The existence of light-like homogeneous geodesics in homogeneous Lorentzian manifolds Zdenˇ ek Duˇ sek Leuven, 2012
Contents Homogeneous geodesics in pseudo-Riemannian manifolds General settings, Geodesic lemma Existence of homogeneous geodesic in Riemannian manifold Homogeneous geodesics in homogeneous affine manifolds The affine method, Killing vector fields Existence of homogeneous geodesics Light-like homogeneous geodesics in Lorentzian manifolds Adapting the affine method The existence in even dimension Invariant metric on a Lie group
Motivation and results ◮ In pseudo-Riemannian geometry, null homogeneous geodesics are of particular interest. Plane-wave limits (Penrose limits) of homogeneous spacetimes along light-like homogeneous geodesics are studied. However, it was not known whether any homogeneous pseudo-Riemannian or Lorentzian manifold admits a null homogeneous geodesic. ◮ An example of a 3-dimensional Lie group with an invariant Lorentzian metric which does not admit light-like homogeneous geodesic was described (G. Calvaruso). ◮ In the present project, the affine method is adapted to the pseudo-Riemannian case. ◮ We show that any Lorentzian homogeneous manifold of even dimension admits a light-like homogeneous geodesic. ◮ In the case of a Lie group G = M with a left-invariant metric, the calculation is particularly easy. As an illustration, we apply the method on an example of a Lie group in dimension 3.
Homogeneous geodesics in Riemannian and pseudo-Riemannian manifolds Let ( M , g ) be a pseudo-Riemannian manifold, G ⊂ I 0 ( M ) be a transitive group of isometries - homogeneous pseudo-Riemannian manifold Let p ∈ M be a fixed point, H be the isotropy group at p - homogeneous space ( G / H , g ) Let ( G / H , g ) be fixed homogeneous space, g , h the Lie algebras of G , H ; m a vector space such that g = h + m and Ad ( H ) m ⊂ m - reductive decomposition (may not exist in the pseudo-Riemannian case) Let g = h + m be a fixed reductive decomposition - the natural identification of m ⊂ g and T p M (via the natural projection π : G → G / H ) - Ad ( H )-invariant scalar product � , � on m
Definition The geodesic γ ( s ) through the point p defined in an open interval J ( where s is an affine parameter ) is homogeneous if there exists 1) a diffeomorphism s = ϕ ( t ) between the real line and the interval J ; 2) a vector X ∈ g such that γ ( ϕ ( t )) = exp ( tX )( p ) for all t ∈ ( −∞ , + ∞ ). The vector X is then called a geodesic vector.
Lemma (Geodesic lemma) Let X ∈ g . The curve γ ( t ) = exp ( tX )( p ) is a geodesic curve with respect to some parameter s if and only if � [ X , Z ] m , X m � = k � X m , Z � for all Z ∈ m , where k ∈ R is some constant. Further, if k = 0 , then t is an affine parameter for this geodesic. If k � = 0 , then s = e − kt is an affine parameter for the geodesic. The second case can occur only if the curve γ ( t ) is a null curve in a ( properly ) pseudo-Riemannian space. Theorem (Kowalski, Szenthe) On every Riemannian homogeneous manifold there exist at least one homogeneous geodesic through arbitrary point.
Homogeneous geodesics in affine manifolds Lemma Let ( M , ∇ ) be a homogeneous affine manifold. Then each regular curve which is an orbit of a 1 -parameter subgroup g t ⊂ G on M is an integral curve of an affine Killing vector field on M. Lemma Let ( M , ∇ ) be a homogeneous affine manifold and p ∈ M. There exist n = dim( M ) affine Killing vector fields which are linearly independent at each point of some neighbourhood U of p. Lemma The integral curve γ ( t ) of the Killing vector field Z on ( M , ∇ ) is geodesic if and only if ∇ Z γ ( t ) Z = k γ · Z γ ( t ) holds along γ . Here k γ ∈ R is a constant.
Existence of homogeneous geodesics Theorem Let M = ( G / H , ∇ ) be a homogeneous affine manifold and p ∈ M. Then M admits a homogeneous geodesic through p. Proof. Killing vector fields K 1 , . . . , K n independent near p , basis B = { K 1 ( p ) , . . . , K n ( p ) } of T p M . Any vector X ∈ T p M , X = ( x 1 , . . . x n ) in B , determines a Killing vector field X ∗ = x 1 K 1 + · · · + x n K n and an integral curve γ X of X ∗ through p . ◮ Sphere S n − 1 ⊂ T p M , vectors X = ( x 1 , . . . , x n ) with � X � = 1. ◮ Denote v ( X ) = ∇ X ∗ X ∗ and t ( X ) = v ( X ) − � v ( X ) , X � X , then t ( X ) ⊥ X and X �→ t ( X ) defines a vector field on S n − 1 . ◮ If n is odd, according to the Hair-Dressing Theorem for sphere, there is ¯ X ∈ T p M such that t ( ¯ X ) = 0. ◮ We see v ( ¯ X ) = k ¯ X ¯ X = k ¯ X , hence ∇ ¯ X .
Existence of homogeneous geodesics We refine the proof to arbitrary dimension: Recall that X �→ t ( X ) defines a smooth vector field on S n − 1 . Assume now that t ( X ) � = 0 everywhere. Putting f ( X ) = t ( X ) / � t ( X ) � , we obtain a smooth map f : S n − 1 → S n − 1 without fixed points. According to a well-known statement from differential topology, the degree of f is odd (integral degree is deg( f ) = ( − 1) n ). On the other hand, we have v ( X ) = v ( − X ) and hence f ( X ) = f ( − X ) for each X . If Y is a regular value of f , then the inverse image f − 1 ( Y ) consists of even number of elements, hence deg( f ) is even, which is a contradiction. This implies that there is ¯ X ∈ T p M such that t ( ¯ X ) = 0 and again, a homogeneous geodesic exists. �
Homogeneous Lorentzian manifolds Proposition Let φ X ( t ) be the 1 -parameter group of isometries corresponding to the Killing vector field X ∗ . For all t ∈ R , it holds φ X ( t ) ∗ ( X ∗ p ) = X ∗ φ X ( t )( p ) = γ X ( t ) , γ X ( t ) . The covariant derivative ∇ X ∗ X ∗ depends only on the values of X ∗ along γ X ( t ). From the invariance of g and ∇ , we obtain Proposition Along the curve γ X ( t ) , it holds for all t ∈ R g p ( X ∗ , X ∗ ) g γ X ( t ) ( X ∗ γ X ( t ) , X ∗ = γ X ( t ) ) , φ X ( t ) ∗ ( ∇ X ∗ X ∗ � ∇ X ∗ X ∗ � p ) = γ X ( t ) . � �
Proposition Let ( M , g ) be a homogeneous Lorentzian manifold, p ∈ M and X ∈ T p M. Then, along the curve γ X ( t ) , it holds ∇ X ∗ X ∗ � γ X ( t ) ∈ ( X ∗ γ X ( t ) ) ⊥ . � Proof. We use the basic property ∇ g = 0 in the form ∇ X ∗ g ( X ∗ , X ∗ ) = 2 g ( ∇ X ∗ X ∗ , X ∗ ) . (1) According to Proposition 2, the function g ( X ∗ , X ∗ ) is constant along γ X ( t ). Hence, the left-hand side of the equality (1) is zero and the right-hand side gives the statement. �
Theorem Let ( M , g ) be a homogeneous Lorentzian manifold of even dimension n and let p ∈ M. There exist a light-like vector X ∈ T p M such that along the integral curve γ X ( t ) of the Killing vector field X ∗ it holds ∇ X ∗ X ∗ � γ X ( t ) = k · X ∗ γ X ( t ) , � where k ∈ R is some constant. Proof. ◮ Killing vector fields K 1 , . . . K n such that { K 1 ( p ) , . . . , K n ( p ) } is a pseudo-orthonormal basis of T p M with K n ( p ) timelike. ◮ Any airthmetic vector x = ( x 1 , . . . , x n ) ∈ R n determines the Killing vector field X ∗ = � n i =1 x i K i . p and R n ≃ T p M . ◮ We identify x with X ∗ x ∈ S n − 2 ⊂ R n − 1 . ◮ We consider x = (˜ x , 1), where ˜ x ∈ S n − 2 ◮ For X ∗ , we have g p ( X ∗ p , X ∗ p ) = 0 and the vectors ˜ determine light-like directions in R n ≃ T p M .
x , 1) ∈ R n ≃ T p M , we denote Y x = ∇ X ∗ X ∗ � ◮ For x = (˜ p . � ◮ With respect to the basis B = { K 1 ( p ) , . . . , K n ( p ) } , we denote the components of the vector Y x as y ( x ) = ( y 1 , . . . , y n ). ◮ Using Proposition, we see that y ( x ) ⊥ x . ◮ We define the new vector t x as t x = y ( x ) − y n · x . ◮ Because x is light-like vector, it holds also t x ⊥ x . ◮ In components, we have t x = (˜ t x , 0), where ˜ t x ∈ R n − 1 . ◮ We see that ˜ t x ⊥ ˜ x , with respect to the positive scalar product on R n − 1 which is the restriction of the indefinite scalar product on R n . ◮ The assignment ˜ x �→ ˜ t x defines a smooth tangent vector field on the sphere S n − 2 . If n is even, it must have a zero value. x ∈ S n − 2 such that for the corresponding ◮ There exist a vector ˜ vector x = (˜ x , 1) it holds t x = 0. For this vector x , it holds y ( x ) = k · x and ∇ X ∗ X ∗ � γ X ( t ) = k · X ∗ γ X ( t ) is satisfied. � �
Corollary Let ( M , g ) be a homogeneous Lorentzian manifold of even dimension n and let p ∈ M. There exist a light-like homogeneous geodesic through p. Proof. We consider the vector X ∈ T p M which satisfies Theorem. The integral curve γ X ( t ) through p of the corresponding Killing vector field X ∗ is homogeneous geodesic. �
Invariant metric on a Lie group Let M = G be a Lie group with a left-invariant metric g . ◮ For any tangent vector X ∈ T e M and the corresponding Killing vector field X ∗ , we consider the vector function X ∗ γ X ( t ) along the integral curve γ X ( t ) through e . ◮ It can be uniquely extended to the left-invariant vector field L X on G . Hence, along γ X , we have L X γ X ( t ) = X ∗ γ X ( t ) . (2) ◮ At general points q ∈ G , values of left-invariant vector field L X do not coincide with the values of the Killing vector field X ∗ , which is right-invariant . ◮ As we are interested in calculations along the curve γ X ( t ), we can work with respect to the moving frame of left-invariant vector fields and use formula (2).
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