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

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Phys 4390: General Relativity Lecture 5: Intro. to Tensor Algebra 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 parame- ter, 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 con- straint. 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 ! x~a is described as an invertible mapping from Rn to Rn: x~a = 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 2 @x~1 @x~1 @x~1 3 @x1 @x2 ::: @xn 2 2 2 a @x~ @x~ @x~ @x~ 6 1 2 ::: n 7 = 6 @x @x @x 7 b 6 . 7 @x 4 . ::: . 5 @x~n @x~n @x~n @x1 @x2 ::: @xn Taking the determinant of this matrix we produce the Jacobian of the coordinate transformation @x~a J~ = j 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 = f~a(~x1; :::; x~n) is @xa 1 J = j j = @x~b J~ @xa @x~a @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 @x~a @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 = dx~a 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 @x~a θ~a = 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 @x~a1 @x~ar θ~a1;:::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 ~a, @φ @φ @xb @xb @φ = = @x~a @xb @x~a @x~a @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 @x~a 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 coordi- nate transformation as @xb1 @xbs e~a ;:::a = ::: eb ;:::;b 1 s @x~a1 @x~as 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 coor- dinate 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: @x~a @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 @x~a1 @x~ar @xd1 @xds ~a1;:::;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. C1). 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 compo- nents. 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.
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