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APPENDIX B on a vector

In this Appendix, we gather mathematical definitions and results pertaining to tensors. The purpose is mostly to introduce the “modern”, geometrical view on tensors, which defines them by their action on vectors or one-forms, i.e. in a coordinate-independent way (Sec. B.1), in contrast to the “old” definition based on their behavior under transformations (Sec. B.2). The reader is assumed to already possess enough knowledge on linear to know what are vectors, linear (in)dependence, (multi), matrices. . . Similarly, the notions of , field, application/function/mapping. . . are used without further mention. In the remainder of these lecture notes, we actually consider tensors on real vector , i.e. for which the underlying base field K of scalars is the set R of real numbers; here we remain more general. Einstein’s convention is used throughout.

B.1 Vectors, one-forms and tensors

B.1.1 Vectors . . . are by definition the elements ~c of a V , i.e. of a set with 1) a binary (“addition”) with which it is an Abelian group, and 2) a multiplication with “scalars”—elements of a base field K—which is associative, has an identity element, and is distributive with respect to both additions on V and on K. Introducing a basis B = {~ei}, i.e. a family of linearly independent vectors that span the whole space V , one associates to each vector ~c its uniquely defined components {ci}, elements of the base field K, such that i ~c = c ~ei. (B.1) If the number of vectors of a basis is finite—in which case this holds for all bases— and equal to some D—which is the same for all bases—, the space V is said to be finite-dimensional and D is its (over K): D = dim V . We shall assume that this is the case in the remainder of this . B.1.2 One-forms . . . on a vector space V are the linear applications, hereafter denoted as h, from V into the base field of scalars K. e The set of 1-forms on V , equipped with the “natural” addition and multiplication, is itself a vector space over the field K, denoted by V ∗ and said to be dual to V . ∗ ∗ If V is finite-dimensional, so is V , with dim V = dim V . Given a basis B = {~ei} in V , one can then construct its B∗ = {j} in V ∗ such that e j j  (~ei) = δi , (B.2) j e where δi denotes the usual symbol. The components of a 1-form h on a given basis will be denoted as {hj}: j e h = hj  . (B.3) e e B.1 Vectors, one-forms and tensors 53

Remarks: ∗ The choice of notations, in particular the position of indices, is not innocent! Thus, if {j} denotes the dual base to {~ei}, the reader can trivially check that e i i c =  (~c) and hj = h(~ej). (B.4) e e ∗ In the “old” language, the vectors of V resp. the 1-forms of V ∗ were designated as “contravariant vectors” resp. “covariant vectors” or “covectors”, and their coordinates as “contravariant” resp. ”co- variant” coordinates. The latter two, applying to the components, remain useful short denominations, especially when applied to tensors (see below). Yet in truth they are not different components of a same mathemat- ical quantity, but components of different objects between which a “natural” correspondence was introduced, in particular by using a as in § B.1.4.

B.1.3 Tensors

:::::::B.1.3 a :::::::::::::::::::::::::::Definition and first results Let V be a vector space with base field K, and m, n denote two nonnegative . The multilinear applications of m one-forms—elements of V ∗—and n vectors—elements of V —into m K are referred to as the tensors of type n on V , where linearity should hold with respect to every argument. The integer m + n is the order (or often, but improperly, ) of the tensor. Already known objects arise as special cases of this definition when either m or n is zero: 0 K • the 0 -tensors are simply the scalars of the base field ;

1 (19) • the 0 -tensors coincide with vectors; 0 0 • the 1 -tensors are the one-forms. More generally, the n -tensors are also known as (multi- linear) n-forms.

2 • Eventually, 0 -tensors are sometimes called “” or “”. Tensors will generically be denoted as T, irrespective of their rank, unless the latter is 0 or 1. A tensor may be symmetric or antisymmetric under the exchange of two of its arguments, either both vectors or both 1-forms. Generalizing, it may be totally symmetric—as e.g. the we shall encounter below—, or antisymmetric. An instance of the latter case is the , which is the only (up to a multiplicative factor) totally antisymmetric D-form on a vector space of dimension D. m ∗ m ∗ n 0 0 Remark: Consider a n -tensor T :(V ) ×(V ) → K, and let m ≤ m, n ≤ n be two nonnegative 0 0 integers. For every m -uplet of one-forms {hi} and n -uplet of vectors {~cj}—and corresponding multiplets of argument positions, although heree we take for simplicity the first ones—the object  T h1, . . . , hm0 , · ,..., · ;~c1, . . . ,~cn0 , · ,..., · , e e where the dots denote “empty” arguments, can be applied to m − m0 one-forms and n − n0 vectors 0 0 to yield a scalar. That is, the tensor T induces a multilinear application(20) from (V ∗)m × (V ∗)n m−m0 into the set of n−n0 -tensors. 1 For example, the 1 -tensors are in natural correspondence with the linear applications from V into V , i.e. in turn with the matrices of order dim V .

(19)More accurately, they are the elements of the double dual of V , which is always homomorphic to V . (20)Rather, the number of such applications is the number of independent—under consideration of possible symmetries—combinations of m0 resp. n0 one-form resp. vector arguments. 54 Tensors on a vector space

:::::::B.1.3 b :::::::::::::::::::::::Operations on tensors The tensors of a given type, with the addition and inherited from V , form a vector space on K. Besides these natural addition and multiplication, one defines two further operations on tensors, the outer or —which increases the rank—and the contraction, which decreases the rank. 0 m m0 0 Consider two tensors T and T , of respective types n and n0 . Their T ⊗T is m+m0 0 0 a tensor of type n+n0 satisfying for every (m + m )-uplet (h1, . . . , hm, . . . , hm+m ) of 1-forms and 0 every (n + n )-uplet (~c1, . . . ,~cn, . . . ,~cn+n0 ) of vectors the identitye e e

0  T ⊗ T h1, . . . , hm+m0 ;~c1, . . . ,~cn+n0 = e e  0  T h1, . . . , hm;~c1, . . . ,~cn T hm+1, . . . , hm+m0 ;~cn+1, . . . ,~cn+n0 . e e e e For instance, the outer product of two 1-forms h, h0 is a 2-form h ⊗ h0 such that for every pair of vectors (~c,~c 0), h ⊗ h0(~c,~c 0) = h(~c) h0(~c 0). In turn,e e the outer producte e of two vectors ~c, ~c 0 is a 2 0 0 0 0 0 0 0 -tensor ~c ⊗ ~c suche thate for everye paire of 1-forms (h, h ), ~c ⊗ ~c (h, h ) = h(~c) h (~c ). m Tensors of type n that can be written as outere productse ofeme vectorse ande n one-forms are sometimes called simple tensors. m Let T be a n -tensor, where both m and n are non-zero. To define the contraction over its j-th one-form and k-th vector arguments, the easiest—apart from introducing the tensor components—is to write T as a sum of simple tensors. By applying in each of the summand the k-th one-form to m−1 the j-th vector, which gives a number, one obtains a sum of simple tensors of type n−1 , which is the result of the contraction operation. Examples of contractions will be given after the metric tensor has been introduced.

:::::::B.1.3 c ::::::::::::::::::::Tensor coordinates j ∗ Let {~ei} resp. { } denote bases on a vector space V of dimension D resp. on its dual V —in principle, they neede not be dual to each other, although using dual bases is what is implicitly always done in practice—and m, n be two nonnegative integers. m+n j1 jn The D simple tensors {~ei1 ⊗ · · · ⊗~eim ⊗  ⊗ · · · ⊗  }, where each ik or jk runs from 1 to D, m form a basis of the tensors of type n . The componentse e of a tensor T on this basis will be denoted i1...im as {T j1...jn }: i1...im j1 jn T = T j1...jn ~ei1 ⊗ · · · ⊗~eim ⊗  ⊗ · · · ⊗  , (B.5a) e e where i1...im i1 im T j1...jn = T( , . . . ,  ;~ej1 , . . . ,~ejn ). (B.5b) e e The possible or antisymmetry of a tensor with respect to the exchange of two of its arguments translates into the corresponding symmetry or antisymmetry of the components when exchanging the respective indices. In turn, the contraction of T over its j-th one-form and k-th vector arguments yields the tensor ...ij−1,`,ij+1,... with components T ...jk−1,`,jk+1,..., with summation over the repeated index `. B.1.4 Metric tensor Nondegenerate(21) symmetric bilinear forms play an important role, as they allow one to intro- duce a further structure on the vector space V , namely an inner product.(22) j ∗ i j Accordingly, let { } denote a basis on the V . A 2-form g = gij  ⊗  is a metric ~ ~ ~ tensor on V if it is symmetric—i.e.e g(~a, b) = g(b,~a) for all vectors ~a, b, or equivalentlye e gij = gji

(21) This will be introduced 4 lines further down as a condition on the with elements gij , which is equivalent to stating that for every non-vanishing vector ~a there exists ~b such that g(~a,~b) 6= 0. (22)More precisely, an inner product if g is (positive or negative) definite, a semi-inner product otherwise. B.1 Vectors, one-forms and tensors 55

~ for all i, j—and if the with elements gij is regular. The number g(~a, b) is then also denoted ~a ·~b, which in particularly gives  gij = g ~ei,~ej = ~ei ·~ej, (B.6) j where {~ei} is the basis dual to { }. ij Since the D × D-matrix withe elements gij is regular, it is invertible. Let g denote the elements jk k ij i 2 ij 2 ij of its inverse matrix: gijg = δi , g gjk = δk. The D scalars g define a 0 -tensor g ~ei ⊗~ej, the inverse metric tensor, denoted as g−1.

Using results on symmetric matrices, the square matrix with elements gij is diagonalizable—i.e.  one can find an appropriate basis {~ei} such that g ~ei,~ej = 0 for i 6= j. Since g is nondegenerate, the eigenvalues are non-zero: at the cost of multiplying the basis vectors {~ei} by a numerical factor,  one may demand that every g ~ei,~ei be either +1 or −1, which yields the canonical form

gij = diag(−1,..., −1, 1,..., 1) (B.7) for the matrix representation of the components of the metric tensor. ij −1 In that specific basis, the component g of g coincides with gij, yet this does not hold in an arbitrary basis.

::::::::::::::::::::::::::::Role of g in i In agreement with the remark at the end of § B.1.3 a, for any given vector ~c = c ~ei the object j g(~c, ) maps vectors into the base field K, i.e. it is a one-form c = cj  , such that i e i e cj = c(~ej) = g(~c,~ej) = g(c ~ei,~ej) = c gij. (B.8a) e That is, a metric tensor g provides a mapping from vectors onto one-forms. Reciprocally its inverse metric tensor g−1 maps one-forms onto tensors, leading to the relation

i ij c = g cj. (B.8b)

Generalizing, a metric tensor and its inverse thus allow one “to lower or to raise indices”, which m m∓1 are operations mapping a tensor of type n on a tensor of type n±1 , respectively. Remarks: ∗ Lowering resp. raising an index actually amounts to an outer product with g resp. g−1 followed by the contraction of two indices. For instance

i outer product i j k contraction i k k ~c = c ~ei 7−→ ~c ⊗ g = c gjk~ei ⊗  ⊗  7−→ c = c gik  = ck  e e e e e where the first and second arguments of ~c ⊗ g have been contracted. ∗ Generalizing the “” notation for the inner product defined by the metric tensor, the contraction is often also denoted with a dot product. For example, for a 2-form T and a vector ~c i j k  j i T · ~c = Tij  ⊗  · c ~ek = Tij c  , e e e where we implicitly used Eq. (B.2). Note that for the dot-notation to be unambiguous, it is better if T is symmetric, so that which of its indices is being contracted plays no role. Similarly, if T denotes a dyadic tensor and T0 a 2-form 0 ij  0 k l ij 0 l T · T = T ~ei ⊗~ej · T kl  ⊗  = T T jl~ei ⊗  , e e e which is different from T0 · T if the tensors are not symmetric. The reader may even find in the literature the notation 0 ij 0 T : T ≡ T T ji , involving two successive contractions. 56 Tensors on a vector space

B.2 0 ∗ i 0∗ j0 Let B = {~ei} and B = {~ej0 } denote two bases of the vector space V , and B = { }, B = { } the corresponding dual bases on V ∗. The basis vector of B0 can be expressed in termse of thosee of i B with the help of a non-singular matrix Λ with elements Λ j0 such that i ~ej0 = Λ j0~ei. (B.9)

Remark: Λ is not a tensor, for the two indices of its elements refer to two different bases—which is emphasized by the use of one primed and one unprimed index—while both components of a 1 (23) 1 -tensor are with respect to the “same” basis. k0 −1 Let Λ i denote the elements of the inverse matrix Λ , that is k0 i k0 i k0 i Λ iΛ j0 = δj0 and Λ k0 Λ j = δj.

k0 ∗ 0∗ One then easily checks that the numbers Λ i govern the change of basis from B to B , namely j0 j0 i  = Λ i  . (B.10) e e Accordingly, each “vector” component transforms with Λ−1:

0 0 0 0 0 0 j j i j1...jm j1 jm i1...im c = Λ i c , T = Λ i1 ··· Λ im T . (B.11)

In turn, every “1-form” component transforms with Λ:

i i1 in h 0 = Λ 0 h , T 0 0 = Λ 0 ··· Λ 0 T . (B.12) j j i j1...jn j1 jn i1...in One can thus obtain the coordinates of an arbitrary tensor in any basis by knowing just the transformation of basis vectors and one-forms.

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(23)Or rather, with respect to a basis and its dual.