Mathematical Tripos Part III General Relativity PDF

Mathematical Tripos Part III General Relativity Harvey Reall  CONTENTS Contents 1 Manifolds and tensors 4 1.1 Introduction 4 1.2 Differentiable manifolds...

Mathematical Tripos Part III General Relativity Harvey Reall  CONTENTS Contents 1 Manifolds and tensors 4 1.1 Introduction 4 1.2 Differentiable manifolds 5 1.3 Smooth functions 8 1.4 Curves and vectors 9 1.5 Covectors 14 1.6 Abstract index notation 15 1.7 Tensors 16 1.8 Tensor fields 20 1.9 Integral curves 21 1.10 The commutator 22 2 Metric tensors 24 2.1 Definition 24 2.2 Lorentzian signature 26 2.3 Curves of extremal proper time 28 3 Covariant derivative 31 3.1 Introduction 31 3.2 The Levi-Civita connection 35 3.3 Geodesics 37 3.4 Normal coordinates 39 4 Curvature 40 4.1 Parallel transport 40 4.2 The Riemann tensor 41 4.3 Parallel transport again 42 4.4 Symmetries of the Riemann tensor 45 4.5 Geodesic deviation 46 4.6 Curvature of the Levi-Civita connection 48 5 Diffeomorphisms and Lie derivative 49 5.1 Maps between manifolds 49 5.2 Diffeomorphisms 51 5.3 Lie derivative 53 Part 3 GR October 12, 2022 1 H.S. Reall  CONTENTS 6 Physical laws in curved spacetime 57 6.1 Motivating principles 57 6.2 Physical laws in curved spacetime 60 6.3 Energy-momentum tensor 63 6.4 Einstein’s equation 67 6.5 Gauge freedom 69 7 Linearized theory 71 7.1 The linearized Einstein equation 71 7.2 The Newtonian limit 74 7.3 Gravitational waves 77 7.4 The field far from a source 83 7.5 The energy in gravitational waves 86 7.6 The quadrupole formula 90 7.7 Comparison with electromagnetic radiation 93 7.8 Gravitational waves from binary systems 94 8 Differential forms 99 8.1 Introduction 99 8.2 Connection 1-forms 100 8.3 Spinors in curved spacetime (non-examinable) 103 8.4 Curvature 2-forms 105 8.5 Volume form 107 8.6 Integration on manifolds 108 8.7 Submanifolds and Stokes’ theorem 109 9 Lagrangian formulation 111 9.1 Scalar field action 111 9.2 The Einstein-Hilbert action 112 9.3 Energy momentum tensor 116 10 The initial value problem 118 10.1 Introduction 118 10.2 Extrinsic curvature 119 10.3 The Gauss-Codacci equations (proofs non-examinable) 120 10.4 The constraint equations 123 10.5 The initial value problem for GR 123 Part 3 GR October 12, 2022 2 H.S. Reall  CONTENTS Preface These are lecture notes for the course on General Relativity in Part III of the Cambridge Mathematical Tripos. There are introductory GR courses in Part II (Mathematics or Natural Sciences) so, although self-contained, this course does not cover topics usually covered in a first course, e.g., the Schwarzschild solution, the solar system tests, and cosmological solutions. You should consult an introductory book (e.g. Gravity by J.B. Hartle) if you have not studied these topics before. Acknowledgment I am very grateful to Andrius Štikonas for producing the figures. Conventions We will use units in which the speed of light is one: c = 1. Sometimes we will use units where Newton’s gravitational constant is one: G = 1. We will use ”abstract indices” a, b, c etc to denote tensors, e.g. V a , gcd. Equations involving such indices are basis-independent. Greek indices µ, ν etc refer to tensor components in a particular basis. Equations involving such indices are valid only in that basis. We will define the metric tensor to have signature (− + ++), which is the most common convention. Some authors use signature (+ − −−). Our convention for the Riemann tensor is such that the Ricci identity takes the form ∇a ∇b V c − ∇b ∇a V c = Rc dab V d. Some authors define the Riemann tensor with the opposite sign. Bibliography There are many excellent books on General Relativity. The following is an incomplete list: 1. General Relativity, R.M. Wald, Chicago UP, 1984. 2. Advanced General Relativity, J.M. Stewart, CUP, 1993. 3. Spacetime and geometry: an introduction to General Relativity, S.M. Carroll, Addison-Wesley, 2004. Part 3 GR October 12, 2022 3 H.S. Reall  4. Gravitation, C.W. Misner, K.S. Thorne and J.A. Wheeler, Freeman 1973. 5. Gravitation and Cosmology, S. Weinberg, Wiley, 1972. Our approach will be closest to that of Wald. The first part of Stewart’s book is based on a previous version of this course. Carroll’s book is a very readable intro- duction. Weinberg’s book contains a good discussion of equivalence principles. Our treatments of the Newtonian approximation and gravitational radiation are based on Misner, Thorne and Wheeler. 1 Manifolds and tensors 1.1 Introduction In Minkowski spacetime we usually use inertial frame coordinates (t, x, y, z) since these are adapted to the symmetries of the spacetime so using these coordinates simplifies the form of physical laws. However, a general spacetime has no symmetries and therefore no preferred set of coordinates. In fact, a single set of coordinates might not be sufficient to describe the spacetime. A simple example of this is provided by spherical polar coordinates (θ, φ) on the surface of the unit sphere S 2 in R3 : z θ φ y x Figure 1. Spherical polar coordinates These coordinates are not well-defined at θ = 0, π (what is the value of φ there?). Furthermore, the coordinate φ is discontinuous at φ = 0 or 2π. To describe S 2 so that a pair of coordinates is assigned in a smooth way to every point, we need to use several overlapping sets of coordinates. Generalizing this example leads to the idea of a manifold. In GR, we assume that spacetime is a 4-dimensional differentiable manifold. Part 3 GR October 12, 2022 4 H.S. Reall  1.2 Differentiable manifolds 1.2 Differentiable manifolds You know how to do calculus on Rn. How do you do calculus on a curved space, e.g., S 2 ? Locally, S 2 looks like R2 so one can carry over standard results. However, one has to confront the fact that it is impossible to use a single coordinate system on S 2. In order to do calculus we need our coordinates systems to ”mesh together” in a smooth way. Mathematically, this is captured by the notion of a differentiable manifold: Definition. An n-dimensional differentiable manifold is a set M together with a col- lection of subsets Oα such that S 1. α Oα = M , i.e., the subsets Oα cover M 2. For each α there is a one-to-one and onto map φα : Oα → Uα where Uα is an open subset of Rn. 3. If Oα and Oβ overlap, i.e., Oα ∩ Oβ 6= ∅ then φβ ◦ φ−1 α maps from φα (Oα ∩ Oβ ) ⊂ n n Uα ⊂ R to φβ (Oα ∩ Oβ ) ⊂ Uβ ⊂ R. We require that this map be smooth (infinitely differentiable). The maps φα are called charts or coordinate systems. The set {φα } is called an atlas. Oα Oβ M φβ φα φβ ◦ φ−1 α Uβ Uα φα (Oα ∩ Oβ ) φβ (Oα ∩ Oβ ) Figure 2. Overlapping charts Remarks. 1. Sometimes we shall write φα (p) = (x1α (p), x2α (p),... xnα (p)) and refer to xiα (p) as the coordinates of p. Part 3 GR October 12, 2022 5 H.S. Reall  1.2 Differentiable manifolds 2. We use the charts to define a topology on M : we say that R ⊂ M is open iff φα (R ∩ Oα ) is an open subset of Uα for all α. 3. Strictly speaking, we have defined above the notion of a smooth manifold. If we replace “smooth” in the definition by C k (k-times continuously differentiable) then we obtain a C k -manifold. We shall always assume the manifold is smooth. Examples. 1. Rn : this is a manifold with atlas consisting of the single chart φ : (x1 ,... , xn ) 7→ (x1 ,... , xn ). 2. S 1 : the unit circle, i.e., the subset of R2 given by (cos θ, sin θ) with θ ∈ R. We can’t define a chart by using θ ∈ [0, 2π) as a coordinate because [0, 2π) is not open. Instead let P be the point (1, 0) and define one chart by φ1 : S 1 − {P } → (0, 2π), φ1 (p) = θ1 with θ1 defined by Fig. 3. y θ1 P x Figure 3. Definition of θ1 Now let Q be the point (−1, 0) and define a second chart by φ2 : S 1 − {Q} → (−π, π), φ2 (p) = θ2 where θ2 is defined by Fig. 4. Neither chart covers all of S 1 but together they form an atlas. The charts overlap on the “upper” semi-circle and on the “lower” semi-circle. On the first of these we have θ2 = φ2 ◦ φ−1 −1 1 (θ1 ) = θ1. On the second we have θ2 = φ2 ◦ φ1 (θ1 ) = θ1 − 2π. These are obviously smooth functions. 3. S 2 : the two-dimensional sphere defined by the surface x2 + y 2 + z 2 = 1 in R3. Introduce spherical polar coordinates in the usual way: x = sin θ cos φ, y = sin θ sin φ, z = cos θ (1.1) Part 3 GR October 12, 2022 6 H.S. Reall  1.2 Differentiable manifolds y Q θ2 x Figure 4. Definition of θ2 These define θ ∈ [0, π] and φ ∈ [0, 2π) uniquely but the map (x, y, z) 7→ (θ, φ) does not define a chart because [0, π]×[0, 2π) is not an open set. Note that points with θ = 0, π or φ = 0 are the points on S 2 with y = 0, x ≥ 0, which form a line of longitude as shown in Fig. 5. Let O be S 2 with this line removed. Then we have a chart ψ : O → U = (0, π) × (0, 2π) ⊂ R2 , given by ψ : (x, y, z) 7→ (θ, φ). z O y x Figure 5. The subset O ⊂ S 2 : points with θ = 0, π and φ = 0 are removed. We can define a second chart using a different set of spherical polar coordinates defined as follows: x = − sin θ0 cos φ0 , y = cos θ0 , z = sin θ0 sin φ0 , (1.2) where θ0 ∈ (0, π) and φ0 ∈ (0, 2π) are uniquely defined by these equations. This is a chart ψ 0 : O0 → U 0 , where O0 is S 2 with the line z = 0, x ≤ 0 removed, see Fig. 6, and U 0 is (0, π) × (0, 2π). Clearly S 2 = O ∪ O0. The functions ψ ◦ ψ 0 −1 and ψ 0 ◦ ψ −1 are smooth on O ∩ O0 so these two charts define an atlas for S 2. Part 3 GR October 12, 2022 7 H.S. Reall  1.3 Smooth functions z O0 y x Figure 6. The subset O0 ⊂ S 2 : points with θ0 = 0, π and φ0 = 0 are removed. 4. Recall the Cartesian product of two sets X and Y is defined as X × Y = {(x, y) : x ∈ X, y ∈ Y }. Given differentiable manifolds M and M 0 of dimensions n and n0 we can make M × M 0 into a differentiable manifold of dimension n + n0 as follows. Let the charts of our two manifolds be φα : Oα → Uα ⊂ Rn and 0 0 φ0α : Oα0 → Uα0 ⊂ Rn. Now define φαβ : Oα × Oβ0 → Uα × Uβ0 ⊂ Rn+n by φαβ (p, q) = (φα (p), φ0β (q)). You can check that the set of φαβ forms an atlas for M × M 0. This can be generalized to more than 2 factors in the obvious way, e.g. we can define the n-torus as T n = S 1 × S 1 ×... × S 1 (n factors). Remark. A given set M may admit many atlases, e.g., one can simply add extra charts to an atlas. We don’t want to regard this as producing a distinct manifold so we make the following definition: Definition. Two atlases are compatible if their union is also an atlas. The union of all atlases compatible with a given atlas is called a complete atlas: it is an atlas which is not contained in any other atlas. Remark. We will always assume that were are dealing with a complete atlas. (None of the above examples gives a complete atlas; such atlases necessarily contain infinitely many charts.) 1.3 Smooth functions We will need the notion of a smooth function on a smooth manifold. If φ : O → U is a chart and f : M → R then note that f ◦ φ−1 is a map from U, i.e., a subset of Rn , to R. Definition. A function f : M → R is smooth if, and only if, for any chart φ, F ≡ f ◦ φ−1 : U → R is a smooth function. Part 3 GR October 12, 2022 8 H.S. Reall  1.4 Curves and vectors Remark. In GR, a function f : M → R is sometimes called a scalar field. Examples. 1. Consider the example of S 1 discussed above. Let f : S 1 → R be defined by f (x, y) = x where (x, y) are the Cartesian coordinates in R2 labelling a point on S 1. In the first chart φ1 we have f ◦ φ−1 1 (θ1 ) = f (cos θ1 , sin θ1 ) = cos θ1 , which −1 is smooth. Similary f ◦ φ2 (θ2 ) = cos θ2 is also smooth. If φ is any other chart then we can write f ◦ φ−1 = (f ◦ φ−1 −1 i ) ◦ (φi ◦ φ ), which is smooth because we’ve just seen that f ◦ φ−1i are smooth, and φi ◦ φ−1 is smooth from the definition of a manifold. Hence f is a smooth function. 2. Consider a manifold M with a chart φ : O → U ⊂ Rn. Let φ : p 7→ (x1 (p), x2 (p),... xn (p)). Then we can regard x1 (say) as a function on the subset O of M. Is it a smooth function? Yes for any other chart φ0 , x1 ◦ φ0−1 is smooth because it is the first component of the map φ ◦ φ0 −1 , which is smooth by the definition of a manifold. 3. Often it is convenient to define a function by specifying F instead of f. More precisely, given an atlas {φα }, we define f by specifying functions Fα : Uα → R and then setting f = Fα ◦φα. One has to make sure that the resulting definition is independent of α on chart overlaps. For example, for S 1 using the atlas discussed above, define F1 : (0, 2π) → R by θ1 7→ sin(mθ1 ) and F2 : (−π, π) → R by θ2 7→ sin(mθ2 ), where m is an integer. On the chart overlaps we have F1 ◦ φ1 = F2 ◦ φ2 because θ1 and θ2 differ by a multiple of 2π on both overlaps. Hence this defines a function on S 1. Remark. After a while we will stop distinguishing between f and F , i.e., we will say f (x) when we mean F (x). 1.4 Curves and vectors Rn , or Minkowski spacetime, has the structure of a vector space, e.g., it makes sense to add the position vectors of points. One can view more general vectors, e.g., the 4-velocity of a particle, as vectors in the space itself. This structure does not extend to more general manifolds, e.g., S 2. So we need to discuss how to define vectors on manifolds. For a surface in R3 , the set of all vectors tangent to the surface at some point p defines the tangent plane to the surface at p (see Fig. 7). This has the structure of a 2d vector space. Note that the tangent planes at two different points p and q are different. It does not make sense to compare a vector at p with a vector at q. For example: if Part 3 GR October 12, 2022 9 H.S. Reall  1.4 Curves and vectors p q Figure 7. Tangent planes. one tried to define the sum of a vector at p and a vector at q then to which tangent plane would the sum belong? On a surface, the tangent vector to a curve in the surface is automatically tangent to the surface. We take this as our starting point for defining vectors on a general manifold. We start by defining the notion of a curve in a manifold, and then the notion of a tangent vector to a curve at a point p. We then show that the set of all such tangent vectors at p forms a vector space Tp (M ). This is the analogue of the tangent plane to a surface but it makes no reference to any embedding into a higher-dimensional space. Definition A smooth curve in a differentiable manifold M is a smooth function λ : I → M , where I is an open interval in R (e.g. (0, 1) or (−1, ∞)). By this we mean that φα ◦ λ is a smooth map from I to Rn for all charts φα. Let f : M → R and λ : I → M be a smooth function and a smooth curve respectively. Then f ◦ λ is a map from I to R. Hence we can take its derivative to obtain the rate of change of f along the curve: d d [(f ◦ λ)(t)] = [f (λ(t))] (1.3) dt dt In Rn we are used to the idea that the rate of change of f along the curve at a point p is given by the directional derivative Xp · (∇f )p where Xp is the tangent to the curve at p. Note that the vector Xp defines a linear map from the space of smooth functions on Rn to R: f 7→ Xp · (∇f )p. This is how we define a tangent vector to a curve in a general manifold. (We restrict to curves without self-intersections, i.e., λ(t1 ) 6= λ(t2 ) if t1 6= t2. A self-intersecting curve can have multiple tangent vectors at p.) Definition. Let λ : I → M be a smooth curve without self-intersections and (wlog) λ(0) = p. The tangent vector to λ at p is the linear map Xp from the space of smooth functions on M to R defined by   d Xp (f ) = [f (λ(t))] (1.4) dt t=0 Note that this satisfies two important properties: (i) it is linear, i.e., Xp (f + g) = Xp (f ) + Xp (g) and Xp (αf ) = αXp (f ) for any constant α; (ii) it satisfies the Leibniz Part 3 GR October 12, 2022 10 H.S. Reall  1.4 Curves and vectors rule Xp (f g) = Xp (f )g(p) + f (p)Xp (g), where f and g are smooth functions and f g is their product. If φ = (x1 , x2 ,... xn ) is a chart defined in a neighbourhood of p and F ≡ f ◦ φ−1 then we have f ◦ λ = f ◦ φ−1 ◦ φ ◦ λ = F ◦ (φ ◦ λ) and hence    µ  ∂F (x) dx (λ(t)) Xp (f ) = (1.5) ∂xµ φ(p) dt t=0 Note that (i) the first term on the RHS depends only on f and φ and the second term on the RHS depends only on φ and λ; (ii) we are using the Einstein summation convention, i.e., µ is summed from 1 to n in the above expression. Proposition. The set of all tangent vectors at p forms a n-dimensional vector space Tp (M ): the tangent space at p. Proof. Consider curves λ and κ through p, wlog λ(0) = κ(0) = p. Let their tangent vectors at p be Xp and Yp respectively. We need to define addition of tangent vectors and multiplication by a constant. let α and β be constants. We define αXp + βYp to be the linear map f 7→ αXp (f ) + βYp (f ). Next we need to show that this linear map is indeed the tangent vector to a curve through p. Let φ = (x1 ,... , xn ) be a chart defined in a neighbourhood of p. Consider the following curve: ν(t) = φ−1 [α(φ(λ(t)) − φ(p)) + β(φ(κ(t)) − φ(p)) + φ(p)] (1.6) Note that ν(0) = p. Let Zp denote the tangent vector to this curve at p. From equation (1.5) we have     ∂F (x) d µ µ µ µ µ Zp (f ) = [α(x (λ(t)) − x (p)) + β(x (κ(t)) − x (p)) + x (p)] ∂xµ φ(p) dt t=0     µ   µ   ∂F (x) dx (λ(t)) dx (κ(t)) = µ α +β ∂x φ(p) dt t=0 dt t=0 = αXp (f ) + βYp (f ) = (αXp + βYp )(f ). Since this is true for any smooth function f , we have Zp = αXp + βYp as required. Hence αXp + βYp is tangent to the curve ν at p. It follows that the set of tangent vectors at p forms a vector space (the zero vector is realized by the curve λ(t) = p for all t). The next step is to show that this vector space is n-dimensional. To do this, we exhibit a basis. Let 1 ≤ µ ≤ n. Consider the curve λµ through p defined by λµ (t) = φ−1 (x1 (p),... , xµ−1 (p), xµ (p) + t, xµ+1 (p),... , xn (p)). (1.7) Part 3 GR October 12, 2022 11 H.S. Reall  1.4 Curves and vectors The tangent vector to this curve at p is denoted ∂x∂ µ p. To see why, note that, using
equation (1.5)     ∂ ∂F (f ) =. (1.8) ∂xµ p ∂xµ φ(p) The n tangent vectors ∂x∂ µ p are linearly independent. To see why, assume that there
exist constants αµ such that αµ ∂x∂ µ p = 0. Then, for any function f we must have
  µ ∂F (x) α = 0. (1.9) ∂xµ φ(p) Choosing F = xν , this reduces to αν = 0. Letting this run over all values of ν we see that all of the constants αν must vanish, which proves linear independence. Finally we must prove that these tangent vectors span the vector space. This follows from equation (1.5), which can be rewritten  µ    dx (λ(t)) ∂ Xp (f ) = (f ) (1.10) dt t=0 ∂xµ p this is true for any f hence dxµ (λ(t))     ∂ Xp = , (1.11) dt t=0 ∂xµ p i.e. Xp can be written as a linear combination of the n tangent vectors ∂x∂ µ p. These
n vectors therefore form a basis for Tp (M ), which establishes that the tangent space is n-dimensional. QED. Remark. The basis { ∂x∂ µ p , µ = 1,... , n} is chart-dependent: we had to choose
a chart φ defined in a neighbourhood of p to define it. Choosing a different chart would give a different basis for Tp (M ). A basis defined this way is sometimes called a coordinate basis. Definition. Let {eµ , µ = 1,... , n} be a basis for Tp (M ) (not necessarily a coordinate basis). We can expand any vector X ∈ Tp (M ) as X = X µ eµ. We call the numbers X µ the components of X with respect to this basis. Example. Using the coordinate basis eµ = (∂/∂xµ )p , equation (1.11) shows that the tangent vector Xp to a curve λ(t) at p (where t = 0) has components  µ  µ dx (λ(t)) Xp =. (1.12) dt t=0 Part 3 GR October 12, 2022 12 H.S. Reall  1.4 Curves and vectors Remark. Note the placement of indices. We shall sum over repeated indices if one such index appears “upstairs” (as a superscript, e.g., X µ ) and the other “downstairs” (as a subscript, e.g., eµ ). (The index µ on ∂x∂ µ p is regarded as downstairs.) If an
equation involves the same index more than twice, or twice but both times upstairs or both times downstairs (e.g. Xµ Yµ ) then a mistake has been made. Let’s consider the relationship between different coordinate bases. Let φ = (x1 ,... , xn ) and φ0 = (x0 1 ,... , x0 n ) be two charts defined in a neighbourhood of p. Then, for any smooth function f , we have     ∂ ∂ −1 (f ) = (f ◦ φ ) ∂xµ p ∂xµ φ(p)   ∂ 0 −1 0 −1 = [(f ◦ φ ) ◦ (φ ◦ φ )] ∂xµ φ(p) Now let F 0 = f ◦φ0 −1. This is a function of the coordinates x0. Note that the components of φ0 ◦ φ−1 are simply the functions x0 µ (x), i.e., the primed coordinates expressed in terms of the unprimed coordinates. Hence what we have is easy to evaluate using the chain rule:     ∂ ∂ 0 0 (f ) = (F (x (x))) ∂xµ p ∂xµ φ(p)  0ν   0 0  ∂x ∂F (x ) = µ ∂x φ(p) ∂x0 ν φ0 (p)  0ν    ∂x ∂ = (f ) ∂xµ φ(p) ∂x0 ν p Hence we have ∂x0 ν       ∂ ∂ = (1.13) ∂xµ p ∂xµ φ(p) ∂x0 ν p This expresses one set of basis vectors in terms of the other. Let X µ and X 0 µ denote the components of a vector with respect to the two bases. Then we have    0µ    ν ∂ ν ∂x ∂ X=X =X (1.14) ∂xν p ∂xν φ(p) ∂x0 µ p and hence ∂x0 µ   0µ ν X =X (1.15) ∂xν φ(p) Elementary treatments of GR usually define a vector to be a set of numbers {X µ } that transforms according to this rule under a change of coordinates. More precisely, they usually call this a “contravariant vector”. Part 3 GR October 12, 2022 13 H.S. Reall  1.5 Covectors 1.5 Covectors Recall the following from linear algebra: Definition. Let V be a real vector space. The dual space V ∗ of V is the vector space of linear maps from V to R. Lemma. If V is n-dimensional then so is V ∗. If {eµ , µ = 1,... , n} is a basis for V then V ∗ has a basis {f µ , µ = 1,... , n}, the dual basis defined by f µ (eν ) = δνµ (if X = X µ eµ then f µ (X) = X ν f µ (eν ) = X µ ). Since V and V ∗ have the same dimension, they are isomorphic. For example the linear map defined by eµ 7→ f µ is an isomorphism. But this is basis-dependent: a different choice of basis would give a different isomorphism. In contrast, there is a natural (basis-independent) isomorphism between V and (V ∗ )∗ : Theorem. If V is finite dimensional then (V ∗ )∗ is naturally isomorphic to V. The isomorphism is Φ : V → (V ∗ )∗ where Φ(X)(ω) = ω(X) for all ω ∈ V ∗. Now we return to manifolds: Definition. The dual space of Tp (M ) is denoted Tp∗ (M ) and called the cotangent space at p. An element of this space is called a covector at p. If {eµ } is a basis for Tp (M ) and {f µ } is the dual basis then we can expand a covector η as ηµ f µ. ηµ are called the components of η. Note that (i) η(eµ ) = ην f ν (eµ ) = ηµ ; (ii) if X ∈ Tp (M ) then η(X) = η(X µ eµ ) = X µ η(eµ ) = X µ ηµ (note the placement of indices!) Definition. Let f : M → R be a smooth function. Define a covector (df )p by (df )p (X) = X(f ) for any vector X ∈ Tp (M ). (df )p is the gradient of f at p. Remarks. 1. Let f be a constant. Then X(f ) = 0 for any X so (df )p = 0 for any p. 2. Let (x1 ,... , xn ) be a coordinate chart defined in a neighbourhood of p, recall that xµ is a smooth function (in this neighbourhood) so we can take f = xµ in the above definition to define n covectors (dxµ )p. Note that   !  µ ∂ ∂x (dxµ )p ν = ν = δνµ (1.16) ∂x p ∂x p Hence {(dxµ )p } is the dual basis of {(∂/∂xµ )p }. Part 3 GR October 12, 2022 14 H.S. Reall  1.6 Abstract index notation 3. To explain why we call (df )p the gradient of f at p, observe that its components in a coordinate basis are  !     ∂ ∂ ∂F [(df )p ]µ = (df )p = (f ) = (1.17) ∂xµ p ∂xµ p ∂xµ φ(p) where the first equality uses (i) above, the second equality is the definition of (df )p and the final equality used (1.8). Exercise. Consider two different charts φ = (x1 ,... , xn ) and φ0 = (x0 1 ,... , x0 n ) defined in a neighbourhood of p. Show that  µ ∂x ν µ (dx )p = (dx0 )p , (1.18) ∂x0 ν φ0 (p) and hence that, if ωµ and ωµ0 are the components of ω ∈ Tp∗ (M ) w.r.t. the two coordinate bases, then  ν 0 ∂x ωµ = ων. (1.19) ∂x0 µ φ0 (p) Elementary treatements of GR take this as the definition of a covector, which they usually call a “covariant vector”. 1.6 Abstract index notation So far, we have used Greek letters µ, ν,... to denote components of vectors or covec- tors with respect to a basis, and also to label the basis vectors themselves (e.g. eµ ). Equations involving such indices are assumed to hold only in that basis. For example an equation of the form X µ = δ1µ says that, in a particular basis, a vector X has only a single non-vanishing component. This will not be true in other bases. Furthermore, if we were just presented with this equation, we would not even know whether or not the quantities {X µ } are the components of a vector or just a set of n numbers. The abstract index notation uses Latin letters a, b, c,.... A vector X is denoted X or X b or X c etc. The letter used in the superscript does not matter. What matters a is that there is a superscript Latin letter. This tells us that the object in question is a vector. We emphasize: X a represents the vector itself, not a component of the vector. Similarly we denote a covector η by ηa (or ηb etc). The idea is that an equation written using Latin indices is basis-independent. More precisely, if we have some equation written with Greek indices, and we know that it is true for any basis, then we can replace Greek indices with Latin indices (e.g. µ → a, ν → b etc). For example, in any basis we have η(X) = ηµ X µ = X µ ηµ Part 3 GR October 12, 2022 15 H.S. Reall  1.7 Tensors hence we can write η(X) = ηa X a = X a ηa. Similarly, if f is a smooth function then X(f ) = df (X) = X a (df )a. Conversely, if we have an equation involving abstract indices then we can obtain an equation valid in any particular basis simply by replacing the abstract indices by basis indices (e.g. a → µ, b → ν etc.). For example if ηa X a = 2 then in any basis we have ηµ X µ = 2. Latin indices must respect the rules of the summation convention so equations of the form ηa ηa = 1 or ηb = 2 do not make sense. 1.7 Tensors In Newtonian physics, you are familiar with the idea that certain physical quantities are described by tensors (e.g. the inertia tensor). You may have encountered the idea that the Maxwell field in special relativity is described by a tensor. Tensors are very important in GR because the curvature of spacetime is described with tensors. In this section we shall define tensors at a point p and explain some of their basic properties. Definition. A tensor of type (r, s) at p is a multilinear map T : Tp∗ (M ) ×... × Tp∗ (M ) × Tp (M ) ×... × Tp (M ) → R. (1.20) where there are r factors of Tp∗ (M ) and s factors of Tp (M ). (Multilinear means that the map is linear in each argument.) In other words, given r covectors and s vectors, a tensor of type (r, s) produces a real number. Examples. 1. A tensor of type (0, 1) is a linear map Tp (M ) → R, i.e., it is a covector. 2. A tensor of type (1, 0) is a linear map Tp∗ (M ) → R, i.e., it is an element of (Tp∗ (M ))∗ but this is naturally isomorphic to Tp (M ) hence a tensor of type (1, 0) is a vector. To see how this works, given a vector X ∈ Tp (M ) we define a linear map Tp∗ (M ) → R by η 7→ η(X) for any η ∈ Tp∗ (M ). 3. We can define a (1, 1) tensor δ by δ(ω, X) = ω(X) (1.21) for any covector ω and vector X. Part 3 GR October 12, 2022 16 H.S. Reall  1.7 Tensors Definition. Let T be a tensor of type (r, s) at p. If {eµ } is a basis for Tp (M ) with dual basis {f µ } then the components of T in this basis are the numbers T µ1 µ2...µr ν1 ν2...νs = T (f µ1 , f µ2 ,... , f µr , eν1 , eν2 ,... , eνs ) (1.22) In the abstract index notation, we denote T by T a1 a2...ar b1 b2...bs. Remark. Tensors of type (r, s) at p can be added together and multiplied by a constant, hence they form a vector space. Since such a tensor has nr+s components, it is clear that this vector space has dimension nr+s. Examples. 1. Consider the tensor δ defined above. Its components are δ µ ν = δ(f µ , eν ) = f µ (eν ) = δνµ , (1.23) where the RHS is a Kronecker delta. This is true in any basis, so in the abstract index notation we write δ as δba. 2. Consider a (2, 1) tensor. Let η and ω be covectors and X a vector. Then in our basis we have T (η, ω, X) = T (ηµ f µ , ων f ν , X ρ eρ ) = ηµ ων X ρ T (f µ , f ν , eρ ) = T µν ρ ηµ ων X ρ (1.24) Now the basis we chose was arbitrary, hence we can immediately convert this to a basis-independent equation using the abstract index notation: T (η, ω, X) = T ab c ηa ωb X c. (1.25) This formula generalizes in the obvious way to a (r, s) tensor. We have discussed the transformation of vectors and covectors components under a change of coordinate basis. Let’s now examine how tensor components transform under an arbitrary change of basis. Let {eµ } and {e0µ } be two bases for Tp (M ). Let {f µ } and {f 0 µ } denote the corresponding dual bases. Expanding the primed bases in terms of the unprimed bases gives µ f 0 = Aµ ν f ν , e0µ = B ν µ eν (1.26) for some matrices Aµ ν and B ν µ. These matrices are related because: µ δνµ = f 0 (e0ν ) = Aµ ρ f ρ (B σ ν eσ ) = Aµ ρ B σ ν f ρ (eσ ) = Aµ ρ B σ ν δσρ = Aµ ρ B ρ ν. (1.27) Part 3 GR October 12, 2022 17 H.S. Reall  1.7 Tensors Hence B µ ν = (A−1 )µ ν. For a change between coordinate bases, our previous results give  0µ   µ µ ∂x µ ∂x A ν= , B ν = (1.28) ∂xν ∂x0 ν and these matrices are indeed inverses of each other (from the chain rule). Exercise. Show that under an arbitrary change of basis, the components of a vector X and a covector η transform as µ X 0 = Aµ ν X ν , η 0 µ = (A−1 )ν µ ην. (1.29) Show that the components of a (2, 1) tensor T transform as µν T0 ρ = Aµ σ Aν τ (A−1 )λ ρ T στ λ. (1.30) The corresponding result for a (r, s) tensor is an obvious generalization of this formula. Given a (r, s) tensor, we can construct a (r − 1, s − 1) tensor by contraction. This is easiest to demonstrate with an example. Example. Let T be a tensor of type (3, 2). Define a new tensor S of type (2, 1) as follows S(ω, η, X) = T (f µ , ω, η, eµ , X) (1.31) where {eµ } is a basis and {f µ } is the dual basis, ω and η are arbitrary covectors and X is an arbitrary vector. This definition is basis-independent because µ T (f 0 , ω, η, e0 µ , X) = T (Aµ ν f ν , ω, η, (A−1 )ρ µ eρ , X) = (A−1 )ρ µ Aµ ν T (f ν , ω, η, eρ , X) = T (f µ , ω, η, eµ , X). The components of S and T are related by S µν ρ = T σµν σρ in any basis. Since this is true in any basis, we can write it using the abstract index notation as S ab c = T dab dc (1.32) Note that there are other (2, 1) tensors that we can build from T abc de. For example, there is T abd cd , which corresponds to replacing the RHS of (1.31) with T (ω, η, f µ , X, eµ ). The abstract index notation makes it clear how many different tensors can be defined this way: we can define a new tensor by ”contracting” any upstairs index with any downstairs index. Another important way of constructing new tensors is by taking the product of two tensors: Part 3 GR October 12, 2022 18 H.S. Reall  1.7 Tensors Definition. If S is a tensor of type (p, q) and T is a tensor of type (r, s) then the outer product of S and T , denoted S ⊗ T is a tensor of type (p + r, q + s) defined by (S ⊗ T )(ω1 ,... , ωp , η1 ,... , ηr , X1 ,... , Xq , Y1 ,... , Ys ) = S(ω1 ,... , ωp , X1 ,... , Xq )T (η1 ,... , ηr , Y1 ,... , Ys ) (1.33) where ω1 ,... , ωp and η1 ,... , ηr are arbitrary covectors and X1 ,... , Xq and Y1 ,... , Ys are arbitrary vectors. Exercise. Show that this definition is equivalent to (S ⊗ T )a1...ap b1...br c1...cq d1...ds = S a1...ap c1...cq T b1...br d1...ds (1.34) Exercise. Show that, in a basis, any (2, 1) tensor T at p can be written as T = T µν ρ eµ ⊗ eν ⊗ f ρ (1.35) This generalizes in the obvious way to a (r, s) tensor. Remark. You may be wondering why we write T ab c instead of Tcab. At the moment there is no reason why we should not adopt the latter notation. However, it is convenient to generalize our definition of tensors slightly. We have defined a (r, s) tensor to be a linear map with r + s arguments, where the first r arguments are covectors and the final s arguments are vectors. We can generalize this by allowing the covectors and vectors to appear in any order. For example, consider a (1, 1) tensor. This is a map Tp∗ (M ) × Tp (M ) → R. But we could just as well have defined it to be a map Tp (M )×Tp∗ (M ) → R, which defines a different type of (1, 1) tensor. The abstract index notation allows us to distinguish these two types easily: the first would be written as T a b and the second as Ta b. (2, 1) tensors come in 3 different types: T ab c , T a b c and Ta bc. Each type of of (r, s) tensor gives a vector space of dimension nr+s but these vector spaces are naturally isomorphic so often one does not bother to distinguish between them. There is a final type of tensor operation that we shall need: symmetrization and antisymmetrization. Consider a (0, 2) tensor T. We can define two other (0, 2) tensors S and A as follows: 1 1 S(X, Y ) = (T (X, Y ) + T (Y, X)), A(X, Y ) = (T (X, Y ) − T (Y, X)), (1.36) 2 2 where X and Y are vectors at p. In abstract index notation: 1 1 Sab = (Tab + Tba ), Aab = (Tab − Tba ). (1.37) 2 2 Part 3 GR October 12, 2022 19 H.S. Reall  1.8 Tensor fields In a basis, we can regard the components of T as a square matrix. The components of S and A are just the symmetric and antisymmetric parts of this matrix. It is convenient to introduce some notation to describe the operations we have just defined: we write 1 1 T(ab) = (Tab + Tba ), T[ab] = (Tab − Tba ). (1.38) 2 2 These operations can be applied to more general tensors. For example, 1 T (ab)c d = (T abc d + T bac d ). (1.39) 2 We can also symmetrize or antisymmetrize on more than 2 indices. To symmetrize on n indices, we sum over all permutations of these indices and divide the result by n! (the number of permutations). To antisymmetrize we do the same but we weight each term in the sum by the sign of the permutation. The indices that we symmetrize over must be either upstairs or downstairs, they cannot be a mixture. For example, 1 T (abc)d = T abcd + T bcad + T cabd + T bacd + T cbad + T acbd.
(1.40) 3! 1 T a [bcd] = (T a bcd + T a cdb + T a dbc − T a cbd − T a dcb − T a bdc ). (1.41) 3! Sometimes we might wish to (anti)symmetrize over indices which are not adjacent. In this case, we use vertical bars to denote indices excluded from the (anti)symmetrization. For example, 1 T(a|bc|d) = (Tabcd + Tdbca ). (1.42) 2 Exercise. Show that T (ab) X[a|cd|b] = 0. 1.8 Tensor fields So far, we have defined vectors, covectors and tensors at a single point p. However, in physics we shall need to consider how these objects vary in spacetime. This leads us to define vector, covector and tensor fields. Definition. A vector field is a map X which maps any point p ∈ M to a vector Xp at p. Given a vector field X and a function f we can define a new function X(f ) : M → R by X(f ) : p 7→ Xp (f ). The vector field X is smooth if X(f ) is a smooth function for any smooth f. ∂ Example. Given any coordinate chart φ = (x1 ,... , xn ), the vector field ∂xµ is defined by p 7→ ∂x∂ µ p. Hence
    ∂ ∂F (f ) : p 7→ , (1.43) ∂xµ ∂xµ φ(p) Part 3 GR October 12, 2022 20 H.S. Reall  1.9 Integral curves where F ≡ f ◦ φ−1. Smoothness of f implies that this map defines a smooth function. Therefore ∂/∂xµ is a smooth vector field. (Note that (∂/∂xµ ) usually won’t be defined on the whole manifold M since the chart φ might not cover the whole manifold. So strictly speaking this is not a vector field on M but only on a subset of M. We shan’t worry too much about this distinction.) Remark. Since the vectors (∂/∂xµ )p provide a basis for Tp (M ) at any point p, we can expand an arbitrary vector field as   µ ∂ X=X (1.44) ∂xµ Since ∂/∂xµ is smooth, it follows that X is smooth if, and only if, its coordinate-basis components X µ are smooth functions. Definition. A covector field is a map ω which maps any point p ∈ M to a covector ωp at p. Given a covector field and a vector field X we can define a function ω(X) : M → R by ω(X) : p 7→ ωp (Xp ). The covector field ω is smooth if this function is smooth for any smooth vector field X. Example. Let f be a smooth function. We have defined (df )p above. Now we let p vary to define a covector field df. Let X be a smooth vector field. Then df (X) : p 7→ (df )p (Xp ) = Xp (f ) hence df (X) = X(f ). This is a smooth function of p (because X is smooth). Hence df is a smooth covector field: the gradient of f. Remark. Taking f = xµ reveals that dxµ is a smooth covector field. Definition. A (r, s) tensor field is a map T which maps any point p ∈ M to a (r, s) ten- sor Tp at p. Given r covector fields η1 ,... , ηr and s vector fields X1 ,... , Xs we can define a function T (η1 ,... , ηr , X1 ,... , Xs ) : M → R by p 7→ Tp ((η1 )p ,... , (ηr )p , (X1 )p ,... , (Xs )p ). The tensor field T is smooth if this function is smooth for any smooth covector fields η1 ,... , ηr and vector fields X1 ,... , Xr. Exercise. Show that a tensor field is smooth if, and only if, its components in a coordinate chart are smooth functions. Remarks. Note that we can regard a function on M as a (0, 0) tensor field. Henceforth we shall assume that all tensor fields that we encounter are smooth. 1.9 Integral curves In fluid mechanics, the velocity of a fluid is described by a vector field u(x) in R3 (we are returning to Cartesian vector notation for a moment). Consider a particle suspended in the fluid with initial position x0. It moves with the fluid so its position x(t) satisfies dx = u(x(t)), x(0) = x0. (1.45) dt Part 3 GR October 12, 2022 21 H.S. Reall  1.10 The commutator The solution of this differential equation is called the integral curve of the vector field u through x0. The definition extends straightforwardly to a vector field on a general manifold: Definition. Let X be a vector field on M and p ∈ M. An integral curve of X through p is a curve through p whose tangent at every point is X. Let λ denote an integral curve of X with (wlog) λ(0) = p. In a coordinate chart, this definition reduces to the initial value problem dxµ (t) = X µ (x(t)), xµ (0) = xµp. (1.46) dt (Here we are using the abbreviation xµ (t) = xµ (λ(t)).) Standard ODE theory guar- antees that there exists a unique solution to this problem. Hence there is a unique integral curve of X through any point p. Example. In a chart φ = (x1 ,... , xn ), consider X = ∂/∂x1 + x1 ∂/∂x2 and take p to be the point with coordinates (0,... , 0). Then dx1 /dt = 1, dx2 /dt = x1. Solving the first equation and imposing the initial condition gives x1 = t, then plugging into the second equation and solving gives x2 = t2 /2. The other coords are trivial: xµ = 0 for µ > 2, so the integral curve is t 7→ φ−1 (t, t2 /2, 0,... , 0). 1.10 The commutator Let X and Y be vector fields and f a smooth function. Since Y (f ) is a smooth function, we can act on it with X to form a new smooth function X(Y (f )). Does the map f 7→ X(Y (f )) define a vector field? No, because X(Y (f g)) = X(f Y (g) + gY (f )) = f X(Y (g)) + gX(Y (f )) + X(f )Y (g) + X(g)Y (f ) so the Leibniz law is not satisfied. However, we can also define Y (X(f )) and the combination X(Y (f )) − Y (X(f )) does obey the Leibniz law (check!). Definition. The commutator of two vector fields X and Y is the vector field [X, Y ] defined by [X, Y ](f ) = X(Y (f )) − Y (X(f )) (1.47) for any smooth function f. To see that this does indeed define a vector field, we can evaluate it in a coordinate Part 3 GR October 12, 2022 22 H.S. Reall  1.10 The commutator chart:     ν ∂F µ ∂F [X, Y ](f ) = X Y −Y X ∂xν ∂xµ     µ ∂ ν ∂F ν ∂ µ ∂F = X Y −Y X ∂xµ ∂xν ∂xν ∂xµ ∂Y ν ∂F ∂X µ ∂F = Xµ µ ν − Y ν ν  ∂x ∂x ∂x ∂xµ µ ∂Y ∂X µ ∂F = Xν ν − Y ν ν ∂x ∂x ∂xµ   ∂ = [X, Y ]µ (f ) ∂xµ where  µ µ  µ ν ∂Y ν ∂X [X, Y ] = X −Y. (1.48) ∂xν ∂xν Since f is arbitrary, it follows that   µ ∂ [X, Y ] = [X, Y ]. (1.49) ∂xµ The RHS is a vector field hence [X, Y ] is a vector field whose components in a coordinate basis are given by (1.48). (Note that we cannot write equation (1.48) in abstract index notation because it is valid only in a coordinate basis.) Example. Let X = ∂/∂x1 and Y = x1 ∂/∂x2 + ∂/∂x3. The components of X are constant so [X, Y ]µ = ∂Y µ /∂x1 = δ2µ so [X, Y ] = ∂/∂x2. Exercise. Show that (i) [X, Y ] = −[Y, X]; (ii) [X, Y + Z] = [X, Y ] + [X, Z]; (iii) [X, f Y ] = f [X, Y ] + X(f )Y ; (iv) [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0 (the Jacobi identity). Here X, Y, Z are vector fields and f is a smooth function. Remark. The components of (∂/∂xµ ) in the coordinate basis are either 1 or 0. It follows that   ∂ ∂ , = 0. (1.50) ∂xµ ∂xν Conversely, it can be shown that if X1 ,... , Xm (m ≤ n) are vector fields that are linearly independent at every point of some open region R, and whose commutators all vanish in R, then, in a neighbourhood of any point p ∈ R, one can introduce a coordinate chart (x1 ,... , xn ) such that Xi = ∂/∂xi (i = 1,... , m) throughout this neighbourhood. Part 3 GR October 12, 2022 23 H.S. Reall 2 Metric tensors 2.1 Definition A metric captures the notion of distance on a manifold. We can motivate the required definition by considering the case of Euclidean space. Let x(t), a < t < b be a curve in R3 (we’re using Cartesian vector notation). Then the length of the curve is Z b r dx dx dt ·. (2.1) a dt dt Inside the integral we see the norm of the tangent vector dx/dt, in other words the scalar product of this vector with itself. Therefore to define a notion of distance on a general manifold, we shall start by introducing a scalar product between vectors. A scalar product maps a pair of vectors to a number. In other words, at a point p, it is a map g : Tp (M ) × Tp (M ) → R. A scalar product should be linear in each argument. Hence g is a (0, 2) tensor at p. We call g a metric tensor. There are a couple of other properties that g should also satisfy: Definition. A metric tensor at p ∈ M is a (0, 2) tensor g with the following properties: 1. It is symmetric: g(X, Y ) = g(Y, X) for all X, Y ∈ Tp (M ) (i.e. gab = gba ) 2. It is non-degenerate: g(X, Y ) = 0 for all Y ∈ Tp (M ) if, and only if, X = 0. Remark. Sometimes we shall denote g(X, Y ) by hX, Y i or X · Y. Since the components of g form a symmetric matrix, one can introduce a basis that diagonalizes g. Non-degeneracy implies that none of the diagonal elements is zero. By rescaling the basis vectors, one can arrange that the diagonal elements are all ±1. In this case, the basis is said to be orthonormal. There are many such bases but a standard algebraic theorem (Sylvester’s law of inertia) states that the number of positive and negative elements is independent of the choice of orthonormal basis. The number of positive and negative elements is called the signature of the metric. In differential geometry, one is usually interested in Riemannian metrics. These have signature + + +... + (i.e. all diagonal elements +1 in an orthonormal basis), and hence g is positive definite. In GR, we are interested in Lorentzian metrics, i.e., those with signature − + +... +. Definition. A Riemannian (Lorentzian) manifold is a pair (M, g) where M is a differ- entiable manifold and g is a Riemannian (Lorentzian) metric tensor field. A Lorentzian manifold is sometimes called a spacetime. Part 3 GR October 12, 2022 24 H.S. Reall  2.1 Definition p Definition. On a Riemannian manifold, the norm of a vector X is |X| = g(X, X) and the angle between two non-zero vectors X and Y (at the same point) is θ where cos θ = g(X, Y )/(|X| |Y |). (These definitions, in terms of the scalar product, agree with the usual definitions of Euclidean geometry.) Remark. On a Riemannian manifold, we can now define the length of a curve in exactly the same way as above: let λ : (a, b) → M be a smooth curve with tangent vector X. Then the length of the curve is Z b q dt g(X, X)|λ(t) (2.2) a Exercise. Given a curve λ(t) we can define a new curve simply by changing the parameterization: let t = t(u) with dt/du > 0 and u ∈ (c, d) with t(c) = a and t(d) = b. Show that: (i) the new curve κ(u) ≡ λ(t(u)) has tangent vector Y a = (dt/du)X a ; (ii) the length of these two curves is the same, i.e., our definition of length is independent of parameterization. In a coordinate basis, we have (cf equation (1.35)) g = gµν dxµ ⊗ dxν (2.3) Often we use the notation ds2 instead of g and abbreviate this to ds2 = gµν dxµ dxν (2.4) This notation captures the intuitive idea of an infinitesimal distance ds being deter- mined by infinitesimal coordinate separations dxµ. Examples. 1. In Rn = {(x1 ,... , xn )}, the Euclidean metric is g = dx1 ⊗ dx1 +... + dxn ⊗ dxn (2.5) (Rn , g) is called Euclidean space. A coordinate chart which covers all of R4 and in which gµν = diag(1, 1,... , 1) is called Cartesian. 2. In R4 = {(x0 , x1 , x2 , x3 )}, the Minkowski metric is η = −(dx0 )2 + (dx1 )2 + (dx2 )2 + (dx3 )2. (2.6) (R4 , η) is called Minkowski spacetime. A coordinate chart which covers all of R4 and in which ηµν = diag(−1, 1, 1, 1) everywhere is called an inertial frame. Part 3 GR October 12, 2022 25 H.S. Reall  2.2 Lorentzian signature 3. On S 2 , let (θ, φ) denote the spherical polar coordinate chart discussed earlier. The (unit) round metric on S 2 is ds2 = dθ2 + sin2 θ dφ2 , (2.7) i.e. in the chart (θ, φ), we have gµν = diag(1, sin2 θ). Note this is positive definite for θ ∈ (0, π), i.e., on all of this chart. However, this chart does not cover the whole manifold so the above equation does not determine g everywhere. We can give a precise definition by adding that, in the chart (θ0 , φ0 ) discussed earlier, g = dθ0 2 + sin2 θ0 dφ0 2. One can check that this does indeed define a smooth tensor field. (This metric is the one induced from the embedding of S 2 into 3d Euclidean space: we will see later that it is the “pull-back” of the metric on Euclidean space.) Definition. Since gab is non-degenerate, it must be invertible. The inverse metric is a symmetric (2, 0) tensor field denoted g ab and obeys g ab gbc = δca (2.8) where δca is the (1, 1) tensor defined in (1.21) with components δνµ. Example. For the metric on S 2 defined above, in the chart (θ, φ) we have g µν = diag(1, 1/ sin2 θ). Definition. A metric determines a natural isomorphism between vectors and covectors. Given a vector X a we can define a covector Xa = gab X b. Given a covector ηa we can define a vector η a = g ab ηb. These maps are clearly inverses of each other. Remark. This isomorphism is the reason why covectors are not more familiar: we are used to working in Euclidean space using Cartesian coordinates, for which gµν and g µν are both the identity matrix, so the isomorphism appears trivial. Definition. For a general tensor, abstract indices can be “lowered” by contracting with gab and “raised” by contracting with g ab. Raising and lowering preserve the ordering of indices. The resulting tensor is denoted by the same letter as the original tensor. Example. Let T be a (3, 2) tensor. Then T a b cde = gbf g dh g ej T af c hj. 2.2 Lorentzian signature Remark. On a Lorentzian manifold, we take basis indices µ, ν,... to run from 0 to n − 1. Part 3 GR October 12, 2022 26 H.S. Reall  2.2 Lorentzian signature At any point p of a Lorentzian manifold, we can choose an orthonormal basis {eµ } so that, at p, g(eµ , eν ) = ηµν ≡ diag(−1, 1,... , 1). Such a basis is far from unique. If e0 µ = (A−1 )ν µ eν is any other such basis then we have ηµν = g(e0 µ , e0 ν ) = (A−1 )ρ µ (A−1 )σ ν g(eρ , eσ ) = (A−1 )ρ µ (A−1 )σ ν ηρσ. (2.9) Hence ηµν Aµ ρ Aν σ = ηρσ. (2.10) These are the defining equations of a Lorentz transformation in special relativity. Hence different orthonormal bases at p are related by Lorentz transformations. We saw earlier that the components of a vector at p transform as X 0 µ = Aµ ν X ν , which is the same as the transformation law of the components of a vector in special relativity. A similar result holds for tensors of other types. Thus, on a Lorentzian manifold, tensor compo- nents w.r.t. different orthonormal bases at p are related in exactly the same way as in special relativity. Definition. On a Lorentzian manifold (M, g), a vector X ∈ Tp (M ) is timelike if g(X, X) < 0, null (or lightlike) if X 6= 0 and g(X, X) = 0, and spacelike if g(X, X) > 0. A vector is causal if it is timelike or null. Remark. In an orthonormal basis at p, the metric has components ηµν so the tangent space at p has exactly the same structure as Minkowski spacetime, i.e., null vectors at p define a double cone, the light cone, that separates timelike vectors at p from spacelike vectors at p (see Fig. 8). Note that causal vectors at p fall into two disconnected sets. timelike null spacelike Figure 8. Light cone structure of Tp (M ) Exercise (Examples sheet 1). Let X a , Y b be non-zero vectors at p that are orthog- onal, i.e., gab X a Y b = 0. Show that (i) if X a is timelike then Y a is spacelike; (ii) if X a is null then Y a is spacelike or null; (iii) if X a is spacelike then Y a can be spacelike, Part 3 GR October 12, 2022 27 H.S. Reall  2.3 Curves of extremal proper time timelike, or null. (Hint. Choose an orthonormal basis to make the components of X a as simple as possible.) Definition. A curve in a Lorentzian manifold is said to be timelike if its tangent vector is everywhere timelike. Null, spacelike and causal curves are defined similarly. (Most curves do not satisfy any of these definitions because e.g. the tangent vector can change from timelike to null to spacelike along a curve.) Remark. The length of a spacelike curve is defined in the same way as on a Riemannian manifold (equation (2.2)). What about a timelike curve? Definition. let λ(u) be a timelike curve with λ(0) = p. Let X a be the tangent to the curve. The proper time τ from p along the curve is defined by dτ q = − (gab X a X b )λ(u) , τ (0) = 0. (2.11) du Remark. In a coordinate chart, X µ = dxµ /du so this definition can be rewritten in the form dτ 2 = −gµν dxµ dxν , (2.12) with the understanding that this is to be evaluated along the curve. Integrating the above equation along the curve gives the proper time from p to some other point q = λ(uq ) as s  Z uq  dxµ dxν τ= du − gµν (2.13) 0 du du λ(u) Definition. If proper time τ is used to parametrize a timelike curve then the tangent ua to the curve is called the 4-velocity of the curve. In a coordinate basis, it has components uµ = dxµ /dτ. Remark. (2.12) implies that 4-velocity is a unit timelike vector: gab ua ub = −1. (2.14) 2.3 Curves of extremal proper time Let p and q be points connected by a timelike curve. A small deformation of a timelike curve remains timelike hence there exist infinitely many timelike curves connecting p and q. The proper time between p and q will be different for different curves. Which curve extremizes the proper time between p and q? This is a standard Euler-Lagrange problem. Consider timelike curves from p to q with parameter u such that λ(0) = p, λ(1) = q. Let’s use a dot to denote a derivative Part 3 GR October 12, 2022 28 H.S. Reall  2.3 Curves of extremal proper time with respect to u. The proper time between p and q along such a curve is given by the functional Z 1 τ [λ] = du G (x(u), ẋ(u)) (2.15) 0 where q G (x(u), ẋ(u)) ≡ −gµν (x(u))ẋµ (u)ẋν (u) (2.16) and we are writing xµ (u) as a shorthand for xµ (λ(u)). The curve that extremizes the proper time, must satisfy the Euler-Lagrange equa- tion   d ∂G ∂G µ − µ =0 (2.17) du ∂ ẋ ∂x Working out the various terms, we have (using the symmetry of the metric) ∂G 1 1 µ = − 2gµν ẋν = − gµν ẋν (2.18) ∂ ẋ 2G G ∂G 1 µ = − gνρ,µ ẋν ẋρ (2.19) ∂x 2G where we have relabelled some dummy indices, and introduced the important notation of a comma to denote partial differentiation: ∂ gνρ,µ ≡ gνρ (2.20) ∂xµ We will be using this notation a lot henceforth. So far, our parameter u has been arbitrary subject to the conditions u(0) = p and u(1) = q. At this stage, it is convenient to use a more physical parameter, namely τ , the proper time along the curve. (Note that we could not have used τ from the outset since the value of τ at q is different for different curves, which would make the range of integration different for different curves.) The parameters are related by  2 dτ = −gµν ẋµ ẋν = G2 (2.21) du and hence dτ /du = G. So in our equations above, we can replace d/du with Gd/dτ , so the Euler-Lagrange equation becomes (after cancelling a factor of −G) dxν dxν dxρ   d 1 gµν − gνρ,µ =0 (2.22) dτ dτ 2 dτ dτ Hence d 2 xν dxρ dxν 1 dxν dxρ gµν + gµν,ρ − gνρ,µ =0 (2.23) dτ 2 dτ dτ 2 dτ dτ Part 3 GR October 12, 2022 29 H.S. Reall  2.3 Curves of extremal proper time In the second term, we can replace gµν,ρ with gµ(ν,ρ) because it is contracted with an object symmetrical on ν and ρ. Finally, contracting the whole expression with the inverse metric and relabelling indices gives d 2 xµ ν µ dx dx ρ + Γ νρ =0 (2.24) dτ 2 dτ dτ where Γµνρ are known as the Christoffel symbols, and are defined by 1 Γµνρ = g µσ (gσν,ρ + gσρ,ν − gνρ,σ ). (2.25) 2 Remarks. 1. Γµνρ = Γµρν. 2. The Christoffel symbols are not tensor components. Neither the first term nor the second term in (2.24) are components of a vector but the sum of these two terms does give vector components. 3. Equation 2.24 is called the geodesic equation and its solutions are called geodesics. Geodesics will be discussed more generally below. 4. We obtain exactly the same equation if we consider curves in a Riemannian manifold, or spacelike curves in a Lorentzian manifold, that extremize proper length. Example. In Minkowski spacetime, the components of the metric in an inertial frame are constant so Γµνρ = 0. Hence the above equation reduces to d2 xµ /dτ 2 = 0, which is the equation of a straight line. Thus, in Minkowski spacetime, timelike curves of extremal proper time are straight lines. It can be shown that these lines maximize the proper time between two points. In a general spacetime, this is true only locally, i.e., for any point p there exists a neighbourhood of p within which timelike geodesics are curves that maximize proper time. Exercise. Show that (2.24) can be obtained more directly as the Euler-Lagrange equation for the Lagrangian dxµ dxν L = −gµν (x(τ )) (2.26) dτ dτ This is usually the easiest way to derive (2.24) or to calculate the Christoffel symbols. Example. The Schwarzschild metric in Schwarzschild coordinates (t, r, θ, φ) is 2M ds2 = −f dt2 + f −1 dr2 + r2 dθ2 + r2 sin2 θ dφ2 , f =1− (2.27) r where M is a constant. We have  2  2  2  2 dt −1 dr 2 dθ 2 2 dφ L=f −f −r − r sin θ (2.28) dτ dτ dτ dτ Part 3 GR October 12, 2022 30 H.S. Reall so the EL equation for t(τ ) is d2 t   d dt dt dr 2f =0 ⇒ 2 + f −1 f 0 =0 (2.29) dτ dτ dτ dτ dτ From this we can read off f0 Γ001 = Γ010 = , Γ0µν = 0 otherwise (2.30) 2f The other Christoffel symbols are obtained in a similar way from the remaining EL equations (examples sheet 1). 3 Covariant derivative 3.1 Introduction To formulate physical laws, we need to be able to differentiate tensor fields. For scalar fields, partial differentiation is fine: f,µ ≡ ∂f /∂xµ are the components of the covector field (df )a. However, for tensor fields, partial differentiation is no good because the partial derivative of a tensor field does not give another tensor field: Exercise. Let V a be a vector field. In any coordinate chart, let T µ ν = V µ ,ν ≡ ∂V µ /∂xν. Show that T µ ν do not transform as tensor components under a change of chart. The problem is that differentiation involves comparing a tensor at two infinitesi- mally nearby points of the manifold. But we have seen that this does not make sense: tensors at different points belong to different spaces. The mathematical structure that overcomes this difficulty is called a covariant derivative or connection. Definition. A covariant derivative ∇ on a manifold M is a map sending every pair of smooth vector fields X, Y to a smooth vector field ∇X Y , with the following properties (where X, Y, Z are vector fields and f, g are functions) ∇f X+gY Z = f ∇X Z + g∇Y Z, (3.1) ∇X (Y + Z) = ∇X Y + ∇X Z, (3.2) ∇X (f Y ) = f ∇X Y + (∇X f )Y, (Leibniz rule), (3.3) where the action of ∇ on functions is defined by ∇X f = X(f ). (3.4) Part 3 GR October 12, 2022 31 H.S. Reall  3.1 Introduction Remark. (3.1) implies that, at any point, the map ∇Y : X 7→ ∇X Y is a linear map from Tp (M ) to itself. Hence it defines a (1, 1) tensor (see examples sheet 1). More precisely, if η ∈ Tp∗ (M ) and X ∈ Tp (M ) then we define (∇Y )(η, X) ≡ η(∇X Y ). Definition. let Y be a vector field. The covariant derivative of Y is the (1, 1) tensor field ∇Y. In abstract index notation we usually write (∇Y )a b as ∇b Y a or Y a ;b Remarks. 1. Similarly we define ∇f : X 7→ ∇X f = X(f ). Hence ∇f = df. We can write this as either ∇a f or f;a or ∂a f or f,a (i.e. the covariant derivative reduces to the partial derivative when acting on a function). 2. Does the map ∇ : X, Y 7→ ∇X Y define a (1, 2) tensor field? No - equation (3.3) shows that this map is not linear in Y. Example. Pick a coordinate chart on M. Let ∇ be the partial derivative in this chart. This satisfies all of the above conditions. This is not a very interesting example of a covariant derivative because it depends on choosing a particular chart: if we use a different chart then this covariant derivative will not be the partial derivative in the new chart. Definition. In a basis {eµ } the connection components Γµνρ are defined by ∇ρ eν ≡ ∇eρ eν = Γµνρ eµ (3.5) Example. The Christoffel symbols are the coordinate basis components of a certain connection, the Levi-Civita connection, which is defined on any manifold with a metric. More about this soon. Write X = X µ eµ and Y = Y µ eµ. Now ∇X Y = ∇X (Y µ eµ ) = X(Y µ )eµ + Y µ ∇X eµ (Leibniz) ν µ µ = X eν (Y )eµ + Y ∇X ν eν eµ = X ν eν (Y µ )eµ + Y µ X ν ∇ν eµ by (3.1) = X ν eν (Y µ )eµ + Y µ X ν Γρµν eρ = X ν eν (Y µ ) + Γµρν Y ρ eµ
(3.6) and hence (∇X Y )µ = X ν eν (Y µ ) + Γµρν Y ρ X ν (3.7) Part 3 GR October 12, 2022 32 H.S. Reall  3.1 Introduction so Y µ ;ν = eν (Y µ ) + Γµρν Y ρ (3.8) In a coordinate basis, this reduces to Y µ ;ν = Y µ ,ν + Γµρν Y ρ (3.9) The connection components Γµνρ are not tensor components: Exercise (examples sheet 2). Consider a change of basis e0µ = (A−1 )ν µ eν. Show that µ Γ0 νρ = Aµ τ (A−1 )λ ν (A−1 )σ ρ Γτλσ + Aµ τ (A−1 )σ ρ eσ ((A−1 )τ ν ) (3.10) The presence of the second term demonstrates that Γµνρ are not tensor components. Hence neither term on the RHS of equation (3.8) transforms as a tensor. However, the sum of these two terms does transform as a tensor. Exercise. Let ∇ and ∇ ˜ be two different connections on M. Show that ∇ − ∇ ˜ is a (1, 2) tensor field. You can do this either from the definition of a connection, or from the transformation law for the connection components. The action of ∇ is extended to general tensor fields by the Leibniz property. If T is a tensor field of type (r, s) then ∇T is a tensor field of type (r, s + 1). For example, if η is a covector field then, for any vector fields X and Y , we define (∇X η)(Y ) ≡ ∇X (η(Y )) − η(∇X Y ). (3.11) It is not obvious that this defines a (0, 2) tensor but we can see this as follows: (∇X η)(Y ) = ∇X (ηµ Y µ ) − ηµ (∇X Y )µ = X(ηµ )Y µ + ηµ X(Y µ ) − ηµ X ν eν (Y µ ) + Γµρν Y ρ X ν ,
(3.12) where we used (3.7). Now, the second and third terms cancel (X = X ν eν ) and hence (renaming dummy indices in the final term) (∇X η)(Y ) = X(ηµ ) − Γρµν ηρ X ν Y µ ,
(3.13) which is linear in Y µ so ∇X η is a covector field with components (∇X η)µ = X(ηµ ) − Γρµν ηρ X ν = X ν eν (ηµ ) − Γρµν ηρ
(3.14) This is linear in X ν and hence ∇η is a (0, 2) tensor field with components ηµ;ν = eν (ηµ ) − Γρµν ηρ (3.15) Part 3 GR October 12, 2022 33 H.S. Reall  3.1 Introduction In a coordinate basis, this is ηµ;ν = ηµ,ν − Γρµν ηρ (3.16) Now the Leibniz rule can be used to obtain the formula for the coordinate basis com- ponents of ∇T where T is a (r, s) tensor: T µ1...µr ν1...νs ;ρ = T µ1...µr ν1...νs ,ρ + Γµσρ1 T σµ2...µr ν1...νs +... + Γµσρr T µ1...µr−1 σ ν1...νs − Γσν1 ρ T µ1...µr σν2...νs −... − Γσνs ρ T µ1...µr ν1...νs−1 σ (3.17) Exercise. Prove this result for a (1, 1) tensor. Remark. We are using a comma and semi-colon to denote partial, and covariant, derivatives respectively. If more than one index appears after a comma or semi-colon then the derivative is to be taken with respect to all indices. The index nearest to comma/semi-colon is the first derivative to be taken. For example, f,µν = f,µ,ν ≡ ∂ν ∂µ f , and X a ;bc = ∇c ∇b X a (we cannot use abstract indices for the first example since it is not a tensor). The second partial derivatives of a function commute: f,µν = f,νµ but for a covariant derivative this is not true in general. Set η = df in (3.16) to get, in a coordinate basis, f;µν = f,µν − Γρµν f,ρ (3.18) Antisymmetrizing gives f;[µν] = −Γρ[µν] f,ρ (coordinate basis) (3.19) Definition. A connection ∇ is torsion-free if ∇a ∇b f = ∇b ∇a f for any function f. From (3.19), this is equivalent to Γρ[µν] = 0 (coordinate basis) (3.20) Lemma. For a torsion-free connection, if X and Y are vector fields then ∇X Y − ∇Y X = [X, Y ] (3.21) Proof. Use a coordinate basis: X ν Y µ ;ν − Y ν X µ ;ν = X ν Y µ ,ν + Γµρν X ν Y ρ − Y ν X µ ,ν − Γµρν Y ν X ρ = [X, Y ]µ + 2Γµ[ρν] X ν Y ρ = [X, Y ]µ (3.22) Hence the equation is true in a coordinate basis and therefore (as it is a tensor equation) it is true in any basis. Remark. Even for a torsion-free connection, the second covariant derivatives of a tensor field do not commute. More soon. Part 3 GR October 12, 2022 34 H.S. Reall  3.2 The Levi-Civita connection 3.2 The Levi-Civita connection On a manifold with a metric, the metric singles out a preferred connection: Theorem. Let M be a manifold with a metric g. There exists a unique torsion-free connection ∇ such that the metric is covariantly constant: ∇g = 0 (i.e. gab;c = 0). This is called the Levi-Civita (or metric) connection. Proof. Let X, Y, Z be vector fields then X(g(Y, Z)) = ∇X (g(Y, Z)) = g(∇X Y, Z) + g(Y, ∇X Z), (3.23) where we used the Leibniz rule and ∇X g = 0 in the second equality. Permuting X, Y, Z leads to two similar identities: Y (g(Z, X)) = g(∇Y Z, X) + g(Z, ∇Y X), (3.24) Z(g(X, Y )) = g(∇Z X, Y ) + g(X, ∇Z Y ), (3.25) Add the first two of these equations and subtract the third to get (using the symmetry of the metric) X(g(Y, Z)) + Y (g(Z, X)) − Z(g(X, Y )) = g(∇X Y + ∇Y X, Z) − g(∇Z X − ∇X Z, Y ) + g(∇Y Z − ∇Z Y, X) (3.26) The torsion-free condition implies ∇X Y − ∇Y X = [X, Y ] (3.27) Using this and the same identity with X, Y, Z permuted gives X(g(Y, Z)) + Y (g(Z, X)) − Z(g(X, Y )) = 2g(∇X Y, Z) − g([X, Y ], Z) − g([Z, X], Y ) + g([Y, Z], X) (3.28) Hence 1 g(∇X Y, Z) = [X(g(Y, Z)) + Y (g(Z, X)) − Z(g(X, Y )) 2 + g([X, Y ], Z) + g([Z, X], Y ) − g([Y, Z], X)] (3.29) Part 3 GR October 12, 2022 35 H.S. Reall  3.2 The Levi-Civita connection This determines ∇X Y uniquely because the metric is non-degenerate. It remains to check that it satisfies the properties of a connection. For example: 1 g(∇f X Y, Z) = [f X(g(Y, Z)) + Y (f g(Z, X)) − Z(f g(X, Y )) 2 + g([f X, Y ], Z) + g([Z, f X], Y ) − f g([Y, Z], X)] 1 = [f X(g(Y, Z)) + f Y (g(Z, X)) + Y (f )g(Z, X) 2 − f Z(g(X, Y )) − Z(f )g(X, Y ) + f g([X, Y ], Z) − Y (f )g(X, Z) + f g([Z, X], Y ) + Z(f )g(X, Y ) − f g([Y, Z], X)] f = [X(g(Y, Z)) + Y (g(Z, X)) − Z(g(X, Y )) 2 + g([X, Y ], Z) + g([Z, X], Y ) − g([Y, Z], X)] = f g(∇X Y, Z) = g(f ∇X Y, Z) (3.30) and hence g(∇f X Y − f ∇X Y, Z) = 0 for any vector field Z so, by the non-degeneracy of the metric, ∇f X Y = f ∇X Y. Exercise. Show that ∇X Y as defined by (3.29) satisfies the other properties required of a connection. Remark. In differential geometry, this theorem is called the fundamental theorem of Riemannian geometry (although it applies for a metric of any signature). Let’s determine the components of the Levi-Civita connection in a coordinate basis (for which [eµ , eν ] = 0): 1 g(∇ρ eν , eσ ) = [eρ (gνσ ) + eν (gσρ ) − eσ (gρν )] , (3.31) 2 that is 1 g(Γτνρ eτ , eσ ) = (gσν,ρ + gσρ,ν − gνρ,σ ) (3.32) 2 The LHS is just Γτνρ gτ σ. Hence if we multiply the whole equation by the inverse metric g µσ we obtain 1 Γµνρ = g µσ (gσν,ρ + gσρ,ν − gνρ,σ ) (3.33) 2 This is the same equation as we obtained earlier; we have now shown that the Christoffel symbols are the components of the Levi-Civita connection w.r.t. a coordinate basis. Remark. In GR we take the connection to be the Levi-Civita connection. This is not as restrictive as it sounds: we saw above that the difference between two connections is a tensor field. Hence we can write any connection (even one with torsion) in terms of the Levi-Civita connection and a (1, 2) tensor field. In GR we could regard the latter as a particular kind of “matter” field, rather than as part of the geometry of spacetime. Part 3 GR October 12, 2022 36 H.S. Reall  3.3 Geodesics 3.3 Geodesics Previously we considered curves that extremize the proper time between two points of a spacetime, and showed that this gives the equation d 2 xµ µ dxν dxρ + Γ νρ (x(τ )) = 0, (3.34) dτ 2 dτ dτ where τ is the proper time along the curve. The tangent vector X a to the curve has components X µ = dxµ /dτ. This is defined only along the curve. However, we can extend X a (in an arbitrary way) to a neighbourhood of the curve, so that X a becomes a vector field, and the curve is an integral curve of this vector field. The chain rule gives d2 x µ dX µ (x(τ )) dxν ∂X µ = = = X ν X µ ,ν. (3.35) dτ 2 dτ dτ ∂xν Note that the LHS is independent of how we extend X a hence so must be the RHS. We can now write (3.34) as X ν X µ ,ν + Γµνρ X ρ = 0
(3.36) which is the same as X ν X µ ;ν = 0, or ∇X X = 0. (3.37) where we are using the Levi-Civita connection. We now extend this to an arbitrary connection: Definition. Let M be a manifold with a connection ∇. An affinely parameterized geodesic is an integral curve of a vector field X satisfying ∇X X = 0. Remarks. 1. What do we mean by ”affinely parameterized”? Consider a curve with parameter t whose tangent X satisfies the above definition. Let u be some other parameter for the curve, so t = t(u) and dt/du > 0. Then the tangent vector becomes Y = hX where h = dt/du. Hence ∇Y Y = ∇hX (hX) = h∇X (hX) = h2 ∇X X + X(h)hX = f Y, (3.38) where f = X(h) = dh/dt. Hence ∇Y Y = f Y describes the same geodesic. In this case, the geodesic is not affinely parameterized. It always is possible to find an affine parameter so there is no loss of generality in restricting to affinely parameterized geodesics. Note that the new parameter is also affine iff X(h) = 0, i.e., h is constant. Then u = at + b where a and b are constants with a > 0 (a = h−1 ). Hence there is a 2-parameter family of affine parameters for any geodesic. Part 3 GR October 12, 2022 37 H.S. Reall  3.3 Geodesics 2. Reversing the above steps shows that, in a coordinate chart, for any connec- tion, the geodesic equation can be written as (3.34) with τ an arbitrary affine parameter. 3. In a spacetime, curves of extremal proper time are timelike geodesics (with ∇ the Levi-Civita connection). One can also consider geodesics which are not timelike. These satisfy (3.34) with τ an affine parameter. The easiest way to obtain this equation is to use the Lagrangian (2.26). Theorem. Let M be a manifold with a connection ∇. Let p ∈ M and Xp ∈ Tp (M ). Then there exists a unique affinely parameterized geodesic through p with tangent vector Xp at p. Proof. Choose a coordinate chart xµ in a neighbourhood of p. Consider a curve parameterized by τ. It has tangent vector with components X µ = dxµ /dτ. The geodesic equation is (3.34). We want the curve to satisfy the initial conditions  µ µ µ dx x (0) = xp , = Xpµ. (3.39) dτ τ =0 This is a coupled system of n ordinary differential equations for the n functions xµ (t). Existence and uniqueness is guaranteed by the standard theory of ordinary differen