The Geometrical Interpretation of Temporal Cone Norm in Almost Minkowski Manifold

Sorin Noaghi

Dedicated to the Memory of Grigorios TSAGAS (1935-2003), President of Balkan Society of Geometers (1997-2003)

Abstract

For almost Minkowski manifolds we prove that the norm determined by a unitary vector field which belongs to the timelike cone is the sum of two fundamental forms induced by the Lorentzian metrics on two submanifolds.

Mathematics Subject Classification: 53C50, 53B30, 53C80. Key words: almost Minkowski manifold, topological cone norm.

1 Introduction

A Lorentz manifold is a pair (M, g) where M is an n + 1 dimensional smooth paracompact manifold and g is a global smooth two-times covariant symmetric tensor field which is nondegenerate and has n − 1 signature.

A time-normalized space-time [M, g, Z] is a Lorentz manifold (M, g) for which a global unitary (i.e., g (Z, Z) = −1) tangent vector field Z of timelike vectors is fixed; this will be denoted by [M, g, Z].

Definition 1 An almost Minkowski manifold is a time-normalized space-time [M, g, Z] provided that the distribution

Δ : x ∈ M → Δ_{x} =^{def} {Y ∈ T_{x}M | g (Y, Z) = 0}

is totally integrable.

Proposition 2 The necessary and sufficient condition that a time-normalized spacetime manifold [M, g, Z] be an almost Minkowski manifold is the existence of a preferential atlas

A= \{(U_{α}, χ_{α}) | α ∈ Γ, χ_{α} (x) = (x^{i}) , i = \overline{1, n + 1} ∂_{n+1} = Z|_{U_{α}}\}, where

\frac{∂g_{in+1}}{∂x^{j}} = \frac{∂g_{jn+1}}{∂x^{i}} ∀i, j ∈ \overline{1, n + 1}.

∗ Balkan Journal of Geometry and Its Applications, Vol.8, No.2, 2003, pp. 37-42.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2003.

Proof. For X,Y belonging to Δ we have

g (X, Z) = g (Y, Z) = 0,

which infers

g (∇_{Y} X, Z) + g (X, ∇_{Y} Z) = 0
g (∇_{X}Y, Z) + g (Y, ∇_{X}Z) = 0,

where ∇ is the Levi-Civita connection of Lorentz manifold (M, g). In the local charts of atlas A where Z = ∂_{n+1}, X = X^{i}∂_{i} and Y = Y^{j}∂_{j} we have

g ([X, Y] , Z) = g (∇_{X}Y, Z) − g (∇_{Y} X, Z) =
= g (X, ∇_{Y} Z) − g (Y, ∇_{X}Z) =
= X^{i}Y^{j} [Γ_{jn+1}^{k}g_{in+1} − Γ_{in+1}^{k}g_{jn+1}] =
= X^{i}Y^{j} (\frac{∂g_{in+1}}{∂x^{j}} − \frac{∂g_{jn+1}}{∂x^{i}} ) .

Therefore [X, Y] belongs to Δ if and only if \frac{∂g_{in+1}}{∂x^{j}} = \frac{∂g_{jn+1}}{∂x^{i}}

□

Remark 3 The existence of almost Minkowski manifolds is obvious, since it is possible to choose Z so that g (∂_{i}, Z) ∂_{i} be irrotational. If (M, g) is stable causal then there exists a real global function f with the gradient ∇f of timelike type, (see e.g. [1],

[6]), and the corresponding 1-form of Z = \frac{1}{\sqrt{−g (∇f, ∇f)}}∇f closed, and therefore it

respects the previous conditions.

Remark 4 If the corank one distribution is not integrable, then any two points can be connected by a curve γ : [0, 1] → M where g (γ′ (t) , Z_{γ(t)}) = 0, according to the Carathéodory theorem, ([3, p.10]).

Definition 5 We define the ordering relation for the elements of T_{x}M, x ∈ M :

X ≤ Y ⇔ Y − X ∈ K_{x},

where K_{x} = {X ∈ T_{x}M | g (X, X) < 0, g (X, Z) < 0} is the interior of the timelike cone of the tangent vectors.

From ([4]) we have:

• (T_{x}M, K_{x}) is a Krein space, ∀x ∈ M

• The map | |_{Z} : T_{x}M → R, |X|_{Z} =^{def} min {λ ≥ 0 | − λZ ≤ X ≤ λZ} is a topological norm of T_{x}M, named the Z-norm of the almost Minkowski manifold [M, g, Z] .

• An easy calculation implies |X|_{z} = |g (X, Z)| + \sqrt{g (X, Z)^{2} + g (X, X)}. (1.1)

Proposition 6 The Z-norm is invariant to a conformal change of Lorentzian metric.

Proof. Let g, ĝ be two conformal metrics (i.e. g = ĝΩ^{2}). The Z-norms expressions for the two almost Minkowski manifolds are, ([5])

|X|_{Z}^{g} = |\frac{g (X, Z)}{g (Z, Z)}| + \sqrt{[\frac{g (X, Z)}{g (Z, Z)}]^{2} + \frac{g (X, X)}{g (Z, Z)}} =

= |\begin{matrix}ĝ & (X, Z) \\ ĝ & (Z, Z) \end{matrix}| + \sqrt{[\begin{matrix}ĝ & (X, Z) \\ ĝ & (Z, Z) \end{matrix}]^{2} + \begin{matrix}ĝ & (X, X) \\ ĝ & (Z, Z) \end{matrix}} = |X|_{g} ; Z

Remark 7 The Z-norm can be defined if the existent condition of the global timelike vector field Z is weakened and replaced with the existence of a line element field which is equivalent to the existence of Lorentzian metrics ([2]).

2 The Z-norm dependence on the first fundamental form of the hypersurface normal to Z

Consider the preferential atlas A of Proposition 2 (this exists, cf. [4]). We note by S the integral submanifold of distribution Δ with p ∈ S. Obviously S is a hypersurface imbedded in M with inclusion map θ : S → M. Let n ∈ T_{q}^{∗}M, q ∈ S be the 1-form n (X) = g (X, Z) , ∀X ∈ T_{q}M. This implies n (θ_{∗}X) = 0, ∀X ∈ T_{q}M and if we denote H_{q} = θ_{∗} (T_{q}S), this is a hyperplane in T_{q}M. If Z is be tangent to θ (S) , then there exist X ∈ T_{q}S \ {0} such that θ_{∗} (X) = Z and −1 = g (Z, Z) = g (θ_{∗} (X) , Z) = 0, which is impossible. Therefore Z is not in the tangent space of θ (S) . If {E_{1}, ..., E_{n}} is a basis in T_{q}S, then {Z, θ_{∗} (E_{1}) , ..., θ_{∗} (E_{n})} is linearly independent and hence is a basis for T_{q}M. The components of g with respect to this basis are

(g_{ab}) = ( \begin{matrix}g (Z, Z) & 0 & ... & 0 \\ 0 & g (θ_{∗} (E_{1}) , θ_{∗} (E_{1})) & ... & g (θ_{∗} (E_{1}) , θ_{∗} (E_{n})) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & g (θ_{∗} (E_{n}) , θ_{∗} (E_{1})) & ... & g (θ_{∗} (E_{n}) , θ_{∗} (E_{n})) \end{matrix} ) =

= ( \begin{matrix}−1 & 0 \\ 0 & [g (θ_{∗} (E_{i}) , θ_{∗} (E_{j})]) \end{matrix} )

Because g has one negative eingenvalue, then θ^{∗}g is positively definite. Let’s consider θ^{∗} : T_{q}^{∗}M → T_{q}^{∗}S, and H_{q}^{∗} = {ω ∈ T_{q}^{∗}M | ω (Z) = 0} .

From θ^{∗}|_{H_{q}^{∗}} : H_{q}^{∗} → T_{q}^{∗}S, being obviously a bijection, we denote its inverse by θ̃_{∗} : T_{q}^{∗}S → H_{q}^{∗}. Therefore there exist two bijections θ_{∗} and θ̃_{∗} between T_{q}S and H_{q} and respectively between T_{q}^{∗}S and H_{q}^{∗}. This map can be extended in a usual way to a map θ̃ of arbitrary tensors on S to θ (S) in M. Since n is normal to hypersurface θ (S), for a given tensor T ∈ T_{s,q}^{r}S we obtain that θ̃(T) has zero transvections with n in all indices:

(θ̃T )_{j_{1}...j_{s}}^{i_{1}..m..i_{r}} n_{m} = (θ̃T )_{j_{1}..m..j_{s}}^{i_{1}...i_{r}} g^{mp}n_{p} = 0.

Denote by h the metric on θ (S), defined by h = θ̃(θ^{∗}g). In the preferential atlas A the components of h are

h_{ab} = g_{ab} + n_{a}n_{b} = g_{ab} + g_{an+1}g_{bn+1} , ∀a, b ∈ \overline{1, n + 1}.

Proposition 8 The (1.1) tensor associate to h having the components h_{a}^{b} is a projection operator, h_{a}^{b} = δ_{a}^{b} + g_{an+1}δ_{n+1}^{b} and

a) The projection of X ∈ T_{q}M onto the subspace H_{q} is

h_{a}^{b}X^{a}∂_{b} = X + g (X, Z) Z.

b) The projection of ω ∈ T_{q}^{∗}M onto the subspace H_{q}^{∗} is

h_{a}^{b}ω_{b}dx^{a} = ω + ω (Z) n.

Proof.

\begin{aligned}h_{a}^{b} & = & h_{ac}g^{cb} = (g_{ac} + g_{an+1}g_{cn+1}) g^{cb} = δ_{a}^{b} + g_{an+1}δ_{n+1}^{b} \\ h_{b}^{a}h_{c}^{b} & = & (δ_{b}^{a} + g_{bn+1}δ_{n+1}^{a}) (δ_{c}^{b} + g_{cn+1}δ_{n+1}^{b}) = h_{c}^{a} \\ X + g (X, Z) Z & = & X^{a}∂_{a} + X^{a}g_{an+1}∂_{n+1} = X^{a} (δ_{a}^{b} + g_{an+1}δ_{n+1}^{b}) ∂_{b} = h_{a}^{b}X^{a}∂_{b} \\ ω + ω (Z) n & = & ω_{a}dx^{a} + ω_{n+1}g_{an+1}dx^{a} = ω_{a} (δ_{b}^{a} + g_{bn+1}δ_{n+1}^{a}) dx^{b} = h_{b}^{a}ω_{a}dx^{b} \end{aligned}

Remark 9 Analogously we can project the tensor T ∈ T \binom{r}{s}_{q} S to

H (T) ∈ \underbrace{H_{q} ⊗ ... ⊗ H_{q}}_{r times} ⊗ \underbrace{H_{q}^{∗} ⊗ ... ⊗ H_{q}^{∗}}_{s factors} =^{def} H_{s}^{r} (q) via

H (T) = T + (−1)^{r+s+1} T (Z^{∗}, .., Z^{∗}, Z, .., Z) \underbrace{Z ⊗ ... ⊗ Z}_{r factors} ⊗ \underbrace{Z^{∗} ⊗ ... ⊗ Z^{∗}}_{s factors}

One can than verify the relation

H (H (T)) = H (T) , ∀T ∈ T \binom{r}{s}_{q} S.

Proposition 10 (S, θ^{∗}g) is a totally geodesic submanifold

Proof. In the above notation, the coordinates of the second fundamental form of S are ([1, p. 46])

χ_{ab} = h_{a}^{c}h_{b}^{d}n_{c; d} = h_{a}^{c}h_{b}^{d}g_{cn+1; d} = 0.

Remark 11 We will denote the covariant differentiation with respect to the Levi-Civita connection of (S, θ^{∗}g) by double stroke.

Then for any tensor T ∈ T \binom{r}{s}_{q} S we have:

H (T)_{j_{1}...j_{s}||m}^{i_{1}...i_{r}} = \overline{T} _{l_{1}...l_{s} ; p}^{k_{1}...k_{r}}h_{k_{1}}^{i_{1}}...h_{k_{r}}^{i_{r}}h_{j_{1}}^{l_{1}}...h_{j_{s}}^{l_{s}}h_{m}^{p} ,

where \overline{T} is an extension of H (T) to a neighborhood of θ (S). This formula is correct because the double stroke of the induced metric is zero and the torsion vanishes,

h_{ab||c} = (g_{ef} + g_{en+1}g_{fn+1}) _{; i}he_{a}h_{b}^{f}h_{c}^{i} = 0
f_{||ab} = h_{a}^{c}h_{b}^{d}f_{;cd} = h_{a}^{c}h_{b}^{d}f_{;dc} = f_{||ba}.

For p ∈ M, we denote by s the 1-dimensional submanifold which passes through p and T_{q}s = ⟨Z_{q}⟩ , ∀q ∈ s. The local imbedding map i : s ↪ M is the inclusion which determines the applications i_{∗,q} : T_{q}s → T_{q}M and i_{q}^{∗} : T_{q}^{∗}M → T_{q}^{∗}s. Denote N_{q} = i_{∗,q} (T_{q}s) , N_{q}^{∗} = {ω ∈ T_{q}^{∗}M | ω (X) = g (X, Z) , λ ∈ R}; it is obvious that i_{∗,q} : T_{q}s → N_{q} and i_{q}^{∗}|_{N_{q}^{∗}} : N_{q}^{∗} → T_{q}^{∗}s are bijections.

For l = [i_{q}^{∗}|_{N_{q}^{∗}}]^{−1} ∘ (i_{q}^{∗}g), we have l (X, Y) = −\overline{X}\overline{Y}, where

X = \overline{X}Z_{q}, Y = \overline{Y} Z_{q}, X, Y ∈ T_{q}s,

l must be a two symmetric 2-form negatively definite on i (s) and for ∀q ∈ θ (S)∩i (s)

T_{q}M = H_{q} ⊕ N_{q}, T_{q}^{∗}M = H_{q}^{∗} ⊕ N_{q}^{∗}.

Proposition 12 The Z-norm on the almost Minkowski manifold [M, g, Z] is the sum of the Riemannian norms associated to the projections onto the submanifolds (θ(h) , h) and (i (s) , −l).

Proof. In (θ (S) , h) the Riemannian norm is

|X|_{h} = \sqrt{h (X, X)} = \sqrt{g (X, X) + g (X, Z)^{2}}, ∀X ∈ H_{q}.

In (i (s) , −l) the Riemannian norm is

|X|_{l} = \sqrt{−l (X, X)} = |\overline{X}| = |g (X, Z)| , ∀X ∈ N_{q}.

If X ∈ T_{q}M = H_{q} ⊕ N_{q}, by Proposition 8 a) we have X = X_{1} + X_{2}, where X_{1} = X + g (X, Z) Z and X_{2} = −g (X, Z) Z;

h (X_{1}, X_{1}) = g (X_{1}, X_{1}) + g (X_{1}, Z)^{2} = g (X, X) + g (X, Z)^{2}
l (X_{2}X_{2}) = −g (X, Z)^{2}

Then

|X_{1}|_{h} + |X_{2}|_{l} = \sqrt{−l (X_{2}, X_{2})} + \sqrt{h (X_{1}, X_{1})} =
= |g (X, Z)| + \sqrt{g (X, X) + g (X.Z)^{2}} =^{1.1} |X|_{Z} .
□

References

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[2] S.W. Hawking, G.F.R. Ellis, The large scalar structure of space-time, Cambridge University Press, 1973.

[3] R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, Math. Surveys and Monographs, 1991.

[4] D.I.Papuc, The geometry of vector bundles endowen with a cone field , Gen. Math. 5, (1997), 313-323.

[5] D.I.Papuc, A new geometry of Lorentzian manifold, Publ. Math. Debrecen Tomus 52, 1998, Fasc 1-2 , 145-158.

[6] H.J. Seifert, Smoothing and extending cosmic time functions, Gen. Rel. Grav. 8, 1977, 815-831.

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