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In a Previous Lecture, We Introduced Pull-Backs of Differential Forms. We Will Now Put This Into Use and Describe Killing Vectors

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In a Previous Lecture, We Introduced Pull-Backs of Differential Forms. We Will Now Put This Into Use and Describe Killing Vectors 43 6. SYMMETRY In a previous lecture, we introduced pull-backs of differential forms. We will now put this into use and describe Killing vectors. Heuristically, these are infini- tesimal diffeomorphisms of the manifold leaving the metric invariant, i.e. infini- tesimal isometries. Suppose that ϕ : M N is a diffeomorphism, where M and N are smooth manifolds. We know from Section 3.2.6 that we can pull-back differential forms on N to differential forms→ on M. With vector fields – indeed dual to differential forms – the induced map goes in the other direction: it maps vector fields on M to vector fields on N. The best way to see this is by viewing a vector field on M as a derivation on smooth functions on M and use the definition of pull-back of a smooth function on N. The pushforward vector field ϕ∗v X(N) of v X(M) by the diffeomorphism ϕ : M N is defined by ∈ ∈ ϕ∗v(f) := v(f ϕ)→ (f C (M)). ◦ ∈ ∞ This is intrinsically coordinate independent, but a bit abstract. Let’s derive the corresponding expressions in tensor notation; let {xi} be coordinates in a neigh- borhood of p on M and {yk} in a neighborhood of ϕ(p) on N. Then, on writing k k i i ϕ∗v =(ϕ∗v) ∂/∂y and v = v ∂/∂x we have k k ∂y i (ϕ∗v) = v . ∂xi ∂yk with the Jacobian matrix ∂xi evaluated at p. Exercise 6.1. Check this last equality by evaluating ϕ∗v on f C (N). ∈ ∞ Remark. Thus, the pushforward is a map ϕ∗ : X(M) X(N) of sections of the tangent bundles TM and TN, respectively. However, a common understanding → of the pushforward is as a map between tangent spaces: ϕ∗ : TxM Tϕ(x)N for x M. The two are related by ϕ∗ v(x) = ϕ∗v(ϕ(x)) where on the left-hand-side ∈ → ϕ∗ acts by matrix multiplication with the Jacobian matrix. With the definition of a differential p-form ω on M asa C (M)-multilinear, skew symmetric map ω : X(M)p C (M) we claim that the following∞ relation exists between pullbacks of forms and pushforwards∞ of vector fields: ∗ → ϕ ω(v1,...,vp) = ω(ϕ∗v1,...,ϕ∗vp). (6.1) This can be seen most easily in tensor notation for which i1 ip ω(v1,...,vp) = ωi ···i v v . 1 p 1 ··· p Exercise 6.2. Check Eq. (6.1) by choosing local coordinates on M and using tensor notation. More generally – by dropping the condition of skew symmetry on forms – one can define the pullback of a (0,p)-tensor T by the same formula (6.1), of course re- placing ω by T. In tensor notation, this becomes the familiar tensor transformation law k1 kp ∗ ∂y ∂y (ϕ T)i1···ip = Tk1 ···kp . ∂xi1 ··· ∂xip 44 6.1. Lie derivatives. We already discussed integral curves of a vector field on M. Let us recall that given v X(M) we can construct a parametrized curve on M whose velocity (tangent vector)∈ is equal to v. This amounts to solving the differen- tial equation: dσi(x) t = vi(σ (x)), (6.2) dt t i i and σt(x) is the integral curve of the vector field v starting at x M, i.e. σ0(x) = x. It defines a natural diffeomorphism by following the point x through∈ the flow defined by the integral curve. At ‘time’ t, we arrive (in a smooth manner) at the point σt(x). The flow {σt}t∈R depends smoothly on t and defines a so-called one-parameter subgroup of diffeomorphisms of M with the defining property: σt(σs(x)) = σs+t(x). This guarantees that σt is invertible – with inverse given by σ−t – and thus a dif- feomorphism. v σt(x) x FIGURE 7. The flow of the vector field v ∗ Exercise 6.3. The diffeomorphism σt : M M induces a map σt : C (M) ∗ ∞ ∗ C (M) on functions (i.e. the pull-back) given by σt(f) = f σt (Note that σt is invertible∞ as well). Show that (6.2) implies that→ ◦ → d σ∗(f) = v(f) (6.3) dt t t=0 for any smooth function f on M. We are interested in how a vector field u changes along the integral curves of another vector field v. We learned from the discussion on connections that com- paring vector fields at different points on a manifold is a subtle business. However, with the notion of pushforward of the previous section, what we can compare is the pushforward of u at σt(x) by the diffeomorphism σ−t and u at the point x (see 45 Figure 8). Infinitesimally, this change defines a vector field called the Lie derivative of u along v defined by d (σ−t)∗u − u Lvu := (σ−t)∗u = lim . dt t 0 t t=0 An equivalent definition is given→ as a vector at x M by ∈ d L u(x) := (σ− )∗ u(σ (x)) , v dt t t t=0 when (σ−t)∗ is understood as a map from Tσt (x) M to TxM. u (σ−t)∗ (σ−t)∗u ( ) u σt x x FIGURE 8. The Lie derivative; the dashed vector is the difference ((σ−t)∗u − u)(x) Exercise 6.4. Use Eq. (6.2) to derive the following expression for the components of the Lie derivative ∂ui ∂vi (L u)i = vj − uj. v ∂xj ∂xj In other words, the Lie derivative on vector fields is nothing else but the Lie bracket encountered before: Lvu =[v, u], which also explains the terminology. Consequently, the Lie derivative of a vector field is again a vector field. We conclude from this that in the particular case that v is a coordinate vector field v = ∂k the Lie derivative is just the partial derivative: i i (L∂k u) = u ,k. 6.1.1. Lie derivative of 1-forms. We will now define the Lie derivatives of differential forms. Let us start with 1-forms. In contrast with vector fields, one naturally uses the pullback instead of the pushforward by σt to compare differential forms at different points on the integral curve of a vector field. We define the Lie derivative of a 1-form α along a vector field v d L α := σ∗α. (6.4) v dt t t=0 46 In tensor notation, we can write j j (Lvα)i = v αi,j + v ,iαj This follows easily by taking the derivative with respect to t, putting t = 0 after- k ∗ ∗ ∂σt wards, of (σtα)i := σtα(∂i) = αk(σt(x)) ∂xi where the index k corresponds to the k transformed coordinate system {σt (x)}. As a consequence of this identity, we can prove Cartan’s identity: y y Lvα = v dα + d(v α). which follows directly from the definitions of the interior product and the exterior derivative. 6.1.2. Lie derivative of covariant tensors. We extend the previous definition of Lie derivative to contravariant tensors – and in particular p-forms – as follows. For T : X(M)p C (M) we have in terms of the pullback: ∞ → d L T := σ∗T. (6.5) v dt t t=0 Exercise 6.5. Show that for 0-tensors (i.e. smooth functions) f on M, this becomes Lv(f) = v(f). Of particular interest is the Lie derivative of the metric tensor g on a Riemannian or Lorentzian manifold: d L g(u, v) = g ((σ )∗u, (σ )∗v) (6.6) v dt t t t=0 In tensor notation, this becomes: k k k (Lvg)ij = v gij,k + v ,igkj + v ,jgik (6.7) Of course, we are not so happy with this expression since is involves partial deriva- tives and is therefore not explicitly covariant. However, it turns out that it is equiv- alent to the following expression: k k k k k (Lvg)ij = v gij|k + v |igkj + v |jgik = v |igkj + v |jgik (6.8) where the covariant derivatives are with respect to the Levi-Civita connection. Again, observe that if v = ∂k only the partial derivative of the components of the metric appear in the Lie derivative of g. Exercise 6.6. Check the equivalence of Eq. (6.7) and (6.8). Remark. In general, one can derive the following formula for the Lie derivative of a (0,p)-tensor T, j j j (LvT)i ···i = v Ti ···i |j + v Tji ···i + + v Ti ···i j. 1 p 1 p |i1 2 p ··· |ip 1 p−1 47 6.2. Killing vectors. In the previous subsection, we introduced the Lie derivative, which describes the action of a vector field on tensors such as another vector field, one forms or the metric. We now restrict to vector fields which are symmetries of the metric. Suppose that σt solves the differential equation (6.2) for a vector field ξ on a Riemannian or Lorentzian manifold (M, g). Then ξ is called a Killing ∗ vector field if σt is an isometry, i.e. it leaves the metric invariant σtg = g. This is completely equivalent to the condition that the Lie derivative of the metric g along ξ vanishes: Lξg = 0. (6.9) 6.2.1. Killing’s equation. The last defining equation for a Killing vector field can be written in tensor notation using Eq. (6.8). The result is known as Killing’s equation and reads ξj|i + ξi|j = 2ξ(i|j) = 0. (6.10) where the round bracket denotes the symmetrization of indices.
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