The Light Cone and the Conformal Sphere: Differential Invariants And

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

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

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

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

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 ∈ R deﬁned as: CL = {u ∈ R|hu, uiJ = 0}. 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 −1  0 1 0 0  J =   .  0 0 1 0  −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   u0 u1 the Lorentzian metric. In particular, if u =   and u · u = 0 u2 u3 q u1 2 2 and if u^ = , where ku^k = u1 + u2, then u2 2 2 2 u1 + u2 = ku^k = 2u0u3 on CL. The Lorentzian group T O(3, 1) = {Θ ∈ M4×4|Θ JΘ = J} can act on the light cone.

Θ(3, 1) × CL CL

(Θ, u) Θu →

→

Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Light cone and conformal sphere We can projectivize the cone to obtain the conformal, or Möbius sphere, M. The action also projectivizes into conformal transformations. The Möbiussphere is topologically equivalent to m a 2-sphere so we can locally represent the sphere by m = 1 . m2 Define Λ : M CL   u0 m→1 Λ(m)   =m ^ , m2 1 1 2 1 2 → 2 where u0 = 2 kmk = 2 (m1 + m2) so that Im(Λ) ∈ CL. This is the standard lift from M to CL associated to the projectivization   u0 u1 u1 Π :   u3 . u  u2  2 u3 u3 → Theresa C. Anderson The Light Cone and the Conformal Sphere: Differential Invariants and their Relations Light cone and conformal sphere

The general form of the Lorentzian group is:

     T 1 2 1 0 0 α 0 0 1 v 2 kvk g1g2g3 =  ξ I 0 0 A 0  0 I v  1 2 T −1 2 kξk ξ 1 0 0 α 0 0 1 This can act on M using Λ and Π :

1 kmk2 kvk2 g · m = (αξ( + v T m + ) + A(m + v)) f 2 2 where α kmk2 kvk2 f = kξk2( + v T m + ) + ξT A(m + v) + α−1 2 2 2

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations Light cone and conformal sphere

Moving frames Are the curves and the invariant curve evolutions of the action related? To find out, first find the moving frame. To do this, we use the normalization procedure from Fels and Olver (1999).

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations Light cone and conformal sphere

The cone case We begin the normalization process by normalizing:     u0 0 g3 ·  u^  = c0 = 0 . u3 1

We can do this since g1g2 are in the isotropy subgroup of e3. From this calculation, we ﬁnd v = − u^ and α = u . We next restrict u3 3 0 0 ourselves to non-degenerate curves, where hu , u iJ 6= 0 always. Since CL consists of two connected components, we also will only 0 2 0 T 0 permit ku kJ = (u ) Ju > 0.

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations Light cone and conformal sphere

First order normalization: we next normalize  0  0 0 ku kJ  gu = c1 =    0  0 and use the structure of the Lie Algebra to ﬁnd

 u u u u  det 3 1 det 3 2  u 0 u 0 u 0 u 0  1  3 1 3 2  A = 0   = (aij ). u3ku kJ    u2 u3 u3 u1  det 0 0 det 0 0 u2 u3 u3 u1 We also ﬁnd 0 u3 ξ1 = − 0 . u3ku kJ

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations Light cone and conformal sphere

Next, we normalize the second prolonged actions

 0 2 ku kJ 00  b  gu = c2 =   ,  0  d where b and d are invariants. By doing so, we ﬁnd the remaining entries for ρ.

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations Light cone and conformal sphere

We can now combine all of the components into the right moving frame:

 T 2  u3 u^ u0ku^k T 2 ρ =  u3ξ u3ξ · v + A u0ξku^k + Av  , u3 2 u3 2 T T u0 2 2 T −1 2 kξk 2 kξk v + ξ A 2 kξk ku^k + ξ Av + u3 where A and ξ are as before.

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations Light cone and conformal sphere

−1 From here we can ﬁnd K = ρx ρ , which contains the invariants. Theorem

 0  0 −ku kJ 0 0 ku 00k2 −(ku 0k 0 )2  J J 0 0 −ku 0k   2ku 0k3 J  K =  J  = (kij ).  k31 0 0 0   00 2 0 0 2  ku kJ −(ku kJ ) 0 0 3 k31 0 2ku kJ

We use the recurrence relation K · cr = −cr+1 + (cr )x and use the moving frame to calculate the third entry of c3 and

0 3 k31 · (ku kJ ) = −c3 .

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations Light cone and conformal sphere

Conformal case Now we will obtain the moving frame for the conformal sphere with a similar normalization technique. Our 0 1 choices in this case are c = c = and c = to match 0 2 0 1 0 with the projectivization.

 T 1 2  1 −m 2 kmk T 1 2 ρ =  ξ A − ξm 2 ξkmk − Am  , 1 2 1 2 T T 1 2 2 T 2 ξ2 − 2 ξ2m + ξ A 4 ξ2kmk − ξ Am + 1

 0 00 m1 m1 det 0 00 0 2 (km k )x m2 m2 where ξ1 = − 2km 0k2 , ξ2 = − km 0k2 and

0 0 1 m1 m2 A = 0 0 0 . km k −m2 m1

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations Light cone and conformal sphere

We now find the Maurer-Cartan matrix for the moving frame, using the recurrence relations. To proceed, we first calculate the infinitesimal action for any k ∈ g and m ∈ M. Similarly, we can find the infinitesimal first and second prolonged actions v · m 0 and v · m 00 by replacing m with m 0 and m 00 respectfully. We next (i) substitute k with K and m with ci and use the recurrence relations.

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations Light cone and conformal sphere

For the conformal sphere,

 0 −1 0 0  m I2 0 0 −1 K =  m  . I3 0 0 0  m m 0 I2 I3 0

3((km 0k2) )2 −hm 0, m 000i 3 I m = x + J − kξk2 2 km 0k4 km 0k2 2 1 1 I m = − (ξm · 3hm 0, m 00i ) − (−m 0 m 000 + m 0 m 000) 3 km 0k2 2 J km 0k2 2 1 1 2 m m We now see that the two nonconstant invariants are I2 and I3 .

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations Light cone and conformal sphere

With both K matrices, we can see the relationship between invariants. For the cone:  0  0 −ku kJ 0 0 ku 00k2 −(ku 0k 0 )2  J J 0 0 −ku 0k   2ku 0k3 J  K =  J   k31 0 0 0   00 2 0 0 2  ku kJ −(ku kJ ) 0 0 3 k31 0 2ku kJ Sphere:  0 −1 0 0  m I2 0 0 −1 K =  m  . I3 0 0 0  m m 0 I2 I3 0 c 0 Notice that I1 = −ku kJ is an invariant of arc-length type, and if 0 ku kJ 6= 0 we can reparameterize. The equality of the other invariants follows through calculation.

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations Light cone and conformal sphere We will show that the two remaining invariants of the cone and the 0 sphere are related if we re-parameterize to let ku kJ = 1 Given an 0 m^ we can multiply by a unique u3 such that ku kJ = 1. Let

CL M   u0 u1→ u1 1 Π(u) = Π( ) = m. u2 u2 u3 u3 → And M CL   u0 m1 → u1 Λ( )   m2 u2 u3 → where u1 = m1u3, u2 = m2u3. Thus 1 u = . 3 km 0k Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations Light cone and conformal sphere

We see that hu 0, u 000i 3 (km 0k2)2 hm 0, m 000i 3 km 00k2 I c = − J = x − J − = I m. 2 2 4 km 0k4 km 0k2 2 km 0k2 2

2 0 0 Finally, since a21 = −u3m2, a22 = u3m1 then

c 0 00 0 0 000 2 I3 = 3m1m2 u3u3 − m1m2 u3 =

m 00 m 00 − 1 2 − ( 0 2) 3 det 0 0 km k x 0 000 m2 m1 1 m1 m1 0 4 − 0 2 det 0 000 2km k km k −m2 m2 - 0 00 0 000 3ξ2hm , m iJ 1 m1 m1 m 0 4 − 0 2 det 0 000 = I3 . km k km k −m2 m2

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations Light cone and conformal sphere

We can ﬁnally relate the invariant evolutions. Theorem Any invariant evolution is of the following type:

ut = r1T1 + r2T2 + r3T3

−1 for the cone where the T ’s are the columns of dΦρ(o)c , where this is the inverse of the diﬀerential of the map Φρ : CL CL with Φρ(u) = ρ · u and mt = s1M1 + s2M2 → −1 where A = M1 M2 for the sphere.

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations Light cone and conformal sphere

Theorem 0 Let the invariant evolutions be deﬁned as above and let ku kJ = 1. Then the projectivization gives

Π∗(ut ) = Π∗(r1T1 + r2T2 + r3T3) = s1M1 + s2M2 = mt , where r1 = s1, r2 = s2, M1, M2, T1, T2, T3 are as above and 0 r3 = (r1)x is the condition required to preserve ku kJ .

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations 2. Used frames to calculate K, which give all diﬀerential invariants 3. Relationship between invariants and evolutions becomes clear after reparameterization 4. What are the limitations and extensions to this technique? 5. Does this technique work in other pairs of geometries whenever we can projectivize from one to the other?

Light cone and conformal sphere

Conclusions and Future directions: 1. Using normalization procedure, found moving frames for the Light cone and Conformal sphere

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations 3. Relationship between invariants and evolutions becomes clear after reparameterization 4. What are the limitations and extensions to this technique? 5. Does this technique work in other pairs of geometries whenever we can projectivize from one to the other?

Light cone and conformal sphere

Conclusions and Future directions: 1. Using normalization procedure, found moving frames for the Light cone and Conformal sphere 2. Used frames to calculate K, which give all diﬀerential invariants

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations 4. What are the limitations and extensions to this technique? 5. Does this technique work in other pairs of geometries whenever we can projectivize from one to the other?

Light cone and conformal sphere

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations 5. Does this technique work in other pairs of geometries whenever we can projectivize from one to the other?

Light cone and conformal sphere

Conclusions and Future directions: 1. Using normalization procedure, found moving frames for the Light cone and Conformal sphere 2. Used frames to calculate K, which give all diﬀerential invariants 3. Relationship between invariants and evolutions becomes clear after reparameterization 4. What are the limitations and extensions to this technique?

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations Light cone and conformal sphere

Theresa C. Anderson The Light Cone and the Conformal Sphere: Diﬀerential Invariants and their Relations