Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics

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296 12 Accessible Points and Sub-Manifolds of Mechanical Systems. Controllability

Bv =





pΓ a(s, S(v(s)+C

S(v(s)+C

))ds



([0,t

],T

≤ δ(Kt

+ Ct

−1

)

= Kt

Hence the completely continuous operator B sends U

into itself and hence,

by the classical Schauder principle, it has a ﬁxed point u

∗

∈ U

.Usingthe

same argument as in the proof of Theorem 12.19, one can easily prove that

m(t)=S(u

∗

)(t) is a solution of (7.46) with m(0) = m

and m(t

)=m



Consider a bounded Hausdorﬀ continuous force ﬁeld a(t, m, X) with con-

vex closed values on M , as above. Following Deﬁnition 11.50 we say that a

trajectory m(t) of the mechanical system with force a(t, m, X)isgoverned by

extreme values of controlling force if a.e. ˙m(t) belongs to Ext a(t, m(t), ˙m(t))

(see Deﬁnition 11.48), i.e. (11.18) is satisﬁed.

Theorem 12.21 Assume that a is Hausdorﬀ continuous and has convex

closed values, and let m

be non-conjugate to m

along at least one geodesic

g(·) joining them.

(i) If a has lower than quadratic growth in velocity, then there exists a

positive number L(m

,g) such that if 0 <t

<L(m

,g), there

exists a trajectory, governed by extreme values of controlling force, for

which m(0) = m

and m(t

)=m

(ii) If a has a quadratic bound and a(t, m) satisﬁes the hypothesis of The-

orem 12.16, then the conclusion of (i) above holds.

(iii) If a(t, m, X) <a(t, m)X and a(t, m) satisﬁes the hypothesis of

Theorem 12.20, the conclusion of (i) above holds for every t

> 0.

Recall that the trajectories governed by extreme values of controlling force

are described by the inclusion (11.18) and, in the case under consideration,

the right-hand side of (11.18) is lower semicontinuous (see Lemma 11.49).

Thus the Theorem follows from Theorems 12.14, 12.16 and 12.20.

Remark 12.22. We refer the reader to [241, Theorems 3.3 and 3.4] where the

following generalization of Theorem 12.16 for right-hand sides with greater

than quadratic growth in velocity is obtained. Let a(t, m, X) either satisfy

the upper Carath´eodory condition and have convex closed bounded values

or be almost lower semicontinuous and have closed bounded values. Let m

and m

be non-conjugate along some geodesic of the Levi-Civit´a connection.

Then ε and C, deﬁned as above, and the compact set Ξ from Lemma 3.51

are well-deﬁned. Suppose that for t ∈ [0,t

], for some t

> 0, and for m ∈ Ξ

the estimate a(t, m, X) <f( X) holds where f :[0, ∞) → [0, ∞)isa

function increasing on [0, ∞]. If f (εt

−1

+ Ct

−1

) ≤ εt

−2

, then there exists a

solution m(t)of(11.17) for which m(0) = m

and m(t

)=m

. The method

of proof is a modiﬁcation of the one given above in this Section.

12.4 Generalizations to Systems with Constraints 297

12.4 Generalizations to Systems with Constraints

In this section, we show how to generalize the existence theorems of Sec-

tion 12.3 to systems with constraints (see Section 11.6). In the framework

of mechanics with constraints, it is more natural to consider the question of

whether or not a submanifold transversal to the union of the least constrained

geodesics leaving a speciﬁed point is accessible from that point. The author

is grateful to Boris D. Gel’man for pointing out this problem.

The main technical trick here is the replacement of the operators S and

Γ by their constraint analogs S

and Γ

introduced in Section 11.8.2.

Let M be a complete Riemannian manifold equipped with a constraint

β.Letm

∈ M. The exponential map exp

: β

→ M can be deﬁned in

the same manner as that for a manifold without constraint. Explicitly, for

X ∈ β

, we set exp

(X)=γ

(1), where γ

(t) is the least constrained

geodesic with γ

(0) = m

and ˙γ

(0) = X. It is clear that exp

is a C

∞

smooth map.

Deﬁnition 12.23. A point m

∈ exp

(β

)isnot conjugate to m

along

the geodesic γ

(where γ

(1) = m

) if the diﬀerential d exp

has maximum

rank at X ∈ β

In particular, this means that the image of exp

is a smooth submanifold

in a neighborhood of m

if m

is not conjugate to m

. Moreover, exp

is a

diﬀeomorphism of a neighborhood of X ∈ β

onto a neighborhood of m

the submanifold.

Assume that m

is not conjugate to m

along a least constrained geodesic

.LetN ⊂ M , m

∈ N, be a submanifold which is transversal to the image

of exp

. (In other words, the sum of the spaces T

N and T

exp

(β

)

coincides with T

M.) An example of such a manifold is an open neighbor-

hood of m

in M.

Theorem 12.24 Under the above-mentioned hypothesis for any K>0 there

exists a constant

L(m

,N,K,γ

) > 0 such that for 0 <t

L(m

,N,K,γ

)

and for any continuous curve u(t) ∈ U

⊂ C

([0,t

],β

), there exists a

vector C

∈ β

satisfying the condition S

(u + C

)(t

) ∈ N. Furthermore,

is unique in a neighborhood of t

−1

X ∈ β

and is continuous in u.

The proof is quite similar to that of Theorem 3.47. The only extra argu-

ment needed is that the manifold N stays transversal to a C

-small pertur-

bation of the image of exp

= S

(·)(1). Below, in this section, we use ˆε and

C in analogy with ε and C from Lemma 3.48.

Let a be a set-valued vector ﬁeld on M and P

: T

M → β

be the

ﬁeld of orthogonal projections. Consider the following set-valued analog of

the constrained Newton law (11.13)

˙m(t) ∈ Pa (t, m(t), ˙m(t)) , (12.6)

298 12 Accessible Points and Sub-Manifolds of Mechanical Systems. Controllability

where the constraint covariant derivative

is deﬁned in Section 11.6.The

inclusion (12.6) arises in constraint analogs of the problems considered in

Sections 11.7 and 12.3, for example, as a discontinuous force acting on the

system, or where the image of P a is formed by all possible values of the control

force. It is easy to see that if a has convex values, the sets Pa(t, m, X) are also

convex and the set-valued vector ﬁeld P a is upper (lower) semicontinuous if

a is upper (lower) semicontinuous, respectively.

Since the norm of the orthogonal projector P equals 1, by replacing S

and Γ by S

and Γ

, respectively, by using the space C

(I,β

) in place

of C

(I,T

M) and by using Theorem 12.24 rather than Theorem 3.47,one

can easily prove the following analogs of the non-constrained theorems from

Section 12.3.

Theorem 12.25 Let a(t, m, X) either satisfy the upper Carath´eodory condi-

tion and have convex closed bounded values or be almost lower semicontinuous

and have closed bounded values. Let the points m

and m

be non-conjugate

along some least constraint geodesic g.

(i) If a has less than quadratic growth in X, for any submanifold N  m

transversal to the image of exp

, there exists a positive number

L(m

,N,g) such that if 0 <t

<L(m

,N,g), there exists an ad-

missible solution m(t) of (12.6) for which m(0) = m

and m(t

) ∈ N.

(ii) Suppose that a has a quadratic bound in X and in addition, for t ∈

[0,l]=I and m ∈ Ξ,where[0,l]=I is some interval and Ξ is the

compact set from Lemma 3.51, for the function a(t, m) from Deﬁnition

12.8 there exists a real number δ such that the estimate a(t, m) <δ<

¯ε

(¯ε+

holds. Then for any submanifold N  m

transversal to to the

image of exp

, there exists a positive number L(m

,N,g) such that

if 0 <t

<L(m

,N,g), there exists an admissible solution m(t) of

(12.6) for which m(0) = m

and m(t

) ∈ N.

(iii) Suppose that a satisﬁes the estimate (12.5) with a continuous function

a(t, m) > 0, that the estimate a(t, m) <δholds on I × Ξ,where

the compact set Ξ is as in Lemma 3.51 and that δ>0 satisﬁes the

inequality δ<

¯ε

(¯ε+

. Then for any submanifold N  m

transversal

to the image of exp

, and for any t

> 0, t

∈ I, there exists an

admissible solution m(t) of the inclusion (12.6) for which m(0) = m

and m(t

) ∈ N.

Remark 12.26. In [241, Theorems 3.7 and 3.8] a generalization of Theo-

rem 12.25 is obtained which is analogous to that of Theorem 12.16, mentioned

in Remark 12.22.

Chapter 13

Some Problems on Lorentz Manifolds

13.1 Introduction to Relativity Theory

In this Section we give a brief introduction to some core notions of relativity

theory. This material suﬃces to understand the language of relativity and to

describe the relativistic problems discussed below. We are mainly interested

in the constructions of general relativity, the formulae of special relativity

arising as consequences of the latter. Since the exposition is intended for

mathematicians, we present it axiomatically, starting from a basic set of pos-

tulates. Such an approach allows us to focus on the mathematical background

of general relativity and on the physical interpretation of the mathematical

developments. We do not touch on the physical background, however it is

used to motivate the above-mentioned postulates. We refer the reader to

[182, 200] for details.

13.1.1 Space-times

Recall that in Deﬁnition 1.50 we introduced the notion of a semi-Riemannian

metric on a manifold M as a family of symmetric non-degenerate but not

necessarily positive deﬁnite bilinear forms ·, ·

on the tangent spaces T

i.e., it is a symmetric non-degenerate (0, 2)-tensor ﬁeld that, like its inverse,

is called a metric tensor (see Remark 1.52). Recall also that in the deﬁnition

of physical equivalence (see Sections 1.4 and 1.5), and in the construction of

the Levi-Civit´a connection, we used only the non-degeneracy of the metric

tensor and did not use its positive deﬁniteness. Thus for all semi-Riemannian

metrics the notion of physical equivalence and the construction of the Levi-

Civit´a connection are well-deﬁned. In particular, this means that the notions

of covariant derivative, parallel translation and geodesic are well-deﬁned on

Y.E. Gliklikh, Global and Stochastic Analysis with Applications

to Mathematical Physics, Theoretical and Mathematical Physics,

DOI 10.1007/978-0-85729-163-9

13, © Springer-Verlag London Limited 2011

299

300 13 Some Problems on Lorentz Manifolds

semi-Riemannian manifolds, and their properties are analogous to those on

Riemannian manifolds.

Among the semi-Riemannian metrics, some play a special role in the math-

ematics of general relativity. Recall that for any non-degenerate symmetric

bilinear form on a vector space there exists a basis (an orthonormal basis)

such that the matrix (g

) of the metric tensor with respect to this basis is

a diagonal matrix with diagonal entries equal to +1 or −1. For positive def-

inite forms, of course, the diagonal entries are all equal to +1. For general

non-degenerate forms the row of signs + and − corresponding to the values

+1 and −1 appearing on the diagonal is called the signature of the form.

Deﬁnition 13.1. A semi-Riemannian metric is said to be a Lorentz metric

if its signature is either (− + ···+) or (+ −···−), i.e., the above-mentioned

diagonal form of (g

) includes only one −1 and all other elements of the

diagonal are +1 (or only one diagonal element is +1 and all other elements

are −1). A manifold on which a Lorentz metric is speciﬁed is called a Lorentz

manifold.

Notice that the case where only one diagonal element is equal to −1 (and all

others +1) can be transformed into the other case simply by multiplication by

−1, and so both cases are equivalent. For a Riemannian metric, multiplication

by −1 results in a negative deﬁnite (and hence, not Riemannian) metric, but

for a semi-Riemannian metric such multiplication does not lead us out of the

class of semi-Riemannian metrics. For the sake of simplicity we choose one of

the above equivalent cases, namely the case with signature (− + ···+), i.e.,

where the diagonal form has only one −1 (and all others +1). The other case

leads to the same theory.

In contemporary physics the notions of space and time, considered sepa-

rately in classical physics, are united into a common continuum called space-

time.

Postulate 1 The physical space-time of our universe is described mathemat-

ically as a 4-dimensional Lorentz manifold.

Among all 4-dimensional Lorentz manifolds, those whose Levi-Civit´acon-

nection satisﬁes the so-called Einstein equation (derived by Einstein and

Hilbert) expressing the connection via the distribution of matter in the uni-

verse, are called space-times. The Einstein equation is a complicated partial

diﬀerential equation. We discuss it brieﬂy in Section 13.1.6 below. In common

with many other PDEs, the Einstein equation has many diﬀerent solutions

depending on the initial data, boundary conditions and other constraints.

Such solutions describe the metric in diﬀerent cases: locally (e.g., in a neigh-

borhood of a certain star), globally, in special cases when certain negligible

inﬂuences can be omitted, and so on.

Notation 13.2 Everywhere in this section, by M

we denote a space-time

under consideration, i.e., a 4-dimensional Lorentz manifold whose metric sat-

isﬁes the Einstein equation.

13.1 Introduction to Relativity Theory 301

In all Sections dealing with general relativity below we use the Levi-Civit´a

connection of a Lorentz metric to determine the covariant derivative, parallel

translation, geodesics, etc.

In a space-time we shall denote local coordinates by the symbols q

, q

and q

. The index 0 denotes the direction corresponding to the −1entry

in the signature (i.e., vectors tangent to this axis have negative squares)

while the others correspond to +1 and so their tangent vectors have positive

squares.

Notation 13.3 In order to avoid confusion we shall use Greek letters for

indices which take values from 0 to 3 and Latin indices for indices which

range from 1 to 3.

Thus, g

αβ

indicates that all coeﬃcients of the Lorentz metric are under

consideration while g

indicates that only those in the so-called “space di-

rections” are under consideration (i.e., i, j =1, 2, 3; here the word “space”

has no physical meaning).

Let us present three examples of space-times. At present the global topo-

logical structure (i.e., the actual view) of physical space-time is not known.

We can deal only with a neighborhood of our galaxy that resembles a part

of a vector space (a chart). In the examples below we obtain various 4-

dimensional manifolds that can be considered as models of physical reality.

We are mainly interested in the metrics on space-times. These metrics will

(according to the general notation of Section 1.5 and of this section) be de-

noted by g

αβ

⊗ dq

Example 13.4. Minkowski space. M

is the standard vector space R

and

the coeﬃcients of the Lorentz metric are of the form g

= −1, g

=1for

i =1, 2, 3 and all others are zero. i.e., ·, ·

= −dq

⊗ dq



i=1

⊗ dq

This is the simplest Lorentz manifold playing the same role as the Euclidean

space among Riemannian manifolds. It corresponds to the case when the

gravitational inﬂuence is so small that it can be omitted and so it is used

in special relativity where only the electromagnetic ﬁeld is under considera-

tion. Notice that for any Lorentz manifold M any tangent space T

M has

the structure of Minkowski space. This is an analog of the fact that on a

Riemannian manifold every tangent space is equipped with the structure of

Euclidean space.

Example 13.5. Einstein–de Sitter space-time. M

is the “upper” half-

space of R

, i.e., M

= R

= {(q

) ∈ R

> 0}. The coeﬃcients

of the Lorentz metric are as follows: g

= −1, g

=(q

)

, i =1, 2, 3,

and all others are zero, i.e., ·, ·

= −dq

⊗ dq

+(q

)



i=1

⊗ dq

.The

Einstein–de Sitter space-time is an example of a so-called Friedman universe.

The level surfaces q

= const have the structure of Euclidean spaces with an

302 13 Some Problems on Lorentz Manifolds

inner product (q

)



i=1

⊗dq

depending on the value of q

(which can be

considered as an analog of the time variable).

Specify a value q

and consider two points x

=(q

)andx

) on its level surface. The Euclidean distance between x

and x

)



− q

)

+(q

− q

)

+(q

− q

)

Consider another value q

∗

. On its level surface the distance between

“the same” points x

∗

=(q

∗

)andx

∗

=(q

∗

)isequalto

∗

)



− q

)

+(q

− q

)

+(q

− q

)

Since q

∗

, the latter distance is greater than the former. This is a model

for the so-called redshift, the experimental fact that all distances are increas-

ing in time. Below in Remark 13.13 it will be shown that along the lines

= c

where c

are constants, the variable q

has

the physical interpretation of time. Notice that the points do not move in

the level surfaces and that the growth of distance is a consequence of the

variation of the metric.

By contemporary physical theory our Universe was born 15–20 billion

years ago from a single point in an event called the Big Bang. Observe that

all distances in level surfaces in Einstein-de Sitter space-time tend to zero as

→ 0. This is a model of the Big Bang. In this model the Universe (i.e., the

level surface) does not shrink into a single point as q

→ 0 but the metric

becomes degenerate.

The Einstein-de Sitter space-time describes our Universe at an early stage

after the Big Bang. There are other models, with a diﬀerent metric, for the

Universe at the present time (Robertson-Walker space-times).

Example 13.6. Schwarzschild space-time. Let μ>0. Consider two 2-

dimensional manifolds: S

, the 2-dimensional sphere, and the manifold A

{(r, t) ∈ R

| r =2μ}.LetM

= A

× S

. Since at any m =(a, s) ∈ M

the tangent space T

= T

⊕ T

,wherea ∈ A

and s ∈ S

,wecan

determine the following Lorentzian metric on M

: ·, ·

Sch

(a,s)

= −(1−

2μ

r(a)

)dt⊗

dt+(1−

2μ

r(a)

)dr ⊗dr+·, ·

where r(a) is the value of coordinate r at a ∈ A

and ·, ·

is the ﬁrst fundamental form of S

at s.

Schwarzschild space-time is M

with the metric ·, ·

Sch

. It describes phys-

ical space-time in a neighborhood of a black hole with mass 8πμ. Notice that

is divided into two components: the points with r(a) > 2μ are said to

be “outside the black hole”, and the points with r(a) < 2μ are said to be

“inside the black hole”. We point out that t plays the role of q

outside the

black hole (i.e., vectors in its direction have negative square) and r plays the

role of q

inside the black hole (for some details, see Remark 13.13 below). In

13.1 Introduction to Relativity Theory 303

the popular scientiﬁc literature this fact is usually expressed by the words:

inside a black hole, space and time exchange places.

13.1.2 World lines. The light cone. Proper time

As a space-time unites space and time into a common continuum, we need to

change the notation usually found in the old physics. For example, in the old

physics a ‘point’ typically means an element of space, a ‘trajectory’ means a

line describing how the position in space depends on time, etc.

Points of a space-time are called events. The interpretation is “something

held at a certain point of space at a certain moment of time”. Instead of the

word “trajectory” we shall use the term “world line”. This is a certain smooth

curve in the space-time that is interpreted as a 1-dimensional continuum

of events. We do not specify a parameter in the world line from the very

beginning, this will be done in a special way a little later. Thus a world line

is a certain 1-dimensional manifold embedded (or immersed) into M

In a tangent space T

at any event m to the space-time M

one can

ﬁnd non-zero vectors whose squares are negative, positive and equal to zero.

Deﬁnition 13.7. The set of vectors in T

whose square is equal to zero

is called the light cone. Vectors lying on the light cone are also called isotropic

or light-like.

A vector X ∈ T

such that X

< 0 is called time-like.Thesetof

time-like vectors is called the interior of the light cone. It is also said that a

time-like vector lies inside the light cone.

A vector X ∈ T

such that X

> 0 is called space-like or is said to lie

outside the light cone.

The interior of the light cone has two non-intersecting components while

the light cone itself, and the set of space-like vectors, are connected. The

physical interpretation of the two components of the interior of the light

cone is that one of them is directed into the future and the other one into the

past. Of course, in order to obtain a consistent theory these directions must

be coordinated at diﬀerent points of space-time in a similar way to the notion

of an orientation on a manifold (see Section 1.6). We shall not consider here

the general construction of such orientation of directions, instead deﬁning

only a certain special case of it that is, in any case, general enough for our

purposes.

Deﬁnition 13.8. A space-time M is called time-oriented if there exists a

smooth vector ﬁeld X on M such that X

< 0 at all events in M. The part

of the light cone in the tangent space at any event that contains X is said to

be “directed into the future”.

Everywhere below we consider time-oriented space-times.

304 13 Some Problems on Lorentz Manifolds

Deﬁnition 13.9. A world line for which all tangent vectors are time-like is

called a time-like world line; a world line for which all tangent vectors are

light-like is called a light-like or isotropic world line; a world line for which

all tangent vectors are space-like is called a space-like world line.

Postulate 2 Only time-like and light-like (isotropic) world lines have a sen-

sible physical interpretation. The time-like world lines describe the “life” of

objects moving slower than the speed of light while light-like world lines de-

scribe the “life” of objects traveling at light speed.

Postulate 2 may be considered as an experimental fact. One may consider

also space-like world lines whose tangent vectors are space-like, but they play

an auxiliary role since they describe the motion of objects moving faster than

light, which is forbidden in the present theory. From Postulate 2 it follows

in particular that the type of world line of a material object cannot change.

The physical interpretation of this fact will be clariﬁed later.

Postulate 3 Along any world line, corresponding to a material object, there

is a vector ﬁeld P , called the 4-momentum, that is tangent to the world line,

directed into the future and which has constant square.

For light-like world lines the above constant square is evidently zero while

for time-like ones it is a negative number (if it were zero, P would be a zero

vector, i.e. directionless, contradicting the fact that it should be directed into

the future).

Deﬁnition 13.10. The positive real number m such that m

= −P

called the mass of the object corresponding to the time-like world line under

consideration.

Thus along any time-like or light-like world line m(·) a unique (up to an

additive constant) parameter η can be introduced such that

dη

m(η)=P .

Postulate 4 If there is no inﬂuence of forces other than gravity, the time-

like (or light-like) world line parametrized by η is a geodesic.

This postulate means that the equation of the above world line is

dη

m(η) = 0 (13.1)

where

dη

is the covariant derivative of the Levi-Civit´a connection, and that

dη

m(η)=P is parallel along m(η).

Along a time-like world line we can construct another important vector

ﬁeld.

Deﬁnition 13.11. The vector ﬁeld V which, at any event of a time-like world

line m(·), is tangent to m(·), directed into the future and has square equal

to −1, is called the 4-velocity of the world line m(·).

13.1 Introduction to Relativity Theory 305

V exists along any time-like world line, i.e., its existence need not be

postulated, unlike the existence of P . Immediately from the deﬁnition it

follows that P = mV .

As above, in the case of P there exists a unique (up to an additive constant)

parameter τ along a time-like world line m(·) such that

dτ

m(τ)=V .

Deﬁnition 13.12. The above parameter τ is called the proper time of the

time-like world line.

The physical interpretation of proper time is that it is the time that is

shown by the watch of an observer whose world line is under consideration.

Every observer has its own proper time, i.e., we all live according to our own

time.

One can easily show that a time-like world line that is geodesic with respect

to the parameter η described above (i.e., such that no force besides gravitation

has an inﬂuence on it) also satisﬁes the geodesic equation

dτ

V =0. (13.2)

Remark 13.13. In Examples 13.4 and 13.5, along all lines of the form q

= c

where c

are constants, the variable q

is a proper

time. Indeed, such a curve is described in coordinates as (q

)sothat

its derivative in q

takes the form (1, 0, 0, 0). Substituting this vector into the

metrics of the above-mentioned examples one easily ﬁnds that its square is

−1.

In Example 13.6 the situation is not so simple. Let s

∗

∈ S

. It is obvious

that the curve (r =const,t,s

∗

) ∈ M

outside the black hole and the curve

(r, t =const,s

∗

) ∈ M

inside black hole are time-like: the squares of their

derivatives in t and r, respectively, are negative, but generally speaking are

not equal to −1. So, changing the parameters, we can ﬁnd the proper times

along those world lines. Thus, the proper time ﬂows along the former lines

outside and along the latter lines inside the black hole. Hence, the space and

time variables are exchanged inside the black hole, as was claimed in the

example above.

13.1.3 Reference frames and 3-dimensional notions

In this section we explain how the above-mentioned 4-dimensional objects

correspond to the 3-dimensional reality around us. Without this interpreta-

tion, the content of the previous sections would be nothing more than an

abstract mathematical construction, while in fact it is a mathematical model

of physics.

Let m(τ) be a time-like world line describing the evolution in the space-

time M of an observer where τ is the proper time. Let m = m(0) be an event