The existence of light-like homogeneous geodesics in homogeneous - - PowerPoint PPT Presentation

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)

• f 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 ⊂ I0(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

• f 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 TpM

(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, Xm = kXm, 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

• ne 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 gt ⊂ G

• n 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 ∇Zγ(t)Z = kγ · Zγ(t) holds along γ. Here kγ ∈ 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 K1, . . . , Kn independent near p,

basis B = {K1(p), . . . , Kn(p)} of TpM. Any vector X ∈ TpM, X = (x1, . . . xn) in B, determines a Killing vector field X ∗ = x1K1 + · · · + xnKn and an integral curve γX of X ∗ through p.

◮ Sphere Sn−1 ⊂ TpM, vectors X = (x1, . . . , xn) with X = 1. ◮ Denote v(X) = ∇X ∗X ∗ and t(X) = v(X) − v(X), XX,

then t(X) ⊥ X and X → t(X) defines a vector field on Sn−1.

◮ If n is odd, according to the Hair-Dressing Theorem

for sphere, there is ¯ X ∈ TpM such that t( ¯ X) = 0.

◮ We see v( ¯

X) = k ¯ X, hence ∇ ¯

X ¯

X = k ¯ X.

Existence of homogeneous geodesics

We refine the proof to arbitrary dimension: Recall that X → t(X) defines a smooth vector field on Sn−1. Assume now that t(X) = 0 everywhere. Putting f (X) = t(X)/t(X), we obtain a smooth map f : Sn−1 → Sn−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 ∈ TpM such that t( ¯ X) = 0 and again, a homogeneous geodesic exists.

Homogeneous Lorentzian manifolds

Proposition

Let φX(t) be the 1-parameter group of isometries corresponding to the Killing vector field X ∗. For all t ∈ R, it holds φX(t)(p) = γX(t), φX(t)∗(X ∗

p ) = X ∗ γX (t).

The covariant derivative ∇X ∗X ∗ depends only on the values of X ∗ along γX(t). From the invariance of g and ∇, we obtain

Proposition

Along the curve γX(t), it holds for all t ∈ R gp(X ∗, X ∗) = gγX (t)(X ∗

γX (t), X ∗ γX (t)),

φX(t)∗(∇X ∗X ∗

• p)

= ∇X ∗X ∗

• γX (t).

Proposition

Let (M, g) be a homogeneous Lorentzian manifold, p ∈ M and X ∈ TpM. Then, along the curve γX(t), it holds ∇X ∗X ∗

• γX (t) ∈ (X ∗

γX (t))⊥.

• Proof. We use the basic property ∇g = 0 in the form

∇X ∗g(X ∗, X ∗) = 2g(∇X ∗X ∗, X ∗). (1) According to Proposition 2, the function g(X ∗, X ∗) is constant along γX(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 ∈ TpM such that along the integral curve γX(t) of the Killing vector field X ∗ it holds ∇X ∗X ∗

• γX (t) = k · X ∗

γX (t),

where k ∈ R is some constant. Proof.

◮ Killing vector fields K1, . . . Kn such that {K1(p), . . . , Kn(p)}

is a pseudo-orthonormal basis of TpM with Kn(p) timelike.

◮ Any airthmetic vector x = (x1, . . . , xn) ∈ Rn determines the

Killing vector field X ∗ = n

i=1 xiKi. ◮ We identify x with X ∗ p and Rn ≃ TpM. ◮ We consider x = (˜

x, 1), where ˜ x ∈ Sn−2 ⊂ Rn−1.

◮ For X ∗, we have gp(X ∗ p , X ∗ p ) = 0 and the vectors ˜

x ∈ Sn−2 determine light-like directions in Rn ≃ TpM.

◮ For x = (˜

x, 1) ∈ Rn ≃ TpM, we denote Yx = ∇X ∗X ∗

• p.

◮ With respect to the basis B = {K1(p), . . . , Kn(p)}, we denote

the components of the vector Yx as y(x) = (y1, . . . , yn).

◮ Using Proposition, we see that y(x) ⊥ x. ◮ We define the new vector tx as tx = y(x) − yn · x. ◮ Because x is light-like vector, it holds also tx ⊥ x. ◮ In components, we have tx = (˜

tx, 0), where ˜ tx ∈ Rn−1.

◮ We see that ˜

tx ⊥ ˜ x, with respect to the positive scalar product on Rn−1 which is the restriction of the indefinite scalar product on Rn.

◮ The assignment ˜

x → ˜ tx defines a smooth tangent vector field

• n the sphere Sn−2. If n is even, it must have a zero value.

◮ There exist a vector ˜

x ∈ Sn−2 such that for the corresponding vector x = (˜ x, 1) it holds tx = 0. For this vector x, it holds y(x) = k · x and ∇X ∗X ∗

• γX (t) = k · X ∗

γX (t) is satisfied.

Corollary

Let (M, g) be a homogeneous Lorentzian manifold

• f even dimension n and let p ∈ M.

There exist a light-like homogeneous geodesic through p.

• Proof. We consider the vector X ∈ TpM which satisfies Theorem.

The integral curve γX(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 ∈ TeM and the corresponding

Killing vector field X ∗, we consider the vector function X ∗

γX (t)

along the integral curve γX(t) through e.

◮ It can be uniquely extended to the left-invariant

vector field LX on G. Hence, along γX, we have LX

γX (t) = X ∗ γX (t).

(2)

◮ At general points q ∈ G, values of left-invariant

vector field LX do not coincide with the values

• f the Killing vector field X ∗, which is right-invariant.

◮ As we are interested in calculations along the curve γX(t),

we can work with respect to the moving frame

• f left-invariant vector fields and use formula (2).

Proposition

Let {L1, . . . , Ln} be a left-invariant moving frame on a Lie group G with a left-invariant pseudo-Riemannian metric g and the induced pseudo-Riemannian connection ∇. Then it holds ∇LiLj =

n

• k=1

γk

ijLk,

i, j = 1, . . . , n, where γk

ij are constants.

• Proof. It follows from the invariance of the affine connection ∇.

An example of a Lie group

Now we illustrate the affine method of the previous section with an example of the 3-dimensional Lie group E(1, 1) with an inv. Lorentzian metric which has no light-like homogeneous geodesic. We choose one of the examples described by G. Calvaruso using the geodesic lemma for reductive pseudo-Riemannian homogeneous

• manifolds. We construct explicitly the vector field ˜

tx, which has no zero value in this case.

The group E(1, 1) can be represented by the matrices   e−w u ew v 1   . Hence, M = E(1, 1) can be identified with R3[u, v, w]. The left-inv. vector fields are U = e−w∂u, V = ew∂v, W = ∂w. We choose the new moving frame {E1, E2, E3} given as E1 = U − V , E2 = −W , E3 = 1/2(U + V ). The pseudo-Riemannian metric g such that the basis determined by the above frame at any point p ∈ M is pseudo-orthonormal basis of TpM with E3 timelike.

The above metric g in coordinates is ds2 = −1 4(3e2wdu2 + 3e−2wdv2 + 10dudv − 4dw2) and the nonzero Christoffel symbols are Γ3

11 = 3

4e2w, Γ1

13 = − 9 16,

Γ2

13 = 15

16e2w, Γ3

22 = −3

4e−2w, Γ2

23 = 9 16,

Γ1

23 = −15

16e−2w. In the frame {E1, E2, E3}: g(E1, E1) = g(E2, E2) = 1, g(E3, E3) = −1, g(Ei, Ej) = 0 and nonzero covariant derivatives (which satisfy the Proposition): ∇E1E2 = −3 4E3, ∇E1E3 = −3 4E2, ∇E2E3 = 5 4E1, ∇E2E1 = 5 4E3, ∇E3E1 = −3 4E2, ∇E3E2 = 3 4E1. We will perform all calculations in this moving frame,

• r with respect to the corresponding pseudo-orthonormal basis

B = {E1(e), E2(e), E3(e)} of the tangent space TeM ≃ R3.

Any x = (x1, x2, x3) ∈ R3 determines LX = x1E1 + x2E2 + x3E3. Light-like vectors X ∈ TeM are x = (sin(ϕ), cos(ϕ), 1), ˜ x = (sin(ϕ), cos(ϕ)) ∈ S1. For LX, it holds ∇LX LX = 2 cos(ϕ)E1 − 3 2 sin(ϕ)E2 + 1 2 sin(ϕ) cos(ϕ)E3, y(x) =

• 2 cos(ϕ), −3

2 sin(ϕ), 1 2 sin(ϕ) cos(ϕ)

• .

We see that y(x) ⊥ x. The projection tx is tx = y(x) − 1 2 sin(ϕ) cos(ϕ) · x = =

• 2 − 1

2 sin2(ϕ)

• ·
• cos(ϕ), − sin(ϕ), 0
• .

◮ tx ⊥ x and ˜

tx ⊥ ˜ x

◮ ˜

x → ˜ tx defines the smooth vector field on S1, which is nonzero everywhere.

◮ There is not any vector X ∈ TeG which satisfies

Main Theorem.

References

Duˇ sek, Z.: The existence of homogeneous geodesics in homogeneous pseudo-Riemannian and affine manifolds, J.

• Geom. Phys 60 (2010).

Duˇ sek, Z.: On the reparametrization of affine homogeneous geodesics, Differential Geometry, J.A. ´ Alvarez L´

• pez and E.

Garc´ ıa-R´ ıo (Eds.), World Scientific (2009), 217–226. Duˇ sek, Z., Kowalski, O., Vl´ aˇ sek, Z.: Homogeneous geodesics in homogeneous affine manifolds, Result. Math. 54 (2009), 273–288.