GRAVITATION F10 S. G. RAJEEV Lecture 18 1. Lie Derivatives 1.1. A vector field is an infinitesimal transformation of points. Given a vector field, from each point x we can move to a nearby point x in the direction determined by it: µ µ µ µ x x x + v (x) → ≡ Ascalarfieldtransformsas µ φ(x) φ(x)+v ∂ φ ≈ µ Thus the infinitesimal transformation of a scalar under a vector field is just φ = vµ∂ φ. Lv µ This is called the Lie derivative of φ by v . The infinitesimal transformation vω of a covariant vector field ωµ can be found L µ from the invariance of the combination ωµdx µ µ ν µ µ ν ω(x)dx = ωµ(x)dx + [v ∂ν ωµdx + ωµ∂ν v dx ] so that its Lie derivative is [ ω] = vν ∂ ω + ω ∂ vν Lv µ ν µ ν µ Acovariantsymmetrictensor(likethemetric)hasaLiederivativethatcanbe calculated similarly: [ g] = vρ∂ g + g ∂ vρ + g ∂ vρ Lv µν ρ µν ρν µ µρ ν It is interesting to rewrite this in terms of the covariant form of the vector field ρ vµ = gµρv [ g] = vρ∂ g vρ∂ g vρ∂ g + ∂ v + ∂ v Lv µν ρ µν − µ ρν − ν µρ µ ν ν µ Recalling the formula for the covariant derivative of a vector field we see that the Lie derivative of the metric can be written as [ g] = D v + D v . Lv µν µ ν ν µ 1.2. If the Lie derivative of the metric by a vector field is zero, it is a Killing vector. Killing vectors describe infinitesimal symmetries of the metric. 1 GRAVITATION F10 2 2. Killing Vectors 2.1. Translations and rotations are the symmetries of the Euclidean met- ric. Infinitesimally, a symmetry of the Euclidean metric is a vector field that leaves its metric unchanged under the transformation xi xi + vi. → ∂ivj + ∂jvi =0 j Aconstantisasolution:translation.alinearfunctionvi(x)=rijx is a solution if rij + rji =0 These are infinitesimal rotations. For example a rotation through an angle θ in the x1x2plane has the effect x1 cos θx1 +sinθx2 → x2 sin θx1 + cos θx2 → − If the angle is infinitesimally small x1 x1 01 x1 + θ x2 → x2 10 x2 − Any anti-symmetrc tensor can be written as the sum of infinitesimal rotations in planes such as this. There are no other solutions to the conditions above. 2.2. Translations and Lorentz transformations are the symmetries of the Minkowski metric. The Killing vectors satisfy ∂µvν + ∂ν vµ =0 The solutions are ν vµ = aµ + λµν x where λ = λ µν − νµ These ten independent solutions can be thought of as translations aµ,rotations λij and Lorentz boosts λ0i. 2.3. Killing vectors are infinitesimal symmetries of the geodesic varia- tional principle. Recall 1 dxµ dxν S = g dτ 2 ˆ µν dτ dτ Under an infinitesimal transformation xµ xµ + vµ the change in the action is → 1 dxµ dxν dvµ dxν dxµ dvν δ S = vρ∂ g + g + g dτ v 2 ˆ ρ µν dτ dτ µν dτ dτ µν dτ dτ 1 dxµ dxν dxρ dxν dxµ dxρ = vρ∂ g + g ∂ vµ + g ∂ vν dτ 2 ˆ ρ µν dτ dτ µν ρ dτ dτ µν ρ dτ dτ GRAVITATION F10 3 1 dxµ dxν = [ g ] dτ 2 ˆ Lv µν dτ dτ Thus a Killing vector leads to a symmetry of the geodesic equation. 2.4. To each Killing vector there corresponds a conserved quantity of the geodesic equation. This is a special case of a much more general theorem of Noether: symmetries in a variational principle lead to conservaton laws. We have d dxν d dxν dxν dxρ g vµ = vµ g + g ∂ vµ dτ µν dτ dτ µν dτ µν ρ dτ dτ For a geodesic d dxν 1 dxρ dxν g = ∂ g dτ µν dτ 2 µ ρν dτ dτ so that d dxν 1 dxρ dxν dxν dxρ g vµ = vµ ∂ g + g ∂ vµ dτ µν dτ 2 µ ρν dτ dτ µν ρ dτ dτ d dxν 1 dxρ dxν g vµ = [ g] dτ µν dτ 2 Lv ρν dτ dτ Thus if v is a Killing vector field and x is a geodesic we have the conservation law d dxν g vµ =0. dτ µν dτ 2.5. Since the tangent vector to the geodesic has constant length we always have one conservation law. We don’t even need a Killing vector for this. This follows from the fact that τ does not appear explictly in the action principle. In the language of mechanics, the Hamiltonian is conserved when the Lagrangian is independent of the evolution parameter (τin our case). dxµ dxν g = H = constant. µν dτ dτ This parameter has the physical meaning of the square of the mass. For a time-like geodesic this constant is positive: the mass is real and non-zero. It is convenient to choose the unit of τ that this is unity; in this case τ is proper time. For null geodesics H will vanish: the mass is zero. Space-like geodesics have H<0 but they don’t have a meaning as the path of any particle. They can be of mathematical interest, however..