is related to the tangent base vectors of Section 3. The Jacobian J and associated functions are defined, though the significance of J is deferred to Section 8. The last three subsections briefly discuss the connection between tensor algebra and special relativity (with a mention of spinor algebra), general relativity, and continuum mechanics. Section 6 introduces the reciprocal (dual) base vectors En which are later called en in the Standard Notation. Of special interest are the covariant dot products among the en and En. It is shown how an arbitrary vector can be expanded onto different basis sets. It is found that when a contravariant vector in x-space is expanded on the tangent base vectors en, the vector components in the expansion are in fact those of the contravariant vector in x'-space, V'i = RijVj. This fact proves useful in later sections which express differential operators in x-space in terms of curvilinear coordinates and objects of x'-space. The reciprocal base vectors U'n of the inverse transformation are also discussed. Section 7 motivates and then makes the transition from the developmental notation to the Standard Notation where contravariant indices are up and covariant ones are down. Although such a transition might seem completely trivial, confusing issues do arise. Once a matrix can have up and down indices, matrix multiplication and other matrix operations become hazy: a matrix becomes four different matrices. The matrices R and S act like tensors, but are not tensors, and in fact are not even located in a well-defined space. The third last subsection discusses the significance of tensor analysis with respect to physics in terms of covariant equations, and the second last broaches the topic of the covariant derivative of a vector field with its associated Christoffel symbols. Finally, the last subsection describes how to expand tensors of any rank in various bases and notations. The focus then fully shifts to curvilinear coordinates as an application of tensor analysis. The final sections are all written in the Standard Notation. Section 8 shows how differential length, area and volume transform under x' = F(x). This section considers the inverse mapping of a differential orthogonal N-piped (N dimensional parallelepiped) in x'-space to a skewed one in x-space. It is shown how the scale factors h'n = describe that ratio of N-piped edges, while the Jacobian J = describes the ratio of N-piped volumes. The relationship between the vector areas of the N-pipeds is more complicated, and it is found that the ratio of vector area magnitudes is . Heavy use is made of the results of Appendices A and B, as outlined below. Sections 9 through 13 use the information of Section 8 and earlier material to derive expressions for all the standard differential operators expressed in general non-orthogonal curvilinear coordinates: divergence, gradient, Laplacian, curl, and vector Laplacian. The last two operators are treated only in N=3 dimensions where the curl has a vector representation, but then the curl is generalized to N dimensions. Section 14 summarizes all the differential operator expressions in a set of tables, and revisits the polar coordinates example one last time to illustrate a reasonably clean and practical curvilinear notation. Section 15 rederives the general results of Sections 9 through 13 using the ideas of covariance and covariant differentiation. These derivations are elegantly brief, but lean heavily on the idea of tensor densities (Appendix D) and on the implications of covariance of tensor objects involving covariant derivatives (Appendix F). Much of our content is contained in a set of Appendices, which are located in a separate document with its own table of contents. Appendix A develops an alternative expression for the reciprocal base vector En as a generalized cross product of the tangent base vectors en, applicable when x-space is Cartesian. This alternate En is shown to match the En defined in Section 6, and the covariant dot products involving En and en are verified. Appendix B presents the geometry of a parallepiped in N dimensions (called an N-piped). Using the alternate expression for En developed in Appendix A, it is shown that the vector area of the nth pair of faces on an N-piped spanned by the en is given by ± An, where An = |det(S)| En , revealing a geometric significance of the reciprocal base vectors. Scaled by differentials so dAn = |det(S)| En(Πi≠n dx'i), this equation is then used in Section 9 where the divergence of a vector field is defined as the total flux of that field flowing out through all the faces of the skewed differential N-piped in x-space divided by its volume. This same dAn appears in Section 8 with regard to the transformation of N-piped face vector areas. Appendix C presents a case study of an N=2 non-orthogonal coordinate system, elliptical polar coordinates. Both the forward and inverse coordinate lines are displayed. The meaning of the curvilinear (x'-space) component V'n of a contravariant vector is explored in the context of this system, and the difficulties of drawing such components in non-Cartesian (curvilinear) x'-space are pondered. Finally, the Jacobian Integration Rule for changing integration variables is derived. Appendix D discusses tensor densities and their rules of the road. Special attention is given to the Levi-Civita ε tensor, including a derivation of all the εε contraction formulas and their covariant statements. It is noted that the curl of a vector is a vector density. Appendix E describes direct product and polyadic notations (including dyadics) and shows how to expand tensors (and tensor densities) of arbitrary rank on an arbitrary basis. Appendix F deals with covariant derivatives and the affine connection Γ which tells how the tangent base vectors en(x') change as x' changes. Everything is derived from scratch and the results provide the horsepower to make Section 15 go. The last two appendices provide demonstrations of most ideas presented in this paper. In each appendix, a connection is first made to continuum mechanics, and then the results are derived both by "brute force" and by the covariant technique enabled by Appendix F. Maple is used to compute the results for several coordinate systems. Appendix G shows how to express the dyadic object (v) in curvilinear coordinates. Appendix H does the same for the object div(T) = ∂jTij. Notations diag(a,b,c..) means a diagonal matrix with diagonal elements a,b,c.. RHS, LHS refer to the right hand side and left hand side of an equation QED = which was to be demonstrated ("thus it has been proved") det(A), AT = determinant of the matrix A, transpose of a matrix A = un = unit vector pointing along the nth positive axis of some coordinate system // indicates a comment on something shown to the left of // Maple = a computer algebra system similar to Mathematica V,a means ∂aV which means ∂V/∂xa, and V;a refers to the corresponding covariant derivative 1. The Transformation F: invertibility, coordinate lines, and level surfaces If x and x' are elements of the vector space RN (N-dimensional reals) , one can specify a mapping x' = F(x) F: RN → RN defined by a set of N continuous (C2) functions Fi , each of N variables, x'1 = F1(x1, x2, x3... xN) x'2 = F2(x1, x2, x3... xN) ... x'N = FN(x1, x2, x3... xN) If all functions Fi are linear in all of their arguments, then the mapping F: RN → RN is a linear mapping. Otherwise the mapping is non-linear. A mapping is often referred to as a transformation. We shall be interested only in transformations which are 1-to-1 and are therefore invertible. For such transformations, x' = F(x) x = F-1(x') , or in an equivalent notation x' = x'(x) x = x(x') In the transformation x' = F(x), if x roams over the entire RN of x-space (the domain is RN), we may find that x' roams over only some subset of RN in x'-space. The 1-to-1 invertible mapping is then between the domain of mapping F which is all of RN, and the range of mapping F which is this subset. As just noted, it will be assumed that x' = F(x) is essentially invertible so x = F-1(x') exists for any x'. By essentially is meant there may be a few problem points in the transformation which can be "fixed up" in some reasonable manner so that x' = F(x) is invertible. The functions Fi must be C1 continuous to support the linearization derivatives appearing in Section 2, and they must be C2 continuous to support some of the differential operators expressed in curvilinear coordinates in Sections 9-14 and the covariant derivative in Section 7 (v). Example 1: Polar coordinates (N=2) (a) The transformation from Cartesian to polar coordinates is given by, x = (x1, x2 ) = (x,y) x' = (x1', x2') = (θ,r) // note that r = x2' x = F-1(x') ↔ x = rcos(θ) x1 = x2' cos(x1') y = rsin(θ) x2 = x2' sin(x1') x' = F(x) ↔ r = x2' = θ = tan-1(y/x) x1' = tan-1(x2/x1) (b) The transformation is non-linear because at least one component function( e.g., r = ) is not of the form r = Ax + By. In this transformation all functions are non-linear. (c) Here is a drawing showing the nature of this mapping: The domain of x' = F(x) in x-space on the right is all of R2, but the range in x'-space is shown in gray. Imitating the language of complex variables, we can regard this gray range as depicting the principle branch of the multi-variable function x' = F(x). Other branches are obtained by shifting the gray rectangle left or right by multiples of 2π. Still other branches are obtained by taking the other branch of the real function r = which produces down-facing rectangles. The principle branch plus all the other branches then fill up the E2 of x'-space, but we care only about the principle branch range shown in gray. (d) This mapping illustrates a "problem point" involving θ = tan-1(y/x). This occurs when both x and y are 0, indicated by the red dot on the right. The inverse mapping takes the entire red line segment into this red origin point, so we have a lack of 1-to-1 going on here, meaning that formally the function F is not invertible. This can be fixed up by eliminating the red line segment from the range of F, retaining only the point at its left end. Another problem is that both the left and right vertical edges of the gray area map into the real axis in x-space, and that is fixed by removing the right edge. Thus, by doing a suitable trimming of the range, F can be made fully invertible. No one has ever had major problems using polar coordinates due to these minor issues. Example 2: Spherical coordinates (N=3) (a) The transformation from Cartesian to spherical coordinates is given by, x = (x1,x2,x3 ) = (x,y,z) x' = (x1',x2',x3') = (r,θ,φ) x = F-1(x') ↔ x = r sin(θ) cos(φ) x1 = x1'sin(x2')cos(x3') y = r sin(θ) sin(φ) x2 = x1'sin(x2')sin(x3') z = r cos(θ) x3 = x1'cos(x2') x' = F(x) ↔ r = x1' = θ = cos-1(z/) x2' = cos-1(x3/ ) φ = tan-1(y/x) x3' = tan-1(x2/x1) (b) The transformation is non-linear because at least one component function( e.g., r = ) is not of the form r = Ax + By + Cz. In this transformation, all three functions are non-linear. (c) Here is a drawing showing the nature of this mapping The domain of x' = F(x) in x-space on the right is all of E3, but the range in x'-space is the interior of an infinitely tall rectangular solid on the left we shall call an "office building". We could regard this office building as depicting the principle branch of the multi-variable function x' = F(x). Other branches are obtained by shifting the building left and right by multiples of 2π, or fore and aft by multiples of π, or by flipping it vertically, taking the other branch of r = . The principle branch plus all the other branch offices then fill up the E3 of x-space, but we care only about the principle branch office building whose walls are mostly shown in gray. (d) This mapping illustrates some "problem points". One is that entire green office building main floor (r=0) maps into the origin in x-space. This problem is fixed by trimming away the main floor keeping only the origin point of the bottom face of the office building. Another problem is that the entire red line segment (θ = 0) maps into the red point shown in x-space. This is fixed by throwing out the back wall of the office building, retaining only a line going up the left edge of the back wall. A similar problem happens on the front wall (θ = π, blue) and we fix it the same way: throw out the wall but maintain a thin line which is the left edge of this front wall (this line is missing its bottom point). Thus, by doing a suitable trimming of the range, F is made fully invertible. Cartesian Space and Quasi-Cartesian Space (a) Cartesian Space. For the purposes of this document, a Cartesian Space in N dimensions is "the usual" Hilbert Space EN in which the distance between two vectors is given by the formula d(x,y) = => [d(x+dx,x)]2 = Σi=1N (dxi)2 metric tensor = diag(1,1,1....1) as discussed in Section 4 below. The θ-r space in the above Example 1 would be a Cartesian space if it were declared that the distance between two points there was D'2 = (θ-θ')2 + (r-r')2, but that is not the usual intent in using that space. As shown below, the metric tensor used there is g = diag (r2,1) and not diag(1,1). One might argue that our Cartesian Space is in fact a Euclidean space (hence EN) having Cartesian coordinates. A non-Cartesian space is sometimes referred to as a "curved space" (non-Euclidean) and the coordinates in such a space as "curvilinear coordinates". An example is the θ-r space above. With the Cartesian Space metric tensor as gC = 1 = diag(1,1....1), the above equations can be written d2(x,y) = gCij(xi-yi)(xj-yj) and [d(x+dx,x)]2 = gCij dxi dxj ≡ (ds)2 where repeated indices are implicitly summed (sometimes called the Einstein convention). (b) Quasi-Cartesian Space. We now define a Quasi-Cartesian Space (not an official term) as one which has a diagonal metric tensor G whose diagonal elements are independently +1 or -1 instead of all +1 as with gC. In a Quasi-Cartesian Space the two equations above become d2(x,y) = Gij(xi-yi)(xj-yj) and [d(x+dx,x)]2 = Gij dxi dxj ≡ (ds)2 and of course this allows for the possibility of a negative distance squared (see Section 5 (i)). Notice that G-1 = G for any distribution of the ±1's in G. As shown later, this means that that covariant and contravariant versions of G are the same. The motivation for introducing this Quasi-Cartesian Space is to cover the case of special relativity which involves 4 dimensional linear transformations with G = diag(1,-1,-1,-1). Pictures A,B,C and D We shall always work with one of four different "pictures" involving transformations. In each picture the spaces and transformations (and their associated objects) have certain names that prove useful in certain situations. The matrices R and S are associated with transformation F as described in Section 2 below, while G and g's are metric tensors. Systems not marked Cartesian could of course be Cartesian, but we think of them as general "curved" systems with strange metric tensors. And in general, all the full transformations might be non-linear. The polar coordinates example above was presented in the context of Picture B. Picture B is the right picture for studying curvilinear coordinates where for example x-space = Cartesian coordinates and x'-space = toroidal coordinates. Picture C is useful for making statements applying to objects in curved x-space where we don't want lots of primes floating around. Pictures A and D are appropriate for consideration of general transformations, as well as linear ones like rotations and Lorentz transformations. In Sections 9-14 Picture M&S (Moon & Spencer) is introduced for the special purpose of displaying the differential operator expressions. This is Picture B with x'→ u and g'→g on the left side. The entire rest of this section uses the Picture B context. Coordinate Lines Suppose in x'-space one varies a single coordinate, say x'i, keeping all the other coordinates fixed. In x'-space the locus of points thus created is just a straight line parallel to the x'i axis, or for a principle branch situation like that of the above examples, a straight line segment. When such a straight line or segment is mapped into x-space, the result is a curve known as a coordinate line. A coordinate line is associated with a specific x'-space coordinate x'i, so one might refer to the " x'i -coordinate line", x'i being a label. In N dimensions, a point x in x-space lies on a unique set of N coordinate lines with respect to a transformation F. Remember that each such line is associated with one of the x'i coordinates. In x'-space, a point x' lies on a unique intersection of straight lines or segments, and then this all gets mapped into x-space where point x = F-1(x') then lies on a unique intersection of coordinate lines. For example, in spherical coordinates we start with some (x,y,z) in x-space and compute the xi' = (r,θ,φ) in x'-space. Our point x in x-space then lies on the r-coordinate line whose label is r, it lies on the θ-coordinate line whose label is θ, and it lies on the φ-coordinate line whose label is φ (see below). In general a coordinate "line" is some non-planar curve in N-dimensional x-space, meaning that a coordinate line might not lie on an N-1 dimensional plane. In the 2D polar coordinates example below, the red coordinate line does not lie on a 1-dimensional plane (line). In the next example of 3D spherical coordinates, it happens that every coordinate line does lie on a 2-dimensional plane. But in ellipsoidal coordinates, another 3D orthogonal system, every coordinate line does not lie on a 2-dimensional plane. Some authors refer to coordinate lines as level curves, especially in two dimensions mapping the real and imaginary part of analytic functions w = f(z) ( Ahlfors p 89). Example 1: Polar coordinates, coordinate lines Here are some coordinate lines for our prototype N=2 non-linear transformation, Cartesian to polar coordinates: The red circle is a θ-coordinate line, and the blue ray is an r-coordinate line Example 2: Spherical coordinates, coordinate lines These coordinate lines are generated exactly as described above. In x'-space one holds two coordinates fixed while allowing one to vary. The locus in x'-space is a line segment or a half line (in the case of varying r). In x-space, the corresponding coordinate lines are as shown. The green coordinate line is a θ-coordinate line, since only θ is varying. The red coordinate line is an r-coordinate line, since only r is varying. The blue coordinate line is a φ-coordinate line, since only φ is varying. The point x indicated by a black dot in x-space lies on the unique set of coordinates lines shown. Appendix C gives an example of coordinate lines for a non-orthogonal 2D coordinate system. Level Surfaces (a) Suppose in x'-space one fixes one coordinate, say x'i, and varies all the other coordinates. In x'-space the locus of points thus created is just an (N-1 dimensional) plane perpendicular to the xi axis, or for a principle branch situation like that above, a rectangle or half strip in the case of r. Mapping this planar surface in x'-space into x-space produces a surface in x-space (of dimension N-1) called a level surface. The equations of the N different xi level surface types are a'i(n) = Fi(x1, x2.....xN) i = 1,2...N where a'i(n) is some constant value selected for fixed coordinate x'i. By taking some set of closely spaced values for this constant, { a'i(1), a'i(2).....}, one obtains a family of level surfaces all of the same general shape which are closely spaced. For some different value of i, the shapes of such a family of level surfaces will in general be different. In general if f(x1, x2.....xN) = k, the set of points x which make this equation true for some fixed k is called a level set, so a level set is a surface of dimension N-1. Thus, all our level curves are also level sets. In the polar coordinates example, since there are only 2 coordinates, there is no distinction between a level surface and a coordinate line. In the spherical coordinates example, there is a distinction. If one fixes r and varies θ and φ over their horizontal rectangle inside the office building, the level surface in x-space is a sphere. If one fixes θ and varies r and φ over a left-right vertical strip inside the office building, the level surface in x-space is a sphere is a polar cone If one fixes φ and varies r and θ over a fore-aft vertical strip inside the office building, the level surface in x-space is a half plane at azimuth φ. (b) In N dimensions there will be N level surfaces in x-space, each formed by holding some x'i fixed. The intersection of N-1 level surfaces (omitting say the x3' level surface) will have all of the x'i fixed except x'3. But this describes the x'3 coordinate line. Thus, each coordinate line can be considered as the intersection of the N-1 level surfaces associated with the other coordinates. One can see this happening on the spherical coordinates example: The green coordinate line is the intersection of two level surfaces: half-plane and sphere. The red coordinate line is the intersection of two level surfaces: half-plane and cone. The blue coordinate line is the intersection of two level surfaces: sphere and cone. 2. Linear Local Transformations associated with F : scalars and two kinds of vectors We now shift to the Picture A context, where x-space is not necessarily Cartesian. Consider again the possibly non-linear transformation x' = F(x) mapping F: RN→ RN. Imagine a very small neighborhood around the point x in x-space, a "ball" around x. Where the mapping is continuous in both directions, one expects a tiny x-space ball around x to map into a tiny x'-space ball around x' and vice versa. Here is a picture of this situation, where everything in one picture is the mapping of the corresponding thing in the other picture. In particular, we show a small vector in x-space called dx which maps into a small vector in x'-space called dx'. Since F was assumed invertible, it must be invertible locally in these two balls. That is, given a dx above, one can determine dx', and vice versa. Anticipating a few lines below, this means that the matrices S and R will be invertible so neither can have zero determinant. How are these two differential vectors related? For a linear approximation, x'i + dx'i = Fi(x + dx ) ≈ Fi(x) + Σk( ∂Fi(x)/∂xk) dxk => dx'i = Σk( ∂Fi(x)/∂xk) dxk The last line shows an equals sign in the limit that dxk is a vanishing differential. Since Fi(x) = x'i , dx'i = Σk(∂x'i/∂xk) dxk = Σk Rik dxk Rik ≡ (∂x'i/∂xk) Doing the same operation in the other direction gives dxi = Σk( ∂xi/∂x'k) dx'k = Σk Sik dxk' Sik ≡ (∂xi/∂x'k) One can regard Rik and Sik as elements of NxN matrices R and S. In vector notation then, dx' = R(x) dx Rik(x) ≡ (∂x'i/∂xk) R = S-1 // dx'i = Rij dxj dx = S(x') dx' Sik(x') ≡ (∂xi/∂x'k) S = R-1 // dxi = Sij dx'j It is obvious that matrices R and S are inverses of each other, just staring at the above two vector equations. One can verify this fact from the definitions of R and S using the chain rule (RS)ij = Σk RikSkj = Σk (∂x'i/∂xk) (∂xk/∂x'j) = Σk = = δi,j We could get rid of one of these matrices right now, perhaps keeping R and replacing S = R-1, but keeping both simplifies expressions encountered later, so for now both are kept. The letter R does not imply that matrix R is a rotation matrix, although it could be. According to the polar decomposition theorem (Lai p 110), any matrix R (detR ≠ 0) can be uniquely written in the form R = RU = VR where R is a rotation matrix (the same one in RU and VR) and U and V are symmetric positive definite matrices (called right and left stretch tensors) related by U = RTVR. Matrix S could of course be written in a similar manner. Matrices R(x) and S(x') are in general functions of a point in space x' = F(x). As one moves around in space, all the elements of matrices R and S are likely to change. So R and S represent point-dependent linear transformations which are valid for the differentials shown. One might wonder at this point how the vector dx is related to its components dxi and the same question for dx'i and dx'i. As will be shown in Section 6 (f), dx = Σndxn un where the un are x-space axis-aligned basis vectors of the form u1 = (1,0,0,..0) dx' = Σndx'n e'n where the e'n are x'-space axis-aligned basis vectors of the form e'n = (1,0,0,..0) If x-space and x'-space were Cartesian, one could write un = and e'n = ', but in general the un and e'n vectors do not have (covariant) unit length, as will be demonstrated later. The reader familiar with covariant "up and down" indices will notice that all indices are peacefully sitting "down" in the presentation so far (subscripts, no superscripts). As we carry out our various developmental tasks, that is where all indices shall remain until Section 7, whereupon they will start frantically bobbing up and down, seemingly at will. [ Since rules are made to be violated, we have violated this one in some examples below where non-standard notation would be hard to swallow. ] Are there any "useful objects" that can be constructed from differentials dx and which might then transform according by R or S? The answer is yes, but first we discuss scalars. (a) Scalars A quantity is a scalar with respect to transformation F if it is the same in both spaces. Thus, any constant like π would be a scalar under any transformation. The mass m of a potato would be a constant under transformations that are rotations or translations. A function of space φ(x) is a "field" and it would be a "scalar field" if φ'(x') = φ(x). For example, temperature would be a scalar field under rotations. Notice that φ is evaluated at x, while φ' is evaluated at x' = F(x). As noted in section (k) below, one could be more precise by referring to the objects described here as a "tensorial scalar" and a "tensorial scalar field". (b) Contravariant vectors If transformation F (possibly non-linear) transforms x-space to x'-space without affecting time, then consider the familiar velocity vector, vi = dxi/dt => v = dx/dt Since dt transforms as a constant (scalar) under our selected transformation type, it seems pretty clear that velocity in x'-space can be related to velocity in x-space using the dx' = R(x) dx rule above: v' = R(x) v Even though the matrix R(x) changes as we move around, this linear transformation R is valid at any point x when applied to velocity. Momentum p = mv would work the same way, since mass m is a scalar (Newtonian mechanics). In contrast, unless R(x) is a constant in space (which would be the case only if F were a linear transformation) x' ≠ R(x) x, so in general x itself is not a contravariant vector although dx is. Any vector that transforms according to V' = R(x)V with respect to a transformation F (such as Newtonian velocity and momentum with respect to rotations) is called a contravariant vector. (c) Covariant vectors Much of physics is described by differential equations involving the gradient operator ( the reason for the overbar is given in the next section) i = i = ∂/∂xi which involves an "upside down" differential. Here is how this operator transforms going from x-space to x'-space, again according to the chain rule (implied sum on k) , 'i = 'i = = = Skik = STik k = STik k => ' = ST One can think of as acting on a scalar field φ(x) = φ'(x'), and then the above becomes 'i φ'(x') = φ'(x') = φ(x) = STik k φ(x) => 'φ'(x') = ST φ(x) Since the differential is "upside down", one might expect to transform according to S = R-1 instead of R, but it is really ST that does the job. One could write ' = S in terms of row vectors. Vectors that transform according to V' = ST(x) V such as the gradient operator are called covariant vectors with respect to transformation F. An example of a covariant vector is the electrostatic electric field obtained from the potential Φ = - Φ i = - iΦ = - ∂Φ/∂xi (d) Bar notation In order to distinguish a contravariant from a covariant vector, we shall (for a while) adopt this bar convention: contravariant vectors shall be written V with components Vi and covariant vectors shall be written with components i. This is why overbars were placed on and i and in the previous section. We call this our "developmental notation", as distinct from the Standard Notation introduced in Section 7. The transformation rules for the two vector types can now be written this way: V' = R V contravariant Rik(x) ≡ (∂x'i/∂xk) R = S-1 ' = ST covariant Sik(x') ≡ (∂xi/∂x'k) = STki(x') One could imagine replacing S with some QT to make the second equation more like the first, but of course then RQT = 1 instead of RS = 1. In the Standard Notation, where there are four versions of the matrix R, we shall see that R → Rij and S → Sij = Rji and S can be removed from the picture (see Section 7 (q) ) . (e) Origin of the names contravariant and covariant A justification of the terms covariant and contravariant is presented at the end of Section 7 (t), since the idea is more easily presented there than here. It seems that these terms were first used in 1851 (a half century before special relativity) in a paper (see Refs.) by J.J. Sylvester of Sylvester's Law of Inertia fame. Sylvester uses the words covariant and contravariant to describe the relations between a pair of "transformations". In much simpler notation than he uses, if those "transformations" (functions) are F(x) and G(x) and if A is an 3x3 matrix, then the pair F(Ax) and G(Ax) are said to be covariant (or concurrent) the pair F(Ax) and G(A-1x) are said to be contravariant (or reciprocal) The idea is that in comparing the way two things transform, if they both move the same way, then it is covariant, and if they move in opposite directions it is contravariant. In Section 7 (t) this idea is applied to the transformation of two "things", where one thing is the component of a vector like Vn and the other thing is a basis vector onto which a vector is expanded. The connection is a bit distant, but the underlying concept carries through. Notations like y = F(Ax) would have mystified Sylvester in 1851, although in this same paper he introduced two-dimensional arrays of letters and referred to them as "matrices". According to a web piece by John Aldrich of the University of Southampton, J.W. Gibbs in 1881 was the first person to use a single letter to represent a vector (he used Greek letters). It was not until 1901 when his student E.B.Wilson published Gibb's lectures in a Vector Analysis book that the idea was propagated to a wider circle. Wilson converted those Greek letters to bolded ones, The Wilson/Gibbs book was reprinted seven times, the last being 1943. In 1960 it continued as a Dover book and is now available online as a public domain document. (f) Other vector types? Are there any other kinds of vectors with respect to a transformation F? There might be, but only the two types mentioned above are of interest to us in this document. They are both called rank-1 tensors, and there are no other rank-1 tensor types in "tensor analysis" (for rank-n tensors, see Section 7 (j)). Some authors refer to the rank of a tensor as the order of a tensor.) In the Standard Notation introduced later, where contravariant vector components are written with indices up and covariant vectors with indices down, and where the notation is so slick and smooth and automatic, one sometimes imagines there are two kinds of vectors because there are two places to put indices, up and down. It is of course the other way around: the up/down notation was adopted because there are two rank-1 tensor types. Two particular (linear) transformation types of interest are rotations and Lorentz transformations, each of which has a certain number of continuous parameters (3 and 6). As the parameters are allowed to vary over their ranges, the set of transformations can be viewed as elements of a continuous group ( SO(3) and SO(3,1) ). Each of these groups has exactly one "vector representation" ( "1" and "(1/2)(1/2)" ). One should not imagine that somehow the "two-ness" of vector types under general transformations F is connected to there being two vector representations of some particular group. It happens that the Lorentz group does have two "spinor representations" (1/2)0 and 0(1/2), but this has nothing at all to do with our general notion of two kinds of vectors. This subject is discussed in more detail in Section 5 (m). (g) Linear transformations For a linear transformation F, the matrix elements of R and S are constants and don't depend on x or x'. The reason is fairly obvious. For linear x' = F(x) (an added constant would make F non-linear ) F(αx' + βy') = αF(x') + βF(y') => x'i = Fi1x1 + Fi2x2 + .... FiN xN // = Fi(x) where the Fij are constants independent of the coordinates, in which case dx'i = Fi1 dx1 + Fi2 dx2 + .... FiN dxN = Σk Fik dxk so R = F and S = F-1 This is the situation with rotations and Lorentz transformations. (h) Vectors that are contravariant by definition A contravariant vector has been defined above as any N-tuple which transforms the same way that dx transforms with respect to F, namely, dx' = R(x) dx. One might state this as { dx', dx } dx' = R(x) dx contravariant vector Suppose we start with an arbitrary N-tuple V and simply define V' ≡ RV. One would have to conclude that the pair { V', V } transforms as a contravariant vector. { V', V } V' ≡ R(x)V contravariant vector Conversely, one could start with some given V' and define V ≡ S(x)V' (recall S = R-1), and again one would conclude that { V', V } represents a vector that transforms as a contravariant vector. We refer to either process as producing a vector which is "contravariant by definition". Creating a contravariant vector in this fashion is a fine thing to do, as long as the defined vector does not conflict with something that already exists. Example 1: We know that if F is non-linear, the vector x does not transform as a contravariant vector, because x' = R(x)x is not true, where x' = F(x). If we start with x and try to force {x', x} to be "contravariant by definition" by defining x' ≡ R(x) x, this x' conflicts with the existing x' = F(x), so the method of contravariant by definition is unacceptable. Example 2: As another example, consider an N-tuple in x'-space of three masses V' = (m1,m2,m3). The transformation is taken in this example to be regular rotations. Since masses are rotational scalars with respect to such rotations, we know that in an x-space rotated frame of reference we would find V = (m1,m2,m3). We could attempt to set up { V', V } as a vector that is "contravariant by definition" by defining V ≡ SV', but this conflicts with the existing fact that V = (m1,m2,m3), so the method of contravariant by definition is again unacceptable. Example 3: This time F is a general transformation and we start with V' = e'n which are a set of axis-aligned basis vectors in x'-space. We define vectors V = en according to en ≡ Se'n. Then { e'n, en } form a vector which is "contravariant by definition" and e'n = R en (R = S-1). Since the newly defined vector en does not conflict with some already-existing vector in x-space, the method of contravariant by definition in this example is acceptable. This is exactly what is done in the next section with the tangent base vectors en. (i) Vector Fields We considered above vectors like position x (and dx) and velocity v and the vector operator , and we referred to a generic vector as V. Many vectors of interest (in fact, most) are functions of x, which is to say, they are vector fields. Examples are the electric and magnetic fields E(x) and B(x), or the average velocity of a small region of fluid V(x) or a current density J(x). Another example is the transformation F(x). We already mentioned scalar fields, such as temperature T(x) or electrostatic potential Φ(x). The way a scalar temperature field transforms going from x-space to x'-space is this T'(x') = T(x) where x' = F(x) If the transformation is a 3D rotation from frame S to frame S', then T' is the temperature measured in frame S' at point x' and T is the temperature measured at the corresponding point x in frame S and of course there is only one temperature at that point so the numbers are equal. In x'-space one needs the prime on T' because the functional form (how T' depends on the x'i) is not the same as that of T (how T depends on the xi). For example, if transformation F is from 2D Cartesian to polar coordinates, then T'(r,θ) = T(x,y) = T(rcosθ,rsinθ) ≠ T(r,θ) Contravariant and covariant vector fields transform as described above, but now one must show the argument for each field in its own space, and again x' = F(x) : V'(x') = R V(x) contravariant Rik(x) ≡ (∂x'i/∂xk) R = S-1 '(x') = ST (x) covariant Sik(x') ≡ (∂xi/∂x'k) = STki(x') Similar transformation rules apply to tensors of any rank. For example, the metric tensor gab (developmental notation) is a rank-2 contravariant tensor field and the transformation rule is this g'ab(x') = Raa'Rbb'ga'b'(x) or g'ab = Raa'Rbb'ga'b' Often the coordinate dependence of g is suppressed, just as it is for R and S, as shown on the right above. Jumping momentarily into Standard Notation, in special relativity one has x'μ = Λμνxν where F = R = Λ is a linear transformation, and one would then specify the transformation of a contravariant vector field as V'μ(x'α) = Λμν Vν(xα) x'μ = Λμνxν (j) Names and symbols The matrix Rik(x) = (∂x'i/∂xk) is called the Jacobian matrix for the transformation x' = F(x) , while the matrix Sik(x') = (∂xi/∂x'k) is then the Jacobian matrix of the inverse transformation x = F-1(x'). The magnitude of the determinant of Jacobian matrix S will be shown in Section 8 (e) to have a certain significance, and that determinant is called "the Jacobian" = det(S(x')) ≡ J(x'). The author has anguished over what names to give the matrices R and S = R-1. One option was to use R = L, where L stands for the fact that this matrix is describing a Local coordinate system at point x, or a Linearized transformation. But L is always used for differential operators, so that got rejected. R is often called Λ in special relativity, but why go Greek so early? Another option is to use R = J for Jacobian, but J looks too much like "an integer" or angular momentum or "the Jacobian". T for Transformation might have been confused with the transformation F. Our chosen notation R makes one think perhaps R is a Rotation, but that won't in general be the case. For the moment we will continue to use R and S, where recall RS = 1. The fact that vectors are processed by NxN matrices R and S puts that part of the subject into the field of linear algebra, and that may be the origin of the name tensor algebra as a generalization of this idea (tensors as objects of direct product algebras). Of course the differential calculus aspect of the subject is already highly visible, there are ∂ symbols everywhere (hence the name tensor calculus). (k) Definition of the words "scalar" and "vector". These words have multiple potential definitions. Although we shall lapse very frequently, the following set of definitions would allow for precision statements: A "scalar" is a single number (or expression), a 1-tuple. A "tensorial scalar" is a scalar that transforms under transformation F as a tensorial scalar, which is also known as a rank-0 tensor. An example would be m' = m, mass with respect to 3D rotations. A "vector" is an N-tuple of numbers. A "tensorial vector" is a vector that transforms under transformation F as either a contravariant vector or a covariant vector, so a tensorial vector is a rank-1 tensor. A "scalar field" is a single function of x ( the x-space coordinates). A "tensorial scalar field" is a scalar field that transforms under transformation F as a tensorial scalar field, which is also known as a rank-0 tensor field. For example, f'(x') = f(x) is a scalar field. A "vector field" is an N-tuple of functions of x -- an N-tuple of scalar fields. A "tensorial vector field" is a vector field that transforms under transformation F as either a contravariant vector field or a covariant vector field, so a tensorial vector field is a rank-1 tensor field. The notion of tensor densities described in Appendix D further complicates the nomenclature. One can have scalar densities and vector densities of various weights. 3. Tangent Base Vectors en and Inverse Tangent Base Vectors u'n This entire section is in the context of Picture A, In the previous picture showing dx and dx', one has much freedom to "try out" different differential vectors. For any dx one picks at point x, one gets some dx' according to dx' = R(x) dx. Consider this slightly enhanced version of the previous drawing (red curves added) The point x in x-space (right side) can be regarded as lying on some arbitrary 1-dimensional curve in RN shown on the right in red. Select dx to be the tangent to this curve at point x. That curve will then map into some (probably very different) curve in x'-space which passes through the point x'. The tangent to this curve at the point x' must be dx' = R(x) dx. A similar statement can be made starting instead with an arbitrary curve in x'-space. The tangent dx' there then maps into dx = S(x') dx' in x-space. The curves are in N-dimensional space and are in general non-planar and the tangents are of course N dimensional tangents, so this 2D picture is mildly misleading. We now specialize such that the red curve on the left is a straight line parallel to an x'-space axis, which means the curve on the right is a coordinate line, Admittedly the drawing does not strongly suggest that the red line segment on the left is parallel to an axis in x'-space, but since those axes are not drawn, one cannot complain too strenuously. (a) Definition of the en ; the en are the columns of S First, define a set of N basis vectors in x'-space which point along the positive axes of x'-space, e'n , n = 1,2...N // (e'n)i = δn,i e'1 = (1,0,0...) etc Assume that the dx' arrow above points in this e'n direction so that dx' = e'n dx'n // no implied sum on n where dx'n is a positive differential variation of coordinate x'n along the e'n axis in x'-space. The corresponding dx in x-space will be, dx = S dx' = S [e'n dx'n] = [ Se'n] dx'n ≡ en dx'n where this last equality serves as the definition of en , en ≡ Se'n Vector en = en(x) points along dx in x-space and is tangent to the x'n- coordinate line there at point x. This vector en is generally not a unit vector, hence no hat ^ . Writing the above in components, dx = en dx'n => dxi = (en)i dx'n . But of course dxi = Sin dx'n , and therefore (en)i = Sin = ∂xi/∂x'n or en = ∂x/∂x'n = ∂'nx => (en)i = ∂xi/∂x'n = Sin This says that the vectors en are the columns of the matrix S: S = [e1, e2, e3 .... eN ] matrix = N columns We shall call these en vectors the tangent base vectors. The vectors exist in x-space and point along the various coordinate lines that pass through a point x. If the points on the x'n-coordinate line were labeled with the values of x'n from which they came, one would find that en points in the direction in which those labels increase. As one moves from x to some nearby point, the tangent base vectors all change slightly because in general S = S(x'(x)) and the en = en(x) are the columns of S. Any set of basis vectors which depends on x in this way is called a local basis. In contrast, the corresponding x'-basis e'n shown above with (e'n)i = δn,i is a global basis in x'-space since it is the same at any point x' in x'-space. Since det(S) ≠ 0 due to our assumption that F was invertible, the tangent base vectors are linearly independent and provide a basis for EN. One can of course normalize each of the en to be a unit vector n according to n = en/ |en|. Here is a traditional N=3 picture showing the tangent base vectors pointing along three generic coordinate lines in x-space all of which pass through the point x: Comment on notation. Some authors refer to our en as gn or Rn or other. Later it will be shown that enem = 'nm where 'nm is the covariant metric tensor for x'-space, so admittedly this provides a reasonable argument for using gn so that gngm = 'nm. But then the primes don't match which is confusing: the gn are vectors in x-space, while ' is a metric tensor in x'-space. We shall be using yet another g in the form g = det(nm) and a corresponding g'. Due to this proliferation of g objects, we stick with en, the notation used by Margenau and Murphy (p 193). A g-oriented reader can replace e → g as needed anywhere in this document. As for unit vector versions of the en, we use the notation n ≡ en/|en|. Morse and Feshbach use an for this purpose (Vol I p 22). A g-person might use n . A related issue is what symbols to use for the "usual" basis vectors in Cartesian x-space. As noted above, we are using un with (un)i = δn,i as "axis-aligned basis vectors" in x-space. If = 1 for x-space, then these are the usual Cartesian unit vectors (see section (c) below). Many authors use the notation en for these vectors which then conflicts with our use of en as the tangent base vectors. Morse and Feshbach use the symbols i, j, k for our Cartesian u1, u2, u3. Other authors use , , so then un = . Often the notation en is used to represent some generic arbitrary set of basis vectors. For this purpose, we use the notation bn. (b) en as a contravariant vector The situation described above was this, dx' = e'n dx'n x'-space // no implied sum on n dx = en dx'n x-space // no implied sum on n and the full transformation F maps dx into dx'. Since dx is a contravariant vector, the linear transformation R also maps dx into dx'. Thus dx' = R(x) dx e'n dx'n = R(x) en dx'n => e'n = R(x) en We can regard the last line as a statement that the vector en transforms as a contravariant vector under F. Written out in components one gets (e'n)i = Rij (en)j => δn,i= RijSjn recovering the fact that RS = 1. This is an example of a vector being "contravariant by definition", as discussed in Section 2 (h). These two expansions are easy to show just by verifying that components of both sides are the same: en ≡ Se'n = Σi Sin e'i since (en)j = Σi Sin (e'i)j = Σi Sin δi,j = Sjn = (en)j e'n ≡ Ren = Σi Rin ei since (e'n)j = Σi Rin (ei)j = Σi Rin Sji = (SR)jn = δj,n = (e'n)j (c) a semantic question: unit vectors Above it was noted that e'1 = (1,0,0....). Should this be called "a unit vector" ? It will be seen below that in fact |e'1| = ≠ 1 where ' is the covariant metric tensor in x'-space, and |e'1| is the covariant length of e'1. So e'n is a unit vector in the sense that it has a single 1 in its column vector definition, but it is not a unit vector in the sense that it does not (in general) have unit magnitude (it would if x'-space were Cartesian with g'=1).We take the magnitude = 1 requirement as the proper definition of a unit vector. For this reason, we refer to the e'n in x'-space as just "axis-aligned basis vectors" and they have no "hats". One wonders how such a vector should be depicted in a drawing, see Example 1 (b) below and also Appendix C (e). Example 1: Polar coordinates, tangent base vectors (a) The first step is to compute the matrix Sik(x') ≡ (∂xi/∂x'k) from the inverse equations: x = (x1, x2 ) = (x,y) x' = (x1', x2') = (θ,r) x = F-1(x') ↔ x = rcos(θ) x1 = x2' cos(x1') y = rsin(θ) x2 = x2' sin(x1') So S11 = (∂x/∂θ) = -rsinθ S12 = (∂x/∂r) = cosθ Sik ≡ ( ∂xi/∂x'k) S21 = (∂y/∂θ) = rcosθ S22 = (∂y/∂r) = sinθ S = // det(S) = -r R = S-1 = The tangent base vectors en can be read off as the columns of S e1 = r(-sinθ,cosθ) = eθ = r θ // = r e2 = (cosθ,sinθ) = er = r // = Notice that eθ in this case is not a unit vector. Below is a properly scaled drawing showing the location of the two x'-space basis vectors on the left, and the two tangent base vectors on the right. As just shown, the length of er is 1, while the length of eθ is 2. The tangent base vectors are fairly familiar animals, since er = and eθ = r in usual parlance. If one moves radially outward from point x, the er base vector stays the same, but eθ grows longer. If one moves azimuthally from x to some larger angle θ+Δθ, both vectors stay the same length but they rotate together staying perpendicular. (b) This is a good place to point out that vectors drawn in a non-Cartesian space can have magnitudes which do not equal the length of the drawn arrows. The "graphical arrow length" of a vector v is (vx2 + vy2)1/2, but that is not the right expression for |v| in a non-Cartesian space. For example, as will be shown below, |eθ'| = |eθ| , so the magnitude of the vector e'θ shown on the left above is in fact |eθ'| = r = 2 and not 1, but the graphical length of the arrow is 1 since e'θ = (1,0). See Appendix C (e) for further discussion of this topic with a specific 2D non-orthogonal coordinate system. (c) In this example, two basis vectors e'n in x'-space on the left map into the two en vectors on the right according to en ≡ Se'n. If one were to apply the full mapping x = F-1(x') to each point along the arrows e'n, for some general non-linear F one would find that these arrows map into warped arrows on the right whose bases are tangent to those of the en. Those warped arrows lie on the coordinate lines. For this particular mapping, e'θ maps under F-1 into the warped gray arrow, while e'r maps into er. Example 2: Spherical Coordinates, tangent base vectors x = (x1, x2, x3 ) = (x,y,z) x' = (x1', x2',x3') = (r,θ,φ) x = F-1(x') ↔ x = rsinθcosφ y = rsinθsinφ z = rcosθ S11= (∂x/∂r) = sinθcosφ Sik ≡ (∂xi/∂x'k) S12 = (∂x/∂θ) = rcosθcosφ S13 = (∂x/∂φ) = -rsinθsinφ S21= (∂y/∂r) = sinθsinφ S22 = (∂y/∂θ) = rcosθsinφ S23 = (∂y/∂φ) = rsinθcosφ S31= (∂z/∂r) = cosθ S32 = (∂z/∂θ) = -rsinθ S33 = (∂z/∂φ) = 0 S = R = where Maple computes R as S-1 and finds as well that det(S) = r2 sinθ The tangent base vectors are the columns of S, so er = (sinθcosφ, sinθsinφ,cosθ) |er| = 1 = h'r eθ = r(cosθcosφ,cosθsinφ,-sinθ) |eθ| = r = h'θ eφ = rsinθ(-sinφ,cosφ,0) |eφ| = rsinθ = h'φ and unit vector versions are then r = (sinθcosφ, sinθsinφ,cosθ) = er = θ = (cosθcosφ,cosθsinφ,-sinθ) = eθ = r φ = (-sinφ,cosφ,0) = eφ = rsinθ The unit vectors can be displayed in this standard picture, Notice that (, , ) = (1, 2, 3) form a right-handed coordinate system at the point x = r. (d) The inverse tangent base vectors u'n and inverse coordinate lines A complete swap x' ↔ x for a mapping x' = F(x) of course produces the "inverse mapping". This has the effect of causing R ↔ S in the above discussion. The tangent base vectors for the inverse mapping would then be the columns of matrix R instead of S. We shall denote these inverse tangent base vectors which exist in x'-space by the symbol u'n. Then: (en)i = Sin = ∂xi/∂x'n // the tangent base vectors as above S = [e1, e2, e3 .... eN ] // are the columns of S (u'n)i = Rin = ∂x'i/∂xn // inverse tangent base vectors R = [u'1, u'2, u'3 .... u'N ] // are the columns of R By varying only xn in x-space holding all the other xi = constant, one generates the xn-coordinate lines in x'-space, just the reverse of the earlier discussion of this subject. Then inverse tangent base vectors u'n will then be tangent to these inverse coordinate lines. An example is given below and another in Appendix C. In section (b) above the vector en transformed as a contravariant vector into an axis-aligned basis vector e'n in x'-space e'n = R en (e'n)i = Rij (en )j (en)i = Sin (e'n)i = δn,i The same thing happens here, only in reverse : u'n = S un (u'n)i = Sij (un)j (u'n)i = Rin (un)i = δn,i where now the un are axis-aligned basis vectors in x-space. A prime on an object indicates which space it inhabits. The inverse tangent base vectors u'n are not the same as the reciprocal base vectors En introduced in Section 6 below. Example 1: Polar coordinates: inverse tangent base vectors and inverse coordinate lines It was shown earlier for polar coordinates that, R = S-1 = so the inverse tangent base vectors are given by the columns of R, u'x = ( -sinθ/r,cosθ) // note near θ = 0 that u'x indicates a large negative slope u'y = (cosθ/r,sinθ) // note near θ = 0 that u'y indicates a small positive slope One expects u'x to be tangent to an inverse coordinate line in x'-space which maps to a line in x-space along which only x is varying, which is a horizontal line at fixed y (red). Looking at the small θ region of the plot on the left below, one sees slopes as just described above. For the polar coordinates mapping discussed above, horizontal (red) and vertical (blue) lines in x'-space mapped into circles (red) and rays (blue) in x-space, and the tangent base vectors in x-space were tangent to the coordinate lines there. If one instead takes horizontal (red) and vertical (blue) lines in x-space and maps them back into coordinate lines in x'-space, the picture is a bit more complicated. Since y = rsinθ, the plot of an x-coordinate line (x is varying, y fixed at yi) in x'-space has the form r = yi/sinθ, where yi denotes some selected y value (a red horizontal line), so plotting r = yi/sinθ in x'-space for various values of yi displays a set of inverse x-coordinate lines (red). Similarly r = xi/cosθ gives some y-coordinate lines (blue). Here is a Maple plot: x'-space (θ,r) x-space (x,y) Another example is given in Appendix C. 4. Notions of length, distance and scalar product in Cartesian Space This section can be interpreted in either Picture B or Picture D where the x-space is Cartesian, G=1. Up to this point, we have dealt only with the vector space RN (a vector space is sometimes called a linear space), and have not "endowed" it with a norm, metric or a scalar product. Quantities like dxi above were just little vectors and x + dx was vector addition. Now, for the first time (officially), we discuss length and distance, such as they are in a Cartesian Space, as defined in Section 1. For RN one first defines a norm which determines the "length" of a vector, the first notion of distance in a limited sense. The "usual" norm is the L2 norm given by norm of x = || x || ≡ ( x12 + x22 + .... + xN2 )1/2 ≡ | x | Now we have a normed linear space. One next defines the notion of the distance between two vectors. Although this can be done in many ways, just as there are many possible norms, for RN the "natural metric" is defined in terms of the above L2 norm, so that distance between x and y = metric = d(x,y) ≡ || x - y || = ( [x1-y1]2 + [x2-y2]2 + .... + [xN-yN]2 )1/2 . Now our space is both a normed linear space and a metric space, a combo known as a Banach Space. One finally adds the notion of a scalar product (inner product) in this way (x,y) ≡ Σixiyi ≡ x y // = Σi,j δi,j xi yj which of course implies this special case, (x,x) = x x = Σixi2 = ||x||2 = | x |2 Our space has now ascended to the higher level of being a real Hilbert Space of N dimensions. All this structure is implied by the notation RN, our "Cartesian Space". The length of the vector dx in RN is given by length of dx = distance between vectors x+dx and x ≡ ds ≡ || dx || = To avoid dealing with the square root, one usually writes (ds)2 ≡ || dx ||2 = Σi(dxi)2 = (dx1)2 + (dx2)2 + ... + (dxN)2 = Σi dxi dxi = Σi,j δi,j dxi dxj As shown in the next section, one can interpret δi,j as the metric tensor in Cartesian Space. The cursory discussion of this section is fleshed out in Chapter 2 of Stakgold where the concepts of linear spaces, norms, metrics and inner products are defined with precision. Stakgold compares our N dimensional Cartesian Hilbert Space to the N=∞ dimensional Hilbert Spaces used in functional analysis, where basis vectors might be Legendre polynomials Pn(z) on (-1,1), n = 0,1,2...∞. He has little to say, however, about curvilinear coordinate spaces in this particular book. 5. The Metric Tensor The metric tensor is the heart of the machine of tensor analysis and we shall have a lot to say about it in this section. Each subsection is best presented in the context of one of our Pictures, and there will be some jumping around between pictures. We apologize for this inconvenience and ask forbearance. Hopefully the subsections below will give the reader some experience with typical nitty-gritty manipulations. One advantage of the developmental notation over the standard notation is that matrix methods are easy to use, and they will be used below. We now go to the Picture D context. Comparison with Picture B shows that primes must be placed on objects F, R and S related to the transformation from x-space to x'-space: The various partial derivatives are determined from their definitions, R'ik ≡ (∂x'i/∂xk) R"ik ≡ (∂x"i/∂xk) Rik ≡ (∂x"i/∂x'k) S'ik ≡ (∂xi/∂x'k) S"ik ≡ (∂xi/∂x"k) Sik ≡ (∂x'i/∂x"k) The unprimed S,R can be expressed in terms of the primed objects this way (chain rule) Rik ≡ (∂x"i/∂x'k) = (∂x"i/∂xa) (∂xa/∂x'k) = R"ia S'ak => R = R" S' Sik ≡ (∂x'i/∂x"k) = (∂x'i/∂xa) (∂xa/∂x"k) = R'ia S"ak => S = R' S" (a) Definition of the metric tensor The metric or distance between vectors x and x+dx can be specified as done in Section 4 in terms of the norm of differential vector dx, metric(x+dx, x) = norm( [x+dx] - x) = norm(dx) ≡ ds with the caveat that this is not an official norm, see section (i) below. The squared distance (ds)2 must be a linear combination of products dxidxj just on dimensional grounds. The coefficients in this linear combination form a matrix called the metric tensor (later we show this matrix really is a tensor) (ds)2 = Σi=1N Σj=1N [ metric tensor ]ij dxi dxj This is a bit of chicken and egg because one is really defining "distance" and "metric tensor" at the same time. Each selection of a metric tensor defines the meaning of distance ds in the space of interest. Suppose the length of a small vector dx in a Quasi-Cartesian x-space is known to be ds. Recall from Section 1 that such a space has a diagonal metric tensor G whose diagonal elements are independently either +1 or -1. How might one express this same ds in terms of the other spaces' coordinates x' and x" ? (see Picture D) Going to x'-space one finds, since dx = S'(x') dx', (ds)2 = ΣiGiidxidxi = Σi Gii (ΣkS'ik dx'k) (ΣmS'im dx'm) = ΣkΣm { Σi Gii S'ikS'im } dx'k dx'm Defining the metric tensor in x'-space to be (comment on the bar below) 'km ≡ ΣiGiiS'ikS'im = Σij S'TkiGijS'jm => ' = S'TG S' one then has, with implied summation on the right, (ds)2 = ΣkΣm 'km dx'k dx'm = 'km dx'k dx'm For the transformation from x-space to x"-space in Picture D, a similar result is obtained, "km ≡ Σi GiiS"ikS"im => " = S"TG S" (ds)2 = "km dx"k dx"m Since (ds)2 is a number which is the same in all three systems (that number is the distance between two points in x-space), the quantity 'km dxk' dxm' is a tensorial scalar. The metric tensor is specific to a space; it is a property of the space; it is part of the space's definition. We have placed bars over the g's anticipating what will soon be shown, that these matrices are "covariant" matrices. Then we won't have to go back and fix things up. To summarize, there are three metric tensors for the three spaces in Picture D : = G ' = S'T G S' " = S"T G S" Concerning the invariance of (ds). In the above dicussion, it was assumed that distance (ds)2 is the same in x'-space as it is in x-space. As will be seen soon, this is equivalent to saying that the covariant dot product of any two vectors gives the same number regardless of which space is used to compute the dot product: A B = A' B'. This in turn implies that |A| = |A'| . In other words, it was assumed above that the dot product of two tensorial vectors is a scalar with respect to the underlying transformation F. In our major application, where x-space is Cartesian and x'-space is that of some curvilinear coordinates, it is a requirement that | A | = | A'| . The length of a physical vector is the same no matter how one chooses to describe that vector. Imagine that A is a velocity vector v. The speed |v| of an object is the same number whether one represents v in Cartesian or spherical coordinates. In special relativity one again wants dot products to be scalars and the notion that (ds)2 is a scalar under Lorentz transformations (that is, dxdx = dx'dx') is a key assumption/requirement of the theory. When it is required that (ds)2 ( or A B or |A|) be a tensorial scalar under transformation F from x-space to x'-space, then the metric tensors of the two spaces must be related by ' = S'TG S'. More generally as shown below, if (ds)2 is required the be a tensorial scalar, then one must have ' = ST S where x'-space and x-space are arbitrary spaces with metric tensors ' and . There are, however, applications of transformations where the scalarity of (ds)2 is not required and in fact it is crucial that (ds)2 can change under a transformation. For example, in continuum mechanics one can consider x-space to be a space describing a flow of continuous matter at some initial time t0 and x'-space to be the same flow at a later time t. A general flow has x' = F(x) where x is the position of a continuum "particle" at time t0 and x' is the position of that same particle at time t. In general F is non-linear. The distance between two differentially spaced particles at the two times is dx and dx', and one has dx' = R dx. The whole point here is that during the flow, the distance vector between two close particles rotates and stretches in some manner, and in general (due to this stretch), |dx| ≠ |dx'| , so (ds)2 is definitely not invariant under the flow (ie, under the transformation F). In this case, the rule ' = ST S does not apply, and one is free to select a metric tensor in each space independently. Since material flows usually occur in Cartesian space, one usually takes g = 1 and g' = 1. In Lai (p 105), the idea that dx' = Rdx translates to dx = FdX, and F is called the deformation gradient and is written F = (x) which is a dyadic like notation discussed in Appendix E and F below. The continuum mechanics application is discussed further in Section (o) below. In general, we shall be assuming that in fact (ds)2 is a scalar in almost everything that follows. (b) Inverse of the metric tensor The inverses of the three metric tensors shall be indicates without an overbar, and we shall eventually show these matrices to be "contravariant" matrices and thus deserve no overbar. We thus now define three new g matrices as these inverses, and compute the inverses: g ≡ -1 = G-1 = G // remember G just has +1 and -1 diagonal elements g' ≡ '-1 = (S'T G S')-1 = S'-1 G (S'T)-1 = R' G R'T g" ≡ "-1 = (S"T G S")-1 = S"-1 G (S"T)-1 = R" G R"T Here are the collected facts from above: g = G g' = R'G R'T g" = R" G R"T S = R' S" = G ' = S'TG S' " = S"T G S" R = R" S' g = 1 'g' = 1 "g = 1 Comment: In the Picture C context but with a Quasi-Cartesian x(0)-space, one could take the second column above and write it this way, g = RGRT = STGS g = 1 (ds)2 = km dxk dxm where now the clutter of primes is gone. If x-space is Cartesian so G = 1, then g = RRT and = STS. But we continue with Picture D. (c) A metric tensor is symmetric Any matrix of the form M = ATDA where D is a diagonal matrix (so D=DT) is symmetric: MT = (ATDA)T = ATDA = M // and similarly with A → AT Since all metric tensors shown above match this form, they are all symmetric: gab = gba for any g (with or without an overbar). (d) det(g) and gnn of a Cartesian-generated metric tensor are non-negative If we arrive at x'-space by a transformation F from a Cartesian x-space (as opposed to a Quasi-Cartesian one), we refer to the metric tensor g' in this x'-space as being "Cartesian generated". In this case G = 1 and the metric tensors above are g = RRT and = STS . Any matrix of either of these forms has positive diagonal elements and positive determinant: (ATA)aa = Σb (AT)abAba = Σb (A)baAba = Σb (Aba)2 ≥ 0 // diagonal elements ≥ 0 det(ATA) = det(AT) det(A) = det(A) det(A) = [ det(A) ]2 ≥ 0 // det ≥ 0 To show these results for the AAT form, just replace A→AT everywhere. Recall that transformation F maps RN → RN so the coefficients of the linearized matrices R and S are real, and elements of the metric tensor must therefore also be real. For a Quasi-Cartesian-generated metric tensor, these proofs are invalid since then g = RGRT and = STGS and G ≠1. (e) Definition of two kinds of rank-2 tensors We now switch to Picture A, Recall the vector transformation rules from Section 2 (d), V' = R V contravariant Rik(x) ≡ (∂x'i/∂xk) R = S-1 ' = ST covariant Sik(x') ≡ (∂xi/∂x'k) = STki(x') which can be written out in components V'a = Raa' Va' contravariant Rik(x) ≡ (∂x'i/∂xk) R = S-1 'a = STaa'a' covariant Sik(x') ≡ (∂xi/∂x'k) = STki(x') A rank-1 tensor is defined to be a vector which transforms in one of the two ways shown above. Similarly, a (non-mixed) rank-2 tensor is defined as a matrix which transforms in one of these two ways: M'ab = Raa' Rbb' Ma'b' // contravariant rank-2 tensor 'ab = STaa' STbb' a'b' // covariant rank-2 tensor and again we put a bar over the covariant objects. Digression: Proof that (A-1)T = (AT)-1 for any invertible matrix A: det(A) = det(AT) cof(AT) = [ cof(A)]T since [cof(AT)]ab = cof ( ATab) = cof(Aba) = [cof(A)]ba = [cof(A)]Tab (A-1)T = { [cof(A)]T / det(A) }T = [cof(AT)]T /det(AT) = (AT)-1 This fact is used many times in the manipulations below. (f) Proof that the metric tensor and its inverse are both rank-2 tensors The above rank-2 tensor transformation rules can be written in the following matrix form (something not possible with higher-rank tensors), M' = R M RT // contravariant rank-2 tensor ' = ST S // covariant rank-2 tensor where recall But we now switch these rules to the Picture D context where F maps x'-space to x"-space, M" = R M' RT // contravariant rank-2 tensor " = ST ' S // covariant rank-2 tensor Consider then this sequence of steps: 1 *G * 1 = 1 * G * 1 (S"R") G (S"R")T = (S'R') G (S'R')T // S"R" = 1 S" (R"G R"T) S"T = S'(R'G R'T) S'T // regroup S" g" S"T = S' g' S'T // since g" = R"G R"T and g' = R'G R'T g" S"T = R" S' g' S'T // left multiply by S"-1 = R" g" = R" S' g' S'T R"T // right multiply by S"T,-1 = R"T g" = (R" S') g' (S'T R"T) // regroup g" = (R" S') g' (R" S')T // (AB)T = BTAT g" = R g' RT // expressions in section (b) above for R and S This last result then shows that g' is a contravariant rank-2 tensor with respect to the transformation F taking x'-space to x"-space. Continuing on, g" = R g' RT g"-1 = (R g' RT)-1 g"-1 = ST g'-1 S // RT,-1= ST etc " = ST ' S // ' = g'-1 and this last result shows that ' is a covariant rank-2 tensor with respect to the transformation F taking x'-space to x"-space. This is why we put a bar over this g from the start. These two metric tensor transformation statements can be converted to the Picture A context, g' = R g RT g'ab = Raa'Rbb'ga'b' // g is a contravariant rank-2 tensor ' = ST S 'ab = STaa'STbb'a'b' // is a covariant rank-2 tensor Since RS = 1, the equations can be inverted to get g = S g' ST gab = Saa'Sbb'g'a'b' = RT ' R ab = RTaa'RTbb''a'b' Two more useful variations of the above are Rg = g' ST Rabgbc = g'abScb S = RT ' abSbc = Rba'bc (g) Metric tensor converts vector types We continue in Picture A. Suppose V is a contravariant vector so V' = RV. Construct a new vector W with the following properties ( see Section 7 (u) concerning "covariant equations") W = V x-space W' = ' V' x'-space Is vector W one of our two vector types, or is it neither? One must examine how it transforms under F: W' = ' V' = (ST S) (RV) = ST (SR)V = ST V = ST W Therefore this new vector W is a covariant vector under F, so it should have an overbar, ≡ V This covariant vector can be regarded as the covariant partner of contravariant vector V. This shows the general idea that applying to any contravariant vector produces a covariant vector! So this is one way to construct covariant vectors if we have a supply of contravariant ones. Conversely, starting with a known covariant vector , one can construct a contravariant vector V ≡ g . Thus, every vector of either type can be thought of as having a partner vector of the other type. An obvious notation is to write as so no extra letter is needed. Then one has = V V = g i = ij Vj Vi= gijj (h) Vectors in Cartesian space Theorem: There is no distinction between a contravariant and a covariant vector in Cartesian space. Proof: Pick a contravariant vector V. Since = 1, ≡ V = V . But is a covariant vector. Since = V , every contravariant vector is also covariant and vice versa. In other words, if g = 1, every vector is the same as its covariant partner vector. The transformation rules in this case are V' = R V ' = ST = ST V Although the vectors V and are the same, eliminating V shows that ' and V' are not the same. One finds that ' = (STS) V' = ' V', so ' = ' V' ≠ V'. (i) Metric tensor: covariant scalar product and norm For a Cartesian space, Section 4 defined the norm as the length of a vector, the metric as the distance between two vectors, and the scalar product (inner product) as the projection of one vector on another. The official definitions of norm, metric and scalar product require non-negativity: | x | ≥ 0, d(x,y) ≥ 0, and x x ≥ 0. For non-Cartesian spaces, the logical extensions of these three concepts can result in all three quantities being negative. Nevertheless, we shall use the term "covariant scalar product" with notation A B as defined below, as well as the notation |A|2 ≡ A A where |A| will be called the length, magnitude or norm of A, even though these objects are not true scalar products or norms. In the curvilinear application of tensor analysis, where x-space is Cartesian, since the norm and scalar product are tensorial scalars, and since they are non-negative in Cartesian x-space, the problem of negative norms does not arise in either space. How do authors handle this problem? Some authors refer to A A as "the norm" of A (e.g., Messiah bottom p 878 discussing special relativity), which is our |A|2. For a general 4-vector A in special or general relativity, most authors just write A A (AμAμ in standard notation), note that the quantity is invariant under transformations, but don't give it a name. Whereas we use the bold for this covariant dot product, most special relativity authors prefer to reserve this bold dot for a 3D spatial dot product, and then the 4D dot product is written with some "less bold dot" such as A.B or A•B. Typical usage then in standard notation would be p•p = pμpμ = p02 - pp (see for example Bjorken and Drell p 281). Without further ado, we define the "covariant scalar product" of two contravariant vectors (a new and different use of the word "covariant", but the same as appears in Section 7 (u) ) as: A B ≡ abAaBb = abBbAa = baBaAb = abBaAb = B A This covariant scalar (or dot) product is more interesting and useful than the object AaBa because the covariant scalar product of two contravariant vectors is a tensorial scalar, as we now show (Picture A) A' B' = 'abA'aB'b = 'ab(Raa'Aa') (Rbb'Bb') = 'ab Raa' Rbb' Aa' Bb' = [ (RT)a'a 'ab Rbb' ] Aa' Bb' = [RT ' R]a'b' Aa' Bb' = a'b' Aa' Bb' = ab Aa Bb = A B Recall that for any contravariant vector B, there is a partner covariant vector a = abBb. Using this partner one can restate the above covariant scalar product as A B = abAaBb = Aa a or, taking instead b = baAa , A B = b Bb = a Ba And finally, if in A B = Aa a we write Aa = gabb , we get A B = Aa a = gabba = gabab where the scalar product is now expressed in terms of the covariant partner vectors and . To summarize, there are four different ways to write this covariant scalar product : A B = abAaBb = Aa a = a Ba = gabab = B A Using the appropriate expressions on the above line, one may conclude that the covariant dot product of any two tensorial vectors is a tensorial scalar. In the special case that A = B, we use the shorthand notation (with caveat as noted above) |A|2 ≡ A A Going back to the a result of section (a), one sees the (ds)2 distance squared in a new light, (ds)2 = 'km dx'k dx'm = dx' dx' = dx dx = a scalar with respect to F so ds is sometimes called "the invariant distance". In special relativity, using the Bjorken and Drell notation noted above where g'μν = diag(1,-1,-,1,-1) and c=1, one writes ( standard notation) (dτ)2 = g'μν dx'μdx'ν = dxμdxμ = dx'• dx' = dx • dx = a Lorentz scalar = (dt)2 - dx dx , xμ = (t,x) and dτ is called "the proper time", a particular case of the invariant distance ds. Notice that (dτ)2 < 0 for a spacelike 4-vector dxμ, meaning one that lies outside the future and past lightcones (|dx| > |dt| ). We now restore to our covariant definition. Going back to Section 3 and the vectors e'n and en, a claim made there can now be verified: |e'n|2 = e'n e'n = en en = |en|2 => |e'n| = |en| (j) Metric tensor and tangent base vectors The context of Picture A continues, Recall this fact from Section 3, S = [e1, e2, e3 .... eN ] where the columns of S are the tangent base vectors. It follows that (see end of section (h)) ' = ST S = [e1, e2, e3 .... eN ]T [e1, e2, e3 .... eN ] so e1e1 e1 e2 e1 e3 ...... e1 eN e2e1 e2 e2 e2 e3 ...... e2 eN ' = e3e1 e3 e2 e3 e3 ...... e3 eN ........ eNe1 eN e2 eN e3 ...... eN eN since, enT em = (en)i ij (em)j = ij(en)i(em)j = en em using the covariant scalar product defined in the previous section. Taking the n,m component of the above matrix equation, one gets 'mn = em en or 'mn = ∂'mx ∂'nx which makes a direct connection between the covariant metric tensor in x'-space and the tangent base vectors en in x-space. A less graphical derivation of this fact is 'nm = (STS)nm = STna ab Sbm = ab San Sbm = ab (en)a(eb)n ≡ en em . Therefore, the tangent base vectors will only be mutually orthogonal when the metric tensor ' of x'-space is a diagonal matrix. We refer to the coordinates of an x'-space having a diagonal metric tensor as comprising an orthogonal coordinate system. At any point x in x-space, the tangents en to the N coordinate lines passing through that point are orthogonal. Most examples below will involve such systems, with Appendix C providing a non-orthogonal example. In particular, 'mn = em en lets us write the length of a tangent base vector in terms of the corresponding diagonal element of ', |en|2 = en en = 'nn => |en| = => n = en / The quantities |en| = are called scale factors and are sometimes written h'n or Q'n or H'n. h'n ≡ Q'n ≡ |en| = As a reminder, had we called x'-space something like ξ-space, there would be no primes on these symbols, but then if ≠1 there would be confusion as to which space the symbols applied. Section (d) above showed that 'nn ≥ 0 when x-space is Cartesian. This is the usual case for the curvilinear coordinates application, and so in this case the scale factors h'n are always real and positive. Note: Some authors refer to the scale factors h'n as the Lamé coefficients, while other authors refer to Rij as the Lamé coefficients which they call hji. (Lame) (k) The Jacobian J The context of Picture A continues, First of all, note that since RS = 1, det(S) = 1/det(R) The Jacobian J(x') is defined as follows, J(x') ≡ det(S(x')) = det(∂xi/∂x'k) = 1/det(R(x(x')) = 1/ det(∂x'i/∂xk) Note 1: Objects which relate to the transformation between x-space and x'-space cannot themselves be tensors because tensor objects must be associated with a specific space, the way V(x) is a vector in x-space and V'(x') is a vector in x'-space. Thus Sij(x') = ∂xi/∂x'k , although a matrix, is not a rank-2 tensor. Similarly, J(x'), while a "scalar" function, is not a rank-0 tensorial scalar. One does not ask how S and J themselves "transform" in going from x-space to x'-space. Note 2: An alternative notation used by some authors is this J(x,x') ≡ det(S(x,x')) = det(∂xi/∂x'k) as if x and x' were independent variables. In our presentation, x' = F(x) is not an independent variable but is determined by F(x). Just as one might write f'(x') = ∂f/∂x', we write J(x') = det(∂xi/∂x'k). The connection would be J(x') = J(x=F-1(x'),x')) = J(x(x'),x'). Note 3: Other sources often use the notation | M | to indicate the determinant of a matrix. We shall use the notation det(M), and reserve | | to indicate the magnitude of some quantity, such as |J| below. The determinant of any NxN matrix S may be written (εabc.. is the permutation tensor, Section 7 (h)), det(S) = εabc...x Sa1 Sb2 ... SxN For our particular S with Sin = (en)i this becomes det(S) = εabc...x (e1)a(e2)b....... (eN)x so J is related to the tangent base vectors by J = εabc...x (e1)a(e2)b....... (eN)x . It was shown in section (f) that ' = ST S and g' = R g RT , these being the transformation rules for covariant and contravariant rank-2-tensors. Therefore det(') = det(STS) = det(ST)det()det(S) = det(S)det(S)det() = J2 det() det(g') = det(RgRT) = det(R)det(g)det(RT) = det(R)det(R)det(g) = J-2 det(g) or det(') = J2 det() => J2 = det(') / det() = [det(S)]2 det(g') = J-2det(g) It is a tradition to define certain scalar (but not tensorial scalar) objects with the same name g and g', g(x) ≡ det((x)) = 1/det(g(x)) // in x-space g'(x') ≡ det('(x')) = 1/det(g'(x')) // in x'-space So that J2(x') = det('(x')) / det((x)) = g'(x') / g(x) Normally the argument dependence is suppressed and one then writes J2 = det(')/ det() = g'/g As explained in Appendix D (a), the equation g' = J2 g says that g, instead of being a tensorial scalar, is a scalar density of weight -2. One must be a little careful to distinguish the scalars g and g' from the tensors gij and g'ij expressed in matrix notation as g and g'. It is convenient to make the following definition, called the signature of the metric tensor, s = sign[det()] Since g = 1, one has det(g)det() = 1 so that sign[det()] = sign[det(g)] . Since det(') / det() = [det(S)]2, one has sign[det(')] = [det()]. Therefore: s = sign[det()] = sign[det(g)] = sign[det(')] = sign[det(g')] = sign(g) = sign(g') Since transformation F is assumed invertible in its domain and range, one cannot have det(S) = 0 anywhere except perhaps on a boundary. Since det(') = [det(S)]2det(), if we assume det() vanishes nowhere in the x-space domain of F, then det(') ≠0 everywhere in the range of F. The conclusion with this assumption is that the signature s is always well-defined. Obviously, the quantities sg and sg' are both positive, and since J2 = g'/g one can write |J| = / = | det(S) | = For the curvilinear coordinates application, x-space is Cartesian, det() = 1, and thus s = 1 and then |J| = = | det(S) | // curvilinear For the relativity application, x-space is Minkowski space with det() = -1 so s = -1 and |J| = = | det(S) | // relativity Here then is a summary of the results of this section: J(x') ≡ det(S(x')) = det(∂xi/∂x'k) = 1/det(R(x(x')) = 1/ det(∂x'i/∂xk) g ≡ det() g' ≡ det(') g' = J2g => g is a tensor density of weight -2 s ≡ sign[det()] = sign[det(g)] = sign[det(')] = sign[det(g')] = sign(g) = sign(g') |J| = / = | det(S) | = Note: Weinberg p 98 (4.4.1) defines g = -det(gij). This is the only one of Weinberg's conventions that we have not adopted, so in this paper it is always true that g ≡ + det(gij) even though this is -1 in the application to special relativity. Carl Gustav Jacob Jacobi (1804 –1851). German, Berlin PhD 1825 then went to Konigsberg, did much in a short life. Elucidated the whole world of elliptic integrals and functions, such as F(x,k) and sn(x;k), which occur even in simple problems like the 2D pendulum. Wiki claims he promoted Legendre's ∂ symbol for partial derivatives (used throughout this document) and made it a standard. Among many other contributions, he saw the significance of the object J which now bears his name: "the Jacobian". The Jacobi Identity is another familiar item, a rule for non-commuting operators [x,[y,z]] + [z,[x,y]] + [y,[z,x]] = 0 which finds use with quantum mechanical operators and matrices, and more generally with Lie group generators. (l) Some relations between g, R and S in Picture C In Picture C, the statement of the rank-2 tensor transformation of g and becomes g = RRT = STS which can be written in a variety of ways, RT = (SR)RT = S(RRT) = S g => R = g ST => 1 = S g ST ST = ST(RTST) = (STS)R = R => S = RT => 1 = RT R In summary: g = RRT RT = S g R = g ST 1 = S g ST = STS ST = R S = RT 1 = RT R The diagonal elements of and g are given by nn = Σn STniSin = Σn (Sin2) = Σi (∂xi/∂x'n)2 gnn = Σn RniRTin = Σn (Rni2) = Σi (∂x'n/∂xi)2 If the x'i are orthogonal coordinates, then nm = h'n2δnm and gnm = h'n-2δnm where the h'n are the scale factors mentioned above in section (j). These scale factors may then be expressed as ( M&F p 23 1.3.4) h'n2 = nn = Σi (∂xi/∂x'n)2 h'n-2 = gnn = Σi (∂x'n/∂xi)2 Example 1: Polar coordinates: metric tensor and Jacobian Picture C continues (so now θ = x1 and r = x2) and the metric tensor for polar coordinates will be computed in two ways. On the last visit to this example ( end of Section 3), it was shown that S = = [ e1, e2 ] e1 = r(-sinθ, cosθ) e2 = (cosθ, sinθ) One way to compute is this: ( 1=θ, 2=r) = STS = = => θθ = r2 rr = 1 Another way is this: = = // det() = r2 Notice that this metric tensor is in fact symmetric, and that one of its elements is a function of the coordinates. The length2 of a small vector dx can be written (ds)2 = km dxk dxm = θθ dθ dθ + rr dr dr = r2 (dθ)2 + (dr)2 The Jacobian is given by J(r,θ) = det(S(r,θ)) = det = -r so |J| = r and g = J2 = r2, = r Example 2: Spherical coordinates: metric tensor and Jacobian As with Example 1, Picture C is used, wherein (x1, x2, x3) = (r,θ,φ) . In our last visit to this example (end of Section 3) it was found that S = The metric tensor is then given by Maple as = STS = det() = r4sin2θ so that 11= rr = 1 h1 = hr = = 1 22= θθ = r2 h2 = hθ = = r 33= φφ = r2sin2θ h3 = hφ = = rsinθ The Jacobian is found by Maple to be, J(r,θ,φ) = det(S) = r2sinθ Differential distance is then (ds)2 = km dxk dxm = (dr)2 + r2(dθ)2 + r2sin2(dφ)2 and if dφ = 0, this agrees with the polar coordinates result. (m) Special Relativity and its Metric Tensor: vectors and spinors In this section the Standard Notation introduced below in Section 7 is used. In that notation Rij is written Rij , contravariant vectors Vi are written Vi, and covariant vectors j are written Vj. It is a tradition in special and general relativity to use Greek letters for 4-vector indices and Latin letters for spatial 3-vector indices. The (Quasi-Cartesian) metric tensor of special relativity is frequently taken as G = diag(1,-1,-1,-1) and the ordering of 4-vectors as xμ = (t,x,y,z) where c=1 (speed of light) and μ= 0,1,2,3 ( Bjorken and Drell p 281). General relativity people often use G = diag(-1,1,1,1) ≡ η instead (Weinberg p 26). Still other authors use G = 1 and xμ = (it,x,y,z) where i is the imaginary i, but this approach does not easily fit into our tensor framework which is based on real numbers. A Lorentz transformation is a linear transformation Fμν x'μ = Fμν xν = Rμν xν => xν is a contravariant vector and a theory requirement is that invariant length be preserved x'.x' = x.x = scalar. Special relativity also requires that the metric tensor G be the same in all frames, since no frame is special, so G' = G. But this says, in our old notation, that R G RT = G. This condition restricts the (proper) Lorentz transformations to be rotations, boosts (velocity transformations), or any combination of the two. In particular, R G RT = G => det(R G RT) = det(G) => det(R) det(G) det(RT) = det(G) => [det(R)]2 (-1) = (-1) => det(R) = ±1 Proper Lorentz transformations have det(R) = det(F) = +1, and here are two examples. First, a boost transformation in the x direction, Fμv = = exp(-ibK1) where (K1)μν = and second, a rotation transformation about the x axis, Fμv = = exp(-irJ1) where (J1)μν = The matrices K1 and J1 are called generators and are a part of a set of six 4x4 matrices Ji and Ki for i = 1,2,3. These 6 generator matrices satisfy a set of commutation relations known as a Lie Algebra, [ Ji, Jj] = +i εijkJk // [ A,B ] ≡ AB - BA [ Ji, Kj] = +i εijk Kk [ Ki, Kj] = -i εijkJk In these commutators, the generators Ji and Ki can be regarded as abstract non-commuting operators, while the specific 4x4 matrices shown above for J1 and K1 are just a "representation" of these abstract operators as 4x4 matrices. The six 4x4 generator matrices Ji and Ki are (g = G = diag(1,-1,-1,-1) ) (Jμν)αβ = i ( gμαδνβ – gναδμβ) (J1)αβ ≡ (J23)αβ = i ( g2αδ3β – g3αδ2β) and cyclic 123 (K1)αβ ≡ (J01)αβ = i ( g0αδ1β – g1αδ0β) and cyclic 123 where (-i)(Jμν)αβ = ( gμαgνβ – gναgμβ) is a rank-4 tensor, antisymmetric under μ↔ν and α ↔ β. An arbitrary Lorentz transformation can be represented as Fμv(r,b) = [ exp {– i ( rJ + bK)} ]μν where the 6 numbers r and b are called parameters (rotation and boost) and this F is a combined boost/rotation transformation (note that eA+B ≠ eAeB for non-commuting matrices A,B). The product of two such Lorentz transformations is also a Lorentz transformation, and in fact the transformations form a continuous group known as the Lorentz Group, which then has 6 parameters. The first two commutators shown above ( all J and all K) are each associated with a 3 parameter continuous group called the rotation group. The abstract generators of this group can be "represented" as matrices of any dimension, and are labeled by a number j such that 2j+1 is the matrix dimension. For example, the 2x2 matrix representation of the rotation group is labeled by j = 1/2, and is called the spinor representation and is associated in physics with the "intrinsic spin" of particles of spin 1/2 such as electrons. The vectors (spinors) in this case have two elements, and (1,0) and (0,1) are "up" and "down". Representations of the Lorentz group have labels {j1, j2}, where j1 is for the J-generated rotation subgroup, and j2 for the K-generated rotation subgroup, and are usually denoted j1j2. Such a representation then has vectors containing (2j1+1)(2j2+1) elements. In the case 1/21/2 there are 2*2=4 elements in a vector, and when these elements are linearly combined in a certain manner, they form the 4-vector object which one writes as Aμ such as xμ. This is the "vector representation" of the Lorentz group upon which is built the entire edifice of special relativity tensor algebra. One can also consider two other representations of the Lorentz group which are pretty obvious: 1/2 0 and 1/2 0. These are 2x2 matrix representations and they are different 2x2 representations. For each representation one can construct a whole tensor analysis edifice based on 2-vectors. Just as with the 4-vectors, one has contravariant and covariant 2-vectors.The two representations 1/2 0 and 1/2 0 are called spinor representations since they are each 2-dimensional. Since there are two distinct spinor representations, one needs some way of distinguishing them from each other. One representation might be called "undotted" and the other "dotted" and then there are four 2-vector types to worry about, which transform this way V'a = RabVb V' = RV contravariant 2-vectors V'a = RabVb V' = RV covariant 2-vectors where now dots on the indices indicate which Lorentz group representation that index belongs to. The 2x2 matrices Rab and R are not the same. A typical rank-2 tensor would transform this way, X' a = Raa'R' Xa'' This then is the subject of what is sometimes called Spinor Algebra as opposed to Tensor Algebra, but it is really just regular tensor algebra with respect to the two spinor representations of the Lorentz group. We have inserted this blatant digression just to show that the general subject of tensor analysis includes all this spinor stuff under its general umbrella. In closing, Maple shows that the metric tensor G is indeed preserved under boosts and rotations. In Maple evalm(Bx &* G &* transpose(Bx)) means Bx G BxT and Maple is just verifying that Bx G BxT = G and similarly Rx G RxT = G : (n) General Relativity and its Metric Tensor In general relativity a Picture of interest is Picture C but the x(0) space is replaced by a Quasi-Cartesian space with coordinates ξi with the metric tensor of special relativity. This ξ-space represents a "freely-falling" coordinate system in which the laws of special relativity apply and the metric tensor is taken to be G = diag(-1,1,1,1) ≡ η. The xi are the coordinates of some other coordinate system. There is some transformation x = F(ξ) which defines the relationship between these two systems. The covariant metric tensor in x-space is written = STGS = STηS. Using the Standard Notation introduced in Section 7 below, this is usually written as = STηS // result from Section 5 (b) gdn = ST ηdnS // where (ηdn)ab = ηab gμν = (ST)μα ηαβ Sβν = Sαμ ηαβ Sβν // Standard Notation as in Section 7 gμν = (∂ξα/∂xμ) ηαβ (∂ξβ/∂xν) gμν = (∂ξα/∂xμ) (∂ξβ/∂xν) ηαβ // p 71 (3.2.7) The last line then defines the gravitational metric tensor in x-space based on the transformation ξ = F-1(x) = ξ(x). ( This and the following references are from the book of Weinberg, see References.) Newton's Second Law ma = F appears this way in general relativity m (∂2xμ/∂τ2) = Fμ - m Γμνλ (∂xν/∂τ) (∂xλ/∂τ) // p 123 (5.1.11 following) where Fμ is an externally applied force, but there is then an extra bilinear velocity-dependent term which represents an effective gravitational force (it acts on mass m) arising from spacetime itself. The object Γμνλ is called the affine connection and is related to the metric tensor in this way. Γμνλ = ½ gμσ( ∂νgλσ + ∂λgνσ – ∂σgνλ ) // p 75 (3.3.7) These brief comments are only meant to convince the reader that the equations of general relativity also have their place under the general umbrella of tensor analysis as discussed in this document. The fact that Γμνλ is not a mixed rank-3 tensor is discussed in Appendix F (f). (o) Continuum Mechanics and its Metric Tensors One can describe (Lai) the forward "flow" of a continuous blob of matter by x = x(X,t) where X = x(X,t0). A "particle" of matter (imagine a tiny cube) that starts at location X at time t0 ends up at x at time t. Two points in the flow separated by dX at t0 end up separated by some dx at t. The relation between them is given by dx = F dX where F is called the deformation gradient (a rank-2 tensor). F describes how a particle starting say with a cubic shape at t0 gets deformed into some parallelepiped (3-piped) shape at t. The finite-time flow x = x(X,t) from time t0 to time t can be thought of as a generic (generally non-linear) transformation of the form x = F(X) as in Section 1 above (but we replace our usual F by F to avoid confusion between two F symbols: F is now the linearization of transformation F at a point x). The two times are regarded as fixed parameters. Both the starting X-space and the ending x-space are Cartesian spaces, since this flow occurs in the physical world! Thus, the metric tensors for x-space and X-space are both 1 for Cartesian coordinates in each of these spaces. This in turn means that the covariant tensor analysis concepts such as the preservation of the length of a vector under the transformation go out the window, and in fact the vector dX typically gets stretched as dX → dx so that | dX | ≠ | dx |. In order to put this flow into the notation of this document, let X → x and x → x' so that continuum mechanics this document (Forward Flow) x, X ↔ x', x x = x(X,t) ↔ x' = F(x) // Lai p70 (3.1.4) dx = F dX ↔ dx' = R dx // as in Section 2 // Lai p86 (3.7.6), p105 (3.18.3) F ↔ R X = Cartesian ↔ = 1 x = Cartesian ↔ ' = 1 B = FFT ↔ ' = RRT // as in Section 5 (l) // Lai p121 (3.25.2) Thus, the deformation gradient F is just the R matrix of the forward transformation x = x(X,t) = F(X). What we might call the "would-be" metric tensor, ' = RRT = STS ( that is, the ' metric tensor that would have resulted in scalars being true scalars under the transformation), appears as B = FFT and this is known as the left Cauchy-Green deformation tensor (manifestly symmetric). Regarding the above as a description of forward flow, one could consider instead the inverse flow process, but with F having the same meaning as in the forward flow, dx = F dX. Then the inverse flow translation table would be this : continuum mechanics this document (Inverse Flow) X, x ↔ x', x X = X(x,t) ↔ x' = F(x) dX = F-1 dx ↔ dx' = R dx // as in Section 2 x = F(X) F-1 ↔ R F ↔ S // S = R-1 X = Cartesian ↔ = 1 x = Cartesian ↔ ' = 1 C = FTF ↔ ' = STS // as in Section 5 (l) // Lai p114 (3.23.2) In this direction the would-be ' tensor corresponds to C = FTF which is the right Cauchy-Green deformation tensor. Given the above flow situation, it is then possible to add two more transformations F1 and F2 which take X-space and x-space to independent sets of curvilinear coordinates X' and x': and we then have an interesting triple application of the notions of Section 1 to a real-world problem. This drawing is the implicit subject of Section 3.29 (p131) of Lai. In (reverse) dyadic notation the deformation gradient is written F = (x) where means (X)so that dx = F dX = (x) dX Fij = (x)ij = ∂j(X)xi = ∂xi/∂Xj The (x) notation is explained in Appendix E, and in Appendix G the object (v) for an arbitrary vector field v(x) is expressed in general curvilinear coordinates. Section 8 discusses how length, area and volume transform under a transformation like F. In that discussion we can regard the Section 8 picture with its "Cartesian-View" x'-space and the skewed N-piped to its right as describing the (inverse) fluid flow situation for a tiny fluid particle. It is shown there that the length, area and volume magnitude ratios are given by (converted to developmental notation), | dx(n)|/ dL'n = h'n = ['nn]1/2 = the scale factor for edge dx(n) | dA(n)|/ dA'n = (1/h'n) |J| = (1/h'n) g'1/2 = ['nn g']1/2 = [cof('nn)]1/2 |dV| / dV' = |J| = g'1/2 // g' ≡ det('ij) = J2 , ' = STS which can be translated into our inverse flow context as follows : | dx(n)| / | dX(n)| = ['nn]1/2 = [(FTF)nn]1/2 = [Cnn]1/2 // Lai p114 (3.23.6) | dan| / | dAn| = [cof('nn)]1/2 = [cof((FTF)nn)]1/2 = ['nn g']1/2 // Lai p129 (3.27.11)* |dv| / |dV| = |J| = [det('ij)]1/2 = [det(FTF)]1/2 = |det(F)| // Lai p 130 (3.28.3) where edge area volume X-space : dX(n) dAn dV time t0 x-space : dx(n) dan dv time t Thus, for example, the volume change of a "flowing" particle of continuous matter is given by the Jacobian |J| = |detF| associated with the deformation gradient tensor F. We put quotes on "flowing" only because this might be a particle of steel that is momentarily flowing a very small amount during an oscillation or in response to an applied stress. * Lai's area ratio is presented in the following unusual manner: | dan| / | dAn| = detF | F-1,Tun| = detS |RTun| where un is our usual Cartesian space unit vector. But detS = J = g'1/2 (Section 5 (k)) and [RTun]i[RTun]i = Rni Rni = (RRT)nn = 'nn => |RTun| = ['nn]1/2 which gives | dan|/ | dAn| = detF | F-1,Tun| = g'1/2 ['nn]1/2 = ['nn g']1/2 in agreement with the value given above. 6. Reciprocal Base Vectors En and Inverse Reciprocal Base Vectors U'n This entire Section uses the Picture A context, (a) Definition of the En Although various definitions are possible, we shall define the reciprocal tangent vectors En in the following manner (implied sum on i) En ≡ g'ni ei = g'ni∂'ix => en = 'niEi // since ' = g'-1 Comment: Notice how this differs in structure from the rule for forming a covariant vector from a contravariant one, n = 'niVi In the last equation, the right side is a linear combination of vector components Vi, while in the previous equation the right side is a linear combination of vectors ei. In this case, i is a label on ei , whereas in the other case i is an index on Vi. Labels and indices are different. Since the tangent base vectors ei are contravariant vectors in x-space (Section 3 (b)), and since En is a linear combination of the ei, the En are also contravariant vectors in x-space. Notice that in the definition En ≡ g'ni ei, these two x-space vectors are related by the metric tensor of the other space. One can express the components of En in two ways, (En)k ≡ g'ni (ei)k = Skig'ni / since (ei)k ≡ Ski, Section 3 = g'niSki = Rnigik // since Rabgbc = g'abScb, end of Section 5 (f) so that (En)i = g'naSia = giaRna // sum on second indices Applying R to both sides of En ≡ g'ni ei gives En transformed into x'-space, E'n = g'nk e'k so (E'n)i = g'nk (e'k)i = g'nkδk,i = g'ni (b) The Dot Products and Reciprocity (Duality) Three covariant dot products are of great interest. The first is this (Section 5 (f) for last step) en em = ij (en)i (em)j = ij Sin Sjn = STni ij Sjm = ( ST S)nm = 'nm The second is En em = ij (En)i (em)j = ij gia Rna Sjm = δj,a Rna Sjm = Rnj Sjm = (RS)nm = δn,m and the third is En Em = ij (En)i (Em)j = ij gia Rna gjb Rmb = δj,a Rna gjb Rmb = Rnj gjb Rmb = Rnj gjb RTbm = (R g RT)nm = g'nm To summarize, en em = 'nm => |en| = = h'n (scale factor) En em = δn,m En Em = g'nm => |En| = Using the transformed E'n defined above, one finds that E'n e'm = 'ij (E'n)i(e'm)j = 'ij g'ni δm,j = 'im g'ni = δn,m which is consistent with the fact that this is a covariant dot product of tensorial vectors: E'n e'm = En em = δn,m The other two dot products above work this same way, so e'n e'm = 'nm E'n e'm = δn,m E'n E'm = g'nm Notes on Reciprocity (Duality) 1. The reciprocal base vectors En are more usually defined as being those vectors which satisfy the equations En em = δn,m where the em are known. Each En vector has N components so the full set of En vectors has N2 components. As n and m take all values, En em = δn,m represents N2 linear equations. A solution exists since S is invertible (the em form a complete set). The solution is unique and in fact gives our assumed definition above En ≡ g'ni ei . Here is a fast solution of this Cramer's Rule problem using matrix notation: En em = δn,m => ij(En)i(em)j = δn,m => (En)i ij Sjm = δn,m Let Ani ≡ (En)i . Then have A S = 1, => A = ( S)-1 = S-1 -1 = R g = g' ST (end Sec 5f). Therefore A = g' ST => (En)i = Ani = g'nj (ST)ji = g'nj (ej)i => En = g'nj ej . QED 2. In general, if one has An am = δn,m, the vectors Am are said to be "reciprocal" to the am and vice versa, so the vectors En are reciprocal to the tangent base vectors en. 3. Some authors refer to An am = δn,m as a "duality relation" and either set of vectors is "dual to" the other set. The En are referred to as the dual vectors to en. 4. If the am are contravariant vectors, then An will also be contravariant and then An am is a tensorial scalar. Therefore if An am = δn,m, then so also A'n a'm = δn,m in x'-space, where am' = Ram and A'n = RAn. For example, E'n e'm = δn,m in x'-space where em' = Rem and E'n = REn . 5. In section (e) we shall encounter another dual pair Un um = U'n u'm = δn,m which is associated with the inverse transformation x = F-1(x'). 6. One major significance of the equation An am = δn,m is that it allows the following expansions: V = Σn kn An where km = V am V = Σn cn an where cm = V Am so that for example am V = am [Σn kn An] = Σn kn am An = Σn kn δm,n = km. These expansions are explored in section (f) below for the two dual sets En, en and Un, un. (c) Covariant partner for En The covariant partner for En is given by (n)i = ij (En)j so that (n)i = ij (En)j = ij gja Rna = δi,a Rna = Rni Thus, one can regard the covariant vectors n as being the rows of matrix R R = = [1, 2, 3 .... N ]T which compare to S = [e1, e2, e3 .... eN ] (d) Summary of the basic facts: (en)k = Skn en em = 'nm |en| = = h'n S = [e1, e2, e3 .... eN ] (En)i = gia Rna En Em = g'nm |En| = R = [1, 2, 3 .... N ]T = g'na Sia en Em = δn,m En ≡ g'ni ei en = 'ni Ei (n)i = Rni e'n = R en where (e'n)i = δn,i // from Section 3 (b) In general, neither set of base vectors -- the tangent {..en .. } or the reciprocal {..En .. } -- is orthogonal, since the metric tensor g' in general is not diagonal. And in general none of these vectors is a unit vector. (e) Repeat the above for the inverse transformation: definition of the U'n Section 3 (d) introduced the inverse tangent base vectors called u'n. The prime indicates that these vectors exist in x'-space. In analogy with what was done above, one can define the inverse reciprocal base vectors U'n according to U'n ≡ gni u'i => u'n = ni U'i // since = g-1 Everything goes along as in the previous sections, but with these changes: g'↔ g R ↔ S en → u'n e'n → un En → U'n E'n → Un Here are the key results, translated from above, U'n ≡ gni u'i (SU'n) ≡ gni (Su'i) => Un = gni ui S = R-1 (U'n)i = gnaRia = g'iaSna // sum on second indices (un)k = Ski(u'n)i = δn,k (Un)k = Ski(U'n)i = Ski gnaRia = SkiRiagna = δk,agna = gnk (n)i = 'ij (U'n)j (u'n)k = Rkn u'n u'm = nm |u'n| = = hn R = [u'1, u'2, u'3 .... u'N ] (U'n)i = g'ia Sna U'n U'm = gnm | U'n| = S = [1, 2, 3 .... N ]T = gna Ria u'n U'm = δn,m U'n ≡ gni u'i u'n = ni U'n ('n)i = Sni un um = nm Un um = δn,m Un Um = gnm un = S u'n where (un)i = δn,i // from Section 3 (translated) It is helpful to keep all these eight vector symbol names in mind (and each has a covariant partner) x'-space x-space 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 and recall that An am = δn,m for each of the four dual pairs (two primed, two unprimed). (f) Expanding vectors on different sets of basis vectors x-space expansions on un and Un Assume that V is some generic N-tuple V = (V1,V2....VN). There are various ways to expand V onto basis vectors. One way is to expand on the axis-aligned basis vectors un, which recall live in x-space, V = V1 u1 + V2 u2 +... = ΣnVnun where Un V = Vn Un = gni ui The components Vn are Un V because Un um = δn,m. From Section 5 (g), one can write Vn = gnmm ( regarded here as a definition of the m) so one finds that V = ΣnVn un = Σn gnm m un = Σnm gmn un = Σnm Um and thus another expansion for V is this V = 1 U1 + 2 U2 +... = ΣnnUn where un V = n Comments: 1. If V is not a contravariant vector, one can still define n = nmVm, but n won't be a covariant vector. A familiar example is that xn is never a contravariant vector if F is non-linear, but we can still talk about the components n ≡ nmxm . In Standard Notation, xn → xn and n → xn and we do not hesitate to use these two objects even though they are not tensorial vectors. 2. If V is a contravariant vector, the expansion above V = ΣnVnun displays the contravariant components of V. The second expansion V = Σnm Um is still an expansion for contravariant vector V, but it displays the components of the covariant vector which is the "partner" to V by n = nmVm. It would be incorrect to write this second expansion as = Σnm Um since that would say V = Σnm Um which is just not true. We comment later on how this situation changes a bit in the Standard Notation. x-space expansions on en and En Another possibility is to expand V on the tangent basis vectors en, and we denote the components just momentarily as αn, V = α1 e1 + α21 e2 +... = Σn αn en Using en Em = δn,m one finds that αn = En V = (En)k Vk = Rnk Vk = V'n // = 1 so AB = abAaBb = AkBk Therefore, the expansion is V = V'1e1 + V'2e2 +... = Σn V'n en where En V = V'n If it happens that the N-tuple V = (V1,V2....VN) transforms as a contravariant vector, then Vn are the contravariant components of that vector, and V'n are the contravariant components of V' in x'-space. On the other hand, if V is not a tensorial vector, so Vn are not components of a contravariant vector, we can still define V'n ≡ Rnk Vk, but then the V'n are not the contravariant components of V'. Writing V'n = g'nm'm the above expansion can be expressed as V = Σn V'n en = Σn,m g'nm'm en = Σm 'm Σng'mn en = Σm 'm Em so that V = '1E1 + '2E2 +... = Σn 'n En where en V = 'n Summary of x-space expansions: 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 Expanding on unit vectors. The covariant lengths of the different basis vectors are given by |en| = |e'n| = |un| =|u'n| = |En| = |E'n| = |Un| =|U'n| = Using these lengths, one can define unit vector versions of all the basis vectors and then rewrite the above expansions as expansions on the unit vectors with lower case coefficients. For example, using n ≡ en/ the third expansion above becomes (script font for unit-vector components) V = V'11 + V'22 +... = Σn V'n n where En V = V'n = V'n An example of a case where this last expansion would be useful is in the use of spherical curvilinear coordinates, where for example 1 = . The N-tuple (V'1, V'2 ... V'N), although related to contravariant vector V (V'n = Rnk Vk) , is not itself a contravariant vector since it does not obey the rule V'n = Rnk Vk . In fact V'n = Rnk Vk => (1/) V'n = Rnk (1/) Vk => V'n = Rnk (/)Vk x'-space expansions Having done x-space expansions, we turn now to x'-space expansions. These can be obtained from the x-space expansions by this set of rules, g'↔ g R ↔ S u'n → en un → e'n U'n → En Un → E'n V'n ↔ Vn 'n ↔ n and here then are the x'-space expansions: 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' = V1 u'1 + V2 u'2 +... = Σn Vn u'n where U'n V' = Vn U'n = gni u'i V' = 1 U'1 + 2 U'2 +... = Σn n U'n where u'n V' = n (g) Another way to write the En The reciprocal base vectors were defined above as linear combinations of the tangent base vectors, all in the general Picture A context, Ek ≡ g'ki ei It is rather remarkable that there is another way to write Ek in terms of the ei that looks completely different. In this other way, it turns out that Ek is expressed in terms of all the ei except ek and is given by (only valid in Picture B where g=1) Ek = det(R) (-1)k-1 e1 x e2 x ......x eN // ek missing k = 1,2,3...N This is a generalized cross product (Appendix A) of N-1 vectors, since ek is missing, so there are N-2 "crosses". The above multi-cross-product equation is a shorthand for (Ek)α ≡ det(R) (-1)k-1εαabc...x (e1)a(e2)b ...... (eN)x // κ and (eκ)K are missing Here ε is the totally antisymmetric tensor with N indices. If κ is the kth letter of the alphabet (k = 2 => κ = b ), then κ is missing from the list of summation indices of ε, and the factor (ek)κ is missing from the product of factors, so there are then N-1 factors. This cross product expression for En is derived in Appendix A. This is all fairly obscure sounding, but can be brought down to earth by writing things out for N = 3, where the formula reduces to this cyclic set of equations, E1 = det(R) e2 x e3 E2 = det(R) e3 x e1 E3 = det(R) e1 x e2 These equations can be verified in a simple manner. To show an equation is true, if suffices to show that the projections of both sides on the three en are the same, since the en form a complete basis as noted earlier. For the first equation one needs then to show that E1 en = det(R) e2 x e3 en for n = 1,2,3 If n=2 or n=3, both sides vanish, according to en Em = δn,m on the left, and according to geometry on the right, which leaves just the case n = 1. In this case the LHS = 1, so one has to show that e2 x e3 e1 = 1/det(R) = det(S) . But e1 e2 x e3 = (e1)i (e2 x e3)i = (e1)i εijk (e2)j(e3)k = εijk (e1)i(e2)j(e3)k = εijk Si1Sj2Sk3 = det(S) QED. The other two equations of the set can be verified in the same manner. Here is a picture (N=3) drawn in x-space for a non-orthogonal coordinate system. The vectors shown here form a distorted right handed coordinate system which has det(R) > 0. The reader is invited to exercise his or her right hand to confirm the directions of the arrows, E1 = det(R) e2 x e3 E2 = det(R) e3 x e1 E3 = det(R) e1 x e2 (h) Comparison of n and En One could create a covariant partner to en which would be n = en as described in Section 5 (g). This n is not the same as the reciprocal base vector En ≡ g'nk ek. The comparison is interesting: (n)i ≡ ik(en)k // matrix acts on vector index (En)i ≡ g'nk (ek)i // matrix acts on ek label If x-space is Cartesian, then n = en as usual, but of course En ≠ en in this case since g' ≠1. We mention this to head off a possible confusion when the Standard Notation is introduced in the next Section and the above two equations become (en)i ≡ gik(en)k // Standard Notation, g lowers an index (en)i ≡ g'nk (ek)i // Standard Notation, k is a label on ek, not an index The mapping to standard notation does not include n → en, for example. One fact about the standard notation is that, unlike the developmental notation, one cannot look at a vector in bold like en and determine whether it is contravariant or covariant. Only when the index is displayed can one tell. One can think of en as representing both its contravariant self and its covariant partner (en is a tensorial vector). (i) Handedness of coordinate systems: the en , the sign of det(S), and Parity Handedness of a Coordinate System. Let bn be a complete set of basis vectors in an N dimensional vector space, where the bn are not necessarily of unit length, and are not necessarily orthogonal. Consider this quantity B ≡ det (b1, b2.....bn ) = εabc..x (b1)a (b2)b.... (bN)x = b1 [b2 x b3......x bN] where the generalized cross product is discussed in Appendix A. This basis bn defines a "coordinate system" in that we can expand a position vector as follows x = Σn x(b)n bn where the x(b)n are the "coordinates" of point x in this coordinate system. We make the following definition: system bn is a "right handed coordinate system" iff B > 0 system bn is a "left handed coordinate system" iff B < 0 One of course wants to show that for N = 3 this definition corresponds to one's intuition about right and left handed systems. For N= 3 , B ≡ det (b1, b2, b3 ) = εabc (b1)a (b2)b(b3)c = b1 [b2 x b3] Suppose the bn are arranged as shown in this picture, where the visual intention is that the corner nearest the label b1 is closest to the viewer. With one's high-school-trained right hand, one can see that b2 x b3 points in the general direction of b1 (certainly b2 x b3 lies somewhere in the half space of the b2, b3 face plane which contains b1), and so the quantity B = b1 [b2 x b3] > 0. So this is an example of a right-handed coordinate system. The figure shown is a skewed 3-piped which can be regarded as a distortion of an orthogonal 3-piped for which the bn would span an orthogonal coordinate system in which one would have 1 = 2 x 3. The x'-space e'n coordinate system is always right handed. In our standard picture of x-space and x'-space, the coordinate system in x'-space is spanned by a set of basis vectors e'n where (e'n)i = δn,i , as discussed in Section 3 (a). This system is "right handed" because B = εabc..x (e'1)a (e'2)b.... (e'N) = εabc..x δ1aδ2b....δxN = ε123...N = +1 > 0 where we use the standard normalization of the ε tensor as shown. Notice that this conclusion is independent of the metric tensor g' in x-space. The x-space un coordinate system is always right handed. The basis un where (un)i = δn,i in x-space is right handed for the same reason as shown in the above paragraph, and for any g. When g=1 in x-space, the un form the usual Cartesian right-handed orthonormal basis in x-space. See Section 3 (c) concerning the meaning of "unit vector". The x-space coordinate en system handedness is determined by the sign of det(S). Our x-space coordinate system of great interest is that spanned by the en basis vectors, where (en)i = Sin as discussed in Section 3 (a). One has B = εabc..x (e1)a (e2)b.... (eN) = εabc..x Sa1 Sb2.... SxN = det(S) Therefore, using σ ≡ sign ( detS ) and J being the Jacobian as in Section 5 (k), system en is a "right handed coordinate system" iff det(S) = J > 0 or σ = +1 system en is a "left handed coordinate system" iff det(S) = J < 0 or σ = -1 The Parity Transformation. The identity transformation F = 1 results in matrix SI = I with detSI = +1. In this case the en form a right-handed coordinate system, and in fact en = un. The parity transformation F = -1, on the other hand, results in SP = -I with det(SP) = (-1)N and en = -un. When N is odd, the parity transformation converts the right-handed un system to a left-handed en system. If S is some matrix which does not change handedness, meaning detS > 0, then S' = SSP does change handedness for odd N, since in this case detS' = detS det SP = (-1)N detS. So given some S' that changes handedness for N=odd, one can regard it as "containing the parity transformation" which, if removed, would result in no handedness change. N=3 Parity Inversion Example. Since x' = -x under the parity transform F = -1, parity is a reflection of all position vectors through the origin. Objects sitting in x-space, such as N-pipeds, whether or not the origin lies inside the object, are "turned inside out" by the parity transformation, but the inside of the object still maps to the inside of the parity transformed object under this transformation. Consider this crude picture which shows on the left a right-handed 3-piped in x-space where e1 and e2 happen to be perpendicular, and the back part of the 3-piped is not drawn. This 3-piped is associated with some transformation S [ (en)i = Sin ] with detS > 0. Now consider S' = SSP = SPS = -S. For this S', the 3-piped appears as shown on the right. The two pipeds here are related by a parity transformation, all points inverting through the origin. On the right, the volume of the 3-piped lies toward the viewer from the plane shown. The circled dot on the left represents the out-facing normal vector of the 3-piped face area which is facing the viewer, and this normal is in the direction – e1xe2. This is called a "near face" in Appendix B since it touches the tails of the en. After the parity transformation, this same face has become the back face on the inverted 3-piped shown on the right, with out-facing normal indicated by the X. The direction of this normal is + e1xe2 . In general, an out-facing "near face" area points in the -En direction, and Appendix A shows that E3 = e1 x e2 / det(S). On the left we have E3 = e1 x e2 / |det(S)| so the face there just mentioned points in the -E3 = – e1xe2 direction. On the right we have E3 = e1 x e2 / det(S') = - e1 x e2 /|det(S)|, so the face there points in the -E3 = +e1xe2 direction. The sign of det(S) in the curvilinear coordinates application. For a given ordering of the x'i coordinates, det(S) will have a certain sign. By changing the x'i ordering to any odd permutation of the original ordering (for example, swap two coordinates), det(S) will negate because two columns of Sik(x') ≡ (∂xi/∂x'k) will be swapped. In the curvilinear coordinates application it is therefore always possible to select the ordering of the x'i coordinates to cause det(S) to be positive. One always starts with a right-handed Cartesian system for x-space, and det(S)>0 then guarantees that the en will form a right-handed system there as well. Since the underlying transformation F is assumed invertible, one cannot have det(S)=0 anywhere in the domain x (or range x') of x' = F(x), and therefore det(S) cannot change sign anywhere in the space of interest. For graphical reasons, we have selected coordinates in the "wrong order" in both the polar coordinates examples (called Example 1) and in the elliptic polar system studied in Appendix C, which is why detS < 0 for both these systems. 7. Translation to the Standard Notation In this Section 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. (a) Outer Products It is possible to form larger tensors from smaller ones using the "outer product" method. For example, consider, Tab ≡ UaVb 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' so in this way a contravariant rank-2 tensor (Section 5 (e)) has been successfully constructed from two contravariant vectors. Similarly, ab ≡ ab => 'ab = STaa' STbb' a'b' so the outer product of two covariant vectors transforms as a covariant rank-2 tensor. (b) Mixed Tensors and Notation Issues Suppose we take the "outer product" of a contravariant vector with a covariant vector, [ ... ]ab ≡ Uab 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 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 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 hand side of the equation, 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 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 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 ( see Section 5 (m) ), dots are placed on certain indices 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. (c) 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, 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 and then we get this translation Wac = UaV XcY → Wabcd = UaVbXcYd // the path taken 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 Sections into this new notation. There are some subtle details that will be discussed in the following sections. (d) Some Preliminary Translations: raising and lowering indices on a vector with g In the entire rest of this entire Section, 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 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 , 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 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 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. (e) 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, Va = gabVb It is 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, which conveniently matches the LHS. Similarly, AaBa transforms as a scalar. (f) 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) 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: Rik ≡ (∂x'i/∂xk) → Rik ≡ (∂x'i/∂xk) 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. 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 V'a = RabVb → V'a = RabVb // contravariant 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. (g) Repeat the above section for S We omit the words and just show the translations (∂xi/∂x'k) → (∂xi/∂x'k) Sik ≡ (∂xi/∂x'k) → Sik ≡ (∂xi/∂x'k) 'a = STabb = Sba b → V'a = SbaVb // covariant (h) About ε and δ The Kronecker δ is sometimes written in different ways to make things "look nice", δab = δab = δba = δba = δa,b Section (m) will show that one can regard the above sequence of equalities as saying gab = gab = gba = gba = δa,b 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 or in an ordinary cross product Aa = εabcBbCc . 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 (d) just as with any tensor. Therefore, in Cartesian space with g = 1, indices on ε are raised and lowered with no consequence, 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 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 is always just the permutation tensor, but in the cross product that is not the case, and one would properly write Aa = εabcBbCc 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. (i) Further development of the Standard Notation This long section contains a veritable grab-bag of important Standard Notation facts. Each such fact is proven and not just quoted. Many of the results presented here anticipate more formal presentations of the same results in later sections. Translation of determinants. Section (g) 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 where ε is the bookkeeping permutation tensor discussed in section (h). Inverse of R and S. Again, section (g) 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 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 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 (1) Tensor g raises and lowers any index. So far the following translation rules have been established: gab → gab ab → gab Rik → Rik Sik → Sik It was shown in developmental notation Section 5 (e) how a rank-2 contravariant tensor transforms. Here then is how that statement translates to the new notation M'ab = Raa'Rbb'Ma'b' → M'ab = Raa'Rbb'Ma'b' contravariant rank-2 tensor (2) 'ab = Sa'aSb'ba'b' → M'ab = Sa'aSb'bMa'b' covariant rank-2 tensor Since g is such a rank-2 tensor, replace M by g to get g'ab = Raa'Rbb'ga'b' → g'ab = Raa'Rbb'ga'b' (3) 'ab = Raa'Rbb'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 (4) Since gab and gab are inverses, it follows that Mab = gaa' Ma'b (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' 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: (4) (3) (2) (5) 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 (1) = [Rac (gcd gdi) Sfb] Mif = [Rac δci Sfb] Mif = [Rai Sfb] Mif inverses = Raa' Sb'b Ma'b' QED Similarly one could define Mab ≡ gaa'Ma'b and one would find that M'ab = Sa'a Rbb' Ma'b' 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 (2). 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' One sees then a family of four tensors associated with M. One is contravariant, one is covariant, and the other two are mixed. Let [----i---] represent a tensor with a certain contravariant index i and dashes indicate other indices which could be up or down. Similarly define [----i---] as another tensor in the same family where the index i that was up is now down. Raising Lowering Rule: gaa' [----a'---] = [----a---] and gaa' [----a'---] = [----a---] 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 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. Contraction Tilt Reversal Rule: [-----a---------a----] = [-----a---------a----] 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 "reverse the tilt" on any pair of contracted indices. The Diagonal g Rule: gab = δab and gab = δab // and same for g' This is proved in section (m) below, but since we are going to need it right now, here is a preview of that proof: gab = gaa' ga'b // gaa'raises the first index of tensor ga'b = δab // because gij and gij are inverses of each other Matrix Multiplication in the Standard Notation. Although there are various forms of matrix multiplication, the most standard form is that obtained when all matrices have a "down-tilt" form. Consider this example, Cab = AacBcb where it is assumed that all three objects are down-tilt rank-2 tensors (or objects like Rac and Sac whose indices behave as if they rank-2 tensors). Although these are "split level" matrices, once can see that the index c has the correct "adjacency" property to justify matrix multiplication. A second requirement is that any matrix summed index must be a genuine contraction with one index up and the other down. The above tensor transformation rule can then be written in this more compact matrix notation, C = AB // all down-tilt An example appears in (1) above: δia = RikSka = ↔ 1 = RS S-1 = R By application of suitable g tensors to the first equation above (or by lowering index a and raising index b), one gets Cab = AacBcb . Application of the Tilt Reversal Rule on index c then gives Cab = AacBcb Again the adjacency and contraction rules are met, so this equation can be represented also by C = AB // all up-tilt and so δia = RikSka = ↔ 1 = RS S-1 = R Comment: Notice that the equation 1 = RS is valid both in the standard notation (providing R,S and 1 all have the same tilt) and in the developmental notation. The upshot is that in Standard Notation, matrix notation can be used if all matrices in the equation being represented are either all down-tilt or all up-tilt. As will be shown later, such matrix equations are all "covariant" in that both sides of the equation have the same tensor transformation property, and this is due to the fact that the matrix summation index is a contraction. One could talk about matrix multiplication in other cases, such as Cab = AacBcb but since index c is not a contraction, if A and B are tensors, then C cannot be a tensor and we don't even want to think about such equations. Transpose of a rank-2 tensor. If A is a rank-2 tensor, the translation mapping (AT)ab = Aba → (AT)ab = Aba seems obvious, and the object AT therefore also transforms as a rank-2 tensor. Once (AT)ab = Aba is established in standard notation, one can apply the metric tensor g to lower either or both of the indices of this equation, to get (AT)ab = Aba , (AT)ab = Aba, and (AT)ab = Aba. Notice in all four equations that the indices on the two sides of the equation are reflected in a vertical axis passing between the indices. This causes a left index to become a right index and vice versa, as one would expect for transposing a matrix. Moreover, on each side of all four equations, each index has the same contravariant/covariant sense. This same argument also applies to R and S even though they are not tensors. The only difference is that the first index of Rba is lowered by g' while the second by g. For example, (RT)ab = Rba where b is a g' type index and a is a g type index, as will be elaborated in section (o) below. Similarly (ST)ab = Sba . To summarize the situation with transposes in Standard Notation: (AT)ab = Aba (RT)ab = Rba (ST)ab = Sba (AT)ab = Aba (RT)ab = Rba (ST)ab = Sba (AT)ab = Aba (RT)ab = Rba (ST)ab = Sba (AT)ab = Aba (RT)ab = Rba (ST)ab = Sba Notice that the rule is not (AT)ab = Aba which would be a straight swap of indices (and would result in AT not being a tensor). The straight swap idea works for the pure contravariant and pure covariant forms of A, but not for the tilted forms! As shown in Theorem 4 below, this tilted transpose form has some interesting implications. Theorem 1: For either the down-tilt or up-tilt version of S, S is a real-orthogonal matrix. This means all of the following: S-1 = ST SST = 1 STS= 1 Proof of theorem: start with (3) above which says gij is a covariant rank-2 tensor, g'ab = Sa'aSb'b ga'b' // next, apply gb'b to both sides (or just raise index b on both sides) g'ab = Sa'aSb'b ga'b' // next, reverse the tilt of the b' index g'ab = Sa'aSb'b ga'b' // next, use the Diagonal g rule in two places δab = Sa'aSb'b δa'b' = Sa'aSa'b // next, use (ST)aa' = Sa'a δab = (ST)aa'Sa'b // next use matrix notation as per above 1 = STS => ST = S-1 => 1 = SST Theorem 2: For either the down-tilt or up-tilt version of R, R is a real-orthogonal matrix. This means all of the following: R-1 = RT RRT = 1 RTR= 1 In other words, the previous theorem also applies to R. Again, this fact is not true of the developmental notation matrix Rab. The proof is very similiar but just different enough to warrant showing it : Proof of theorem: start with (3) above which says gij is a covariant rank-2 tensor, g'ab = Raa'Rbb'ga'b' // next, apply gb'b to both sides (or just lower index b on both sides) g'ab = Raa'Rbb'ga'b' // next, reverse the tilt of the b' index g'ab = Raa'Rbb'ga'b' // next, use the Diagonal g rule in two places δab = Raa'Rbb'δa'b' = Raa'Rba' // next, use (RT)a'b = Rba' δab = Raa'(RT)a'b // next use matrix notation as per above 1 = RRT => RT = R-1 => 1 = RTR Comment: In the developmental notation, neither R nor S is a real-orthogonal matrix, unless by accident, but in the tilted standard notation they both are! Theorem 3: For either the down-tilt or up-tilt version of S, (a) S = RT and R = ST (b) Sab = Rba and Sab = Rba ( reflect indices in vertical line between them) Proof of theorem: From the inverse discussion at the start of this section, S = R-1 for any index positions in the standard notation, including the up-tilt and down-tilt positions. From Theorem 2, RT = R-1 in either up-tilt or down-tilt forms. Therefore S = RT (hence ST = RTT= R) in either up-tilt or down-tilt form. Thus S = RT => Sab = (RT)ab = Rba S = RT => Sab = (RT)ab = Rba QED An implication of this theorem is that one can completely eliminate references matrix S in tensor analysis and that is what is usually done. This is like replacing S with R-1 in the devepmental notation. Theorem 4: For a standard notation tilted matrix, det(A) ≠ det(AT). This surprising result points out a potential hazard of using matrices in the standard notation, and perhaps is an indication of why people avoid matrix concepts and just write out all the components. For a traditional matrix Aab the determinant is given by either of these forms (figure on rows or columns) det(A) = εab..A1aA2b.... = εab..Aa1Ab2 ... det(AT) = εab..AT1aAT2b.... = εab..Aa1Ab2...... = det(A) In the tilted standard notation, however, one has det[Aij] = εab..A1aA1b.... = εab..Aa1Ab2 ... det[(AT)ij] = εab.. (AT)1a(AT)1b.... = εab...Aa1Ab2 but this is not the same as either of the det[Aij] forms! Here is a simple example: det(A) = εab..A1aA1b... = = A11 A22 – A21 A22 det(AT) = εab...Aa1Ab2 = = A11 A22 – A21 A12 ≠ det(A) The point is that the standard notation transpose rule (AT)ij = Aij doesn't just swap the indices, it also changes the tilt. That means that the columns of one determinant matrix are not the same as the rows of the other and that is why det(A) ≠ det(AT). Corollary: RTR = 1 does not imply that det(R) = ± 1. The usual proof goes that det(RTR) = det(RT)det(R) = det(R)det(R) = [ det(R) ]2 = 1, but of course the part saying det(RT) = det(R) is no longer valid. Therefore, the fact that RTR = 1 does not lead us to the false conclusion that the Jacobian of Section 5 (k) is somehow forced to be ± 1 ! So although R and S are real orthogonal matrices, they are not "rotation matrices". Orthogonality rules. The first two rules are easy to show, RTR = 1 => (RT)abRbc = δac => RbaRbc = δac RRT = 1 => Rab(RT)bc = δac => RabRcb = δac Lowering a and raising c on both sides and reversing the b tilt then gives the other two rules RbaRbc = δac RabRcb = δac These orthogonality rules are derived in a slightly different manner and order in section (r) below. Variations on the relation between g and g'. Item (3) 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' 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' 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 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. (j) 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' 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 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 Section 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' 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 One could then define (Rdp)abcde ; a'b'c'd'e' ≡ Raa' Rbb' Rcc' Rdd' Ree' so that A'aB'bC'cD'dE'e = (Rdp)abcde ; a'b'c'd'e' Aa'Bb'Cc'Dd'Ee' 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' 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 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. 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. (k) The Contraction Tilt-Reversal Rule In some complicated combination of multiple tensors and perhaps some R and S objects, 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----] 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----] 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 A notable example of course is this: AaYa = AaYa = " A Y " // or perhaps " A.Y " as noted in Section 5 (i) (l) 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 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----] where now the only X's left are for the other indices. Again we look at our canonical example, AaYa = AaYa = A.Y 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 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) It is similarly possible to build larger tensors from smaller ones, for example Zabcde = Va We gab Lc which goes under the same rubric "outer product" mentioned earlier. (m) 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 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 Comparison shows that gbc = δbc As a sanity check, consider Va = gabVb Applying our Contraction Tilt-reversal Rule, this can be written Va = gabVb but this is = δabVb = Va 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 (n) 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 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. (o) 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) (p) 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) (q) 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 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 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 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 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 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) 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 can now be written V'a = RabVb V'a = RabVb 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 becomes T'abcde = Raa' Rbb' Rcc' Rdd' Ree' Ta'b'c'd'e' // no pairs contracted 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. (r) Orthogonality Rules 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 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. (s) 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 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] R = [1, 2, 3 .... N ]T → Rij = [(e1)k, (e2)k, (e3)k .... (eN)k]T The relationship between en and en is very simple, En ≡ g'ni ei → en = g'ni ei and en = g'ni ei 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 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| = 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 . 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 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 Summary table. The summary table given at the end of Section 6 (e) was this x'-space x-space 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 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 Section 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 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 x'-space expansions Similarly, the x'-space expansions of Section 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 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 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 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 Section 5 (k). (t) Comment on Covariant versus Contravariant Consider this expansion for a vector V in x-space, V = Vncn Vn = V cn where cn is some basis having dual basis cn where as usual cn cm = δnm. Imagine taking Vn → V'n = Rnm Vm and ci → ci' = Qij cj. What Q would cause the following to be true? V = Vncn = V'nc'n In other words, how does one transform that basis cn 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'nc'n = [Rnm Vm][ Rnj cj] = (Rnm Rnj) Vm cj = δmj Vm cj = Vj cj = Vncn Compare then the transformation of Vn with that of the basis cn: V'n = Rnm Vm cn' = Rnm cm 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 cm . If one starts over with Vn components and the cn "dual" (reciprocal) expansion vectors and asks for a solution to this corresponding problem, V = Vncn = V'nc'n one finds not surprisingly that the dual basis must vary as cn' = Rnm cm and then one has V'n = Rnm Vm cn' = Rnm cm 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 cm 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 cn one has V'n = Rnm Vm cn' = Rnm cm If the basis cm is varied as shown here, then the dual basis cm varies as shown above and V remains invariant. Comparing now the way the Vn transform with the way the basis vectors cm 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 cm . (u) 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) 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') 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".) 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, being careful to give new summation indices unique names: Q'adc(x') = H'ab(x')T'bc(x') B'd(x') // x'-space equation [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) where we have reordered the R factors to (1) match the LHS as much as possible, and (2) to group any pairs that came from a contracted index. 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) so that (Raa'Rdd'Rcc') Qa'd'c'(x) = (Raa' Rdd' Rcc') Ha'b'(x) Tb'c'(x) Bd'(x) Now apply to both sides the factor RaA RdD RcC and sum on a,c,d. RaA RdD RcC (Raa'Rdd'Rcc') Qadc(x)] = RaA RdD RcC (Raa'Rdd'Rcc') Ha'b'(x) Tb'c'(x) Bd'(x) or (RaA Raa')( RdD Rdd')( RcC Rcc') Qadc(x) = (RaA Raa')( RdD Rdd')( RcC Rcc') Ha'b'(x) Tb'c'(x) Bd'(x) Then use a section (r) orthogonality rule in each (R R) to get (δAa')( δDd')(δCc') Qa'd'c'(x) = (δAa')( δDd')(δCc') Ha'b'(x) Tb'c'(x) Bd'(x) or QADC(x) = HAb'(x) Tb'C(x) BD(x) Finally, rename the indices A,D,C,b' to be a,d,c,b Qadc(x) = Hab(x) Tbc(x) Bd(x) // x-space equation 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) 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) 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') . 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 Section 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 where ∂μ means gμα∂α, the contravariant form of the gradient operator. The components are then where c is the speed of light and of course E and B are the electric and magnetic fields. Two of Maxwell's equations are (in SI units where ε0μ0= 1/c2) ∂νFμν = μ0 Jμ Jμ = (cρ,J) while the other two are ∂αFμν + ∂μFνα + ∂νFαμ = 0 or ∂αFμν + cyclic = 0 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 J'μ = (cρ',J') ∂'αF'μν + ∂'μF'να + ∂'νF'αμ = 0 or ∂'αF'μν + cyclic = 0 Objects like ∂νFμν and ∂αFμν are true rank-3 tensors because the transformation F is linear. The next section shows the pain encountered when F is not linear. (v) 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 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). 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 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 where Γcab is a certain function of the metric tensor g. One then finds that V'b;a = Rad Rbc V d;c or [∂'aV'b – Γ 'kab V'k] = Rad Rbc [∂dVc – Γkcd Vk] (*) 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 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 Section 5(n). 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 and therefore neither of the Christoffel symbols Γdab or Γcab transforms as a tensor in this case. See Appendix F for more detail. (w) 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 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 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 . In the case of rank-2 tensors, a product like uiuj = uiuj is called a dyadic (see App. 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 . Appendix E 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 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).