Phys 4390: General Relativity Lecture 5: Intro. to Tensor Algebra

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PHYS 4390: GENERAL RELATIVITY LECTURE 5: INTRO. TO TENSOR ALGEBRA

Vectors are sufficient to describe Newtonian mechanics, however General Rela- tivity will require more general objects than vectors, namely tensors. To introduce these, we will first discuss the tensor algebra through an abstract approach, and afterwards introduce the conventional approach based on indices. While much of our discussion of tensors will appear to be in Rn, tensors may be defined on any differential manifold. This is due to the fact that a differential manifold is an object that appears locally like Euclidean space but is globally different.

1. Curves and Surfaces on a Manifold 1.1. Parametric Approach. A curve may be described with one single parameter, u, and deﬁnes coordinates in an n-dimensional manifold, xa = xa(u), a = 1, ..n. A surface has two parameters, and may be written as xa = xa(u, v) and in general a m-dimensional surface has m degrees of freedom, and is deﬁned by xa = xa(u1, u2, ..., um) We may call a surface a subspace, and a surface with m = n − 1 degrees of freedom a hypersurface.

1.2. I. nstead of a parametric approach, we can describe a surface as a set of constraints. To see how, consider a hypersurface, xa = xa(u1, ..., un−1), taking each coordinate xa we could eliminate u1 to un−1 leaving one function that must vanish f(x1, x2, ..., xn) = 0 Thus, the parametric approach for a hypersurface is equivalent to imposing a constraint. As an example in R2 the constraint x2 + y2 − R2 = 0 describes a circle ( i.e., S1). Adding more constraints reduces the dimension of the subspace. An m-dimensional subspace must have n − m constraints f 1(x1, ..., xn) = 0 . . f n−m(x1, ..., xn) = 0. This can be shown to be equivalent to the parametric approach. 1 2 PHYS 4390: GENERAL RELATIVITY LECTURE 5: INTRO. TO TENSOR ALGEBRA

2. Coordinate Transformations The reason we want to work with tensors is due to the principle of covariance, any physical law or quantity must be unchanged in any coordinate system. To describe physics in General Relativity we will use tensor equations, because these will be coordinate invariant. For this reason we must discuss coordinate transformations. A change of coordinates xa → xã is described as an invertible mapping from Rn to Rn: xã = f a(x1, ..., xn), a = 1, ..., n. Differentiating each new coordinate with respect to (w.r.t) to each xa we produce a matrix related to the coordinate transformation  ∂x˜1 ∂x˜1 ∂x˜1  ∂x1 ∂x2 ... ∂xn 2 2 2 a ∂x˜ ∂x˜ ∂x˜ ∂x˜  1 2 ... n  =  ∂x ∂x ∂x  b  . . .  ∂x  ......  ∂xñ ∂xñ ∂xñ ∂x1 ∂x2 ... ∂xn Taking the determinant of this matrix we produce the Jacobian of the coordinate transformation ∂xã J˜ = | | ∂xb noindent As we have required that the coordinate transformation is invertible, this quantity will be non-zero. The Jacobian of the inverse transformation xa = fã(˜x1, ..., xñ) is ∂xa 1 J = | | = ∂x˜b J˜ ∂xa ∂xã ∂x ∂˜x In fact, the matrix ∂x˜b is the inverse of ∂xb . so that ∂˜x ∂x = In, the identity matrix 1. We may introduce a helpful definition to write the above matrix equation in terms of indexed components.

a Definition 2.1. The Kronecker delta δ b is defined as a a δ a = 1, δ b = 0, a 6= b Thus we have ∂xã ∂xc = δa . ∂xc ∂x˜b b To compute the differentials of the original coordinate system in terms of the differentials of the new coordinates, use the chain rule n X ∂xa ∂xa dxa = dx˜b = dx˜b ∂x˜b ∂x˜b b=1 ∂xa The matrix ∂x˜b and its inverse play an important role distinguishing between two types of tensors: covariant and contravariant tensors.

1To prove this, consider the composition of the coordinate transformation with its inverse, and use the chain rule. PHYS 4390: GENERAL RELATIVITY LECTURE 5: INTRO. TO TENSOR ALGEBRA 3

3. Contravariant Tensors Suppose we have two points lying a particular coordinate system, such that they are separated by an infinitesimal displacement vector PQ. For two different coordinate systems PQ = dxa = dxã using the chain-rule we have a a ∂x b dx˜ = b dx ∂x˜ P here the subscript P on the bracket indicates that the components of the matrix are evaluated at the point P As we are considering an arbitrary point P , we will not use this until needed. Generalizing this identity we have the required prototype definition Definition 3.1. A contravariant tensor of rank 1, θa transforms under a coordinate transformation as ∂xã θã = θb ∂xb ˜ dxa As θ is associated with the point P it is distinct from the tangent vector du on a curve xa(u) since one is a coordinate dependent object at P and the other is a tangent vector along the curve. More generally, we can consider contravariant tensors of higher rank 2. Con- sider the tensor product of two rank 1 contravariant tensors Xa and Y b, we have something that resembles a matrix, XaZb, this is in fact a rank 2 contravariant tensor.

Definition 3.2. A contravariant tensor of rank r, θa1,...,ar transforms under a coordinate transformation as ∂xã1 ∂xãr θã1,...ar = ... θb1,...,br ∂xb1 ∂xbr 4. Covariant Tensors Taking a scalar we may express it in the new coordinate system as φ = φ(xb(˜xa)) , then taking the derivative w.r.t.x ã, ∂φ ∂φ ∂xb ∂xb ∂φ = = ∂xã ∂xb ∂xã ∂xã ∂xb we have found something that looks like our contravariant transformation rule, but involving the inverse coordinate transformation. Yet again, generalizing this definition we have

Definition 4.1. A covariant tensor of rank 1 ea transforms under a coordinate transformation as ∂xb e˜ = e a ∂xã b more generally we have the rank s covariant tensors define as

2Notice that a rank zero tensor φ is a scalar, such that φ˜ = φ. 4 PHYS 4390: GENERAL RELATIVITY LECTURE 5: INTRO. TO TENSOR ALGEBRA

Definition 4.2. A covariant tensor of rank s, ea1,...,as transforms under a coordinate transformation as ∂xb1 ∂xbs eã ,...a = ... eb ,...,b 1 s ∂xã1 ∂xãs 1 s A helpful phrase to remember the position of the index is ”Co goes below” The a ∂ differentials dx and derivative operators ∂xa lie at the core of this distinction between covariant and contravariant tensors, for this reason, we always write coordinate components with an upper index.

5. Mixed Tensors We can now consider objects with both contravariant and covariant parts in terms of indices. For example, a type (1,2) tensor would be: ∂xã ∂xe ∂xf X˜ a = Xd bc ∂xd ∂x˜b ∂x˜c ef from this example we have a general definition Definition 5.1. A tensor of type (r,s), T a1,...,ar transforms under a coordinate b1,...,bs transformation as ∂xã1 ∂xãr ∂xd1 ∂xds ã1,...,ar c1,...,cr Tb ,...,b = ...... Td ,...,d 1 s ∂xc1 ∂xcr ∂x˜b1 ∂x˜bs 1 s At this point, we can demonstrate the coordinate invariance of tensor equations. Suppose we have two tensors rank 2 covariant tensors such that Xab = Yab, then ∂xa ∂xb we may transform both sides by multiplying by ∂x˜c ∂xd to get ∂xa ∂xb ∂xa ∂xb X = Y → X˜ = Y˜ ∂x˜c ∂xd ab ∂x˜c ∂xd ab cd cd

6. Tensor Fields So far we have been speaking about tensors defined at a single point, this can be expanded to consider tensor-fields which are defined over a region of the manifold, so that T a1,...,ar (P ) would define a tensor-field evaluated at the point P. We will b1,...,bs focus primarily on tensor fields which transform at a given point as we have stated, however they have the added benefit of being smooth (i.e. C∞).

7. Operations on Tensors Since we can add vectors and matrices and these are special cases of tensors, we would like tensors in general to have this property, thus if we have two tensors with the same upper and lower indices, we may add them together to create a new tensor, i.e., a a a X bc = Y bc + Z bc is a sensible equation under the transformation rules. Similarly scalar multiplication should hold as well, a a X bc = cY bc. PHYS 4390: GENERAL RELATIVITY LECTURE 5: INTRO. TO TENSOR ALGEBRA 5

We will introduce two new operations on the indices of tensors, however we ﬁrst must discuss symmetry properties of more familiar tensors: matrices (or, type (0,2) tensors - a similar rule will hold for type (2,0) tensors):

Deﬁnition 7.1. A type 2 tensor is symmetric if Xab = Xba

Deﬁnition 7.2. A type 2 tensor is anti-symmetric if Xab = −Xba 1 with a little combinatorics this implies a symmetric tensor has 2 n(n + 1) indepen- 1 dent components and an anti-symmetric tensor has 2 n(n − 1) independent components. Given a general type (0,2) tensor we can compute the symmetric and anti- symmetric part of the tensor as 1 X(ab) = 2 (Xab + Xba), 1 X[ab] = 2 (Xab − Xba).

With a little algebra, it is easy to see that Xab = X(ab) + X[ab]. We may generalize these operations to arbitrary (0, s) tensors

Deﬁnition 7.3. The symmetrization of the tensor Xa1,...,as is " # 1 X X = X (a1,...,as) r! aσ(1),...aσ(s) σ∈Sn where Sn denotes the group of permutations of the set of integers 1 to n, and σ is an element of Sn.

Deﬁnition 7.4. The anti-symmetrization of the tensor Xa1,...,as is " # 1 X X = (−1)|σ|X (a1,...,as) r! aσ(1),...aσ(s) σ∈Sn where Sn denotes the group of permutations of the set of integers 1 to n, σ is an element of Sn, and |σ| records the number of transpositions in the set {1, 2, ...n} to produce the eﬀect of σ.

Example 7.5. For a rank three tensor Xabc the anti-symmetrization is 1 X = (X − X + X − X + X − X ] [abc] 6 abc acb cab cba bca bac 8. Contracted Tensors an Compount Tensors Previously we took two rank 1 tensors and produced a rank 2 tensor. In general a type (p, q) tensor multiplied by a type (r, s) tensor will produce a (p + r, q + s) tensor: a a a Example 8.1. Consider Y b and Zcd then we may produce X bcd = Y bZcd Alternatively we can reduce the rank of tensors by the contraction operation Deﬁnition 8.2. Give a type (r, s) we may produce a type (r − 1, s − 1) tensor by contraction T a1,...,ar−1 = T a1,...ai−1,c,ai+1,...,ar b1,...,bs−1 b1,...bj−1,c,bj+1,...,br where i ∈ [1, r] and j ∈ [1, s] 6 PHYS 4390: GENERAL RELATIVITY LECTURE 5: INTRO. TO TENSOR ALGEBRA

a b Example 8.3. Taking the product of Xbcd and δ a we may contract an upper and a b a lower index to produce: X bcdδ a = X acd = Ycd

9. Relationship to Index-Free Notation So far we have tried to maintain a distinction between geometrical objects, like tensors, and their components in a coordinate system. However, it is clear from our prototype definitions for contravariant and covariant tensors that they can be related by treating the contravariant vector field as an operator on real-valued functions. If we consider a parameterized curve on the manifold xa(u), its tangent vector takes the following form dxα(u) d dxb ∂xa(u) = xa(u) = du du du ∂xb dxb we can treat the components du as components of some vector V defined as d dxb ∂ V = = du du ∂xb ∂ The derivative operators { ∂xb } are then the basis of the tangent vectors (type (0,1) tensors). In component form, an arbitrary vector is ∂ V = Xa = Xa∂ ∂xa a ∂ where we often employ the shorthand ∂a = ∂xa = ∂xa . How does this behave under a coordinate transformation? Supposingx ã = f a(xb) is a coordinate transformation ∂ ∂xã ∂xc ∂ X˜ a∂˜ = X˜ a = Xb a ∂xã ∂xb ∂xã ∂xc ∂xã ∂xc ∂ = Xb ; ∂xb ∂xã ∂xc using the identity introduced after Definition 2.1, we find that ˜ a ˜ c b c X ∂a = δ bX ∂c = X ∂c. Regardless of the coordinate system V defines the same object. Remark 9.1. Notice that the basis elements transform in the opposite nature to the components. In our notation we will ”see” the vectors as a list of components, so when we say a tangent vector X we mean the contravariant components Xa. Definition 9.2. The vector space of all contravariant vectors at a point p in a manifold M is known as the tangent space at P, and is denoted as Tp(M). Generally the tangent space will be different from the underlying manifold. Eu- clidean space and Minkowski spacetime have tangent spaces that ”lie” in their respective manifolds. It might be helpful to think of a tangent plane on a sphere - locally it will approximate the surface of the sphere, but on much larger scales it is a bad approximation. Definition 9.3. The vector space of all covariant vectors at a point p in a manifold ∗ M is defined as the cotangent space at P, and is denoted as Tp (M). PHYS 4390: GENERAL RELATIVITY LECTURE 5: INTRO. TO TENSOR ALGEBRA 7

This space may be considered as a dual to the tangent space, where each covariant vector acts as a mapping from Tp(M) → R. From the proto-defintion of a contravariant tensor of rank 1, it is clear that the basis of coordinate differentials a ∗ {dx } will form a basis for Tp (M) Remark 9.4. As with tangent vectors, when we speak of a cotangent vector (or one form) we mean the covariant components Xa. More generally the vector space of all type (r,s) tensors at a point p of a manifold M is denoted as Tp(r, s)(M) and its basis may be construct from the tensor product ∂ b of r-copies of { ∂xa } and s-copies of {dx }. 10. Lie Bracket If we are given two vector fields (type (0,1) tensor-fields) there is an operation that produces a new vector field, the Lie bracket, or commutator. Definition 10.1. Given vector fields X and Y the Lie Bracket is [X, Y] = XY − YX Given an arbitrary function f it is straightforward to apply the vector fields to the real-valued function

b a b a a b [X, Y]f = (X ∂bY − Y ∂bX )∂af − X Y (∂a∂bf − ∂b∂af) As we are working with the coordinate basis, partial derivatives commute 3. and so this becomes: a b a b a [X, Y] = Z = X ∂b(Y ) − Y ∂b(X ) The Lie bracket satisﬁes the following identities, • [X, X] = 0 • [X, Y] = −[Y, X] • [X, [Y, Z]] + [Z, [X, Y]] + [Y, [Z, X]] = 0 The last is called the Jacobi identity, and most professors in diﬀerential geometry believe students should have to prove it at least once4

11. Terminology: Co vs Contra (variant) To talk about the relationship between the two types of tensors, we are going to consider the coordinate transformationx ˜a = f a(xb) as a Rn one-to-one and onto n n mapping from the manifold M = Rx to N = R x˜, i.e., f : M → N This mapping induces two linear maps on the tangent space and cotangent space ∗ respectively, namely the push forward, f∗ and pullback f :

f∗ : TpM → Tf(p)N ∗ ∗ ∗ f : Tf(p)N → Tp M.

3There are other bases for the tangent space where derivatives in the directions of the basis vectors do not commute- for the curious: google non-coordinate bases 4Usually they will assign this as a question. I on the otherhand will count any proof of this submitted with Assignment 2 as a bonus question 8 PHYS 4390: GENERAL RELATIVITY LECTURE 5: INTRO. TO TENSOR ALGEBRA

The diﬀerence between covariant and contravariant tensors is then a matter of the direction of the derived linear map for a coordinate transformation. This abstract deﬁnition can be restated in more concrete terms • Covariant: Any characteristic under measurement which increases/decreases with an increase/decrease of the underlying parameter. As an example, if we measure a sharp spatial gradient in temperature, say 5Km−1; as distance increases, the temperature will increase as well, thus it is covariant. • Contravariant: Any characteristic which for an increase/decrease of the underlying parameter decreases/increases. if we know that the tempera- −1 1 ture derived through the 5Km gradient is covariant, then temp has the opposite variation - it will be decrease as distance increases, and hence it is contravariant.