Light cone and conformal sphere The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Theresa C. Anderson April, 2010 Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Light cone and conformal sphere Motivation: The recent concept of the group-based moving frame as an equivariant map, introduced by Fels and Olver, has been a useful aid in the quest for differential invariants. The moving frame is used to construct the Maurer-Cartan matrix K; which according to a theorem of Hubert, contains all generators for the differential invariants of a curve. Here we employ the normalization technique of Fels and Olver to construct group-based moving frames for the light cone and the conformal 2-sphere, and we use the moving frame to find the differential invariants. Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Euclidean space admits a natural action from the Euclidean group, the group of rigid transformations (rotations and translations). Light cone and conformal sphere We first recall the following definition and theorem: A classical moving frame on a n-manifold is a set of n vectors along a curve which is invariant under an action by the group (such as a rigid motion in the Euclidean case). Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Light cone and conformal sphere We first recall the following definition and theorem: A classical moving frame on a n-manifold is a set of n vectors along a curve which is invariant under an action by the group (such as a rigid motion in the Euclidean case). Euclidean space admits a natural action from the Euclidean group, the group of rigid transformations (rotations and translations). Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations An important concept that we will need later is the following: Definition (K) A map I : J (R; M) R is a differential invariant if I (g · u(k)) = I (u(k)), where J is the jet space. ! Light cone and conformal sphere A new concept of the moving frame, introduced by Fels and Olver (1997), called a group-based moving frame, was essential to the development of the rest of this research. Definition (k) A right moving frame ρ, is a map ρ : J (R; M) G that is equivariant, which means ! ρ(g · u(n)) = ρ(u)g -1; where M is the manifold and g · u(k) is the prolonged action on the kth jet. Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Light cone and conformal sphere A new concept of the moving frame, introduced by Fels and Olver (1997), called a group-based moving frame, was essential to the development of the rest of this research. Definition (k) A right moving frame ρ, is a map ρ : J (R; M) G that is equivariant, which means ! ρ(g · u(n)) = ρ(u)g -1; where M is the manifold and g · u(k) is the prolonged action on the kth jet. An important concept that we will need later is the following: Definition (K) A map I : J (R; M) R is a differential invariant if I (g · u(k)) = I (u(k)), where J is the jet space. ! Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Light cone and conformal sphere -1 If ρ is a right moving frame, then ρs = Kρ, and ρ is a left -1 -1 moving frame where (ρ )s = ρ A. A theorem of Hubert states that the entries of the matrix K generate all differential invariants of the curve, that is, any differential invariant is a function of the entries of K and their derivatives. Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Light cone and conformal sphere Projective/centroaffine example (Calini, Ivey, Mari-Beffa, 2009): The action in the projective space is: 1 1 PSL(2) × RP RP au + b g · u(s) ! cu + d 0- 1 -u ! u 2 ! u 01=2 ρ = 00 1 00 -u 0- 2 u 3 u - u 3 2u 0 2 2u 0 2 -1 0 -1 ρx ρ = K = 1 2 S(u) 0 u 000 3 u 00 2 Where S(u) = u 0 - 2 ( u 0 ) is called the Schwarzian derivative. Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Light cone and conformal sphere 2 A similar procedure can be done for centroaffine space, in R . The action in this space is given by: 2 2 PSL(2) × R R aγ + bγ g · γ(s) 1 ! 2 : cγ1 + dγ2 We find that ! 0 0 ! γ2 γ1 ρ = α α -γ2 γ1 and 0 β K = ; α 0 0 00 0 00 0 0 -γ2γ1 +γ1γ2 where α = γ1γ2 - γ2γ1 and β = α : Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Light cone and conformal sphere In the projective moving frame, there is only one invariant, S(u), with a more complicated form than the invariants of the centroaffine transformation; additionally the centroaffine matrix has two functionally independent invariants. However, curves in both spaces can be related by what is called the projectivization map 2 1 Π : R RP given locally by ! x x v = : y y ! Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Lemma If ut = ux r, where r is a function of the Schwarzian and its derivatives, then ut = ux r is invariant. Additionally, any invariant evolution must be of that type. Lemma If r1 and r2 are functions of α, β and their derivatives, then γt = r1T1 + r2T2 is invariant and every invariant evolution under the centrofine group action is of this one form. ρleft = T1 T2 Light cone and conformal sphere Definition A differential equation ut = F (x; u; ux ;::: ) is invariant under a group action if whenever u is a solution, g · u is also a solution. The equation is then called an invariant evolution of u. Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Lemma If r1 and r2 are functions of α, β and their derivatives, then γt = r1T1 + r2T2 is invariant and every invariant evolution under the centrofine group action is of this one form. ρleft = T1 T2 Light cone and conformal sphere Definition A differential equation ut = F (x; u; ux ;::: ) is invariant under a group action if whenever u is a solution, g · u is also a solution. The equation is then called an invariant evolution of u. Lemma If ut = ux r, where r is a function of the Schwarzian and its derivatives, then ut = ux r is invariant. Additionally, any invariant evolution must be of that type. Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Light cone and conformal sphere Definition A differential equation ut = F (x; u; ux ;::: ) is invariant under a group action if whenever u is a solution, g · u is also a solution. The equation is then called an invariant evolution of u. Lemma If ut = ux r, where r is a function of the Schwarzian and its derivatives, then ut = ux r is invariant. Additionally, any invariant evolution must be of that type. Lemma If r1 and r2 are functions of α, β and their derivatives, then γt = r1T1 + r2T2 is invariant and every invariant evolution under the centrofine group action is of this one form. ρleft = T1 T2 Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Theorem 2 Let γ(t; x) be a C centroaffine curve, let γt be an invariant evolution of centroaffine curves. Let Π be the projectivization map as previously defined and assume the evolution preserves α = 1: (r2)x Then r1 = 2 and π∗(γt ) = π∗(r1T1 + r2T2) = ut = ux r, where r = r2. Light cone and conformal sphere Finally, the projectivization map takes these invariant evolutions to each other. Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Light cone and conformal sphere Finally, the projectivization map takes these invariant evolutions to each other. Theorem 2 Let γ(t; x) be a C centroaffine curve, let γt be an invariant evolution of centroaffine curves. Let Π be the projectivization map as previously defined and assume the evolution preserves α = 1: (r2)x Then r1 = 2 and π∗(γt ) = π∗(r1T1 + r2T2) = ut = ux r, where r = r2. Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Light cone and conformal sphere The Light cone, an important concept in general relativity is a 4 four-dimensional subspace of u 2 R defined as: CL = fu 2 Rjhu; uiJ = 0g: where we define the inner product on this space as the Minkowski length (a degenerate length), 2 T hu; uiJ = u · u = kukJ = u Ju; where 0 0 0 0 -11 B 0 1 0 0 C J = B C : @ 0 0 1 0 A -1 0 0 0 The Lie group O(3; 1) is defined by the preservation of J: Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Light cone and conformal sphere We can see immediately that kukJ is a degenerate metric, called 0 1 u0 Bu1C the Lorentzian metric.