Introduction to Tensor Calculus Kees Dullemond & Kasper Peeters c 1991-2010 This booklet contains an explanation about tensor calculus for students of physics and engineering with a basic knowledge of linear algebra. The focus lies mainly on acquiring an understanding of the principles and ideas underlying the concept of ‘tensor’. We have not pursued mathematical strictness and pureness, but instead emphasise practical use (for a more mathematically pure resum´e, please see the bib- liography). Although tensors are applied in a very broad range of physics and math- ematics, this booklet focuses on the application in special and general relativity. We are indebted to all people who read earlier versions of this manuscript and gave useful comments, in particular G. B¨auerle (University of Amsterdam) and C. Dulle- mond Sr. (University of Nijmegen). The original version of this booklet, in Dutch, appeared on October 28th, 1991. A major update followed on September 26th, 1995. This version is a re-typeset English translation made in 2008/2010. Copyright c 1991-2010 Kees Dullemond & Kasper Peeters. 1 The index notation 5 2 Bases, co- and contravariant vectors 9 2.1 Intuitive approach ............................. 9 2.2 Mathematical approach .......................... 11 3 Introduction to tensors 15 3.1 The new inner product and the first tensor ............... 15 3.2 Creating tensors from vectors ....................... 17 4 Tensors, definitions and properties 21 4.1 Definition of a tensor ............................ 21 4.2 Symmetry and antisymmetry ....................... 21 4.3 Contraction of indices ........................... 22 4.4 Tensors as geometrical objects ....................... 22 4.5 Tensors as operators ............................ 24 5 The metric tensor and the new inner product 25 5.1 The metric as a measuring rod ....................... 25 5.2 Properties of the metric tensor ....................... 26 5.3 Co versus contra ............................... 27 6 Tensor calculus 29 6.1 The ‘covariance’ of equations ....................... 29 6.2 Addition of tensors ............................. 30 6.3 Tensor products ............................... 31 6.4 First order derivatives: non-covariant version .............. 31 6.5 Rot, cross-products and the permutation symbol ............ 32 7 Covariant derivatives 35 7.1 Vectors in curved coordinates ....................... 35 7.2 The covariant derivative of a vector/tensor field ............ 36 A Tensors in special relativity 39 B Geometrical representation 41 C Exercises 47 C.1 Index notation ................................ 47 C.2 Co-vectors .................................. 49 C.3 Introduction to tensors ........................... 49 C.4 Tensoren, algemeen ............................. 50 C.5 Metrische tensor ............................... 51 C.6 Tensor calculus ............................... 51 3 4 1 The index notation Before we start with the main topic of this booklet, tensors, we will first introduce a new notation for vectors and matrices, and their algebraic manipulations: the index notation. It will prove to be much more powerful than the standard vector nota- tion. To clarify this we will translate all well-know vector and matrix manipulations (addition, multiplication and so on) to index notation. Let us take a manifold (=space) with dimension n. We will denote the compo- ~ nents of a vector v with the numbers v1,..., vn. If one modifies the vector basis, in ~ which the components v1,..., vn of vector v are expressed, then these components will change, too. Such a transformation can be written using a matrix A, of which ~ ~ the columns can be regarded as the old basis vectors e1,..., en expressed in the new ~ ~ basis e1′,..., en′, v1′ A11 A1n v1 . · · · . . = . . (1.1) v A A v n′ n1 · · · nn n Note that the first index of A denotes the row and the second index the column. In the next chapter we will say more about the transformation of vectors. According to the rules of matrix multiplication the above equation means: v1′ = A11 v1 + A12 v2 + + A1n vn , . .· .· · · · .· . (1.2) v = A v + A v + + A v , n′ n1 · 1 n2 · 2 · · · nn · n or equivalently, n v1′ = ∑ A1νvν , ν=1 . (1.3) n vn′ = ∑ Anνvν , ν=1 or even shorter, n N vµ′ = ∑ Aµν vν ( µ 1 µ n) . (1.4) ν=1 ∀ ∈ | ≤ ≤ In this formula we have put the essence of matrix multiplication. The index ν is a dummy index and µ is a running index. The names of these indices, in this case µ and 5 CHAPTER 1. THE INDEX NOTATION ν, are chosen arbitrarily. The could equally well have been called α and β: n N vα′ = ∑ Aαβ vβ ( α 1 α n) . (1.5) β=1 ∀ ∈ | ≤ ≤ Usually the conditions for µ (in Eq. 1.4) or α (in Eq. 1.5) are not explicitly stated because they are obvious from the context. The following statements are therefore equivalent: ~v = ~y v = y v = y , ⇔ µ µ ⇔ α α n n (1.6) ~v = A~y vµ = ∑ Aµνyν vν = ∑ Aνµyµ . ⇔ ν=1 ⇔ µ=1 This index notation is also applicable to other manipulations, for instance the inner product. Take two vectors ~v and w~ , then we define the inner product as n ~ ~ v w := v1w1 + + vnwn = ∑ vµwµ . (1.7) · · · · µ=1 (We will return extensively to the inner product. Here it is just as an example of the power of the index notation). In addition to this type of manipulations, one can also just take the sum of matrices and of vectors: C = A + B Cµν = Aµν + Bµν ⇔ (1.8) ~z = ~v + w~ z = v + w ⇔ α α α or their difference, C = A B Cµν = Aµν Bµν − ⇔ − (1.9) ~z = ~v w~ z = v w − ⇔ α α − α ◮ Exercises 1 to 6 of Section C.1. From the exercises it should have become clear that the summation symbols ∑ can always be put at the start of the formula and that their order is irrelevant. We can therefore in principle omit these summation symbols, if we make clear in advance over which indices we perform a summation, for instance by putting them after the formula, n ∑ Aµνvν Aµνvν ν ν=1 → { } n n (1.10) ∑ ∑ AαβBβγCγδ AαβBβγCγδ β, γ β=1 γ=1 → { } From the exercises one can already suspect that almost never is a summation performed over an index if that index only ap- • pears once in a product, almost always a summation is performed over an index that appears twice in • a product, an index appears almost never more than twice in a product. • 6 CHAPTER 1. THE INDEX NOTATION When one uses index notation in every day routine, then it will soon become irritating to denote explicitly over which indices the summation is performed. From experience (see above three points) one knows over which indices the summations are performed, so one will soon have the idea to introduce the convention that, unless explicitly stated otherwise: a summation is assumed over all indices that appear twice in a product, and • no summation is assumed over indices that appear only once. • From now on we will write all our formulae in index notation with this particular convention, which is called the Einstein summation convection. For a more detailed look at index notation with the summation convention we referto[4]. We will thus from now on rewrite n ∑ Aµνvν Aµνvν , ν=1 → n n (1.11) ∑ ∑ AαβBβγCγδ AαβBβγCγδ . β=1 γ=1 → ◮ Exercises 7 to 10 of Section C.1. 7 CHAPTER 1. THE INDEX NOTATION 8 2 Bases, co- and contravariant vectors In this chapter we introduce a new kind of vector (‘covector’), one that will be es- sential for the rest of this booklet. To get used to this new concept we will first show in an intuitive way how one can imagine this new kind of vector. After that we will follow a more mathematical approach. 2.1. Intuitive approach We can map the space around us using a coordinate system. Let us assume that we use a linear coordinate system, so that we can use linear algebra to describe it. Physical objects (represented, for example, with an arrow-vector) can then be described in terms of the basis-vectors belonging to the coordinate system (there are some hidden difficulties here, but we will ignore these for the moment). In this section we will see what happens when we choose another set of basis vectors, i.e. what happens upon a basis transformation. In a description with coordinates we must be fully aware that the coordinates (i.e. the numbers) themselves have no meaning. Only with the corresponding basis vectors (which span up the coordinate system) do these numbers acquire meaning. It is important to realize that the object one describes is independent of the coordi- nate system (i.e. set of basis vectors) one chooses. Or in other words: an arrow does not change meaning when described an another coordinate system. Let us write down such a basis transformation, ~ ~ ~ e1′ = a11 e1 + a12 e2 , (2.1) ~ ~ ~ e2′ = a21 e1 + a22 e2 . This could be regarded as a kind of multiplication of a ‘vector’ with a matrix, as long as we take for the components of this ‘vector’ the basis vectors. If we describe the matrix elements with words, one would get something like: ~ ~ ~ ~ ~e projection of e1′ onto e1 projection of e1′ onto e2 ~e 1′ = 1 . (2.2) ~e ~ ~ ~ ~ ~e 2′ projection of e2′ onto e1 projection of e2′ onto e2! 2 . Note that the basis vector-colums are not vectors, but just a very useful way to . write things down. We can also look at what happens with the components of a vector if we use a different set of basis vectors. From linear algebra we know that the transformation 9 2.1 Intuitive approach e2 e'2 0.8 1.6 v=( 0.4 ) v=( 0.4 ) e1 e'1 Figure 2.1: The behaviour of the transformation of the components of a vector under the transformation of a basis vector~e = 1~e v = 2v .