Causal Structures on Smooth Manifolds

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Copyright IC> IFAC Nonlinear Control Systems, SI. Petersburg, Russia, 200 I

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CAUSAL STRUCTURES ON SMOOTH MANIFOLDS Victor R. Krym * Nickolai N. Petrov ** * Department of Inorganic Chemistry St.-Petersburg State Agricultural University St.-Petersburg av., 2, 189620 St.-Petersburg, Russia. E-mail: [email protected]. ** Department of Operations Research Faculty of Mathematics and Mechanics St.-Petersburg State University Bibliotechnaya pl., 2, 198904 St.-Petersburg, Russia.

Abstract: We study differential equations of motion on Lorentz manifolds and consequences for the causal structure (local ordering) on smooth manifold with some differential system. Local ordering exists near the point x E M if there is a foliation related with the main differential system. Examples are studied when no local ordering can be defined. Copyright @ [FAC 2001 Keywords: dynamical polysystems, causal structures, optimal control, nonholonomic geometry

1. SMOOTH MANIFOLDS OF KINEMATIC

1.

TYPE Differential systems (distributions) on smooth manifolds are standard objects of the theory of optimal control. Nonholonomic calculus of variations is used in thermodynamics, quantum theory, mechanics and other fields . Causal structure (Krym , 1999) is one of the most fundamental physical structures. In this paper we discuss causal relations on manifolds with some fixed differential system.

U

UEI

U=M,

2. VU E I Vx E U the restriction P(U)x : TxU L is a non-degenerate linear mapping,

-+

3. VU, V E I Vx E U n V the compose mapping P(U)x 0 p(V);1 : L -+ L preserves order in L.

Then M with the family P is called manifold of kinematic type L . The family P is called kinematic atlas on M. In each tangent space T€ M of the manifold of kinematic type L a cone p(U)~I(Q) is defined . Due to (3) this cone does not depend on the choice of the domain U containing~ , thus notation U will not be used. Define Q€ := P€-I(Q) as the cone of future for the origin in T€M, _Q~ = P€-l( -Q) as the cone of past.

In 1967 H. Busemann (Busemann, 1967) and RI. Pimenov (Pimenov, 1968; Pimenov, 1991) expanded causal relations for topological spaces. Causal structure of the general relativity theory was preserved as the most significant example of partial ordering. In a linear space L order < is defined by an open convex cone Q C L . Local ordering of this type can be defined on a smooth manifold. Definition 1. Let M be a smooth manifold, (L , <) be a linear kinematic of the same dimension. Let families of connected open sets I and continuous mappings (P(U»UEI be defined on M such that for any domain U E I P(U) : TU -+ L and

Example 1. Let M be a parallelizable manifold of dimension n , Xl, ... , Xn be continuous vector fields linearly independent in any point of M . Let L be a linear kinematic of the same dimension, YI , . . . , Yn be a basis in L. The mapping P : n

TM

-+

L such that P(Y) =

L k=l

215

ykyk. where

y

=

n

_

~ yk Xk, is defined for the entire tangent

Ak

k=l

3

axi

= LAi8k" + Y

i=O

bundle TM . Triplet (M, P, L) is a manifold of kinematic type.

ax 4 fl..k' k

= 0, ... ,3

vy

Hence it is the 4-potential of the electromagnetic field. Physical dimension of the fifth coordinate x4 is V·m.

Let (M, P) be a manifold of kinematic type L. Time orientation of the manifold M is a continuous vector field X on M such that V~ E M X(~) E Qf.. In particular, X does not vanish anywhere. Time orientation exists on any manifold of kinematic type.

The distribution A can be also defined by the four basis vector fields

a

a

ek = ax k - Ak ax 4 '

k

= 0, . .. ,3

Note that [ei,ejJ = -Fij , where Fi} = ~ is the tensor of the electromagnetic field.

It is well known that the collapse of black holes leads to singularities of space-time (Hawking and Ellis, 1973; Kronheimer and Penrose, 1967) (a point is deleted from a domain with compact closure). This classical physics theorem follows also from the existence of the orientation of time.

- ~:;

An inner product L(x, u) = (u, u}x with the Lorentz signature can be defined for horizontal vectors u E TxM5 . The Euler - Lagrange equations of geodesics maximizing the Lorentz length functional are

Pseudo-Riemannian manifolds of the signature (+, -, .. . , -) are called Lorentz manifolds. Causal structure on a 4-dimensional Lorentz manifold is exactly causal structure of the general relativity. Lorentz manifold with the time orientation is called space-time. Time orientation is a continuous timelike vector field X, i.e. (X, X) > 0 on M . Lorentz manifold with the time orientation is a manifold of kinematic type. The cone of future for each tangent space is defined by Qf. = {u E Tf.M J (u, u}f. > 0, (u, X(~)}f. > O}. In a normal ball neighbourhood U of the point ~ points ~ and x E U can be connected by a geodesic. For P(U)x choose parallel transfer from x to ~ along this geodesic.

doL aL ) aom ( dt auk - ax k

3

+ P4 L

.

Fjku}

=0

j=O

Assuming ao i: 0 we obtain equations of regular geodesics which are identical with the physical equations of motion of a charged particle. Some extremals of the length functional correspond to ao = O. The equations of these abnormal geodesics are 3

LFjku j=O

j

= 0,

k

= 0, ... ,3

°

If det F i: 0 then u = and this is the constant path in a fixed point. If det F = 0, abnormal geodesics can connect distant points on a manifold.

Smooth manifolds with inner product of signature (+, -, ... , -,0, . .. ,0) and time orientation can be another example of the manifolds of kinematic type. Yet local ordering, i.e. causal relation, may not exist. The reason of the phenomenon is nontriviality of the following distribution: If. = {v E Tf.MJVu E Tf.M (u,v) = O}, If. C CIQf.. If this distribution is holonomic and Cl-smooth then local ordering exists on the manifold with this signature. A distribution is holonomic ifI there is a foliation (Tamura, 1979) such that the field of tangent planes of this foliation coincides with the distribution.

2. LOCAL ORDERlNG

Definition 2. Let (M, P) be a manifold of kinematic type L, -y : [tl' t2J --+ M be a piecewise smooth path. Path -y is oriented in future ifI Vt E [tl' t2J -y'(t) E Q-y(t). If -y is not differentiable at t both left and right derivatives at t have to satisfy this condition. A path -y is oriented in past ifIVt E [tl' t2J -y'(t) E -Q-y(t). A path -y is timelike ifI it is oriented in future or in past.

Optimal control theory gives an interesting example of a nonholonomic variational problem. Let U be a coordinate neighbourhood on M 5 , (x O, ... , x 4 ) : U --+ ]R5 be an appropriate chart. Let A be a 4-dimensional distribution defined by the normal covector field n = Aodxo + Aldx l + A2dx 2 + A dx 3 + dx 4

Let (M, P) be a manifold of kinematic type L. A point ~ E M is called a point of type 1 ifI there is its neighbourhood U such that any timelike curve in U has no self-intersection points. All other points of the manifold M are called points of type 11. So ~ E M is a point of type H ifI in any neighbourhood of ~ there is a closed timelike path. Local ordering can be defined near any point of type I. Here we discuss the question when a point ~ E M is a point of type I or H. Since it is a local question M is considered to be a neighbourhood of ~ in Rn. Assume also that there is a ~\iemannian inner product in L.

3

To ensure invariance the smooth structure on M 5 should be restricted (this approach is close to the definition of a fibre bundle). Let for each coordinate transformation x >-> y the Jacobi ax ' ·t = 0, ... , 3, ax" . satls . fy ay" matrIX = 0, ay" = ±l. Then for the new coordinate system

216

The future cone Q S;; L is an open convex cone in L. Let Lo be the maximal linear subspace contained in Cl Q and L~ - subspace orthogonal to Lo. Let al, ... ,ak be a basis of Lo . Using the Minkowsky sum we obtain

F(y )

Define a polysystem D = {±ft, ±/2}. The origin 0 is the target point. It may be shown the polysystem D is L-locally controllable but it is not N-locally controllable one. Let us consider causal structures on the manifolds of kinematic type. The following lemma can be used to prove Theorems 1,2.

where Q* C L~ is a closed convex cone which does not contain straight lines. There is a hyperplane Ho C L~ such that Ho n Q* = {O}. Let 0' E L~ be a normal to Ho lying in half-space containing Q*.

Lemma. Let P : TU ..... L be an element of the kinematic atlas, Q~ := P€-l(Q). Then for any vector q E Q~ there is a neighbourhood V of the point ( and an open convex cone K* C Q~ such that q E K* and II;l(K*) C Qx for all x E V. Here IIx : TxU ..... T(,U is the parallel transfer from the point x to (.

Consider a hyperplane H C L defined by the normal vector 0'. It is obvious that H is a support hyperplane to Cl Q. Also clear that Lo C Hand HnQ* = {O}. Here we remind some definitions from the theory of dynamical polysystems. A polysystem in the domain U is a finite set of continuous vector fields in U. We write f E C* iff f E C and the Cauchy problem x(t) = f(x(t»), x(o) = ( has the unique solution for any point ( E U. Any smooth vector field belongs to the class C·. The set of all vector fields in U belonging to C· is not a linear space.

3. POINTS OF TYPE 11 Let us return to the question of classification of points on the manifold. Note that for dim Lo = 0 any point of the manifold M is a point of type I.

Theorem 1. Let al,' .. , ak be a basis of L o, vector fields hex) := Px-l(ai), i = 1, ... , k, of the class C* are defined on the domain U CM , a point ( E U and the polysystem D := {±fi i = 1, ... , k} is L-locally controllable. Then ( is a point of type II.

I

A trajectory of the polysystem D is a continuous mapping x : [0, T] --t U such that for some partition = to < tl < .. . < tm = T of the segment [0, T] the restriction of x(·) for any subsegment [ti' ti+l], i = 0, ... , m-I, is an integral curve of a member of D.

°

The condition fi E C* in this theorem is essential. Example 3. Let L = ]R2 and the cone Q is an open half-plane. Let the normal to the hyperplane defining this half-plane be the vector (0,1). Assume that this vector is directed inside Q. For the manifold M = ]R2 with the chart (x, y) choose a kinematic atlas with a single map P : T]R2 ..... L -l f(x, y») h 1 ' were e fi ned by P(x,y):= f(x,y)

A polysystem D is L-locally controllable at the point ( (Haynes and Hermes, 1970) iff for any neighbourhood V C U of ( there is a neighbourhood W of ( such that for any point yEW there exists a trajectory x : [0, T] --t V of the polysystem D such that x(o) = y, x(T) = (.

d

°

L-local controllability is used in the theorem l. Well studied N-local controllability is a sufficient condition for L-Iocal controllability. Complete s
=

(1 -

f(x, y) := 3y2/3. Note that detp-l = 1 + f2 > O. The map P defines a field of cones Q(x,y) = p(-l )(Q). Vectors ±(1,0) E CIQ are mapped by x,y p-l to vector fields ±(1, f) ~ C* on M, and the normal vector (0,1) E L is mapped to the vector field (- f, 1). Let D be a polysystem of vector fields ±(1, f) and consider the time minimization problem for D with the target point ( = (0,0). The Bellman function e : ]R2 --t ]R~ of this polysystem, Le. the minimum time of transition from the point (x, y) to the origin, is e(x, y) = lyl/31 + Ix - yl/31· Therefore this polysystem is N-locally controllable in the origin. Yet any path oriented in future and beginning in each point of any curve y = (x - C)3 intersects this curve "from bottom to top" . Hence any point of the plane is a point of type I.

A polysystem D is N -locally controllable at a point ( (Petrov, 1968) iff for any T > there is a neighbourhood V C U of ~ such that for any point y E V there exists a trajectory x : [0, T] --t V with T < T and x(o) = y, x(T) = (.

]R2

y~0

(=

ClQ=Lo+Q*

Example 2. Let us consider the plane chart (x , y) and two vector fields It and /2 = (y, -x), where

0,

= { 3y 2/3, Y ~ 0

Here is a simple example of a smooth N-locally controllable polysystem generated by a kinematic atlas. Example 4. Let L = lR 3 and cone Q be the

with the (O,F(y»

open half-plane defined by the normal vector

217

(1,1 , -2) directed inside Q. Let M be a half-space { (x , y, z) Ix > o} with a kinematic atlas of a single map P : T M

-+

is a chart y : W -+ !Rn such that y-image of each integral submanifold of this distribution is {y E !Rn IYk+1 = Ck+I, "" Yn = en} where k = dimL o. Consider a hyperplane dey( Se) in IRn. This hyper-

L defined by

p-l (x ,y,z) -_

(~~ ~1) 2

00 x

n

plane is defined by the equation

The plane Lo is the linear envelope of vectors Q := (2,0,1) and (3 := (1,1 ,1). The map P defines the field of cones Q(x,y,z) := p(~~y .z )(Q) and vectors Q and {3 E Cl Q are mapped by p-I to h := (1 , 0, x 2 ) and h := (0, l,x2) . Since the Lie bracket [h,h](x,y,z) = (0,0,2x) and vectors h(x,y, z) , h(x, y, z) are linearly independent in each point of M then the polysystem D := {h , - h, h - h} is N-locally controllable in each point ~ E M.

L

AiYi

= 0.

i=k+1 n

Denote Ld := {y E !Rn

I L

AiYi = d} where i=k+1 d E IR. The sets y- I (Ld) are leafs of some foliation in W of codimension 1. Let Sx be a holonomic distribution with these leafs as integral submanifolds. Obvious that Px-1(L o ) C Sx for all x E W .

Let Qi be the set of unit vectors contained in Q* . Define the field of hyperplanes Hx := Px(Sx) in L for x E W. Since He n Q* = {O} then He is a support hyperplane to Cl Qi and due to continuity H x is a support hyperplane to Cl Qi for all x E U, U c W being some neighbourhood of~ . Therefore the hyperplane Sx is support to Cl Px( Q) for all x E U.

Theorem 2. Let aI, . .., ak be a basis of the subspace Lo and D := {±fi J;(x) = Px-I(ai), i = 1, ... , k} be a Coo -smooth po/ysystem in the neighbourhood U of the point~ . Let 2l(D) be a minimal Lie algebra containing D and 2lx (D) := {j(x) E Tx U I f E 2l(D)} . If21~ ( D) n pt(Q) # 0 then ~ is a point of type II.

I

So, in a sufficiently small neighbourhood of the point ~ integral submanifolds of Sx are intersected by the curves oriented in future in the same direction . Therefore no one of these curves can intersect itself and hence ~ is a point of type I. 0

Proof. Let W cUbe a neighbourhood of the point ~ . There is a vector field g E 2l(D) such that -g(~) E Qe. The vector field g is the result of several bracket operations with vector fields ±h , ... , ±fk. Due to Lemma above there are an open convex cone K* C Qe and a convex neighbourhood V C W of ~ such that -g(~) E K* and ll;I(K*) C Qx for all x E V. Let x : [0, T] -+ V, x( 0) = ~, be an integral path of the vector field g , then llx(x(t») E -K* for all t E (0, T] . The path x(t) can be approximated by a trajectory y : [0, T] -+ V of the polysystem D ending in - K*. This trajectory can be approximated by a curve z : [0, T] -+ V oriented in future and ending in _Ko . The point ~ is reachable from z(T) along a segment of a straight line oriented in future and satisfying the differential inclusion :i; E ll; I (K*). This segment lies in the domain V because V is a convex neighbourhood of the point ( . Hence there is a closed timelike path in W. Since W C V is an arbitrary neighbourhood of ( , the point ( is a point of type n. 0

If the distribution P;l(Lo) is smooth then for dim Lo = 1 any point ~ is a point of type 1. If this distribution is not smooth, it can be nonholonomic even for dimension 1 and the problem becomes more complicated. REFERENCES Busemann, H. (1967) Timelike spaces. Rozprawy Matematyczne, (53), 3-50. Hawking, S. and C. Ellis (1973) . The Large Scale Structure of Space-Time. Cambridge. Haynes, C .W. and H. Hermes (1970). Nonlinear controllability via Lie theory. SIAM J. Control, 8, 450-460. Kronheimer, E.H. and R. Penrose (1967). On the structure of causal spaces. Proc. Cambridge Phil. Soc ., 63, 481-501. Krym, V.R. (1999). Smooth manifolds of the kinematic type. Teor. Matem. Fisika, 119 , 264-281. (Russian) Petrov, N.N. (1968). Local controllability of autonomic systems. Diff Uravneniya, 4, 12181232. (Russian) Petrov, N.N. (1969) . A solution of a problem of the control theory. Diff. Uravneniya, 5, 962963. (Russian) Pimenov, R.I. (1968) . Spaces of kinematic type. Zap. Nauchn . Sem. LOMI, 6, 3-496. (Russian) Pimenov, R.L (1991). Foundations of the '..:'heory of Temporal Universe. Syktyvkar. (Russian) Tamura, I. (1979) The Topology of Foliations. Moscow. (Russian)

4. POINTS OF TYPE I Now prove a sufficient condition of local ordering ,n ear the point ( .

Theorem 3. If the distribution Px-I(L o ) is smooth and holonomic in some neighbourhood of Proof. In some neighbourhood U of the point ~ there is a holonomic distribution S of co dimension 1 such that Px-I(L o ) C Sx and Sx is a support hyperplane to the cone P;I(Q) in TxM for all 'x E U. Let Se := Pe-I(H) where H is a support hyperplane in L to Cl Q = Lo + Q* such that Lo C Hand H n Q* = {O} (these objects were defined above) . Since the distribution Px-I(L o ) is holonomic , in some neighbourhood W of ~ there is a chart y : W -+ lRn such that y-image of

218