Tensor calculus. 1. S.S. Chern, W.H. Chen, and K.S. Lam, Lectures on differential geometry (World Scientific,

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1 Tensor calculus The purpose of these notes is to summarize the standard definitions and notation; they are far from mathematically complete, although some exercises have been included to fill in some gaps. Most modern mathematics texts shy away from the index notation which is so essential in relativity. Two good references are 1. S.S. Chern, W.H. Chen, and K.S. Lam, Lectures on differential geometry (World Scientific, 2000), 2. W.M. Boothby, An introduction to differentiable manifolds and Riemannian geometry (Academic Press, 2003). Tangent and cotangent vectors Let {x a, 1 a n} be an arbitrary smooth system of coordinates in some region U R n, where smooth means that each function x a : U R is infinitely differentiable. Fix a point p U. Definition: Let t γ(t) be a smooth curve in U with γ(0) = p. In the coordinates x a, the curve will be given parametrically by (x 1 (t),..., x n (t)). The tangent vector dγ/ to the curve at the point p has the components {dx a / t=0 : 1 a n} in this coordinate system. Example: In spherical polar coordinates in R 3, the tangent vector to the curve x(t) = (r(t), θ(t), φ(t)) has the components (dr/, dθ/, dφ/) t=0. Note that the parametrization of the curve is important: If s(t) is a smooth function with s(0) = 0, then dγ/ds = (dγ/)(/ds), so reparametrization rescales the tangent vector. Definition: Let f : U R be differentiable at p. The derivative of f along γ at t = 0 is = f p dx a = t=0 x a a f(p) dxa t=0 (0). a The function t (f γ)(t) is a real-valued function of one real variable; its derivative is a scalar (real number), independent of the local coordinates. If we do the computation in another system x a of coordinates, 1 we must get the same result: = f (p)d xa t=0 x a (0). 1 To say that x a is another system of coordinates means that we have n C functions x a = f a (x 1,..., x n ) defined in a neighborhood of p whose Jacobian is non-singular. We generally just write x a (x 1,..., x n ) for these functions, and write the inverse functions as x a ( x 1,..., x n ). 1

2 By the chain rule, we have f x = f x b a x b x a (1) d x a = xa dx b x b = xa dx c x c (2) where x a /x b is the (a, b) entry of the Jacobian and x a / x b is the (a, b) entry of its inverse. We therefore have = f d x a x a = f x b x a dx c x b x a x c = f dx c x b δb c = f dx b x b The relations in equations (??) and (??) are identical to those we ve already encountered for covariant and contravariant vectors under a change of basis. To see this, suppose the point p has the local coordinates (x 1 0,..., x n 0), and consider the curve t γ(t) = (x 1 0, x 2 0,..., x b 0 + t,..., x n 0). The derivative of γ at t = 0 is a tangent vector at the point P with all components vanishing except for a 1 in the b th position. The corresponding tangent vector should therefore be the basis vector e b at p. If we compute / for any smooth function f, we get = f t=0 x a (p)dxa (0) = f x a (p)δa b = f x b (p). This suggests that we identify tangent vectors with differential operators: we can write = f dx a x a = dxa x b (f), and we can identify the tangent vector to the curve with the operator dγ = dxa x, b In this expression, the numbers dx a / are the components of the vector dγ/, while the basis vectors are { a or /x a : 1 a n}. Thus the general expression for a tangent vector 2

3 at p takes the form V = V a a (p). The action of V on the function f is just V (f) = v a a (f), the right hand side being evaluated at p. Exercise: Given an n-tuple of numbers {V a }, if p has the local coordinates x 0, find a parametric curve t x(t) with x(0) = x 0, such that V a = dx a / t=0. Exercise: Show that V is a linear operator; that is V (c 1 f 1 + c 2 f 2 ) = c 1 V (f 1 ) + c 2 V (f 2 ), where c 1, c 2 are constants, and f, g are differentiable at p. Exercise: (For the mathematically inclined:) A curve γ(t) with γ(0) = p will be called a smooth curve at p. If µ(t) is another smooth curve at p we ll write γ p µ if for all functions f differentiable at p, / t=0 = d(f µ)/ t=0. Show that p is an equivalence relation; in particular, in any local coordinate system, their components are identical. The set of all equivalence classes is called the tangent space at p and denoted T p R n. For the dual space, define an equivalence relation on the set of all differentiable functions at p by requiring f p g / t=0 = d(g γ)/ t=0 for all curves γ at p. So f p g a (f) p = a (g) p, a. The set of equivalence classes is called the cotangent space at p and denoted Tp R n. Define the vector space operations as follows: Starting with [f], [g] Tp, show that the operations c[f] = [cf] and [f] + [g] = [f + g] are well-defined. What are the components of [f + g] in the local coordinates? Check that V (f), defined above, is really V ([f]) (just depends on the equivalence class). This makes Tp R n into a vector space. We know that the tangent vectors are linear operators on functions from a previous exercise. So if V = [γ], U = [µ], we define cv and V + U by their pointwise action on cotangent vectors: that is, (cv )([f]) = cv ([f]) = cv (f), for any f [f] (U + V )([f] = U([f] + V ([f]) = U(f) + V (f) for any f [f] This defines the vector space structure on T p T p. Definition: The equivalence class [f] T p is denoted by df(p) and called the differential of f at p. The natural basis for Tp will be that dual to the basis { a }, and it s easily found. We re looking for some functions {f a } with the property that a (f b ) = δa. b The obvious candidates are f a = x a. So the natural dual basis of Tp is the set {dx 1, dx 2,..., dx n }, and the general element of the cotangent space at p can be written uniquely in the form φ = φ a dx a. 3

4 Remark: The differential of f at p is not a scalar; it s a linear function on vectors, to wit 2 : df = a f dx a df(v ) = a f dx a (V ) = a f dx a (V b b ) = V b a f dx a ( b ) = V b a fδ a b = V a a f If one writes the components of V as x a, then df(v ) = ( a f) x a. Summary: In a local coordinate system containing the point p of R n, elements of the tangent space T p R n can be written uniquely in the form V = V a a (3) This is a linear operator on smooth functions at p: for any such function f, V (f) = V a ( a f)(p). The operators { a p : 1 a n} form a basis of T p R n, called the natural basis associated to the given coordinate system. Elements of the cotangent (dual) space T p R n can be written uniquely as φ = φ a dx a, (4) These are real-valued linear functions on the tangent space: φ(v ) = φ a V a. The operators {dx a : 1 a n} form the dual basis: dx a ( b ) = δ a b. In a different coordinate system, we have while x = xb a x a x, and Ṽ a = xa b x V b (5) b d x a = xa x b dxb, and φ b = xa x b φ a (6) The various tensor product spaces are now formed in the usual way. Tensor fields Definition: The set of all tangent vectors at all points p U R n is called the tangent bundle of U, denoted T U. In this case, T U U R n. (Globally, the tangent bundle may 2 The points at which these expressions are to be evaluated will generally be taken as understood, and omitted from the notation 4

5 not be a simple product space. For instance, T S 2 S 2 R 2.) The tangent bundle has a natural projection π : T U U given by π(v (p)) = p. It maps each vector to the point at which it lives. T p U = π 1 (p) is called the fiber of this bundle over p. Since the fiber π 1 p is a vector space, the tangent bundle is called a vector bundle over U. The bundles defined below are also vector bundles. Definition: A vector field on U R n is a smooth function V : U T U with the property that V (p) is a vector at p : π V = id U. In a local coordinate system, the components of the vector field V will be smooth functions of the coordinates: V = V a (x) a. As a general rule, it will be clear from the context that we re talking about a vector field, and the coordinate x will be suppressed. Definition: Similarly, we have the cotangent bundle T U with its projection π. A cotangent vector field is a function φ : U T U such that π φ = id U. A cotangent vector field is usually called a differential form on U. Its components are also given locally by n smooth functions φ a (x). Definition: The various tensor bundles r (T U) s (T U) are defined similarly. Examples: 1. γ = x a a is the Euler vector field on R n. It has the useful property that if f is homogeneous of degree r, f(λx 1,..., λx n ) = λ r f(x 1,..., x n ), then γ(f)(x) = rf(x). To see this, differentiate the expression for f with respect to λ and then set λ = If φ(x) is a differential form, then φ is said to be exact if φ = df for some function f. Show that if φ is exact, then a (φ b ) = b (φ a ). The differential form in R 2 given by φ = ydx + xdy x 2 + y 2 is not exact, even though y (φ x ) = x (φ y ). Why? Nevertheless, we often write φ as dθ. 3. The metric tensor g of Minkowski space has the constant components g ab = Diag{1, 1, 1, 1} in any inertial frame. In an arbitrary coordinate system, these go over to the components x c x d g ab = g cd x a x. b 4. If f is a function, then the gradient of f is the vector field given in local coordinates by a f = g ab f b. So the scalar product of f with the vector field V is g( f, V ) = g ab a f V b = g ab g ac c f V b = δ c b c fv b = df(v ). 5

6 5. In the usual basis in R n, a vector field with constant components is a parallel vector field in the sense that it can be generated by taking a single vector and propagating it over all the space keeping it parallel to itself. But in a curvilinear system, such a vector field is not necessarily parallel. For instance, the direction of the vector field /θ (cylindrical coordinates in R 2 depends on location. In fact, since x = r cos θ, y = r sin θ, θ = x θ x + y θ y = r sin θ x + r cos θ y. Derivatives of tensor fields A construction like div(v ) = a V a, pretty as it might appear, is not well-defined, except in systems obtained from the usual Cartesian one by non-singular linear transformations. In a general curvilinear system, we have Ṽ a x d ( ) = xb x a x d x b x V c c = xa x b V c x c x d x + xb 2 x a b x d x b x V c c If the transformation were linear, then the second term on the right would vanish, but it won t in general, and therefore, the quantities V a /x d are not the components of a rank (1, 1) tensor. And neither is its trace div(v ). The divergence of a vector field can be defined properly, but it takes more work. What we ll need to do is to add a correction term which cancels the unwanted second term. This modified derivative operator will be called the covariant derivative and will be discussed elsewhere. For precisely the same reasons, the d Alembertian (wave) operator φ = a a φ must also be modified for general coordinates. Exercise: In R 2, in Cartesian coordinates, the divergence of the Euler vector field γ = x x + y y is 2. Find V r /r + V θ /θ where (r, θ) are polar coordinates. [Answer: 1] Covariance under smaller groups The transformation laws worked out above are valid for general smooth coordinate transformations on the set U. The set of all such transformations forms a group under composition. We ve just seen that there are objects that don t behave well under this group of transformations. But if we make the group smaller, then we get more invariant or covariant objects. Example: special relativity We ve gone to a lot of trouble to make sense out of the four components of the energymomentum vector of a particle. But our conclusions are only valid in an inertial frame. 6

7 That is, if we were to do an arbitrary (even linear) transformation of coordinates, in the new frame, there might not even be a timelike basis vector; that is, there would be no time component in the new system. (Or there could be 4 time components!) There would be no way to talk about the mass or 3-momentum of the particle except by referring things back to an inertial frame. So if we want to think about the velocity of an observer as being a vector with components α(v)(1, v), we have to be in an inertial frame. When we say that V is a 4-vector we mean that V is to be regarded as covariant under the Lorentz group, rather than (immensely) larger group of general coordinate transformations. For instance, the relativistic mass of a particle is Lorentz covariant, but not generally covariant quantity. In an arbitrary coordinate system, it would still be true that P P = P a P a = g ab P a P b = m 2 0, but the decomposition as m 2 0 = m 2 p 2 would be meaningless since the individual components no longer have any physical interpretation. Similarly, the components of the electromagnetic field tensor F = F ab dx a dx b can be separated out in an inertial frame into those of the electric and magnetic field vectors; but this is not so in a general coordinate system. In general relativity, unfortunately, we cannot restrict ourselves to global inertial frames, because they don t exist. In accordance with our experience, we can introduce local inertial frames in each tangent space. (These should better be called infinitessimal inertial frames.) Thus we ll have a certain sort of local Lorentz covariance, which manifests itself in certain types of gauge transformations or local symmetries. But this is a far cry from the Lorentz covariance of special relativity. For example, in general relativity it is entirely non-trivial to define things like the mass and linear momentum of an extended object. 7