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SINGULARITIES IN THEORIES OF GRAVITATION

THEODOSIOS CHRISTODOULAKIS

A thesis submitted for the degree of

Doctor of Philosophy (in Mathematical Physics)

in the University of London

Mathematics Department Imperial College of Science and Technology London S.W.7. DEDICATED TO MY PARENTS

• ACKNOWLEDGEMENTS

I wish to express my deep gratitude to my supervisor,

Dr. Patrick Dolan, whose constant guidance and encouragement

throughout the course of this work was invaluable.

I also thank Mrs. T. Wright for typing this thesis. • ABSTRACT

This work is divided into four Chapters. Each chapter is

more or less self-contained. At the end a conclusion is given,

which may as well serve as an introduction for the reader whose

interest lies in the physical interpretation of the results contained

in the work . A chapter consists of two parts (marked as A and B) each of which is divided into various sections (marked as a,b,c...).

Sections (a) of each part of each chapter serve as a specialized introduction to the subject that follows.

The references are quoted inside parentheses and in addition an alphabetical list of the references quoted, as well as relevant references, is given at the end of each chapter.

Part A of Chapter I is concerned with the question of limits of space-times, and in part B the global properties of the space of all Lorentz metrics are considered. In a particular topology it then proven that the space of the solutions of Einstein's Field

Equations contains no isolated points.

Chapter II is concerned with causality: In part A, a differential structure is induced in a full Causal space, and in part B a boundary for such a space is constructed, using only the causal relation.

Chapter III deals with the b-boundary techniques: Part A contains the standard definitions while in B an attempt is made to reformulate b-boundary techniques in a way which makes use of structures of the space-time manifold only.

In Chapter IV the problem of combining General Relativity with

Quantum theory is considered. Part A is a general discussion on the problem, while part B contains two applications of the ADM formalism

G. . to the ROBINSON-BERTOTTI Universe and to the GODEL Universe. TABLE OF CONTENTS

Page

I. LIMITS AND TOPOLOGIES OF SPACE-TIMES. 1

A. Limits of space-times. 1

B. Topologies on space-times. 15

II. SPACE-TIME STRUCTURE. 35

A. Causal and differential structure 35 SUPPLEMENT 57

B. The Causal Boundary of Space-Time. 60

III. SPACE-TIME b-BOUNDARIES. 76

A. Definition and properties. 76

B. Further properties and examples. 95

IV. QUANTUM GRAVITY. 113

A. General discussion and examples. 113

B. Quantum Cosmology. 132

V. CONCLUSIONS AND SPECULATIONS. 160 •

CHAPTER I

LIMITS AND TOPOLOGIES OF SPACETIMES

A. Limits of Spacetimes

(a) A lot of statements are in common use, concerning the limit of

some family of solutions of Einstein's field equations as some

free parameter approaches a certain value. In all such statements

there is a serious ambiguity, because they usually refer to a particular

coordinate system; by changing it one obtains an entirely different

limit space. On the other hand, the concept of a limit applied to a

spacetime is a useful one. It seems thus worthwhile to try to

formulate some unambiguous definition of this notion.

By a space-time we understand a (connected, Hausdorff) 4-dimensional 09 manifold with a (C ) metric gab of signature (+,-,-,-). Consider a

one-parameter family of such space-times (the results can very easily

be generalized to many parameter families and perhaps to families

depending on arbitrary functions). That is to say, for each value of

a parameter 1 (>0) we have a space-time M1 and a metric gab (1) on M1.

We want to find the limits as 1--?'0. It would be simpler to consider

the gab(1) as a one parameter family of metrics on a fixed manifold M;

this amounts in specifying when two points and p ►~M1 1 (1)dl') p1pM1 1 are to be considered as representing the "same point" of M. But,

providing this information is not appropriate, since we are interested

in finding all limits and studying their properties, because it always

involves singling out a particular limit. An example will clarify

this point: Consider the family of metrics

2 ) ds2 = (I - 2 )dt2 -(I- -I dr2 - r2(de2 + sin2 edd2) (1 ) 13r 13r

depending on the single parameter 1(=m I"3). In the form (I) clearly

there is no limit as 1 0. If, however, we apply the coordinate

transformation r = ir, t = 1-It, p = 1-Ie then (I) becomes

ds2 = (1 2 - ?)dt2 -(1 2 - 2?) -Idr - r2(dp2 + 1-2sin2(1p)dt2) r r

The limit as 1 0 now exists and gives the metric:

ds = - -2„-dt2 + r- dr2 r2(dp2 + p2ds2) r 2

This is a nonflat solution of Einstein's equations discovered originally

by Kasner and obtained by Robinson as a limit of the Schwarzschild

solution. On the other hand, if we apply to the metric (I) the co-

ordinate transformation x = r+1-4, p=1-4e, we obtain flat space in

the limit 1-->0. Thus one cannot simply speak of "the limit of the

Schwarzschild solution as 1 >O" for the limit one obtains depends on

the choice of coordinates. The essential difference between the

various limits above consists in different identifications of the M1.

We seek to find a way to express the idea that (M1 , gab(1)) depend

smoothly on 1 (This is essential in order to define limits) without

at the same time prejudicing the particular limit we are to obtain.

Let us assume that the different M1 may be put together to make a

smooth (Hausdorff) 5-dimensional manifold M. Each M1 is to be a

4-dimensional submanifold of M. The parameter 1 now appears as a

scalar field on M and gab(1) on M1 is uniquely extended to a tensor AB field gAB on M, which is assumed to be smooth. The signature of g

is (0,+,-,-,-,); in fact, the singular direction is precisely the

gradient of 1, i.e. we have gA6vB1=0. M contains all information ab about the original collection (M1, g (1)) but does not define a

preferred correspondence between different Ml's (such a correspondence

could be defined by giving a vector field on M, nowhere vanishing and

nowhere tangent to the M1: and plICMI t are in correspondence p1EM1 if a trajectory of this vector field joins p1 and pi.. But no such vector field is in the structure of M). Now the problem of finding

limits of the family (M1, gab( 1 )) reduces to that of placing a suitable boundary on M. We define as a limit space of M a 5-manifold M' with boundary GM', equipped with a tensor field g'AB, a scalar field 1', and a smooth, one-to-one map Y of M onto the interior of M' such that

the following three conditions are satisfied:

1. Y is an isometry, i.e. Y takes gAB into g'AB and 1 into 1'.

2. GM' is the region given by 1' = 0. Furthermore it is required

that GM' be connected, Hausdorff and non-empty.

3. g'AB has signature (0,+,-,-,-,) on 8M'.

The first condition ensures that M' really represents M with a boundary attached; the second ensures that the boundary represents a limit as

1-'30; the third condition ensures that the metric on the boundary is nonsingular. At this point we must mention a complication with regard

to Hausdorffness. Although the M1's and M are Hausdorff we cannot

impose this to the limit spaces M' if we are to be able to deal with pathological situations;. and indeed pathological situations are the

main reason for introducing boundaries and taking limits.

The above definition, although corresponding to our intuitive idea

of a limit of a collection of spacetimes, is not very useful for actually writing down limits. For this reason, we will try to show how certain structures on M may be used to characterize the limit spaces. By a

family of frames in M we mean an orthonormal tetrad w(1) of vectors

tangent to M1 and attached to a single point p1&M1, for each 1> 0,

such that the w(1) vary smoothly along the smooth curve in M defined by

the points pl. If M' is any limit space of M, we may ask whether or

noithe given family of frames assumes a limit, i.e. approaches a frame

w(0) at some point pof9M' as 1--3 O. In general, of course, the answer

will be no. The reverse, however, is always true: given a limit space

M', we can find some family of frames which does have a limit in M'. — 4 —

Let M' be a limit space of M, and let w(1) be a family of frames assuming

a limit as 1-- 0. Let us represent points in MI in a neighbourhood of

p1 in terms of the system of normal coordinates based on w(1). In terms

of these coordinates, the components of the metric tensor in M1 approach

a limit as 1 --. 0, and the limiting components are precisely the components

of gab(0) in 9M` in a neighbourhood of po. Thus, the family of frames

• w(1) uniquely determines M`, at least in a sufficiently small neighbourhood

of po. This is a computational technique for finding all limit spaces.

Each M' is characterized by some family of frames for which the components

of the metric in the corresponding normal neighbourhoods approach a limit

as 1 - 0. All that has been done, so far, is the formulation of the

usual definition of a limit (in terms of coordinates) into a different

language. But, to obtain useful results we have to consider the global

properties of limits, and it is here that our formalism simplifies matters

considerably. Let MI and M2 be two limit spaces of M. We say, that

Ml is an extension of M2 if there exists a smooth mapping of M1 onto M2

which preserves the metric gAB and leaves invariant each point of M.

The above discussion implies that, when M1 is an extension of M2, there

1 amd M exists a family of frames in M which assumes a limit in both M 2° From a theorem concerning the rigidity of Lorenz metrics rand the above

definition it follows that if M1 is an extension of M2 and M2 is an • extension of M1, then M1 = M2. Now let M' be any limit space of M, and let N denote the disjoint union of all extensions of M`, where, in this

union, we identify corresponding points of M. We now define an equiva-

lence relation in N. If pl E BMI, p2 E 9M2, where M1 and M2 are extensions of M', write p1 = p2 if there exist a family of frames in M

which, in MI has a limit at p1 and, in M2 has a limit at p2. From the

above discussion of normal neighbourhoods we see that, whenever p1 = p2

there exist neighbourhoods of p1 and p2 which are also identified. Thus,

fem" ,see page 10 the set of equivalence classes form, in a natural way, a limit space M.

M is, by construction, an extension of every extension of M'. But,

as we have stated above, two limit spaces, each of which is an extension

of the other, are equal; so M is unique and has no proper extension.

We have thus outlined the proof of the following:

Theorem 1

Every limit space M' has a unique extension M such that (1) M

has no proper extension, and (2) M is an extension of every extension

of M'. In particular, every family of frames either defines no limit

space, or else defines a limit space which is "maximal" in the above

sense. A simple example can be used to clarify matters. Consider

the Reissner-Norsdstrom solution of fixed "mass" mo and varying "charge" 1

(Fig.1). When l---ā 0 we obtain the Schwarzschild solution (Fig.?). It

is evident from these two figures that something drastic happens when

1 >0. The region inside the "throat" of the R-S solution appears to • become swallowed up in the limit, and does not appear in the S. solution.

Let us try to formulate precisely and answer the question: Do the

points between r = r and r = 0 (shaded in Fig.1) disappear or not in the

limit 1---? 0? Since we deal with a particular limit, we have first to

choose an appropriate family of frames. As such, we choose a frame

which is centered at the point p in Fig.1 for each value of 1, and

such that two of the spacelike vectors of the tetrad point along the two

2-spheres of spherical symmetry. Now consider a selection of points 44 gifM1 lying in the shaded region. It is a well-defined question to

ask whether or not the curve in M defined by the q1 has a limit in the

maximal limit space defined by our frame at p. To have the answer we

refer to each q1 a broken geodesic and see if there is a limiting

broken geodesic. The answer is no. The corresponding geodesic in

the Schwarzschild solution always runs into the singularity at r=0. Thus, in a well-defined manner, the "throat" of the R -N solution

"squeezes up" as 1 -- 0 and eventually swallows all points to the future of the horizon r = r_.

FIGURE 1 - The R-N solution. Each point represents a 2-sphere

of spherical symmetry. Their radii define a scalar

field on the diagram. The horizons are at

r =mO± MŌ ?2

tzo

FIGURE 2 - The Schwarzschild solution Note that, among all possible frames in the Reissner-Nordstrom solution, the one we have used above is preferred in that it permits a simple discription in terms of the Killing vectors. Thus, the Schwarzschild solution is, in a certain sense, the "canonical" limit of the Reissner-

Nordstrom solution as the charge parameter goes to zero.

(b) A property of a family (M1, gab(0) of spacetimes is called

hereditary if all the limits of the family also have that property.

In the following we shall try to classify certain properties of space- times according to whether or not they are hereditary. In many cases the answer is obvious, but there are a few surprises.

Suppose there exist some tensor field, constructed from the

Riemann tensor and its derivatives, which vanishes in each member of ab the family (M1, g (1)). Then, since gab is to be smooth on each limit space, our tensor field must also vanish on the boundary of each limit space. Einstein's source-free equations (Rab = 0) and the condition of conformal flatness (Cabal =0) are of this type, and so they are hereditary properties of spacetimes.

Consider next the type of the Weyl tensor. It is well known that, associated with each of the six types of the Weyl tensor there is an algebraic expression in the Weyl tensor, which vanishes whenever the

Weyl tensor is of the corresponding type. Conversely, if one of these expressions vanishes, then the Weyl tensor is necessarily of that type or of one of its specializations. Thus, although the type of the Weyl tensor is not a hereditaryproperty, properties such as "at least as specialized as type. . ." are.

From the topological properties of the underlying manifold, practically none is hereditary. In fact, generally, no property of space-time is hereditary if it can be violated by merely removing some region from the manifold. For example, neither the homology nor the homotopy groups are hereditary: they can be either enlarged or diminished in the limit. The existence of spinor structures is hereditary. This follows from the characterization of these structures in terms of neighbourhoods of certain 2-spheres immersed in the spacetime.

Quite unexpectedly, the absence of spinor structures is not hereditary.

This is so because this property can be destroyed by removing a suitable region (for example, all of the manifold except a small Euclidean neighbourhood) from the spacetime.

Absence of closed time-like curves is hereditary. Indeed, if

AMS has a closed timelike curve, then we may find a closed timelike curve in each Ml for suitable 1. The presence of such curves is not hereditary. The presence or the absence of a Cauchy surface, of asymptotic simplicity, and of geodesic incompleteness are not hereditary.

Regarding Killing vectors, the situation is somewhat more ab complicated. Let (M1, g (1)) be a family of spacetimes each of which has two Killing vectors. It might be thought that limits of the family may not have two Killing vectors, for, as 1 ---> 0, the Killing vectors could approach each other and thus define only a single Killing vector in the limit. We shall show that such a situation cannot arise.

Consider a family of frames in M. For each point of the pleM1 curve in M, defined by the points at which each orthonormal tetrad is attached, let V1 denote the 10-dimensional vector space consisting of all pairs (4a,Fab), where ~a is a Killing vector at p1 and Fab is a skew tensor at p1, both in M1. Given a Killing vector field in M1, its value and the value of its derivative at p1 defines a point of V1, and so the set of Killing vector fields defines a vector subspace K1 of V1. The dimension of K1 is n, where n is the number of independent gab(1)). Killing vector fields in the (M1, But the collection of all n-dimensional subspaces of a 10-dimensional vector space is compact (it is called the Grassmann manifold G(n,10)l. Hence, if Vo denotes the vector space corresponding to p0, there must be some n-dimensional sub-

space Ko of V0 which is an accumulation space of the Kl.

Note that, even though the metric approaches its limit smoothly,

Ko in general is not a limit point of K1 's. Hence we required it to

be only an accumulation point of the K1 's. We will show that each

element (1a, Fab)o of Ko defines a Killing vector field on GM'.

Choose any closed curve co in BM A , beginning and ending at po. It a, Fab)o suffices to prove that, under Killing transport around co, (1

remains unchanged. Let cl be a curve in M1, beginning and ending

at pl, and such that cl approaches c0 in the limit, and let (4a,Fab)1EK1 (1a, Fab)0. (1 ,Fab)0 accumulate at But now c1 --->c (I0, ,Fab)/ -- , and the change ,, ( a Fab)1 on Killing transport around cl is 0.

Therefore, =0 and htt (ia, Fab) defines a Killing ( , Fab)o v vector in GM'. Thus, we conclude that, if the (M1, gab(1)) have an

n-parameter group of motions, then each limit has at least an n-parameter • group of motions.

By similar arguments it is easy to prove that properties of Killing

vector fields, such as hypersurface orthogonal and commutativity-with

other Killing vector fields are also hereditary.

As an illustrative example of all the above let us consider limits

of the Weyl solutions. Each limit must be a source-free solution of

Einstein's equations with spinor structure, no closed timelike curves,

and at least two Killing vectors, one of which is hypersurface orthogonal r and commute with the other. A spacetime with these properties need not,

a priori, be a Weyl solution. Thus, it is possible to find wide classes

of new solutions to Einstein's equations as limits of known solutions.

In particular, we may call a family of solutions closed if it contains

all its limits; otherwise it is called open. For example, the plane

wave solutions are closed, while the Weyl solutions are, presumably,

open. — 10--

Theorem 2

Let M and M' be connected spacetimes, and let w be an orthonormal

tetrad at pEM, and w' at p'EM'. Then there is at most one isometry

of M into M' which takes w into w'.

Let (ma, ma, . . .ma) be any collection of non-zero vectors at p. 1 2 n 411 We construct a broken geodesic as follows: Let cl be the geodesic

which passes through p and whose tangent at p is m1. Choose an

affine parameter on Cl such that r= 0, mi star= 1 at p, and let p1

denote the point on c1 a unit affine distance from p. By parallel

transporting the 71-I vectors (m2,...mn) along cl to p1 and repeating

the process we find p2, etc., after the nth step we obtain a point pn.

The only restriction on the initial n-tuple is that it should be such

that, at each step in the construction, the corresponding geodesic is

extendable at least a unit parameter distance. It is obvious that,

with an appropriate choice of the initial n-tuple, we may arrange

pn =

Let a and m be two isometries from M into M' each of which

takes w into w'. The m and 0 have the same action on any n-tuple

(ma, ma,...ma) and so m(q) and i(q) are defined by broken geodesics

in M' with the same set of n initial vectors. Therefore 0(q)41(q)-VOM

and the theorem has been proven.

This theorem, by referring points of any connected spacetime to

the tangent space of a point, allows us to compare spacetimes by working

to first order at a single point. This was essential in the proof of

Theorem 1. Now we can formulate precisely our previous assertion that

Lorentz metrics are rigid. By that we mean that, once we specify how

the tangent space of a particular point pEM is to be mapped into the

tangent space of a particular point p'EM' the isometry W is uniquely — 11 — determined. Thus, given the action of m "to first order" at p, the requirement that CU be an isometry determines its behaviour everywhere. In contrast to Lorentz metrics, manifolds without any further structure are completely non-rigid. In fact, it is well-known that, given any (connected, Hausdorff) manifold M and 2m points pl,p2,..pm, gl,g2,...gm of M, all distinct, then there exist a diffeomorphism of M onto itself which takes pl to ql, p2 to g2,....pm to qm.

We now give a more general notion of rigidity. By a geometrical structure we mean a general statement of the types of fields under consideration, that is, the number of connections, the numbers and valences of tensor fields, and more generally, the types of geometrical objects. For example, "a Lorentz metric", "a Lorentz metric and a skew covariant tnesor", and "three linearly independent vectors and a connection", are geometrical structures. By a realization of a geometrical structure we mean a (connected, Hausdorff) manifold equipped with fields of the type described by the geometrical structure. Let

M and M' be manifolds with realizations m and 0', respectively, of a given geometrical structure. By an isometry of (M,0) into (MI A') we mean a diffeomorphism of M onto a subset of M' which takes ® into N'.

We can now define the notion of rigidity (applied to a geometrical structure and not to a realization):

A geometrical structure is said to be rigid of order n (n=0,1,2..) A if, given any two isometries N and UI of a realization (MA) into a realization (M',O') of this geometrical structure such that the value and the first n derivatives of m coincide with those of ID at some point of M, the m = M.

It follows immediately from this definition that the group of isometries of any realization of a rigid geometrical structure into itself forms a Lie group. The converse is not true, since we can have a Lie group of transformations on a manifold (which is non-rigid, thought - 12 - of as a realization of the null geometrical structure). We now give some examples: Rigid of order zero: m linearly independent vector fields on an m-manifold, of order one: a nonsingular metric; a systematic connection, of order two: a conformal structure. Not rigid to any order: m vector fields; a symplectic structure. On

5-manifolds, in particular, "a metric of signature along with a family of frames" is rigid.

(d) We now present a brief discussion on Killing transport:

Let M be a connected spacetime with metric gab. Let 1a be a Killing vector field -on M, and set

Fab ā b = F[ab

We then have

d V = (I) CcFa b Rcabdl

Rearranging the indices in (I), and using the fact that Fab is skew, we obtain Id vaFbc - Rbcad

Let 'ha be the tangent vector to some curve w beginning at the point p.

The above equations, contracted with na, yield

a naVaIb = Fabn

a _ da n V aFbc RbcadI n

Equations (II) give the values of (4a, Fab) along w in terms of their values at p.

More generally, given any pair (4a, Fab} (ot necessarily corresponding to some Killing vector) at p, we may always define such a pair at each point of w, via (II). We call this operation Killing transport.

In general, if we apply Killing transport to some pair (1a, Fab) along — 13 _

a closed curve beginning and ending at p, then, on returning to p,

the new pair (4`a, F`ab) will not coincide with our original pair.

Suppose, however, that there exists a Killing vector on M whose

value and derivative at p is (4a, Fab). Then, evidently, for every closed curve, we shall have (iia, F'ab) _ (la, Fab). Conversely, if

) is given at p and if, for every closed curve beginning and (Ia, Fab ending at p, we have (Ila, F _ (~a, Fab), then there exists a

Killing vector on M whose value and derivative at p is precisely

(4a' Fab)* Let V denote the 10-dimensional vector space of all pairs

(Ia, Fab) at p. Each closed curve, beginning and ending at p,

defines a linear transformation on V. That is, we have a "Killing

holonomy group" at p. The fixed points under this group correspond

precisely to the Killing fields on M. In particular, this group permits

us to make the useful distinction between global Killing vectors, which

are well defined over the entire manifold, and local Killing vectors,

which are defined in a neighbourhood and which, when extended over the

entire manifold, become many-valued. Local Killing vectors may be

defined as the fixed points under the subgroup of the "Killing holonomy

group" obtained by permitting only closed curves through p which may

be contracted to a point.

We now mention some corollaries of the above discussion:

(a) If a Killing vector and its derivative both vanish at a single

point, then this Killing vector vanishes everywhere.

(b) Let M be a spacetime, and suppose that there is a Killing vector

4a defined on some open subset U of M. Then fa and all

its derivatives approach finite values on 9U.

Corollary (b) provides a useful test for the extendability of a space-

time. One way to establish the nonexistence of an extension of a

given spacetime is to find some scalar invariant which becomes infinite L4 — in the region across which we want to carry out the extension.

Corollary (b) asserts that, in the construction of such invariants, we may include Killing vectors and their derivatives.

Since Killing vector transport is essentially tied up with the rigidity of the metric and since conformal metrics are also rigid, we might expect to be able to define conformal Killing transport. Let

a be a conformal Killing vector, and set

l Vab = Fab } 2 mgab

ka =V D where is skew. Doing the same thing as above, we find that, for Fab any curve w with tangent vector na,

naVaIb = na(Fab + Zgab0)

naVaM = naka

alr = na(Rbcad d + k[bec)a)

naVakb = na(wd .Lab + + 2Rd(a DLab Fb).

ab = Rab 6 abR. These equations define conformal Killing where L transport of the 4-tuple (4a, Fab, M, ka). The same remarks concerning the corresponding holonomy group (now acting on a 15-dimensional vector space) and its fixed points apply here, too. The fact that we need specify two derivatives of Ia at a point in the conformal case is a reflection of the fact that conformal metrics are rigid of order two.

We now mention two applications of conformal Killing transport:

(a) From equations (III) it is immediately clear how to write down

the general conformal Killing vector in flat spacetime. Introduc- ing Minkowskian coordinates xa, then, when R = 0, we may integrate abcd •

equations (II) beginning with the last :

ka = ka

Fab + 7.- kbxa) Fab

( = kaxa +

.1 = ka4 (x bxb) -~ 2kc(xc )xa + + mxa + I a Fba 2 a

where Ia ,Fab ,(ū and ka are constant tensors (fifteen numbers to

define a conformal Killing vector).

(b) Conformal Killing transport also provides an elementary proof of

the well-known fact that a spacetime whose Weyl tensor vanishes

is, locally, conformally equivalent to flat spacetime. Indeed, when

0, equations (III) imply that the spacetime has, locally, Cabcd fifteen conformal Killing vectors. Select one of these Killing vectors • corresponding to a 4-tuple (4a ,0,0,0) at p, and then choose the

conformal factor so that the norm of the corresponding conformal Killing

vector is constant. Finally, we remark that there is a corresponding

projective Killing transport. The basic equations are identical with

(II), except that Fab need no longer be skew.

B. Topologies on Spacetimes

(a) In the early years of General relativity, the main questions

people were dealing with, were local in character, e.g. the suitability

of describing the gravitational field by a metric, the structure and

consequences of the Field equations, etc. It was presumably felt that

while global questions could possibly be of some physical interest, they

could safely be postponed to a later stage. It seems that we are now

involved in this later stage. Perhaps the most important reason for

the recent emphasis on global questions and properties was the realisation - 16 - that to establish certain results - in particular, the various singularity theorems - necessarily involves the ability of exercising some control over the admissible global behaviour of the space-time.

Today, except for controlling misbehaviour, global methods have been extended and comprise a small but vigorous branch of relativity.

The fruits of these efforts are numerous: We have become more sensitive to global possibilities and pitfalls. (When will a global conditions be needed? Which intuitive ideas have been formulated precisely? What are the consequences of imposing various conditions?

How restrictive are the conditions?). We have reached a better understanding of the variou-s levels of structure of a space-time and their interaction (which properties of a space-time involve only the casual structure, which only the topology, etc.? Do the casual relations determine the topology?). We have been led to deeper and more economical treatments of known results (the notion of a "hereditary" or "generic" property of space-times, for example, offers a potentially valuable approach to singularities). Finally, we have come to appreciate that the largescale structure of space-time may have unexpected physical significance (do the homology or the homotopy groups, the Stiefel Whitney classes or the spinor structure have implications for elementary particle physics?).

(b) It is thus very interesting to investigate some aspects of the

space of all Lorentz metrics. We will try to analyze the topological structure of this space. In fact we shall consider three possible topologies on it. In one of these topologies we then will prove that the subspace of the space of all Lorentz metrics consisting of the solutions of Einstein's field equations has no isolated points.

Also, we will motivate the definition of a generic property (in a given topology) and shall examine several properties as regard to whether they — 17 — are generic or not. We begin by considering the different topologies on the space of all Lorentz metrics. These differ in how many derivatives of a metric have to be "near" to those of another metric for the two metrics to be considered "near" to each other and in what region they are required to be near. The derivatives of a tensor field (such as a Lorentz metric) on a manifold M are most elegantly described by the bundle of jets over M. We shall use this rigorous approach to establish the result mentioned above. For the present a less sophisticated one suffices: It is well known that one can always put a positive definite metric on a manifold M. Let e be such a metric. This metric can be used to define covariant derivatives of tensor fields on M and also to measure the magnitude of these tensor fields and their derivatives.

Thus we have a precise notion of "near" (with respect to the metric e).

There are three main possibilities:

1). The metrics can be required to be near only on compact regions of

the manifold (Fig.3). 7.5

FIGURE 3 - A neighbourhood B(U, G, g) of the metric g in the compact

open topology consists-of all gi which lie within G(x) of g over U of M.

The behaviour near infinity is unrestricted. More precisely, if g iS a Lorentz metric, U a compact set of M and Gi(Q C i . r) a set — 18 — of continuous positive definite functions on M, the neighbourhood

B(U, 6i, g) of g can be defined as the set of all Lorentz metrics whose ith derivatives differ from those of g by less than Ci on U(O C i r). The set of all such _ B(U, GI, g) for all

U,Ei and g form a sub-basis for the Cr compact-open topology for

Cr(L) (the space of all Lorentz metrics whose rth derivative exists and is continuous); i.e. the open sets in this topology are unions and finite intersections of the B(U, 6i, g).

2). The requirement that the sets U should be compact can be removed

and U can then be taken to be M. This means that "nearby" metrics must be "near" everywhere and thus have the same limiting behaviour at infinity (Fig.4). One may call this the Cr open topology for Cr (L).

FIGURE 4 - A neighbour.hood.B(U,Gi,g) of the .metric

topology consists of all metrics g' which lie: within

G(x) of g:over U of M. Note that U can equal M, and

G can go to zero at infinity.

3). Define the set F(U,Gi,g) as the set of all metrics whose ith

derivatives differ from those of g by less than Ei and which coincide with g outside the compact set U. The. neighbourhood B(Ei,g) is then defined as the union of the F(U,Gi,g) for all compact sets U. — 19 —

(Fig.5). The neighbourhoods B(Gi,g) form a sub-basis for the (Whitney)

Cr fine topology on Cr(L).

FIGURE 5 - The neighbourhood of g consists of all g coinciding

with g outside U and differing from g by less than

G over U.

The Cr compact-open topology is coarser than the Cr open topology, which is coarser than the Cr fine topology. That is to say, there are more open sets in the fine topology than in the open topology and still more than in the compact-open topology.

As well as the Cr topologies, one can use the Sobolev Wr topologies.

These differ from the Cr topologies in that instead of requiring the difference between the derivatives to order r of two "nearby" metrics to be pointwise small, they require the integrals of the squares of these differences to be small (of course, the squares and the integrals are defined with respect to the positive definite metric e on M).

Clearly, a Cr tensor field on M is also a Wr field. For the converse, a fundamental lemma of Sobolev shows that,in four dimensions, r+3 a W field is a Cr field. This means that a Wr topology is coarser than the corresponding Cr topology which, in turn, is coarser Wr-r3 than the corresponding topology (coarser in the sense given above.

Note that there is also the dual sense of finer which holds in the reverse — 20 —

order of the above sequences). The Wr topologies are used in a funda-

mental way in the Cauchy problem for General relativity.

Having put a particular topology on the space of all Lorentz metrics,

one can give a precise meaning to the notion of a stable or generic

property of a metric g: A property P of the metric g is stable or

generic in a given topology on Cr(L) if in that topology there is an

open neighbourhood of g, every metric of which has the property P, i.e,

if every sufficiently "nearby" metric has that property. The motivation

for considering stable properties is the following: A physical theory is

a correspondence between certain physical observations and a methematical model (a manifold with Lorentz metric in this case). But for reasons practical and fundamental (uncertainty principle) physical observations can only be of limited accuracy. Thus the only physically significant properties are those which are stable (in a given topology).

Other unstable properties will not have any physical relevance but may be of mathematical inconvenience in that they may provide counter- examples to general theorems one would like to prove about all metrics in a certain region of Cr(L). One thus is led to the notion of

"generically holding" on a certain region of Cr(L) (applied to a theorem as well as a property). By this is meant that the theorm holds on an open dense subset of that region of Cr(L), i.e. it fails only for some particular metrics contained in that region. Thus for physical purposes it is sufficient to prove that a theory holds generically

(because the metric corresponding to the mathematical model for the space-time is defined with only limited accuracy).

and not in others. Which of these topologies is of physical interest will depend on the nature of the property under consideration. Roughly speaking, if one is concerned with structure in a bounded region of space-time then the appropriate topology is the compact-open, but if one is interested in — 21 —

statements about the existence or non-existence of something everywhere

in space-time, one should use the open or the fine topology as this

gives him command of the limiting behaviour near infinity. As we have

seen, the compact-open topology is coarser than the open topology which

is coarser than the fine topology. Thus it is a stronger requirement

on a property for it to be stable in the open topology than in the fine

topology and still stronger for it to be stable in the compact-open

topology.

As an illustration let us discuss stable causality. Ordinary

causality may be defined as the absence of closed timelike curves. The

physical motivation for imposing ordinary causality is the following:

If there were closed timelike curves in the space-time, one could in

theory travel round them and arrive in one's past. The logical diffi-

culties associated with such time travel are fairly obvious: For

example, one could kill one of one's ancestors, thus preventing his own existence. These difficulties could be avoided only by an abandonment of the idea of free-will: by saying that one was not free to behave

in an arbitrary manner if one travelled into the past. This is not some-

thing which it is very easy to accept, however, and it seems more reasonable to assume that there are no closed timelike curves. As well as actually closed timelike curves, it seems reasonable to exclude "almost closed" timelike curves, i.e. to require that there should be no point p such that every sufficiently small neighbourhood of p intersects some timelike curve more than once. A metric having that property is said

to satisfy strong causality. However, even strong causality is not enough to insure that space-time is not on the verge of violating causality,as is shown by the example of Figure 6, in which a strip of

two-dimensional Minkowski space has been identified along the edges to form a cylinder and three "baffles" have been cut out of the space to

— 22 —

prevent there being any closed or almost closed timeline curves.

i (Lerit L Sjr

FIGURE 6

Nevertheless there are timelike curves which pass arbitrarily close to other timelike curves which then come arbitrarily close to the first curves. As a matter of fact, as we see from the example in Fig.6, we can create a whole hierarchy of causality violations by cutting an increasing number of "baffles" out of the space. Indeed, Brandon

Carter has rigorously shown that there is a whole hierarchy of higher causality conditions (see Chapter II for details). This hierarchy is more than countably infinite: one can define an + I)th causality condition, an (o + 2) th and so on.

However, in the light of the previously developed notion of topologies on the space of all Lorentz metrics, one can define an ultimate causality condition which is stronger than all the above hierarchy conditions and which corresponds to space-time not being on the verge of violating causality: A metric g is said to satisfy the stable causality condition if,in a given topology on C°(L), there is an open neighbourhood of g no metric of which has closed timelike curves. Of the three topologies which we have placed on the space of all Lorentz metrics only the open and the fine topology can 23 —. be used in the above definition of stable causality. Indeed, in the compact-open topology any open neighbourhood of any metric g can contain metrics with closed timelike curves. This is because a neighbourhood of g consists of all metrics which are near g on a compact set U. However, outside U they can differ by an arbitrary amount and so can admit closed timelike curves. From the remaining two topologies the open topology has some obvious advantages: Since it is coarser than the fine topology, stable causality defined with respect to the open topology will be stronger than stable causality defined with respect to the fine topology. Also it seems more physical than the fine topology since to use the fine topology we have to make exact measurements in order to establish the coincidence of the metrics outside the compact set U. The definition of stable causality with the open topology has the further advantage of being related to another physically significant property, the existence of cosmic time functions.

By this is meant a smooth function t which increases along every future directed timelike or null curve . The spacelike surfaces of constant value of such a function can be regarded as surfaces of simultan- eity in the universe though, of course, they are not unique. One can show (Hawking, S.W. (1968), Proc.Roy.Soc.A/308,433) that a metric admits such cosmic time function if,and only if, it is stably causal in the

Co open topology. Nevertheless, as we shall see, far more useful results can be proved using the fine topology. The region in C°(L) on which stable causality holds is the interior of the region on which ordinary causality holds. Since the region on which ordinary causality is violated is open, the union of this region with the region on which stable causality holds is an open dense set in C°(L). Thus, it is generic for a metric either to violate causality or to be stably causal.

One would like to prove that it is generic for a causal metric to be i -- 24 --

stably causal. In the next section we shall prove something stronger:

It is generic for a causal conformal structure to be stably causal.

(d) Let us now concentrate in the Ck (Whitney) fine topology on Ck(L).

In fact, the set of all C2 Lorentz metrics on a non-compact four-

manifold will be given the Whitney fine C2 topology. A rigorous mathe-

matical framework will emerge within which it is possible to discuss the

global properties of space-time manifolds in general, and the singularity

theorems in particular.

(I) Let us first present the necessary mathematical framework:

Throughout the rest of this section all manifolds are Cc°, Hausdorff

and paracompact. M denotes a fixed, non-compact, four dimensional

manifold without boundary. Let S-4 M be the vector bundle of twice-

covariant symmetric tensors, and L M the open sub-bundle of indefinite

quadratic forms of signature -2. The set of all Ck sections of L is

denoted Ck(L); any element of this set is called a Ck Lorentz metric.

It is this set we wish to topologize. Because of the fact that many

interesting geometric objects occurring in relativistic space-time are

completely determined bytiteicconformal structure$,it is very convenient

to topologize the set of conformal structures as well. If we delete

the zero section from S and denote the resulting bundle by S we can

find the obvious retraction r of S onto a bundle Q whose fibres

are diffeomorphic to the nine-sphere: simply identify, in each fibre,

all elements which are positive multiples of one'another (in local co-

ordinates r is given by (u1,sab) 4 (ui,sab/s), where a b and

2 1 /2 s = ( E (s ) ) ). Q is a C°° bundle over M, and r is a 111 4.s n mn

C°° open surjection. The set r(L) is an open sub-bundle of Q which

we denote by C. For any Lorentz metric g, g: = rg is a section

of C; if h is another Lorentz metric then h = g if and only if, 25 - h and g are conformally related. For any wG Ck(C), there exists k an hGC (L) such that h = w; to see this, let e be any positive- definite metric on M; at each peM, define h(p) to be the unique element of e-norm I in r-I(w(p)).

We now briefly review jet bundles, and the Whitney toplogies.

(For a neat, precise introduction with proofs see Mather, J. Annals of Mathematics 89,254 (1959)). Let E--- M be any bundle, and let jk(M,E)-►MxE be the bundle of k-jets of local Ck maps from M to E.

Recall that if f is such a map and p is in the domain of f,

k(f)(p) denotes the equivalence class of all local Ck maps g which are k-tangent to f at p. (The Taylor expansion series for f and g agree up to and including the k-th derivaties in any (and thus all) local coordinate systems at p and f(p)). By restricting attention to the subset of all local Ck sections of E, we obtain a closed sub- k bundle J (E) called the bundle of K-jets of sections of E. If a

Ck section f is a geometric object field with components f°°° e e e relative to some local coordinates (u'), the local coordinates of f(f)(p) are given by :

f.°o f°.° .e. (p), (p),°..,f (p). .0. oe.,il where &usually the comma denotes partial differentiation.

Any f G Cr(E) determines a Cr-k section of Jk(E) defined by p — jk(f)(p) and called the k-jet extention of f. Notice, for future reference, that J°(M,E) = MxE, J°(E) = E.

Let U be open in Jk(E). Define N(U) = f GCk(E): jk(f)(M) C U

The Whitney fine Ck topology on Ck(E) is generated by the sets

N(U), where U ranges over the open sets of Jk(E).

For k j r <> 0 the Whitney fine Cr topology is well-defined on

Ck(E) and is strictly coarser than the Whitney fine Ck topology,

on Ck(E). From now on, (f'k(E) stands for the set Ck(E) endowed

with the Whitney fine Ck topology. The motivation for introducing

this topology is the following: Suppose a collection of observers

is scattered throughout M, and that each observer has instructions to

measure the values and the first derivatives of some field f, in

his neighbourhood. The values reported, with judicious error bounds,

determine an open set UcJk(E) with jk(f)(M) C U. Notice also

that if the set U has been made as small as possible, then any other

field f~ satisfying jk(fI )(M)C U must be regarded as a legitimate

perturbation of the "real" field f. We summarize below some facts we shall need concerning r- k(E): (1.1) For non-compact M, /-k(E) is not first-countable. Thus we

must use nets or filters to talk about convergence. In fact,

convergence of a sequence ffi occurs only under the very restrictive

condition that there exist a compact K M nutside of which fi = fj

for sufficiently large i,j, and inside of whichjk(fi)2 converges

uniformly. This establishes the equivalence of the definition of the

fine topology given above with the one given previously (page',9 ). An immediate consequence is that if E is a vector bundle, scalar k multiplication is not continuous in r (E); so it is not a topological

vector space. It is, however, a topological module over the ring of

real-valued Ck functions on M. This is proven in: Mather, J. Annals

of Math. 89,254 (1969). Another consequence is that one (or many)-

parameter families of metrics encountered in relativity theory do not

determine continuous curves (or surfaces) in rk (L). This is part

of the problem of finding limits of space-times and is dealt with in

part A of this chapter.

(1.2) Let E and F be bundles over M; denote by pl and p2 the

projections of J0(F) to M and F respectively. 27

Let : Jk(E)• -:› J°(F) be a continuous map inducing the identity on M(for any s G Ck(E), pIo 6 o jk(s) = idM). Then the map

6: (`k(E) — j"0(F) defined by setting ss(x) = p2o 6(js)(x))k( for any s G rk(E) is continuous: if U is open in J°(F), then 6-1(N(U)) = N(1 -1 (U)) which is open, since 6 is continuous. In many cases of interest, 6 is a continuous partial differential operator.

This will be shown in II for the case of Einstein's field equations.

All the above is shown in Fig.7.

FIGURE 7

(1.3) Let E and F be bundles over M, and let w: E -=›- F be a

Cr (r te, k) map inducing the identity on M. Then the map w : k(F) defined by W: s -> wos for s G J'k(E) is continuous. This is obvious (see Figure 8). — 28

Pp-

poi

P Wo PF FIGURE 8

(1.4) Let L, C and r be as above. Then the map r : ' k(L) Īk(C)

defined by 7. : g--> g = rg is continuous. For k =0 r is open.

Proof: The first part is a direct application of property (1.3).

We have only to show openness. Let U be open in J°(L) = L. Then r(U) is open in C. Let g satisfy g(M) c r(U). We must find g G C°(L) such that g(M) G U and g = rg. Let g' be any C° metric such that g~ = g. The problem reduces to that of finding a positive continuous function 1 such that lg'(M) C U. For each p G M, there exists a positive number a(p); a(p)g'(p) G U f Lp, where L denotes the fibre of L over p. By continuity considerations, there is a neighbourhood V(p) such that V q G V(p) —} a(p)g'(q)

G U (1Lp. Thus we obtain an open cover of M, for which there exists a locally finite open refinement fV. : i G I and a subordinate } partition of unity I fi : i 6 I . For each i choose a V(p) such that Vi C V(p), and let bi be the constant function with value a(p). Put 1 = b f. ; then g = ig' is the desired metric. i 1 - 29 -

(1.5) (Definition). Let g E.0°(C), p C M, T = the tangent space

at p. Define t(g) = E(Z C T:g(Z,Z) > 0 for any (and thus all) g 'G r -1 (g) 3. Let h, g 6 C°(C). We write h(p) 4C g(p) if, and only if t(h) t(g); we then define 5 < g if, and only if 5(p)4( g(p) Ife M. If r 4:g we define (h,g) = : 1k 6 C°(C): h <`k <,3 . This set is called an interval, and the topology on

C°(C) generated by taking the intervals as the open sets, is called the interval topology (see Geroch, R.P. J.Math.Phys. 11,437 (1970).

Intuitively, k 6 (h,g) if, and only if, at each p 6 M, the null cone of k lies strictly between those of h and g.

(1.6) An interval is an open set in 'Q(C).

Proof: Let and k be as above. We have to find a neighbourhood of R in the C° fine topology contained in (h,g). Since J°(C) 2'C, it suffices to find an open set U C C containing k(M) with the property that any El(p) 6 U satisfies h(p) m(p) < g(p). This, however, is trivial, using the local product structure of C.

The converse of (1.6) is also true (see Lerner, D. Thesis,

University of Pitsburgh, 1972). This establishes the complete equivalence of the C° Whitney fine topology with the interval topology.

(II) Consider the map Ric : J2(L) J°(S) sending the 2-jet of

a Lorentz metric to the 0-jet of its Ricci tensor (if g is a

are the Christoffel symbols of g in local section of L and rbc some chart, then the curvature tensor of g has components

The Ricci tensor is given by Rabcd = 2 r b fd, c) + 2 r b [d I-1) s . a = R bca' and the scalar curvature by R = gbeRbc)aChoose a chart Rbc with coordinates (u'). If p is in the chart and g is a local section at p, then the local coordinates for j2(g)(p) are just i - - 2 ,cd(u )), while those of Ric(j (g)(p)) (u (P)' 9ab(u~)' gab,c(u ), gab are (u'(p), Rbc(u')). Notice that the necessary derivatives have - 3 0 - already been taken in forming the bundle J2(L), and that Ric is a purely algebraic map which is obviously continuous. By property (1.2) of the previous section the map g .- Ric(g) is continuous from r2 (L) to ,r•'° (S) , and sends a metric to its Ricci tensor. Similarly, for the map g.— Rg. Using the fact that r°(S) is a C°(M) module, we see that the map T: g .,.. T(g) = - (Ric(g) - Rg) from 2 2(L) to p'° (S) is also continuous.

The set Y = T(j'2 (L)) is the complete set of continuous energy-momentum tensors on M for which Einstein's field equations have. solutions. Although the determination of the detailed structure of Y

(under what conditions can we solve the field equations on some manifold

M?) is one of the major problems in General Relativity, virtually nothing is known about it, even for the case M = R. Some possibilities are :

(a) Y is open: this would be very powerful. For any exact

solution to T(g) = Tp, exact solutions would exist for any

T' sufficiently C° close to To.

(b) yy1t(Y) # 0 : there would then exist an open set W C j"°(S)

in which Y would be dense.

For any s G W, one could obtain approximate solutions of the equation

T(g) = s to any desired degree of accuracy.

to be unsolvable.

As far as the author is aware, nothing is known as regards the truth or falsity of any of the above.

Less interesting perhaps, but still of considerable use is the following result:

Theorem 1. Y contains no isolated points.

Proof: For any To = 1-(g0), we must find a net Ti Y/ i To

— 31 —

with Ti To. Let si be any net of functions converging to 3 2s. 0 in C2(M), Putting fi = e 1 we have T(figo) T(go) = To.

Now, all we need, is to verify that the net si e,lik can be chosen so

that T(figo) # To for V i. Define, for any C2 function s ,

the functions V1 (s) = abgs, a~, b and V s ) = goab s'- ab' where as 1 0 2( usual the comma denotes partial differentiation and the semi-colon

covariant differentiation with respect to go. It is easy to see that

a necessary condition for T(e2sgo) = T(go) is that V1 (s) + V2(s) = 0.

The map D: s V1 (s) + V2(s) from C2(M) to C°(M) is continuous,

so that D-1(0) is closed in C2(M). Clearly .J t(D-1(0)) _ 0, so

that C2(M)/D-1 (0) is open dense. Thus any net in C2(M)/D-1(0)

converging to 0 will suffice.

Although the above theorem looks weaker than statements (a), (b),

consequences. Firstly (translating the theorem) one can say that if

T(go) = To then any neighbourhood of To contains a T' # To for

which T(g') = T' for some g'.

Secondly, in the sense of part A of this chapter concerning

limits of space-times, Theorem 1 above provides a very powerful existence

theorem: Given a space-time which is an exact solution to Einstein's

field equations one can always find by some limiting process at least

one more space-time representing an exact solution to the equations.

Alternatively, we can say that no solution of Einstein's field equations

is closed ("closed" here is taken in the sense of part A, page 9 ,

i.e. a solution is called closed if it has no limits or otherwise if

it contains all its limits. This definition is an obvious consequence

of the definition given on page 9 concerning families of solutions).

Yet another way of expressing Theorem l above is as a statement about

the solutions of Einstein's field equations:

Solutions of the field equations appear either as families (i.e. — 32 — differing from each other by the value of one or more parameters), or else as groups of more than one, the various members of the same group being the limits of one another.

From the above discussion and part A follows that Theorem 1 is not only an existence theorem of academic interest, but also a useful guide for seeking new solutions to the field equations.

(III) On page 20 we have defined stable or generic properties and on

page 22, stable causality. Since the existence of closed tome- like curves is also a property of g, and since the map r is open, stable causality is well-defined in j'k(C) and is stable there. A -1 conformal structure g is said to be causal if some metric in r (g) is not causal. Otherwise, i.e. if every metric in r-1 6) is causal, g is said to be causal. Letting A be the acausal, r°(c) - A the causal, and SC the stably causal conformal structures respectively, we have that

(a) A is open: This is obvious since A forms an interval and as

we have proved in (1.6) an interval is an open set in r 0(C).

(b) SC # 0 on any non-compact M: On such an M, there always exists

a function f with no critical points (df(x) # 0, Yx). If

m is a positive-definite metric on M, and X is the (nowhere

vanishing) gradient of f with respect to m, the Lorentz

metric g defined by:

g(Y,Z) = -m(Y,Z) + 2m(X,Y)m(X,Z) m(X,X)

possesses a cosmic time function (f) and is therefore stably

causal (see page 23 ).

set I: g:g h lies entirely in SC, and any interval containing

5 intersects this set. 33

Among other things, this insures that there are no isolated points in f"(C) - A. This proves the promised (on page 2q) result: It is generic for a causal conformal structure to be stably causal. — 34 —

REFERENCES

Carter, B. (1971), G.R.G. Journal, 1, 349.

Geroch, R.P. (1970), Jour.Mathem.Phys., 11, 437.

Hawking, S.W. (1968), Proc.Roy.Soc., A308, 433.

Kasner, E. (1921), Am.Journal of Math., 43, 217.

Lerner, D. (1972), Thesis, University of Pitsburg.

Mather, J. (1969), Annals of Mathematics, 89, 254.

Steenrod, N. (1951), "The topology of fibre bundles", Princeton Univ. Press. — 35 --'

CHAPTER II

SPACE-TIME STRUCTURE

A) Causal and Differential Structure

(a) The conventional development of general relativity begins with two hypotheses. Firstly, that space-time can be represented roughly as a four dimensional differential manifold and, secondly, the so-called chronometric hypothesis: that there is a metric tensor

(gab in local coordinates) defined on the space-time manifold and that the time interval between the neighbouring events xa and b)1/2. xa dxa is given by (gabdxadx From these two hypotheses all the geometric structures on space-time, its causal, conformal and projective structures can be very easily derived. Then two more hypotheses, the so-called geodesic hypotheses, are needed to determine the behaviour of light signals and the motion of particles in free fall.

However the above outlined approach has been frequently criticized on the following grounds:

I. Whereas adopting this approach we have the advantage of obtaining

the whole mathematical structure of general relativity from just two axioms, we also have, from the physical point of view, the great disadvantage that the physical meaning of the so derived various geometrical structures is obscured. A classical example of this obstruction is the various attempts to modify the theory by rejecting the metrical structure but retaining the conformal and projective structures. In the light of the above such attempts are condemned to lead at the best case only to minor modifications and are sometimes without meaning, since all three structures are derived from the above two axioms.

II. There is an underlying contradiction between the chronometric

and the geodesic hypotheses at least in the physical interpretation: — 36 —

Once the metric coefficients gab have been defined by the chronometric hypothesis, there is no overwhelming physical justification for postu- lating that these coefficients also determine the motion of freely falling particles and light signals.

III. It has been shown by Marzke and Wheeler and by Kundt and Hoffman

that it is possible to construct a standard clock from the paths of freely falling particles and light rays. Thus the geodesic hypotheses alone imply a physical interpretation of the metric in terms of time, independently of the chronometric hypothesis. This means that either the chronometric hypothesis is redundant or it is reduced to a statement of equality between the gravitational time, measured by geodesic clocks, and atomic time, measured by standard clocks. Such a statement is a very strong one, and in any case is out of place in general relativity or in any theory of gravitation, since to make it one should first have at one's disposal a theory embracing both atomic and gravitational phenomena, that is to say, a unified theory. To the present such a theory does not exist.

Another kind of objection is that related to the mathematical construction of the theory rather than its physical interpretation. it is usual to impose certain additional restrictions on a model of space-time in order to consider it as a physically reasonable one.. The weakest such restriction is that there should not be closed nonspacelike curves, known as the weak causality condition. Further restrictions, e.g. the strong causality condition, can also be justified on physical grounds. Now the metric tensor gab cannot be defined independently of the background manifold, and yet even without any causality condition, the metric, which is an entirely local structure, imposes certain restrictions on the global topology of the manifold (e.g. there are very few manifolds on which it is possible to define a homogeneous and spatially isotropic metric). With the various causality conditions

the situation is even worse: for instance, it has been shown by

Geroch that, under certain conditions, a 4-geometry cannot have two

slices (roughly spacelike sections) with different topologies unless

it is acausalm This is a restriction on the topology of space-time

which cannot be formulated until the metric has been defined, and yet

the metric cannot be defined until the background manifold has been

specified.

Of course, in practical terms, this is not a problem. The

field equations for the metric can, in any case, be solved only locally:

the global topology of space-time is then treated as an extension

problem. It does, however, lead to certain difficulties, for instance

in the definition of a singularity. There is a perfectly clear

intuitive idea of what is meant by a singularity and by such statements

as "in the neighbourhood of a singularity", but in general relativity,

because a singularity is a place (in some sense) where the theory

breaks down, singularities can only be discussed in terms of geodesic

incompleteness; this is an inevitable consequence of the fact that

general relativity is essentially a local theory in the sense that it

says something about the geometry of space-time in the locality of

nonsingular points. This situation is very unsatisfactory both because in this way singularities become a global problem, whereas one

intuitively feels that they should be a local phenomenon, and because

for defining singularities via geodesic incompleteness we need

elaborate structures, such as the projective or the Weyl structure,

and yet it seems that it should be possible to give a definition in

terms of more primitive structures: that, for instance, a singularity should be a point where the differential structure of the background manifold breaks down. This, however, is not a useful definition if 38 the background manifold is fixed and independent of the physical processes happening on it.

Various attempts have been made to give a'local characterization of singularities by adding a boundary, consisting of all singular points on a geodesically incomplete space-time. Geroch, for example, does this by identifying certain classes of incomplete timelike geodesics. The trouble with this approach is that there are many such possible boundaries for a given incomplete space-time and no outstanding reason for selecting one over another. Furthermore, this approach makes an unreasonable distinction between singular and nonsingular points. Indeed, the topological structure of space-time is defined in terms of, for instance, the world lines of freely falling particles at singular points, whereas at nonsingular points the differential and topological structures are taken as absolute and fixed. It would seem more reasonable to define all the necessary geometric structures of space-time, both at singular and at non-singular points, in terms of more primitive concepts, such as the trajectories of freely falling particles and light signals, and then to define as singular a point of space-time at which one of these geometric structures breaks down.

Even if this approach did not yield any drastically new techniques for analyzing singularities, it would at least give a criterion for choosing among the different possible boundaries on an incomplete space- time the most physically reasonable one.

It is well-known that the different structures on space-time such as the projective, conformal, Weyl and metric can be built up successively from a few axioms: each one of these is physically well- motivated and it reflects the reason for believing in some geometric structure derived from the chronometric hypothesis. The basic mathematical structures must be taken to be simple physical objects. The criterion for accepting a particular axiom is that it should have a • — 39 —

simple and fairly obvious physical interpretation and involve concepts

exclusively introduced in previous axioms. In the following we will

try to present such an approach.

(b) First of all we have to choose the basic mathematical structures.

As such we choose two of the simplest objects in physics: The paths

of freely falling spherically symmetric, non-rotating particles, on

which it is assumed there is some continuous idea of time, and light R signals between events. In a previously made effort the notion of a

light ray in space-time rather than light signal between events is

taken as. fundamental but then we have to postulate that the light rays

are at the same time null geodesics of the metric, whereas in our

approach this can be deduced from statements about the emission and

absorption of light. Furthermore, the only global restrictions imposed

on space-time are introduced at the beginning through axioms Ia and Ib

which deal with the relation between the causal and topological struct-

ures. In this way we can avoid the embarrassing feedback of local

structures imposing restrictions on a previously defined global topology.

Further, axioms Ia and Ib express properties which can be required for

any reasonable model of space-time, even one with singularities. Thus

we can take as the definition of a singularity the breakdown of any of

the properties expressed in the other axioms. This definition is not

dependent upon the global or the projective structures.

We proceed with some definitions: s Definition I

A space-time is a pair (M,r) where M is a point set whose

elements are called EVENTS and r is a set consisting of subsets of M,

each subset having the structure of a Co one-dimensional manifold,

homeomorphic with R. The elements of r are called PARTICLES; it is

assumed that there is at least one particle through each event. The « — 40 —

Co structure on a particle is interpreted as a continuous idea of time; the homeomorphism with R excludes the possibility of closed particle world lines. Each particle has two possible orientations. A particular choice of orientation of rErdefines an antireflexive

linear ordering on ' denoted by ; i f XE( and y 1 then x 1C{ fir y <

• of the particle Definition II (a) If a particular choice of orientation has been made for each particle r er, and if x and y are any two events, then a trip from x to y is a sequence of events zo = x, z1...zn = y, together with a sequence of particles n suchsuch that for i=0, 1,...,n-1 : z1+- i+l and zi zi , 4<%'1+1 (b) x chronologically precedes y (written x C'( y) if there is a trip from x to yo A trip can be interpreted as the world line of a massive particle which undergoes a finite number of collisions but is otherwise freely falling. To give physical meaning to the above, we have to postulate : Axiom 1(a) (first causality axiom It is possible to choose the orientation of each erso that m M not m (( m. The physical interpretation of axiom 1(a) is that on each particle there is a natural arrow of time and that no particle can enter its own past by undergoing a finite number of collisions. With the above choice of orientation, ({ is an anti-reflexive partial ordering on M. Definition III (a) If xfM then I+(x) = [YÇM/x y , I (x) = fy(M/y « x (b) x( y (x causally precedes y) if I}(x)> I+(y) and I-(x) C I-(y). - 41 -

FIGURE 9 — 42 —

iI+(x)/413 U I (x)/xEM 3

(d) x4y (h.otismos relation) if x Cy but not x «<*y.

Axiom 2.

If x(M and ~~ j then ' /1 I4-(x) and 60I-(x) are open in 5 •

Physically this axiom says that if there is a trip from x to y

then there is always a faster one. We see that the ,topology not

only induces the right topology on each but also resembles an 6 idea of space-time topology: Suppose xn is a sequence of events 3 converging to x in thej,topology. Let be some particle through

x with fixed parametrization (i.e. timescale) and (tl,t2) be an

arbitrarily small time interval in , continaing x; then an infinite

number of the xn will be in the future of t and the past of t2. 1 The time interval (on ) needed to travel from to xn and back K ' to can be made as small as we want since the notion of time is K continuous on Thus there is an obvious physical interpretation K o of the convergence in ,JAL: (See Fig.9).

Definition IV

If xEM and ye, then x almost causally precedes y (written

as x A y) if, V z(I-(x) I+(z) I4-(y). Equivalently, Vzfe(y)

I (z) I-(x).

The point of this is that no real physical measurement is ever

made at a single event: An event should really be thought of as the

limit of its neighbourhoods. If x almost causally precedes y,

then every neighbourhood of x contains events which chronologically

precede events in any neighbourhood of y, and no real physical experiment

would reveal that x and y are not causally related. In the light of this it is reasonable to require : — 43 —

Axiom I(b) (second causality axiom)

If xeM and y(M, then x A y and y A x ; x = y.

This axiom is a somewhat stronger one than I(a). With this axiom M is made a causal space and A acquires some physically reasonable properties. One of these properties is that M is

Hausdorff in the 4il topology. Indeed, if x and y are distinct events, then either not x A y or not y A x. Suppose not x A y. - Then there are z1EI (x) and z2fI+(y) with not z1 z2 and not z2 (< zl. Then I+(zl) and I (z2) are disjoint open sets containing zi and z2 respectively and thus Hausdorffness has been proven. A further property is that

I+(x) = I(y)~ x=y, I-(x) = I (y) : x=y. Indeed, suppose

I4 (x) = I+(y), then x A y and y A x, so x = y. A further property ~ x, is that 4: is full: That is V xfM there is yM such that y ~, and if yl < x, y2 <.< x then there is z x, such that yl <4( z and y2 z. Indeed, let xiM and choose ► with x(o Since is homeomorphic with R, there is A y with y x. If yl « x, y2 x, then I+(yl) (J I(y2) fl ' is open in y and contains x. Hence there is z as above.

It is now clear that (M, , 4k ) is a full causal space. It is an easy matter to prove that the following conditions are fulfilled:

(I) x < x

(II) x a* y and y< z=4;0. x ( z (III) x 4 y and y< x x= y

(IV) not x‹( x

x z (V) x <{ y and y <( z l{ (VI) (a) x y and y 4. z x 4 z (b) x y and y < z.x<.< z

(VII) x+y if x ( y and not x ,<< y by 01.4F61 Olt ). -44-

FIGURE 10 - U is not a past and future reflecting neighbourhood of x. — 45 —

(c) We.now proceed to introduce light propagation into our full causal spaces Guided from the principal property of light we might define that there is a light signal from x toy if yEI+(x), or equivalently I+(x) I}(y) and not x (C y (emission definition).

However, it also seems natural to say that there is a light signal from x to y if xeI (y) (absorption definition). That is, the light signals absorbed at y were emitted on the boundary of the past of y. One possible definition, which maintains the symmetry between emission and absorption, is the following: There is a light signal from x to y if either I+(x) 7I +(y) or I (x) I (y) and not x « y. But then suppose that x*M and yEM, with

I+(x) I (y) but not x

In general ī-(y) will intersect ' in an open set of which x is not a limit point. There will thus be a finite interval on r which is visible at y. (See Figure 10). It is equally possible that there may be single events,visible by an observer on a particle over a finite length of time. If either the emission or the absorption definition alone is adopted, then one, by similar arguments, is led to a situation where a clock (that is an admissible parametrization) on one, particle is seen to vary discontinuously by an observer on another particle: This situation is unacceptable, at least for nearby particles. The way out of this is to require Axiom 3

Every event has a future and past reflecting neighbourhood; that is V x(M there is Nx,an open neighbourhood of x, such that if y, z fNx, then I}(y) I(z) ....;~ I (z) 7.2.)I -(y) and

, I (Y) I (z) =* I{(z) I+(y)

Now it is possible to define light signals, at least locally: If y(Nx for some x, then there is a light signal from x to y if, and .- 46

only if x fy. Later on we shall see how this extends to a global definition and the paths of light signals emerge. Let be a particle and let for each 4( Ux denote a D future and past reflecting neighbourhood of x of the form I-(m1 ) I+(m2) with m1,m2f Let tit A y . be the union of all such neighbourhoods. Ux for all events 4( . U is an open neighbourhood of . If z(U1 then ziUx for some xoE t 0 Define f+(z) = inf xEr x/x <4; z 3 and f (z) = supxf )(ix' z}; inf and sup infer to the linear ordering <41, on t . f+(z) and ((z) certainly both exist and belong to Ux (see Figure 11).

Also I+(z) C I+(f+(z)) and r(z) I (f (z)) but not z {«f+(z) and not f-(z)<. z. Ux is a future and past reflecting neighbourhood; 0 hence z 4 f+(z) and f (z) * z. Thus we have proved that for any

particle t there exist two functions f÷: Ub " and f : Ue (called the message functions) such that z + f+(z) and r(z) 4 z. These functions are determined uniquely by the above horismos relations and describe the process of seeing and be seen by an observer moving along y at nearby events. The following can be proved: (See Supplement (a)).

V irir f+ : U1 -, y and f : Ut are open and

continuous maps. It is then easy to show that the j, topology is the smalest topology which makes all message functions continuous: That is, makes space-time "look" continuous for every observer in free fall. This can be taken as a further confirmation of the fact that the ~4

topology is the appropriate one for space-time; it embodies the most physically reasonable idea of continuity. (d) So far nothing has been said about the dimension of space-time. The simplest way to fix it is to require the following: Axiom 4(a) (first dimension axiom)

", there is "" such that the four Given r2 E r 47

/ '1?

FIGURE 11 - Ut is a past and future reflecting neighbourhood of . — 48 — message functions defined by and , together with the homeo- l ¥2 morphisms A'1 . R and ( 2 --3r R, define a one-to-one map from a neighbourhood of every point in (IJ /I Ur ) nr( ri (2) onto an 1 2 open set in R4. Further, every event belongs to (U~, f U1)^,,( atlU 12) 7 2 for some such pair + Not every pair of particles can 0 1 , d'2 fr • be expected to define such a map: In Minkowski space, for example, two coplanar timelike straight lines do not define a one-to-one map into R4, but a pair of skew timelike lines do. Combining this axiom with the Hausdorffness of space-time in the,topology and the fact that the four message functions are open and continuous maps, we have:

Theorem 1

M is a Co four-dimensional real manifold.

We now proceed to the differential structure of M. The local coordinates are defined by the four message functions of the nearby particles. These are sometimes called the radar coordinates. Each event within the above neighbourhood is parametrized by the time at which a signal is sent from one of the particles to the event and the time at which the echo from the event is received back at the particle.

Consider a particle and suppose a second particle dl ~`2 intersects U . Then the two functions fl y : 1 and ~2 2 --a l f-1 12 : K2 are continuous and, in fact, strictly monotonic; they describe the relationship between a clock on and a clock on jl Now each particle is homeomorphic with R, 12,as seen at Il. that is, on each particle there is a continuous idea of time: All clocks used to measure time have to be continuously related. Each homeomorphism with R induces a differential structure on the particle; each clock defines a preferred class of clocks which are differentially related. Suppose that it is possible to choose a differential structure on each particle by, for instance, using the same type of clock on each -49_.

particle, in such a way that the message functions between every pair

of nearby particles are differentiable, all of class CN say. This

is one possible interpretation of the physical statement: "Space-time

looks as if it has the Nth degree of smoothness In the following we

will try to show that, under certain additional axioms, space-time is

a differential manifold which is as smooth as it looks.

Intuitively, every particle through an event defines a direction

in space-time at this event. In order to introduce a notion of contin-

uously varying direction we make the following definition:

Definition V

A set C = fi, t1 /1fL Y of particles with given parametriz- o ation ti : Yl ...... R is a C -congruence on U M if:

(I) V xiU there is exactly one ( yx,tx)fC such that xf r x 0 and tx(x) = 0.

(II) For each r in some neighbourhood of 0 in R, the map U -- M: 1 ( x• ."* tx is a C°-morphi sm. r)

Physically, a C°-congruence is a fluid. Now suppose a

preferred differential structure has been chosen on each particle.

Let C be a C°-congruence, each of whose elements has a differential parametrization. It is reasonable to suppose that, in addition to being continuous in the above sense, C also "looks" continuous: Any observer on a nearby particle , with parametrization t: R, can assign two parameters to each element ix of C which describe the relative velocity of r x( at xf i+x) with respect to 0 physically these are the red shift of eirx and the doppler shift of signals bounced off The information is contained in the J. derivative v1 and v2 of the two functions: tofXot-1 and tofXot;1 at 0. If C is to appear continuous to the observer on ir , then these two parameters must vary continuously over Ug. Hence we — 50 — require:

Axiom 5•

If e l r and C is a C°-congruence on V C U1 , then the two functions vl:V —° R x --->tof+ot-1 : i x x ?i 0

v2:V _-~ R : x ....~ Ctof-ot 1 x x are continuous.

Now a second definition of dimension can be given: This is, roughly, that there should be four independent directions at each event. A set Çe1 e2...ee ? of C°-congruences on a neighbourhood

U of x is independent at x if the set of events which can be reached from x by going first along an element of el then along one of e2, etc., is a Co submanifold of U, of dimension n. A set of directions at x, that is, particles through x, is independent if each particle can be extended to a C°-congruence on a neighbourhood of x, and these C°-congruences are independent. Thus we arrive at:

Axiom 4 0) (second dimension axiom):

There are four independent directions (in the above sense) at every point.

Axiom 4(b) implies the existence of C°-congruences. Furthermore, it implies the ability of extending a particle through an event, to a

Cp-congruence on a neighbourhood of that event, and that can be interpreted as saying that there are no gravitational singularities at the event (because the gravitational field is related to the divergence of congruences of particles).

It now follows that:

Theorem 2 1 M is a C -manifold if all the message functions between particles are of class Cl. — 51

Define the set C of differentiable functions on M, by saying that f:M belongs to Cl if: M (i) fit R is of class Cl for every particle y (ii)The derivatives of f along the particles of any C°-congruence

on U (with respect to the parameters on the particles of the congruence) define a C° function on U.

At local coordinates, use radar coordinates together with differentiable parametrizations. Then it follows from Axiom 5 that the local coordinates belong to C1 . In these local coordinates, a coordinate path U is mapped onto an open subset U of R4, and a C°-congruence on U is mapped onto a family of Cl curves in U, with continuously varying tangent vectors. Using Axiom 4(b), together with arguments from elementary calculus, it can be shown that any function is. f'ECM represented on U by a function of class Cl (mapping U to R).

Further any function of class Cl on U trivially induces a function on U belonging to CM Thus, since any function which agrees with an element of CM on a neighbourhood of each point of M must also belong to C. Theorem 2 has been proven. The CN structure on M 1 can now be defined inductively: C -congruences are defined on M and modified versions of Axioms 4(b) and 5 are introduced, and hence the

C2 structure is introduced. The process can be repeated successively, terminating only when N equals the differentiability of the message functions between particles; thus showing that M is as smooth as it looks.

(e) The causal structure on M can be used to define a conformal structure on M (M is now taken to be as smooth as is necessary, although limited by the smoothness of the nessage functions). Consider a particle t f r and a point (event) xtff of M. Choose a particular differentiable parametrization t : .---> R on j, , such that t(x) = 0. 52 —

Now define the function g : Uv -- R by g(z) = t(f+(z) X t(f (z).). Thus, g(z) = 0 if and only if zfU . and z4x or x$►z. That is

and £ztJ/gz) = 0 3. =I+(x) el I-( x) (1 Up . But i+(x) 11 U, I-(x) /7 Ug are three dimensional submanifolds of M, everywhere except at x (because they are everywhere the surfaces on which the. radar coordinates defined by ) are constant). Also g(z) = 0

cannot define a hypersurface at x itself, or there would be points

other than x in every neighbourhood of x, belonging to both I+(x)

and I-(x), thus contradicting Axiom 3. Hence, in local coordinates

at x, g,a(x) = 0 and g,ab(x) defines a tensor. Differentiating

g twice along Y gives gabvaVb = 2, where Va is the tangent vector

to at x, Hence gab # 0. Furthermore, it can be easily seen ff that, up to a scalar factor, gab is independent of both e and t.

Any particle (with given parametrization) through x is a curve whose tangent vector Va at x must satisfy gabVaVb.> 0. Any curve through x which lies in i+(x) or I-(x) must have a tangent

vector n nanb = 0. Any point in a curve through x in I+(x) a, gab is a limit of a sequence of events which can be reached from x by trips, that is, in physical terms, by massive particles undergoing a finite number of collisions. Thus a vector at x satisfying gabnanb=0 is either a limiting (average) velocity for trips through x or minus such a velocity. Now, since in nature it seems to be true that limiting velocities of massive particles are finite, it is reasonable to suppose that no vector na can be a limiting velocity for both future and past directed trips. Hence, we impose:

Axiom 6

gab is everywhere nonsingular. gab is the conformal metric on M : It is defined at all nonsingular points, up to a scalar factor. It is easy to see that the only possible signature of gab is ( + - - -). — 53 —

It will now be shown that, with an obvious definition of light paths, light must propagate along smooth curves which are conformal null geodesics.

Definition VI

(a) (non-local definition of light signals):

If x,y(M, then there is a light signal from x to y if there is a sequence of events zo = x,z1 , z2,...zn = y such that for i = 1,2,...n - 1.

1) and z. for some Nz1 zi-1ENz ENNzi

2) z. 4z. and ziitizi+l and z. #z

(b) A light path L is a connected set of events such that V z(L

theyeis a neighbourhood Uz of z such that Uz (1 L is linearly ordered by 4' and is not contained in any larger subset of

Uz linearly ordered by

Thus there is a light signal from x to y if there is a sequence of points between x and y, with light signals between them according to the local definition and if this sequence does not bend. A light path from x to y is, roughly speaking, the set of all events between x and y through which the light signal can travel. Both these definitions are a little more restrictive than those given by Kronheimer and

Penrose. This is inevitable since the causal structure, defined in our approach, does not in general coincide with the natural conformal structure of M, in the sense of Kronheimer and Penrose, as a conformal manifold (since M may not be globally future and past distinguishing).

In this case the chronology relation is the same but the horismos one is not. E.g. consider the pseudo-Riemannian manifold obtained by removing the set (x0,x1 )/x1 = 0, 0 c xo 4 I from Minkowski I space. Taking the particles as timelike straight lines the procedures given above lead to the correct conformal structure on M. However, — 54 — if zl is the point (-1,-1) and z2 is the point (1,2) then they satisfy zl'+z2, even though there is certainly no light signal from zl to z2 (and furthermore z1 does not belong to a future and past reflecting neighbourhood of z2).

Now we give some lemmas

Lemma 1

Let y9, xENy for some N and Ay. Then, if U is any neighbourhood of y, there exist z(U, with z/y, such that x47 and z y.

Similarly, if W is any neighbourhood of x, there exist z/x in W such that x$z and zy.

Proof: M is locally Euclidean and thus locally compact. Choose 1`y <' U , Ny with yey and Vy = zi/ml 4;., z

4,1 with y

I Ii I(zn) Vy = In is compact and nonempty (since x ! zn and a trip from x to zn must contain a point of I /7 II(zn) /7 Vyo

In is thus a nested sequence of nonempty compact sets. So there is 00 at least one z.' (,I o Certainly zit' and x .e„ z. Let f be n=0 n a message function from U X Vy to ' . Suppose yam<

We now remark that:

1. I+(x) /1 I (y) = z('1/ xi'z and z+iy

2. If z( i+(x)/1 I (y) and xek or ziz4y then 4.'+I (x)ni-(y).

Lemma 2 If x(M,y(Nx for some Nx, and xray, I+(x) aI (y) fl V is a smooth path with endpoint y for some neighbourhood of y. Further- more, light paths are one dimensional submanifolds of M.

The first part means that nearby points are "seen" in a definite direction.

Lemma 3

Vx (M (NA i+(x))r1,x and (Nx 11 I-(x))•t+x are null hypersurfaces in Nx (with respect to the conformal structure for sufficiently small Nx).

Lemma 4 - A light path is a null curve.

Thus we finally arrive at :

Theorem 3

Light rays are conformal null geodesics. We can now define a new horismos relation on M by xDy if and only if there is a light signal from x to y and not x #y. The causal structure of M associated with ' ' coincides with its natural causal structure as a conformal manifold.

From this point the treatment of Ehlers, Pirani and Schild can be used to build the projective, Wyel and metric structures on M. An axiom concerning the relation between nearby gravitational clocks leads to the metric structure and one equivalent to the law of inertia defines the projective structure. If we further wish to introduce a Riemann structure on M, we have to appeal to the more detailed structure of space-time, as expressed with its energy-momentum tensor.

Of course, in the entire approach the axioms become more and more — 56 — complicated and it seems as though unnecessary complications are added to a previously simple theory. But, what really happens is that, primitive geometrical structures are associated with simple physical concepts and more specialized geometrical structures to more sophisticated physical concepts. The process must be thought of as revealing complications that are inherent in the apparent simplicity of the chronometric and geodesic hypotheses. 7—

SUPPLEMENT

(a) (Proof of the statement made in page 46.)

Take as a basis for the topology of Uy , open sets of the form

$ and there is ly such = I+(y1) (1 I-(y2) , where yi,y 2f 1 that yl,y 2e+y. Certainly, if y3 < +(y3) lies strictly y1,y2 > , then f between f+(y1 ) and f+(y2) on e.. (see Fig .12a) . Als o by the same argumen as that used in the proof of the previous statement, every even on 0

lying strictly between f+(y1 ) and f+(y2), is the image under f+ of

some event on Iy lying strictly between y1 and y2. Thus the

> is the open interval in y bounded image under f+ of ` y1,y2 4 by f+(y1 ) and f+(y2). This proves f+ is an open map. The proof

for f- is similar. -1 Let (xl,x2) be an open interval in e . Then f+ ( (xl,x2) ) _

U / 1 I-(x e" ,r i- For if (see Fig.12b) 2/ s. (x1) ), which is an open set in Ili,.

z Ue ( I(x2) n, I(x1 )), then z U x for some xoE~' , f+(z) lies 0 certainly in the closed interval E1 2 3 in Y . Suppose f+(z)=x1; ,x then I (xi) I (z) and every neighbourhood of Z intersects I (xi),

IMIPM0111017011. contradicting not zf I-(x1 )0 Suppose f+(z) = x2; then not

zf I-(X2), which is also a contradiction. Hence f+(ilk• Ī) I (x2),NA

I (x1) ) (xl,x2). Conversely, if z 1Jt and f(z) (x1,x2),

then z ( f+(z) and f+(z), < x2, so z41- (x2). If zEI-(x1 ),

then I-(x1 ) I-(z) and so not fl-(z)4*( x2: a contradiction.

Thus zI). That is z~iJg and f+(z)6 (x1 ,x2) implies

zE ( Ui,f1 I (x2) f' I-(xl) ). This completes the proof that ._...~.. f+( (xl,x2)) (U /1 I(x2)r,I (x 1)), and thus proves that f+ is continuous. Similarly for f-. - 5a _ J

FIGURE 12a

•

FIGURE 12b --59-

(b) We now give the proofs of the unproved lemmas in pages. 5:4,.5 5

Proof of Lemma 2

Choose a particle through y with parametrization y + t: Y-=y R such that t(y) = 0. Define the function g: I (x) -► R by g(z) = t(f+(z)).t(f (z)),zfI+(x). Then I-(y) /] I(x) =

ziI+(x) /3(z) = 0 and z< y . I+(x) and I-(y) can never + cross, so I-(y) /1 I (X) CizfI+ , 0/g a(z) = 0 , where xa are local coordinates on the intersection of a neighbourhood of y with I (x). Now g,ab(y) (a 3x3 matrix) must have rank at least 2, since by axiom 6 gab is everywhere nonsingular (and hence at y). A So g,7=0 and g,2=0, say, define a one-dimensional submanifold L of

I+(x) in a neighbourhood U of y; L contains I (y) /1 I+(x)t1 U.

If U is taken to be small enough, L is homeomorphic with an open interval. By lemma 1, there is a point z#y of Lt1 U such that x 'z y, Suppose there is an open interval Z in L between z and y with 4Z10=> not w4y. Let m be an end point of this interval.

By lemma 1, there are two sequences uni and vn 3 in L which both converge to m and which satisfy, V n, x+un#m, mf vnfy, un,

Now it is not possible that, but for a finite number of elements, both sequences lie on the same side of m in L or L would have a null tangent at m which would be both future and past directed. Thus it is not possible that m is an end point of Z, and consequently Z does not exist. Hence, if L' is the part of L between x and y, then I (y) t1 II-(x)/1L' is dense in L'. But I (y)li I+(x) is closed so L' I (y)I) I+(x) and the first part of the lemmas is proven.

A slightly extended version of the above argument shows that if L is a light path and zEL, with x3z, say, and xiNz fJ L, then in some neightbourhood W of z, L is exactly W/1 I+(x)A(j+(z) (JI -(z)) -6Q_

and that this set is a one dimensional submanifold of W. This proves

the second part of the lemma.

Proof of Lemma 3

As above, it can be shown that if zfOix %J I+(x)) f• x, then

i-(z)/1 I+(x) is a smooth curve in some neighbourhood of z: so

Nx /I (x) must be null at z (provided that Nx is small enough

for I+(X) to be differentiable everywhere in Nx e » x.

Proof of Lemma 4

If L is a light path and zfL, then bylemma 1 there is a

sequence 14: L with AN and xn .. z as n C xn Hence L intersects i+(z) at points arbitrarily close to z and

so must be tangent to I+(z) at z. Thus L must be a null curve.

3) The Causal Boundary of Space-Time

(a) Although causality is regarded to be a basic structure almost

in any discussion of laws of nature, it was normally considered only in philosophical discussions rather than in physical computations or proofs of theorems. This was so because the whole of classical physics is connected with Special Relativity and thus the causal structure, being simple and standard, was uninteresting. This situation is immediately changed if one considers General Relativity in the large. The vast set of possible space-time makes the causal structure a powerful tool for seeking correlations between them. Many physical reasons force one to look at the global behaviour of space-times. For example, it seems hardly possibly to fit irreversible thermodynamics into space- times where some word-lines are loops. Therefore much work has been done in the last years in order to analyze causality. Kronheimer and

Penrose (Proc.Cambr.Phil.Soc. 63. 481 (1967)) described the ordering properties of causality on a very general level without any restirction to a structure induced by a previously defined indefinite metric on the -- 61 —

manifold, and this was in contrast to the classical treatment. As

we have seen in part (A) of this chapter, by a succession of axioms,

it is possible to get the differential and then the topological structures from the causal structure. So it is reasonable to ask if

we can attach a boundary to a general space-time V, using only the causal structure of it. If the answer is positive we shall call this

the causal boundary. Such a construction has been successfully carried

out for special space-times by Penrose (Battelle Rencontres, Benjamin,

New York 1969) and others. In what follows we shall give a construction for general V. We shall also see that in a certain sense V gives the full information about the causal structure of V. This is very different from what is happening with other known boundaries (g-boundary, b-boundary, etc.) where certain classes of curves are identified, and thus lots of information is lost.

The idea of V is to assign a future and a past endpoint to any non-extendible causal line. Several structures of V can be extended to V := V (JV:

(1) V is a pointset defined by giving a prescription as to which endpoints have to be identified (see Section (d)).

(2) A relation J is indtroduced on V which includes the old CO causal relation J} on V. e may be greater than JT (see

Section (c)).

(3) A topology is defined in terms of the ordering relation. It

induces the original topology on V (see Section (e)).

(4) The notion of spacelike, timelike, and null surfaces can be

extended to V.

When we have defined structures on V, we can investigate their properties: — 62 --

(1) Under what conditions does J4- define a partial ordering at

all? If and only if V is stably causal (see Theorem c.2.). oe (2) WheJ'is J+ equal to J on V? If and only if V is

future connected (see theorem c.3.).

We can also consider: what do the various properties of V imply for the structure of V:

How can the basic conditions for the causal ordering of V

(stable causality, existence of Cauchy surfaces, etc.) be characterized in terms of V? (See theorem c.2, c.3).

It should be pointed out that the aim of this discussion is neither to give a generalization of General Relativity based on an abstract causal ordering nor to introduce the points of V as events, however singular or infinite they may be. We nerely examine those properties and structures of V which are correlated to significant properties of V irself. And now we give some conventions and abbrev- iations which will be made in what follows:

J(p) : The timelike and null ("causal") future of p.

I+(p) : The timelike future of p.

C(p) : The null future of p.

J}(g:p) : The causal future of p with respect to the metric gab, c-lines : Non-extensible causal lines (lines ordered by J+).

(b) Whereas the existence of a Lorentzian metric implies uniquely

a local causal (conformal) structure, there are different possibilities for the global structure. One of the following conditions is normally used to restrict the global behaviour:

Time orientability: There exist a continuous division of the light- cones into two classes. (Thus we are able to speak formally of future and past; we do not raise the problem of time arrows). — 63 —

Chronology (Kronheimer and Penrose) : No timelike curve meets any

point twice. (No possible observer has a "periodic life").

Causality (Kronheimer and Penrose) : No timelike or null curve meets

any point twice. (The local causal ordering is not globally violate).

Strong causality (Hawking) : Every point has a normal neighbourhood

which is not met twice by any timelike or null curve. (The topology

of the manifold is induced by the causal ordering; the interiors of

the pasts and the futures of all the points on the manifold form a

sub-base for the mentioned topology (Alexandrov Topology)).

Future (past) connectivity (Kronheimer and Penrose) : The light-cones

bound the futures (pasts) of all points. (The acausal (space-like)

region for any point is simply connected).

Stable causality (Hawking) : Another Lorentzian metric g exists so

that all null curves with respect to g are timelike curves with respect

to g (in the following abbreviated by : g >g ), and g ,g are causal. (For a more precise definition see Chapter I, part (8), page 22 ).

Global hyperbolicity (Leray) : The intersection of any future and any

past is causal and compact. (Then some existence and uniqueness

theorems for hyperbolic differential equations hold).

In order to derive some.intuitive characterizations of these postulates and to achieve a finer classification of the global causal structure it is suitable to introduce a "boundary". As we have said before, for special spacetimes, this has been done by Penrose (Battelle

Rencontres, Benjamin, New York, 1968) and others in order to describe particle horizons, to classify singularities, etc.

The main idea is to assign a past and a future endpoint h(œ), h(-0.) to every c-line h. There are many different possibilities to connect this boundary V with the causal and topological structure of the space-time V: — 64 —

(I) V is for illustration only, there is no connection of structures.

(II) h(o) gets a past only: I (h(c0)) = I(h(t)) (t is a

parameter along .h), analogously I+(h(- o)). The relation I+ restricted to V remains unchanged. Since J+(h(t)) = I+(h(t)) t'+ t for any timelike h, we have to add some null generators to I (h(o)) in order to get a suitable J-(h(03)) rather than to take J (h(t)): r(h(-)) = I (h(o)) 141(t)/I-(1( o)) = I (h(cO)); 1 is a null-line .

(III) The future of h( o) should include 1.rtJ+(h(t)). To save the transitivity we are forced to introduce a new relation J+ whose restriction to V does not need to be J} any longer.

is answered by theorem c.l. below, which is based on two lemmas.

Furthermore, a characterization of stable causality in terms of V is derived and the smallest relation fulfilling the conditions for J+ is constructed.

Lemma I If we define J*(p) = 11 J+(g:p), the following statements i> g are valid:

(1) 3+(p) and J-(p) are closed for all p(V.

(2) Pn -- ► p = / J+(Pn) J+(P) ; ')3- ))3 ~J (P)-() • n

(3) If 3+ is an ordering on V, V is strongly causal.

(4) If J+ is an ordering on V and V is strongly causal, then

V is stably causal.

(5) If V is stably causal, then J+ is an ordering.

Proof

(1) q(J+(p) there is a null geodesic l:lez:' J+(p), oil vio.

if qp there exists an rE 1 /7 J (q),r&q. As for any g) g a neighbourhood U of q exists so that U J+(g:r)~ J+(9:p). — 65. --

qf J+(p) and the statement is proven. Similarly for J (p).

(2) ge113 (p pn6 J (q) ' p• J (q) = J-(q) qt.J+(p) n

and the first part of the statement is proven. The proof of

the second part is analogous.

(3) Assume that J} is an order ing and V is not strongly

causal; we will be led to a contradiction: Indeed, let p be

some point and U be a normal neighbourhood of p which has the form:

- U = J(p+) IA J+ (p ) for some p ,p+eV. Let pngn be a sequence

of maximal pointsets linearily ordered by J+ which are all leaving

and re-entering U so that pn .-* p and qn p. The points of

s re-entrance have an accumulation point r on 6. r can be joined with p by a null curve L. pfJ (r) = J (r) (by virtue of (0)).

The last means that rf J(pn) and by virtue of (2) rfJ+(p); thus

J is no ordering, which is a contradiction. So V must be strongly causal and the statement is proven.

(4) Let U be some set which is met by any c-curve at most once.

Thus, J , modified by widening the light-cones within U, remains an ordering and J+ (p) = J+(p) still holds for any p. This procedure can be repeated for a countable covering of V, thus leading to a global widening of the light-cones. This proves the statement.

(5) Is evident in view of the correctness of (4).

Lemma II

3+ is the smallest ordering, not smaller than J+, with closed

J (p) and J (p) for all poi.

Proof: Assume there is a relation J'+ : J+(p)4 J'+(p) = J'+(p) J+(p)

and for some p there is a qfJ (p) - J'+(p), p and q lie spacelike to each other with respect to the closed, hence stably causal +. relation. J' Therefore we can stick together some neighbourhoods of p and q without destroying stable causality. But after this the

— 66—

modified space becomes acausal with respect to J : J+(p)g: J+(p),

which is a contradiction. Thus the lemma has been proven. Similarly A for J. Theorem c .1.

The smallest ordering on V (if there is any) which is not

smaller than J+ and fulfils condition III of Section (b)) is

(h(t),k(t) denote parametrized c-lines):

el eJ+(p):= .s J+(g:p) t k(co)j 3 k(t)(J+(g:p) 1E k(-00) j g )g V k(t)6J+(9 p)] pV

J+(h(')):= 'J+(h(t))

1+(h(-00)):=,,11 J+(9 h(t)) V l'k(oo) 3 for g g

some u3 V k(-,o) ! Vk(t)EJ+(g:h(u)) for some u

Roughly speaking is the intersection over all orderings induced

by metrics 4.4 d, g, if one adds a future (past) endpoint of a c-line k,

if J f V contains at least one point (all points) of k. Proof: The demanded smallest ordering must have closed intersections

with V: consider a null generator I of J+(p): either

1 meets p (4 1 , J+(p)) or one can find a timelike k which

lies in J+(p) and goes asymptotically to 1 (=* I+(k(-oo)) = I+(1(-era))

V1141. 1 JJ (k(-0)) and k(-co) < J+(p) t. le:J+(p)), Evidently co J+(p) V=J+(p) (l V (pg and J+ is the smallest extension of If

onto V. Therefore the theorem is proven (by virtue of lemma II). co Now we can ask under what conditions an ordering J+ does

exist? The answer is provided by the theorem below:

Theorem c.2. A On an extended space-time V one can introduce a partial

ordering J,+ if and only if V is stably causal. — 67 —

Proof: Theorem c.1 above establishes the equivalence of the existence 4:0+ of J to the existence of J on V. Then lemma 1 (step (5)

in particular) establishes the equivalence of the existence of J+ on

V to V being stably causal, thus proving the theorem.

At last one can ask under what conditions the introduction of m J does not change the ordering J

Theorem co3„ 00 The equation J+(p) it V = J+(p) holds for all p if and only

if V is future connected.

Proof: As a consequence of lemma II the above equation is equivalent

to J (p) = J+(p) for all pe But this holds if and only if

the condition of future connectivity : (J+(p))° = C (p) holds for all p.

(d) Bearing in mind that we are mainly concerned about V and that

the construction of V should essentially give us information about V, there is no reason to distinguish between endpoints whose futures and pasts, restricted to V, are identical: 02 co 40 ao (x) p=q 441 J+(p)/) V=J+(q) /1 V and J-(p) /! V = J (q) I.V 444 This holds for any p,q~11, as the intersection of J- with V deter- v.+ mines uniquely what endpoints belong to J-. (x) can be chosen as a suitable convention for identifying two points of V. Without this $0 identification the property of J+ to be a partial- ordering is lost.

As for the possibility of other, stronger identifications, we remark that it is very limited. Indeed, p)(0,...) and k(co) cannot be 00 oto identified if J+(h(oo)» JJ+(k(4,0)) without destroying either the property of J to be the smallest relation in the sense of theorem c.l or the transitivity. By the same argument, h(o.) and k(-40) cannot 00 CO 02 0, be identified unless: J+(k(-, ) c Jt(h( 00)) and J (k(-oo)) 24"(h({4)),

With respect to (x) we have the following: 68

Theorem d.1. co After the004. identification (x), J+ is a partial ordering. Proof (,a): p4 J' (p) f -(p) and the transitivity are evident. 410, t (b) p = J+ (p) f J (p) is evident for any p,~VV , and- for p(V this is a consequence of: J-(h(°°)) = J (k(a0)) .= J+(h(oo)) = J+(k(ao)) and of : k(-4:4,) EJ- Ch(**)) before identifying some endpoints on V implies that 00 oa J- (k(-00)) is a proper subset of J (h (oo)) . We now give some examples which will clarify matters:

Example I . V is the two-dimensional Minkowski space with the line x=0, t 0 removed. (See Fig.13). We define h-:= ct= :12x; t < and k:=1 x=0; t, 0 p

Now we have: -4 ~+ J (h-(40)) = J (k(-m)) = Q;0r1x.

J (h±( )) _ {t <0; 0; . ~< t2 x2 _ J (k(-1o)) =[t <0; x # 0; t2, x2 .

FIGURE 13 — 69 —

+ h ( 4$,), h( 00) and k(-c) are different points in the sense of

(x), but we would not get into any trouble if we identify all of them.

Example II

Remove the origin of the two-dimensional Minkowski space and

take the 3-sheeted covering (three sheets: 1,2,3; along the positive

x-axis they are cut and glued together in the following way: The lower half-plane of 1,2,3 with the upper half-plane of 2,3,1, correspondingly).

Remove it> 13 of 2 and 't ( -13 of 3 and identify these edges. This space is stably causal. The endpoints of the lines Ex = 0 are I six different points in the sense of (x). This time we do not cure the pathology, as in the previous example, by identifying all of the six points. In fact, matters are getting worse if we do this, because on then J+ is acasual or not well defined.

Example III

Remove Lx = t; x> 0 from two-dimensional Minkowski space and identify both edges with the long edges of the subset U = fx. t.x + 1; -x 03 , but these are obviously separated by the set fitted in. xe m OYed

FIGURE 14 — 70 —

Examples II and III above indicate that a convention flo 00 OG 00 (xx) P=CI <40 J+( p) /7 v= J+(q) /1 V O or J-(p) /I V.J-(q) / V# m cannot be used for identification of points in V, although is valid on V,

Another remark is that by extending other structures, e.g. metric structure, it often happens that one gets boundaries which are not in I-I correspondence to our causal boundary. So the metric boundary of the causal part of the Taub-NUT space, for example, consists of closed null lines which have to be identified in odder to • get the causal boundary. This metric boundary is neither conformally invariant nor leads to a causal space, but one needs it to get the important analytic extension. In many cosmological models the "big bang" is metrically a point, but causally a pointset. This is reflect- ed in the fact that, in these models, if one adds even one point, which. is metrically not a radical alteration, one loses all information about particle horizons, which is a severe change in the causal structure of the boundary.

(e) We now proceed to introduce a topology on V. One can find

some rather simple definitions for topologies on V considered as an ordered pointset, but then it proves extremely difficult to prove some simple properties. This may be caused by the fact that V is incomplete from the viewpoint of the theory of ordered sets: c-lines on V may have no endpoints. So we shall introduce a topology by the same procedure we have used in extending the relation the sets I+, I form a sub-base for the topology of V. After having defined I+(h(-00)) _ /I+(h(t)) and I-(h(+o)) = I (h(t)) according to condition II, Section (b), we add an endpoint q to a + 1 set I (p) (*), if and only if any c-line with q as endpoint is contained in I-(p) from one point on. -- 71 -

(xxx) The family of all sets: (p V) I'+(p) := ~ q~V ~ h(co) = q t Vt?. tl h(t),EI+(p}1

I'-(p) := 41/41 h(-co) =q -- t1ER: Vt< tl h(t).6I (p) A form a sub-base for the topology of V. Note that h(0o) may be in V.

Now we can prove some nice properties:

Theorem eal 00 Any J+(p) or J-(p) (pill) is closed. OP Proof: The complement of J+(p) is equal to the union of all I' (q) Ave + where q (p), or q lies on a null generator of the boundary

of J'(p) (q may be an endpoint of such a null generator). So the

complement of J (p) is open. Thus J+(p) is closed and the theorem ao is proven. Similarly for J-(p).

Theorem e.2 A V has a countable base.

Proof: V has a countable dense subset pno For all pEV:

I` .EI` (p) , i.e. the I'}(pn) form +(p) _ (Ī ~I' + (pn )I pn +

a countable sub-base. The set of finite intersections of a countable

family is countable: a countable sub-base generates a countable base.

So V has a countable base, and the theorem is proven.

Theorem e.3

Let pn p, then the following are true:

(a) `Ī n J+(pn)4%3 +03) and V a J (pd (p) v=1 n=v v=1 n=v

(b) ēi lē J+(pn) C J+(P) and ~ I/ J-(pn) G J-49 v=1 n=v v=1 n=v

(c)t) f I'+(pn) 7 I`+(p) and V n ic (pn) ,›I'-(p) v=1 n=v v=1 n=v

(d) VP-1-(pn ) I'+(p) and 4 VI` (pn)::,I1 -(p) v=1 n=v v=1 n=v -72 —

Proof: (c) .:4. (d) and (b) (a) are evident (De Morgan rules).

A violation of (b) world imply that for any metric -g) g there are infinitely many pn elements of the g-past of a point q outside the g-future of p. Even if pn, the closure of I' (q) contains infinitely many pn, which leads to a contradiction because

J+(p) is closed (Theorem e.1) and q (p)., Thus (b) is correct.

A violation of (c) would imply that for some q(I'}(p), I' (q) does not contain infinitely many pn, yet I' (q) is a neighbourhood of p. This is a contradiction, since pn .._4.p, and thus (c) is correct. We may remark that the converse is also true in the most important cases: when NV or the pn are lying on a c-line. As an illustration take Example 1 of Section (d) (see Figo13): for the sequence pn:= 5t=o; x= - 1 , p:= e(o) (a),(b),(c),(d) above hold, but pn . does not converge to p, because f t < x 40 3 L fl(oo) i 1:= t=x-a; a e.;" 03 is a neighbourhood of p which does not contain any pn.

'FIGURE 15 -73=

Theorem e.4

V is a Hausdorff space, ca /44 Proof: By definition of ~J{ I'+(q) G I'+(p) J+(q) J+(p)„

We cannot have pa,* p and pn q, because the formulae of theorem e.3., cannot, in view of the above, hold simultaneously for p and q, if p#q.

Theorem e.5

There exist a function r on V which is strictly monotonic go on each set linearly ordered by J+, and which is continuous along each .-1 ine.

Proof: In view of theorem c,2 in Section (c), which establishes the

equivalence of the existence of J+ as an ordering on V and stable causality on V, the proof of the above theorem is given by the proof of the existence of cosmic time-functions in Hawking S.W. (1968),

Proc.Roy`Soc. A308, 433.

Relevantly, Examples II and III of Section (d) show that it is generally not possible to find a continuous time-function on V.

(f) On our space V we can introduce the concept of spacelike and

timelike sets in a very natural way:

Definition O CO

J is timelike in pfV4 J+(p) /1 VAS J (p) /1 V O V is past spacelike in p , J (p) = p 4. m V is future spacelike in p.•- J+(p) = p 44 V is null otherwise.

Now we can characterize some causal properties of V:

Theorem f.1

V is globally hyperbolic if and only if V contains no

timelike point. 00 00 Proof: Assume the contrary: consider a qiJ+(p) and an rgJ (p). ?_4 -

J-(q)il e(r). is not compact (in V), This contradicts the global hyperbolicity and thus proves the theorem.

Corollary:

A space is globally hyperbolic if and only if there exists a time function r so that any non-extensible line intersects any surface

(r=constant) (global Cauchy surface).

This well known result turns out to be a simple consequence of theorems e.5, and f.l. - 75 -

REFERENCES

Ehlers, J., Pirani, F.A.E., Schild, A. (1972) "The Geometry of free

fall and light propagation" in General Relativity,

Papers in Honour of J.L. Syng, Oxford U.P.

Geroch, R.P. (1967), Thesis, Princeton University.

Hawking, S.W. (1971), "Stable and generic properties of space-times",

G.R.G. Journ., 1, 393.

Kronheimer, E., Penrose, R. (1967), Proc.Camb.Phil.Soc., 63, 481.

Kundt, W., Hoffman, B. (1962), in Recent Developments in General

Relativity, Pergamon Press.

Marzke, R., Wheeler, J.A. (1964) in Gravitation and Relativity,

Benjiamin Press.

Penrose, R. (1969), Battelle Rencontres, Benjiamin Press.

Zeeman, E.C. (1964), Journ.Math.Phys., 5, 490. - 76 -

CHAPTER III

SPACE-TIME b-BOUNDARIES

A) Definition and Properties

(a) One central issue of General Relativity theory is connected

with singularities. Since the early days of General Relativ- ity it was known that certain solutions of the Einstein's field equations possess "singularities", although the meaning of the term was somewhat obscure. Certainly from the term "singularities" were excluded the so-called "co-ordinate singularities", occurring because of a bad choice of the coordinate system. Initially, it was thought that perhaps the existence of singularities was a direct consequence of the symmetry assumptions and thus they would disappear in more realistic solutions. Indeed, all theorems, predicting the occurrence of singularities, before 1965 were involving symmetry assumptions.

In 1965, Penrose gave the first theorem (the collapsing star theorem) predicting the occurrence of a singularities and not involving symmetry assumptions. Since then a number of other important theorems

(Hawkings 1967, 1966; Geroch 1968) established the occurrence of singularities under a fairly wide variety of conditions. It became thus certain that singularities were a quite general feature of solutions of Einstein's field equations. The next logical step was the development of a genuine definition of a singularity. That the very definition of a singularity in General Relativity involves difficulties unknown in other branches of physics is well known and will not be re-discussed here in detail (see, for example, T. Christodoulakis,

D.I.C. Thesis, Imperial College, 1977, Chapter III, part A). Although geodesic incompleteness is useful to be taken as a definition of a singularity (in fact all the above mentioned theorems prove geodesic — 77 —

incompleteness), it is nevertheless inadequate. This is so because there are geodesically complete space-times which contain inextensible curves: of bounded acceleration and finite length. (Such an example is given in Geroch, R. Annals of Physics, 48, 526). It is reasonable to consider these space times as singular although they are g-complete.

The aim however was not only to define singular and non- singular space-times but above all to localize singularities in such a way that statements like "the curvature goes to infinity near a singularity" become meaningful. If there was a reasonable way to do this, singularity would cease to mean essentially "something goes wrong" and would become a term which carries information about the nature of the breakdown. To achieve this, similar constructions were carried out by Geroch (Geroch, R. (1968) J.Math.Phys. 9, 450) and

Hawking (Hawking, S.W. 'Singularities and the geometry of space-time.'

Unpublished essay submitted for the Adams prize, Cambridge University,

1966), essentially along the following lines: given an incomplete space-time, a topological space-the g-boundary - is defined, whose points are equivalence classes of incomplete geodesics. Space-time together with its g-boundary is a topoligical space and points of the g-boundary may be thought of as singular points of the space-time.

There were, however, two main points of critisism. Firstly, the construction was based on geodesics, which was problematic as mentioned above. Secondly, the way of forming the equivalence classes`was not uniquely determined. B.G. Schmidt (G.R.G. Journal, Vol.I, No.3

(1971) page 269) proposed a new boundary which did not suffer from the above mentioned defects. Also, this boundary seems to be the natural generalization onto spaces with indefinite (Lorentzian) metric, of the usual boundary defined via Cauchy sequences (Cauchy completion) for spaces with positive definite (Riemannian) metric. This approach — 78 —

to singularities is based on the structure in the bundle of linear frames L(V4) over a space-time V4 induced by parallel displacement in the space-time. It is a well known result of Modern Differ- ential Geometry (see, for example, Hicks, N.J. (1965) Notes on Differ- ential Geometry. D. van Nostrand Company Inc., Princeton, New Jersey) that a connection in a manifold gives a natural parallelization of the bundle of frames, which contains all the information about the connection. This parallelization can be used to endow L(V4) with a

Riemannian metric. This, in turn, determines on L(V4) a metrical structure (in the topological sense) and thus we can find the unique completion L(V4) via Cauchy sequences. To "project the boundary down" one shows that the action of the general linear group G1(4,R) on L(V4), mapping a frame at a point of V4 to another frame at the same point, can be extended onto L(V4). The set of orbits of

G1(4,R) in L(V4) is the b-boundary V4 of V4(b for bundle). A unique topology is determined on V4 and V4 by the requirement that the projection map should be continuous and open.

This boundary certainly contains the g-boundary, as one can easily find that incomplete geodesics determine boundary points. But it is "bigger" that the g-boundary in the sense that incomplete curves other than geodesics also determine boundary points. More precisely: an inextensible curve in V4 determines a point of the b-boundary V4 if and only if its "Euclidean Length" measured in a parallely propagated frame is finite. An equivalent formulation is that y defines a point in V4 if a curve in Minkowski space whose tangent vector has the same components in a parallely propagated frame as that of is extensible. Which of these curves determine 6 the same point in V4 is difficult to answer. In fact, the identti- fication problem will be dealt with in Part B of this chapter. What

— 79 —

one can see right from the beginning is that the identification via

bundle is the most natural within this framework, since, as we shall

see, the b-boundary of a Riemannian space coincides with its usual

metric boundary.

One could call all points of V4 singularities. But this

would result in calling singularity a regular point of some extension

of V4. Therefore Schmidt proposed: Definition. A singularity of

V4 is a point of the b-boundary V4 which is contained in the

b-boundary of every extension of V4. Of course, it is understood

that the above definition will be practically useful only if criteria

can be found which allow one to make the distinction implied by the

definition without carrying out the extensions. Also the above

approach to singularities, although elegant and fitted for general

discussions, proves difficult to apply in particular examples.

(b) Since, as we have seen in the previous section, the concept

of linear connection is fundamental for the construction of

the b-boundary, we shall present a brief discussion on linear connect-

ions in this section. The essential idea of the structure "linear

connection" is the concept of parallel propagation of vectors. In

Euclidean space parallel transport gives an isomorphism between the

tangent spaces of any two points. A linear connection on a manifold

also defines an isomorphism between the tangent spaces of any two

points, but in this case the isomorphism is path dependant. This

property, together with the conditions that the isomorphism is

independant of the parameter used to parametrize the path, defines

the linear connection.

In spite of the fact that this definition is nearest to the

geometrical idea, the approach via the existence of a covariant

directional derivative is more suitable for what follows. In this — 80 —

content a linear connection is defined by an operator ' which

assigns to each pair of vector fields X and Y on a manifold M

another vector field VXY satisfying certain well known proper-

ties (see, for example, Hawking and Ellis, The large scale structure

of space-time. š2.9).

Yet another description of a linear connection is given in

terms of the bundle of frames. As it is exactly this description on which all the results of this chapter rely, it will be developed in the rest of this section, as far as necessary for the subject.

Let M be an n-dimensional manifold (The Hausdorff property is included in the definition of a manifold. Also all structures are considered to be C 454 ). A linear frame u at a point x of

M is an ordered base Xi,.. Xn of the tangent space Tx(M) of x.

On L(M), the set of all linear frames at all points xfM, the general linear group Gl(n,R) acts on the right in the following way:

(ak 1 )(Gl(n,R) maps the frame (Xi) at x into the frame (ak1 Xi) at x, i, k=1,2,...n. T;he projection f:L(M) „Jo M maps the frame at x into x. A differentiable structure is induced in a natural way on L(M) from M, such that L(M) becomes a principal bundle over M with structure group Gl(n,R).

Now, let us suppose that a linear connection is given on M.

A curve u(t) in L(M) is called horizontal if the frame X1(t)

Xn(t) are parallel along the curve n(u(t)), with respect to the given linear connection. From the properties of parallel displacement in M we deduce that the tangent vectors at u of all horizontal curves through ufL(M) form a vector subspace Hu of dimension n of the tangent space Tu(L(M)) at •u. Since parallel displacement in M is a vector-space isomorphism, it is obvious that the distribution H=(H uEL(M) a T(L(M)) is right invariant under the action of u) — 81 — the structure group G1(n,R). Also, since GL(n,R) acts transitively and freely on the fibres 11 4(x) of L(M), H contains no vector besides the null vector tangent to some fl-1(x),0. Thus H defines a linear connection on L(M).

Using the linear connection given on M one can define certain vector fields and one-forms on L(M). The standard horizontal vector fieldsBi,i=l,...n are defined as the unique horizontal vector fields satisfying nx((Bi)u) = Xi if u=X1,X2, ... Xn. (b.l) = 31, 1 More generally, if 2,... Rn,then B(1) is the unique horizontal vector field with

d x((B( )u) = lltXi if u = X1,X2,...Xn (b.2)

Clearly, for g# G B(š) never vanishes on L(M). From the definition it follows that every integral curve of B(s) is projected onto a geodesic in M and that every lift of a geodesic in M is an integral curve of some vector field B(s ) on L(M). The action of CL(n,R) defines vertical vector fields (independent of the connection) corresponding to elements of the Lie algebra gl(n,R) in the following way: If E(gl(n,R), it uniquely defines a 1-dimensional subgroup a(t) of Gl(n,R). If we denote the action of Gl(n,R) on L(M) by Ra, a unique vertical vector field is defined on L(M) by + = --- (b.3) (E)u ( dt Ra(t)u)t = 0

The mapping E ,--3 E is an isomorphism from the Lie algebra gl(n,R) into the Lie algebra of vector fields on L(M), since as Eki mentioned above Gl(n,R) acts freely on L(M). Let be a basis for gl(n,R). Then, it is obvious that the vector fields Eki,Bi define a parallelization of L(M). - 82 -

The 1-forms dual to these vector fields are defined as

follows: the connection 1-forms W are given by: k~ + Xv = W k(X) Eik (b.4)

where Xv denotes the vertical component of X. Independent of

any particular basis of gl(n,R) a 1-form W with values in

gl(n,R) is defined as

(0(X) = Wki(X) Eik (b.5)

The horizontal subspace is characterized by

f o (X) = 4 4;►' X f Bu (b.6)

We can also define the canonical 7-forms ei as follows:

ft (Xu) = ei(Xu)X1 if u = X1,X2,...Xn (b.7)

Again, independent of a particular basis, a vector valued 7-form

9- with values in Rn is defined by

i e=e ei (b.8) where ei is the natural basis of Rn. As mentioned, the vector fields and '•forms are by definition dual in the following sense:

e1 (Bk) = o (b.9) kl Wk (Br) =

1 rik , e (Ek ) = 0 UI 1k(EiJ) _ (b.lo)

The vector fields and I -forms have the following transformation properties under the action Ra of the general linear group Gl(n,R):

(R ) E = ad(a-1)E (Ra) {, ad(a- ) (b.11) a

-1 (Ra),(B(, ) = B(a ) (Ra)Mo = a-~e (b.12)

- 83 -

These relations are immediately deduced from the definitions: for

example: B( 'g )u is horizontal at u, then (Ra)xB(1;) is

horizontal at Rau, thus it exists such that (Ra)xB(s )u

B(T ')R u Also, since the distribution H is invariant under a Gl.(n,R), we have x(B( )u) = C, (B(I')R u). From these two we a obtain X = ' 'kak1 Xi if a = (ak').and therefore f' = a-1

The qualitative result which will be used later is the fact that

(Ra )x E.' and (R ) B. p ,s .1 and B J axi are linear com inatior crEJ l.

with coefficients constants on L(M).

The next step in developing the theory would be to express

d kz, de' in terms of the fundamental 1-forms and thus to define

curvature and torsion. This can be found in Bishop, R.L. and

Crittenden, R.J. (1964), The Geometry of Manifolds, Academic Press,

New York.

define a Riemannian metric

g (X,Y) = ei(X)s-i(Y) + Gc1ki(X) Wki(Y) i i,k

on. L(M). In other words, g is the Riemann metric with respect +. to which E.', Bi are an orthonormal vector field basis for L(M).

Let us suppose L(M) is connected; otherwise we can take a connected

component of L(M). The Riemannian metric (c.l) defines on L(M)

a distance function (u,v) which is equal to the infimum of the p length of all piecewise differential curves with endpoints u and v

(the length is taken with respect to the metric (c.1); also if L(M)

is not connected, the two components are isometric). Therefore L(M)

is made into a metric space in the topological sense. That is, the

open Balls B(u,r):= viL(M) f fp (u,v) < r constitute a -- 84 —

basis for the topology of L(M) induced by the topology on M. Like any metric space L(M) determines a unique (up to isometries) complete metric space L(M) in which L(M) is dense. Points of L(M) which do not belong to L(M) are equivalence classes of Cauchy sequences in L(M) with no limit in L(M); more precisely, if un, vn are

Cauchy sequences in L(M) with no limit in L(M) (that is, there is no wfL(M) such that P (un,w) = 0 and ?(vn,w) = 0 as n .....1► co), they belong to the same equivalence class if p (un,vn) = 0 as

The set of these points is denoted by L(M) and is called the boundary of L(M) in L(M). To get a boundary for M one has to extend the action of Gl(n,R) from L(M) to L(M).

Basic to this extension is the following:

Lemma c.1

The bundle metric (c.l) satisfies

0 ( W1(a)g(X,X) g((Ra)xX,(Ra)xX) ~~ W2(a)g(X,X) (c.2)

Proof: Because of the compactness of the space of all tangent

vectors with g(X,X) . = 1 the result is trivial even if W1 ,W2 depend on the point uEL(M) as well as on a. Moreover, using the qualitative result that Ej1,Bi transform as (Ra)xEj1 and (Ra)xB. respectively with coefficients constant on L(M) (see

(b.11) and (b.12) above) we deduce that W1, W2 are constants on

L(M). This in turn leads to the conclusion that

Cl p (u,v)

Om. be denoted by Ra o Thus Gl(n,R) is a topological transformation — 85 —

.011010110110 group on L(M). Now we can get a boundary for M. Let M be the

01001/0101. set of orbits of the transformation group Gl(n,R) on L(M) and It

the mapping from L(M) onto M which maps a point in L(M) onto the

orbit of Gl(n,R) through this point. On M a topology is defined as follows: 01601010.11, a subset 0 of M is open if and only if r774 (0) is open in L(M) (this is the finest topology in which 0 is continuous). Obviously fl(L(M)) is isomorphic and thus can be identified with M. The set

O - M = fl(L(M)) = M - M is called the "b-boundary" of M determined by the linear connection.

The space M = M Lrm is called "manifold with b-boundary". The only remaining drawback is that the above construction is based on the metric (c.l) which is not uniquely determined by the connection since it remains the arbitrariness in the choice of the basis for gl(n,R) and Rn defining Edi and Bi respectively. But all the distance functions, determined by different choices of the above basis are uniformly equivalent, and this removes the drawback since the different M and M are identical (up to isomorphisms).

(d) The definition of the b-boundary was presented in the previous

section, in a quite formal way. Its importance will be best understood by means of an interpretation of the boundary points given in this section.

Since points of the b-boundary of M are equivalence classes of Cauchy sequences in L(M), an interpretation of the length of a curve in L(M) is needed in terms of structures on M. For the case of horizontal curves this is fairly obvious: let u(t), tE f0,1J be a horizontal curve in L(M) and ū(t) its tangent vector, then with respect to the bundle metric (c01), the quantity — 86 —

n L = ( oi(u)e161))1" dt (d.l) 0 1=1

is the length of u(t), since Wu) = 0 (because u is horizontal).

If u(t) is the frame X1 (t),....Xn(t) parallel along x(t):= fl (u(t))

then by definition of the ei

x = ff(u) = e'(u)X. (d.2)

i That is to say e (u) are the components of the tangent vector x

of x(t) in the frame Xi(.t),i=1,...n. The length (d.l) can there-

fore be thought of as the "Euclidean length" of x(t) measured as if

the frame Xi(t),i=1,...n were orthonormal. Since it was shown in

section (b) that the lift of a geodesic in M is a horizontal curve

in L(M), the above interpretation proves that incomplete geodesics

do determine points on the b=boundary. It is also obvious that other

inextensible curves also define points of the b-boundary. In fact a

general characterization is as follows:

Consider an inextensible curve x(t) in M and a frame u(t)

parallel along this curve; construct a curve (t) in Rn (endowed

with the trivial connection), and a frame u(t) parallel along the

curve (t), such that x(t) has the same components in the frame

(t) as x(t) does in the frame u(t). Such a curve (t) always AJ exists and clearly x(t) defines a point in M if and only if x(t)

is extensible in Rn.

If the connection is induced by the Lorentz metric of a space- 4 time V then the background for constructing (t) is not Rn but

Minkowski space and it is obvious that for example timelike curves of

finite total length and bounded acceleration determine points of O.

It is obvious. from the interpretation of the boundary points given

above that : — 87

Theorem del

If the bundle metric is complete, then the connection is geodesically complete.

The reverse, however, is not true as the previously mentioned example, given by Geroch, shows.

The following theorem shows that we can concern ourselves exclusively with horizontal curves, when seeking for points of the b-boundary;

Theorem d.2

If ivni• is a Cauchy sequence defining a point of the b-boundary, then in the equivalence class of {vn there is at lease one more Cauchy sequence , lying on a horizontal curve.

Proof: Since {vn)'determine a boundary point, there is a curve v(t),

titG,i) with vn = v(tn) and lim to = 1. There is no to with the property that v(t) is contained in one fibre for t to, since this would imply that has a limit in that fibre, as the fibres are complete (the fibres are homogeneous spaces and as it is well known a homogeneous Riemann space is complete). Thus x(t) = IT (v(t)) is an inextensible curve in M. Let uo be a frame at x(0) and u(t) the horizontal curve with u(0) = uo and

AO f`(u(t)) = x(t); choose a frame uo at the origin of Rn (endowed with the trivial connection), and construct a curve x(t) in Rn whose tangent vector has the same components in a frame parallel to ūo as

X(t) does in the frame u(t). Consider the following mapping :

0: 17-1(x(t)) -- 11_1 (x(t)) defined by m(u(t)) = u(t)

W(Rau(t)) = Rad(t) (d.3) where obviously ū(t) is the lift of x(t) through 0 and Ra denotes the action of Gl(n,R) on the bundle of frames over Rn. — 88 —

Then, it is clear that 0 is an isometry of the Riemann metrics

induced by the bundle metrics on the fibres f-1(x(t)) and - rj (x(t)). Indeed, it is obvious that W maps the vector fields

nv i Eji on J -1 (x(t)) onto the corresponding vector fields E. on

r/-1(x(t)) and u(t) isometrically onto u(t). From the above N and the fact that (Ra)x and (Ra)x act in the same way on u(t)

and ū(t) follows that @ is an isometry. More precisely: if 00 U(t) = BIM) then ū(t) = B( (t)) by definition of u(t), and

the transformation properties of B(š), B(š) (see Section (b) -1 equations (b.11) and (b.12)) imply (Ra)xu(t) = B(a !;(t)) and

(Ra)xu(t) = (a-1 (t)). Hence the vector field B(a-1 (t)) on

Tā-1(x(t) is mapped by ® isometrically onto B(a (t)) on

ti 1 (x(t)). By the definition of x(t) we see that v(t) is if contained in the fibres 17-1 (x(t)) and of course has, by hypothesis, finite total length. By means of the isometry @ we deduce that

(t) has finite length in rl-l( (t)) and thus (t) is extensible

in Rn. Thus, by definition, u(t) determines a point of the b-boundary. Hence, the required Cauchy sequence is tit= :u(tn) and the theorem is proven. We can also prove that:

Theorem d,3

The b-boundary of a Riemann space is identical (up to homo-

morphism) with its metrical boundary.

Proof: As we will see, (see page1O3 ) it is sufficient to consider the

sub-bundle of orthonormal frames 0(M). 0(M) is endowed with a

distance p0(u,v) which is the restriction to 0(M) of the distance p(u,v) defined on L(M) via the bundle metric (c.l). By (d.l)

we see that every horizontal curve in 0(M) has length equal to its

projection and every non-horizontal curve has length longer than its

projection. If we denote by r the distance function determined — 89 —

on the Riemann space itself by the positive definite metric, then it

is clear from the above that P0(u,v) _ ~'(x,y) where of (7-1(x) and 71 v( ( (y), holds for every x,y in the Riemann space. This, however,

implies that there exists a homeomorphism between the b-boundary of the

Riemann space and its metrical boundary, and thus proves the theorem.

The above theorem justifies our previously made assertion that

the b-boundary stands for a generalization of the Cauchy completion of

Riemann spaces. 11 (e) Let us now consider some properties (Topological) of M=M M.

1) M is second countable. proof: L(M) endowed with the metric (c.l) is a Riemann space, and as such is paracompact and satisfies the second axiom of countability (see

for example Kobayashi, S. and Nomizu, K. (1963), Foundations of Differ- ential Geometry, Vol.1, page 271, Interscience, New York). By construct-

ion, this is also true for L(M) and thus for M as well.

2) M is connected, locally connected, and arcwise connected. 06.11..... Proof: This is so because L(M) stares these properties.

3) The topological space M is in general not locally compact as

the following construction shows (see Fig.l6):

FIGURE 16 — 90 —

Example A: Take R2 with its Euclidean metric. The set

A = (x,y) I Y= sinl,x#0 V (Oa) 1 fl 4 j3/1.c‘ is obviously closed, and therefore M:=R2-A is a manifold with a positive definite, flat metric. Let us consider the point p:=(0,2).

Let us call M°the connected component of M containing the point p.

Since M°is a Riemann space (has a positive definite metric), by virtue of Theorem d.3. we can find its b-oboundary via Cauchy completion.

Thus we get ° M = (x,y) ~ y = sini,x#0 II fp,-1)1

Obviously, the point (0,-1)(M° has no neighbourhood with compact closure, and thus M° is not locally compact.

The separation properties of M are also very interesting because they are related to the notion of "causal singularities" (see 'The causal structure of Singularities', H. Seifert, Lecture notes on Mathe- matics, Vol.570, page 539, for further reading). L(M) is always

Hausdorff, since it is a metric space. It is obvious from the construction of M that when M has a positive definite metric, then. M is also Hausdorff (it is homeomorphic to L(M) as theorem d.3. implies).

For the case of an indefinite metric on M, we have:

4) M is not in general Hausdorff (a topological space is Hausdorff

or a T1-space if for any two different points of the space neighbourhoods of these points can be found having empty intersection).

Proof: We shall prove this by means of a counterexample.

Example B: Consider the flat metric ds2=exdx2 on R1. It is clear that the linear connection on R1 to which this metric corresponds is incomplete.0 Indeed, the length of the geodesic from x=0 to = - Oa is L = ex/2dx = 2. The mapping x --y► x+l is an affi nē --'.._trans_ formation. Taking x modulo 1 we get an incomplete flat connection on the circle Si,. The bundle of frames L(S1 ) (or better a .connected — 91 —

component of it) is differomorphic to S1 XR1 and every fibre is incomplete (see Fig.17).

sxR7

FIGURE 17

It is clear that all incomplete curves belong to the same

class, that is define the same point of the boundary. Thus the b-

boundary of S1 with this connection consists of just one point, say w.

Now M`S1 w and clearly M is not a Tl-space (Hausdorff).

Indeed, if we take the pair of points (x,w) with x being any point

of S1 then, as M is the only neighbourhood (open set) of w, we

see that there is no neighbourhood of w not containing x. M is

however a TG-space, that is a topological space in which one of any two points has a neighbourhood not containing the other. With respect

to this the following holds:

Theorem e.l

Suppose { un is a Cauchy sequence in L(M) without limit in l L(M) such that fl({unl)is contained in a compact subset of M. Then there exists x(M such that f l(x) is incomplete and M is at most a 11)-space.

— 92 —

Proof: Since n- {un}is contained in a compact subset of M,un has an infinite subsequence iun i such that lim fl (unf.= xEM. Suppose. n+ ~o -1 17 (x) is complete, hence closed (in L(M)). Then p (un , j~ -1(x)) is always finite (in fact positive) and we can find points vnj 11-1(x)

with (un , -1(x)) = ,vP (un n). But lim (un) = x which 0 n n..~►+~ca

together with the previous relation implies lim 17(un,vn) = 0. This n40,104 means that ivn andiunil(and thuTn1)belong to the same class, i.e.

define the same boundary point. But, by hypothesis ~ un ~ has no limit

in L(M), so, this is also true for i vn}. But vntz fl-1(x), and this is a contradiction since -1(x) is supposed closed. Thus fI -1 (x) is not closed in L(M) and thus is incomplete.

This completes the proof of the first part of the theorem.

For the second part let f'l -1 (x) be an incomplete fibre and v a -1 limit point of a Cauchy sequence ivni in rt 00 without limit in

1 (x). Every open set in LL(M) containing v also contains vn I for some (and in fact all but a finite number) n, thus contains some • points of 's' -1 (x). This proves that L and thus M is at most a TO-space. The proof is thus completed. The following example shows that incomplete fibres do occur

for Lorentz metrics.

Example C

Take two-dimensional Minkowski space with (x,t) the usual

coordinates. The null cone of the origin divides M into four

parts. These are more suitably characterized by means of the co-

ordinates U=x+t, V=x-t. They are (see Fig.18) :

— 93 —

FIGURE 18

u ,v) t U> O, v ,> O MI I = (U,v) I U >o , m( 3 - (u,v) I U < O, V { 0 = {(Uv) u <0 , MIII S , MTV The Lorentz transformations about the origin act freely on MI. Take a discrete subgroup - which is necessarily infinite cyclic - and identify points in the same orbit. The result is a flat Lorentz metric on a cylinder. We denote this space by VI (see Fig.19). Obviously the b-boundary of VI consists of two circles CI, C2 corresponding to the two null half-lines L7 =i U) 0, V = 01 and L2 4 U = 0, V) O in M, and a point w corresponding to the origin in M (see Fig.19).

The action of the Lorentz group shows: Every open set in VI containing w also contains C1 and C2 ; every open set containing pCi contains points of every open set containing C2 and vice versa. Thus VI is at most a T0-space.

VI is extensible through C1 or C2 or C1 and C2 simultaneously - 94 —

(with the abandonment of Hausdorffness in this case). In the extensions

of V1 through C1 or C2 each circle becomes a null geodesic of

the corresponding extended space with exactly the properties of

Example B.

FIGURE 19 — 95 —

B) Further Properties and Examples

(a) In section (d) of the preceding part (A) were given criteria

as to when a curve in L(M) defines a boundary point and when

two Cauchy sequences (and thus the corresponding curves on which they

lie) belong to the same class, i.e. define the same boundary point.

The same problems will now be considered in terms of M only.

Let ''y : CO3'1) --> M be an inextensible curve in M which has finite length as measured in a parallely propagated frame, thus

defining a point, say p in M. One could argue that this is so

when limt ~(t) = p. But, as we have seen, M is not always Hausdorff

(e.g. Example C of section (e) in Part (A) of this chapter). Thus,

such a statement does not always make sense in M. Therefore one is

forced to accept the more complicated definition:

Definition a.l

If Ar is a curve as described above, then we say that it .

defines the point pfM if a horizontal curve u(t) belonging to

fl -1 ( e ) (alternatively called an h-lift of ) satisfies :

(lim u(t)) = pa The limit is understood to be taken in L(M).

Indeed, by assumption on ar each h-lift of Jr has finite generalized

affine length and thus limu(t) exists in L(M). Furthermore, t—'91 f' (lim u(t)) is independent of the particular h-1 ift of Y used, t-->1 that is all h-lifts of ( belong to the same class Indeed, let u(t), v(t) be horizontal curves in /7 -1(), and (un):= u(tn),(vn):=

v(t ),t (0,1) limit) = be Cauchy sequences without limit in L(M). n n 1

We have to show that p (u ,v ) = 0. As we know Gl(n,R) acts freely n ,,,,.~n co and transitively on the fibres of L(M) and thus for corresponding un , vn it always holds that un = Ravn. From application of inequality

(c.3) (page 84) we find that C,7 (un,vn p(Raun,Ravn) C2 r (un,vn) P ) 1 t ♦

— 96 —

also holds. Since all distances are positive or zero, the above

two relations imply that p (un,vn) = 0 and thus the claim is 1 -~9. justified. 1

Definition a.l. is also valid for OM with p y M as we can see by inspection. Of course in that case the requirement of

inextensibility of must be dropped. J Let us turn now to the identification problem. This was

also dealt with in section (d) of part (A) of this chapter, as far as

L(M) is concerned. Indeed, we have defined that two sequences ellymmer un,vn(L(M) without limit in L(M) define the same point of L(M) if

p (un,vn) =0 as n --- . d How does this definition projects down to M. One could argue that two inextensible curves x1: 10,7) i=1,2 having

finite generalized affine length define the same point, say p of M

if lim " (t) = p. But again the possibility of non-Hausdorff M t-° makes the above definition inadequate. Moreover M has in general

an indefinite metric, in contrast to the positive definite metric of

L(M), and this. poses additional problems, as a distance function cannot

be defined on M. By means of the below given theorem a criterion is

presented as to when the curves in M define the same point of the

b-boundary.

Theorem a.l

Let the curves y1: C,'i) --~ M, i=1,2 be as described above.

We shall say that these two curves define the same point p of the

b-boundary M of M if :

There exist sequences n n=1,2..0 in foj) with ti = 0 and (tn) lim (tn) = 1, 1=1,2, n -~► ~o r 2 curves tE n::j (o .-- M with (0) = v (tn 0~(1) = 2 ), n ,i) (t2 and a frame uof 1(y1(0)) having the properties: U - 97 -

i) If L( ern) is the length of irrn (see Section {d), Part (a),

Page 86 ) measured in the parallely propagated frame along

cr n obtained by parallel transport of uo along till (t1 ) 0 7 d n and then along idln, then lim(L(tn)),.= 0 n -~

ii) If uo,n is the frame in rj1(41(0)) obtained by parallel transport of u oalong the closed curve n: [o,2+tn + t ] —'tM with

ti.1(t) t 1)

2(t 1) tE (1,1+t2 J Wn(t) =

en(2+t2 - t) t( (+t, 2 +tn J n

2+t1-t) tf (2+t2, 2+t2 + ti L l(2+tn n n n n then the sequence (a}7 (G1(n,R) defined by u fl has 2,see o, n an accumulation point aofGl(n,R)e

FIGURE 20 _98—

An illustration of the hypotheses of the above theorem, as far as M is concerned, is given in Fig.20.

Proof: We have to find Cauchy sequences {un3(v} L(M) without limit in L(M), definin the same point of L(M), that is satisfying p (un,vn) = 0 as n , and such that

n ( } , e. '7( 6 ). We shall actually construct two {unit 0 ivn 2 such Cauchy sequences. Let u(t) be an h-lift of /'1 with u(0)=u0

and v(t) be an h-lift of 2. Let (uni,ivn) be the sequences O x Y v(t2 - 1 2 2 defined by ui:= u(tI) E fl- ( 1( t1)), vn:= ) f ( Y (t ). 7 By hypothesis for , , 2 {UnVn are sequences in L(M) without limit in L(M). Let dn(t) be an h-lift of ern with dn(0) = un

and dn(I) = vn (these two choices are possible since en(0) =

n(') = 2(tn)). Now property (i) of the theorem translates in L(M) as : ii (L(dn )) = 0. But p (un ,vn) = (dn(0),dn t (T)) ,,< L(dn)

by definition of Thus p (un,vn) = 0 as n It remains p . Ga. to be proved that etuni,ivni are Cauchy sequences. Indeed, property inī (M) (ii) of the theorem translates/into t e requirement of the existence of an accumulation point aJGl(n,R) of the sequence janjn=7,2..Gl(n,R) defined by uo,n = Ra uo where uo,n is the frame in jj- ( x(0)) n y obtained by goingfong the curve wn(L(M) which is the h-lift of the closed curve binel with wn(0) = . T11( tn(0)) = fl ( (r1(0)) = -It 1 ri ( y (0)) and consists of the h-lifts already constructed for the curves composing (Ai n. That is wn is as follows: wn: LO,2~t-i-t- J ---t L(M).

- 99 --

t f [0,7.1

t (I ,1+tn wn(t) _ dn(2+tn - t) t (~t. 2+tn3

u(2+tn +tn -t) t (+tn , 2+tn + tan J

The existence of the accumulation point ao of the above defined sequence essentially means that wn ultimately becomes closed fans as n .- „O and an ...._, ao, which in turn says that the particular do used in the construction of wn becomes irrelevant. Thus for the points dn(0) = un, dn(7) = vn we can assume that for n large enough, say n > No, (un ) = 0 and © (vn ,v ) = 0 as ,un 1 2 t 2 n,,n2 , , with n7,n2 > No. But exactly this is the definition O of a Cauchy sequence. Thus ('uni,ivnip are indeed Cauchy sequences.

Noting again that the assumptions derived from the above construction are independent of the particular h-lifts used (see Proof of this given on page 95 ), the proof is completed. Now we can state that :

Definition a.2.

Two curves in M define the same point of the b-boundary if and only if the hypotheses of theorem a.7 are fulfilled by these curves.

As with Definition a.T, Definition a.2. is also applicable for points pf M with p94, as long as the inextensibility requirement for the curves is relaxed. jj 7 x If M is a Riemann space then the fibre of L(M) - (I (0)) is always complete, hence always compact. This means that the sequence

lies in a compact subgroup of G1(n,R). Thus there is lanin=7... always an accumulation point ao. So we see that for M a Riemann space condition (ii) of theorem a.7, is redundant. — 100 —

(b) Although the criteria developed in the preceding section (a)

are avowedly difficult to handle in practice, they help one to find an answer to the following important question:

The bundle of linear frames L(M), which is needed for the construction of the b-boundary, is a space of much higher dimensionality than M itself, in fact it is a 20-dimensional space for M, a 4-dimensional space-time. So this space is difficult to handle in practice. But there always exist sub-bundles of L(M) of lower dimension. The question one is faced with is under what conditions the boundary of M calculated using these sub-bundles (by a construction analogous to the usual one on L(M)) coincides with its b-boundary. This section is devoted to the formulation and the answer of the question.

To begin with one has to present the notions of G-structures and reducible connections. Let G be a closed subgroup of Gl(n,R) and j; G ---- ' Gl(n,R) the natural injection. A principal bundle

(G(M),M,p) with structure group G is called a G-structure over M if there exist a Ce' map k: G(M) --. L(M) such that (k,j) is a bundle morfism of G(M) into L(M) over M, that is if p = flok and k(Rsv) = Rj(s)k(v). (b.1) where v(G(M), s(G and Rs denotes the action of s on v. Thus the map k is an imbedding and can be considered as a natural injection of G(M) in L(i), k(L(M)) being a closed submanifold of L(M). If there is a connection form W` on

G(M) satisfying W' = kt W (b.2) where W is the connection form on L(M) (see Section (b), Part A of this chapter), then tij on

L(M) is said to be reducible on G(M). If g is the Ricmann metric on L(M) defined by (c.1) in Section (C)_of—part" , then analogously we can define a metric _g on G(M), using of course the connection

6v' instead of w . Let d be the distance function on 6(MM defined analogously o 15 04 L(M). Then G(M) is made into a metric space - 101 -

and, in perfect analogy with what we have done with L(M), we can form

the Cauchy completion G(M) of G(M) and define the space MG rz G(M)/G .

Analogously to 17 (see page(85) we have the mapping p G(M) MG mapping a point v(G(M) into the orbit of G passing through v. We can now state and prove the following theorem:

Theorem b.1.

Let G be a closed Lie subgroup of Gl(n,R). If (G(M),M,p) is a G-structure on M to which the connection on L(M) is reducible, then the space MG is homeomorphic to the space M.

Proof: First of all we note that it is sufficient to prove that the boundaries M and MG are homeomorphic since fl(L(M) and p(G(M)) are both homeomorphic to M and thus homeomoprhic to each other. As we have already seen (see page8 5 ), the spaces M, MG obtained are independent of the particular choice of the bases of the

Lie algebras of Gl(n,R) and G, and the bases of Rn used to obtain the connection forms W , (4 and the 1-forms which are needed for the construction of the metrics g and gG on L(M) and G(M) respectively. It is therefore no restriction to assume that the bases on

Rn are the same, and the basis of the Lie algebra of G is that induced on Lie(G) from the basis of the Lie algebra of Gl(n,R) by means of the injection j. From the above made assumptions and (b.2) it follows that gG ktg and consequently, by definition of p and d, it follows that

p(k(vl), k(v2)) < d(vi,v2) for v1,v2(6(M). (b.3 )

Thus the mapping k is uniformly continuous, and we can extent it to k , k : G(M) *•-4• L(M) . Since also the action of Gl(n,R) and G on

L(M) and G(M) respectively is uniformly continuous and thus extendable (see page 84 ), the mappings v k(Rsv) and v .-.. Rj(s)k(v),

— 102 — vfG(M), s(G are continuous and coincide on G(M), thus are equal on ormworIP G(M) (by virtue of (b.l.)). This suggests the definition of a mapping MG ----- M ~ . x ...... -> fl ok(p -1(x) ) (b.4) from the space MG to the space M.

We mean to prove that i as defined above is a homeomorphism.

By inspection we see that i I_ :p(G(M)) —* jj(L(M)) p(G(M)) is a homeomorphism. Indeed, since both p(G(M)) and fl(L(M)) are homeomorphic to M then i 1_ is induced by the identity map p(G(M)) idM and thus, it is a homemorphism. So i, as defined in (b.4), can be thought of as an extension of idM, since by definition it satisfies ion = i7 ok. In order to prove that i, as defined in

(b.4), is a homeomorphism, we have to show that it is continuous, bijective and open.

(a) i is continuous: Since p, fl are open by definition of the

topologies of MG, M and k is continuous as an extension of the uniformly continuous map k (see (b.3) above), it follows that i is continuous.

(b) i is bijective: We must prove that i is surjective and

injective.

(1) i is surjective: If uofk(G(M))4L(M) then, by uniqueness

of the h-lifts, an h-lift u(t) with u(0) = uo lies entirely in k(G(M)). This, together with the fact that (k,j) is a bundle morphism over M shows that for every y(M there is an h-lift u in 71101, 011401.0.. k(G(M) with f'(lim(u(t)) = y which proves that i is surjective. t $1

(2) i is injective: If xl,x2{ MG with i(x1 ) = 1(x2) = ytM we

have to show that x1 = x2. If v1,v2 are horixontal curves — 103 — in G(M) with p(limm(v1(t)))= x1, p(liimm(~v2(t))) = x2 then ul ; = k(v1), u2:= k(v2) are hrcurves in k(G(M)) which, by hypothesis, define the same point u( 11 (y). That means that, since k is uniformly continuous, the conditions of theorem a.'1. (which hold for u1,u2) are also holding for v1,v2, and that means that x1 = x2. This proves i is injective.

prove that i(U) is open in M. For this, it suffices to prove that ri -7(i(U)) is open in L(M). Let V:= rt 4(i(U)) = k(P ~( U )) and supposePp V is not openP in t ~ M).) This means that there is a sequence unEL(M)/V converging to uo(V. Again, since .~... k is uniformly continuous the sequence vnfG(M)/p ~ (U ) defined bby

d by uo = k(vo) and v p -1(U). un:= k(vn) converges to vo, define o But this means that p -1(U) is not open in G(M). But this, since p is open and continuous, means that U is not open in MG which contradicts the assumption that U is open in MG. Thus fvJCG(M)i p - (U) does not converge to any v fp '(U) and consequently o ge to uo(V. This means that V is un L(M)/V does not conver open in L(M) and so i(U) = fl(V) is open in M. But this last proves that i is open. This completes the proof of theorem baro

We are now in a position to prove rigorously the statement

(made in page 88 and also in Hawking and Ellis "The large scale structure of space-time") that in calculating the b-boundary we can use the bundle of orthonormal frames 0(M) instead of the bundle of linear frames L(M). Indeed, the orthogonal group 0(n) is a closed subgroup of Gl(n,R) and thus (0(M),M, ['l'), _with structure group ------0(n) can be made into an 0(n)-structure on M. Indeed let Gl(n,R), 0(n), L(n) be represented by matrices with respect to the natural basis of Rn, where L(n) stands for the group of translations. — 104 —

Then j : 0(n) r--- Gl(n,R)

(0) --- ► (0)+(L) where (0) is an orthogonal nXn- matrix and (L) is any nXn matrix, is the natural injection of 0(n) into Gl(n,R). There is also an imbedding k : 0(M) -- L(M) such that (k,j) is a bundle morphism and thus

(0(M),M, fly ) is an 0(n)-structure on M. This means that the requirements of theorem b.1, are satisfied and thus the space MO(n) is homeomorphic with the space M. r Another example is the holonomy bundle (H(M),M, fl ) whenever the holonomy group H is closed in Gl(n,R). The h-bundle is constructed as follows: Let potM and tic) be a frame at pc,. Then the fibre f (po) consists of all frames u at po which can result from uo by parallely transporting it along closed curves beginning and ending at pc). As is evident from this definition, the structure group H of the holonomy bundle is the holonomy group of the connection.

When H is closed in Gl(n,R), theorem a.1, shows that the space MH is homeomorphic to M.

spaces. It is easily checked that in the case of Riemann spaces if (M,g) is the product space (Mix M2x...Mm,g1 x g2x...gm), then M = M1 x M2 x ..Mm since in this case according to theorem d.3. of Part A (page 88 ) M,Mi, i = 1.,m are homeomorphic to the Cauchy completions of M,Mi, i =1,2..m for which this property holds, as it is well known. The question now arises if an analogous property holds for the case of Lorentz spaces. This question is answered affirmatively by theorem c.1, below. Though there are not realistic space-times which are product spaces, the importance of this theorem in constructing examples and/or counter-examples is obvious. Furthermore, this theoem can be taken as a further evidence that the b-boundary is a natural generalization of the Cauchy completion of Riemann spaces. The proof — 105 — of the theorem will be similar to that of the previously stated theorem b.7. and theorem c.l. can be considered as an application of theorem b.1. Let us now state the theorem:

Theorem Ci.

Let (Ml,g1 ) be an m-dimensional L-space 1 < m (n, (M2,g2) be an (n-m)-dimensional R-space, (M,g) the (topological) product j—spac.e of (M1, g1) and (M2,g2) . Then M is homeomorphic to M1 x M2.

For simplicity in the statement and the proof of the theorem we considered the (topological) product of only two spaces. The general- isation to more than two spaces is straightforward. Proof: Let (0(M),M, rt), (0 1(M1 ),M1, no be the orthogonal bundles of (M,g) respectively (Ml,g1 ) 1=1,2. Let also P(M):=01 (M1 )x02(M2), p:= x 11 2. Then (P(M),M,p) is a principal bundle over M with structure group 01 x 02. This bundle can be made into an 01 x 02-structure on M. Indeed, if 0,01,1=1,2 are represented by matrices with respect to the natural basis of Rn, Rm, Rn-m correspondingly, then

01 x0- -~— 0

ū • sl 0 (si ,s --> 0 s2 is the natural injection of 01 x02 into 0, where sl is an mgm orthogonal matrix, s2 is an (n-m)x(n-m) orthogonal matrix and the final matrix on the right is an nXn orthogonal one. There also exist an imbedding k of P(M) into 0(M) such that (k,j) is a bundle morphism

(see (b.!) in the previous section). Thus (P(M),M,p) becomes an

x0 -structure on M. Let Pr1:P(M) (M),1=1,2 be the natural 01 2 .--9►01 projections and W the torsion free connection forms on 01 (M) 1 respectively. __Iher_a torsion free connection form- 40 is induced on P(M) defined by

(41 1 = Prt 1a + Prt UJ (c.7) l~ 1 2 2 — 106 —

Now according to (b.2) in the previous section, to on P(M) induces a torsion free connection w on 0(M) defined by (see (b.2) on page 100 )

W' = kt00 (c.2)

We thus see that W coincides with the Levi-Civita connection on 0(M). If gP, gOl are the Riemann metrics on P(M),01 (M) respectively, defined analogously to the metric g (see section (c) of

Part A, page 83 ) it is obvious from (c.'I) and (c.2) above that

0 t 0 2 gP = Pr` g.l +Pr g (c.3) 1 2 0 if dP, d 1 are the distance functions defined on P(M),01 (M) by 0 means of the Riemann metrics gr,g 1 respectively (see section (c) of part A, page 83 ) it follows from (b.3) on page 101 that

d01 (Pr 1=1,2 (c.4) 1 (x), Pr1(y)) < dP(x,y) x,yfP(M)

But from (c.3) above also follows that

01 0 dP(x,y) < d `(Pr1 (x),Pr1 (y)) + d 2(Pr2(x),Pr2(y)) x,y(P(M) c.5)

From (coo) it follows that there is a uniformly continuous extension of Prl, Pr1:01 (M) - P(M). From (c.5) we see that Pr1 also has a uniformly continuous extension Pri : P(M) ---3 01 (M) . The last two relations mean that the map Pr1 x Pr2 is a uniformly continuous homeomorphism from 0 (M) onto 01(M1 ) x 01 (M2).

Pr1 x Pr2(R3(sl,s2)x) = (RslPr1 (x), Rs2Pr2(x )) (c.6)

The last relation suggests the definition of the map -- 107 -

f: x ....~, r7 1 x jl 2(P r 1 x Pr 2( f1 -I (x))), where (1 : 0(M) s M is the map assigning to each pl0(M) the orbit of 0 through u, : O1 (M) Ml 1 =1,2 similarly. 1 Since Pr xPr is bijective, relation (c.6) above shows that f is 1 2 also bijective. Since r71, and are, by definition of /2 fl the topologies on M1,M2 and M respectively, continuous, f is also continuous. f is also open (the proof is the same as that for i in theorem (b.T).on page lol). Thus f is a homeomorphism and the theorem has been proven.

As an application of the above theorem (c.1) we can prove that n-dimensional Mirikowski Mn space is b-complete, without referring to the maximal isometry group. Indeed, Mn = M2 x Rn-2 and since the two spaces on the right are b-complete, theorem (c.1) above shows that Mn is also bundle-complete.

As another application we can easily calculate the b-boundary of the space-time given in the second example of R.K. Sachs ("Space-time b-boundaries", Comm. in Math.Phys. 33, 215 (1973)).

Indeed, let N be R3 with the origin deleted, and h be a

Riemann metric on R3 which is Ce. on N. Let M = NxR, with projections S:M ---j N and T:M -►- R. Define a metric g on M by g:= SNh-dTxdTG Supply (M,g) with the natural orientation, the natural time-orientation and the Levi-Civita connection D. Then

(M,g,D) is a space-time. Sachs, after a rather tedious and incomplete proof, shows that M is homeomorphic with R4. This, however is an immediate consequence of theorem (c.1). Moreover, it is obvious that the b-boundary of M consists of just the missing points (a,a,o)xR -- 108 -

(d) The structure which gives rise to the definition of the

b-boundary is the connection on M. Therefore it is natural

to ask whether mappings of one space-time into another which preserve

the connection are extensible to the corresponding space-times with

b-boundaries. This section is devoted to the formulation and answer

of this question. It will be shown that affine mappings, or isomor-

phisms of the connections of one Lorentz space into another are continu-

ously extensible to the space-times with b-boundaries.

Let (M,g) and (M',g') be space-times, and let f be a diffeo-

morphism of M onto M'. As it is well known f induces a mapping

of L(M) onto L(M').

L(M) L(M') L(f) = (d:) u — L(f)(u):= fxu

The mapping f is called an affine mapping of M onto M' or isomorphism of the connection VO on (M,g) onto the connection $Q ' on (M',g') if L(f)t W ' .10 (d.2) Lemma d.7

For each isomorphism f of the connections on (M,g),(M',g') - - there exists an extension f : M which is a homeomorphism.

If f,h are two such isomorphisms of the connections, for which

wnr. ev pwa foh is defined, then fvb.= foh.

We can remark that in particular isometries of (M,g) onto

(M`)g') are extensible to homeomorphisms of M onto M'.

Proof: By definition of L(f), we have

L(f)(Rsu)= Rs(L(f)(u)) u(L(M),sCG1(n,R) (d.3) where Rs denotes the action of s on L(M),L(M') respectively. By the definition of the 1-forms e, e` on L(M), L(M') respectively, — 109 —

(see section (b), part A) we see that, since (d.3) holds,

L(f)te' = e , (d.4) r If gL, gL are the Riemann metrics on L(M), L(M') respectively, defined as in section (c) of part A, then (d.2) and (d.4) imply that

tgL' L L(f) = g (d.5)

The last relation shows that L(f) is an isometry of the metric space

(L(M),d) onto the metric space (L(M'),d'), where d,d' are the distance functions determined on L(M), L(M') by the Riemann metrics r gL, gL respectively (see section (c), part A).

As such, L(f) is extensible to a uniformly continuous mapping

L(f) : L(M) — ► L(M') between the Cauchy completions of L(M),L(M').

Since L(M) is dense in L(M), (d.3) gives

L(f)(Rsu) = Rs(L(f)(u)) ufL(M),s¢Gl(n,R) (d.6)

This last relation gives rise to the definition of a map

ht -- M' f : - ,.. ,..,—. ,.. p w---~ f(p):= l7 '(L(f)( (p)) d.7)

From the properties of rust of being continuous and open, and relations (d.5), (d.6) above it follows that f is bijective, continuous and open, thus f is a homeomorphism. Since obviously the restriction of f onto M coincides with f, f is the desired extension and the lemma has been proven.

Now we can prove the following theorem:

Theorem d.I. The groups A(M) of affine transformations of (M,g) onto itself and I(M) of isometries of (M,g) act as topological transformation groups on M, that is

— 110 —

A(M} xM --- M I(M) x M

i : i : (f,P) f(p) (f>P) -- f(P)

are continuous.

Proof: If A(M) x L(M) - * L(M) then i =1 ojo r1 -* j: onswir■ (f>p) .. L(f)(P)

and since IT is continuous and open, it suffices to prove that j is continuous. But this is true, since L(f) is continuous.

Thus i is continuous. ix is also continuous, being the restrict-

ion of i to I(M) x M . Thus the theorem has been proven.

Finally, as an application of the above, it will be shown that

the b-boundary of an ordinary cone constitutes a real singularity,

according to Schmidt's definition (see page.79 ).

The example is taken from Schmidt and Hajicek "The b-boundary

of tensor bundles over a space-time, Comm. in Math.Phys., 23, 285,

(1971). Of course, there, it is only proven that the b-boundary is

degenerate and not that it consists a real singularity.

The ordinary conus is a 2-dimensional manifold M together

with the positive definite metric ds2 = dr2 + r2 d62, with coordinate

r, š, r ) 0, 0 < 6 <60, so and 0 identified (see Fig.21). Since

M is a Riemann space the space M is homeomorphic to L(M) (see

theorem (d.3) on page 88). Thus the b-boundary p of this space-

time consists of just the cusp point r=0 of the conus. To be

accurate this point is represented by the surface S = liO, with

m as above. We see that p although metrically is a point,

analytically is a surface. This means that something is going wrong

with p. Indeed, we shall prove firstly that p is degenerate and

afterwards that p is a real singularity. Let pn = (rnAn) be the — 111 — points in M with rn = 1/n, 6n = 0, and un = (Xn,Yn) be frames at pn with Xn = (1,0), Yn = (0,n). Then un converge to .27 uof ( (p).. Consider the frames vn obtained by parallely transporting un along the lines r=rn. Obviously vn also converges

to uo and thus p (un,vn) = 0 as n- a,. But, by construction,

vn = Rsun for every n, with sf0(2) non-trivial. From the last

two relations we conclude that uo = R for sf0(2) non-trivial, s uo and this p is degenerate. If 6t is the isometry 6 --> 6+ t

then for any t obviously L(6t) (u)ju if 40(M) but L(C )(uo) = u0.

But, as it is well known (see e.g. Kobayashi and Nomizu (1963) Foundat-

ions of Differential Geometry, Vol.I, Interscience), two isometrics

f,h of the connection of a pseudo-Riemannian space are equal if there exist at least one point ufL(M) with L(f)(u) = L(h)(u).. Therefore

the existence of uo in the above example shows that there does not

exist an extension of M through the point p preserving the given

isometrics. Thus, this point represents a real singularity according

to Schmidt;s definition. Pc n o)

FIGURE 21 — 112 — REFERENCES

Bishop, R.L,, Crittenden, R.J. (1964), "The Geometry of Manifolds",

Academic Press.

Geroch,R.P.(1968), Ann. of Phys., 48, 526.

Geroch, R.P. (1968), J.Math.Phys., 9, 450.

Hawking, S.W. (1966), Essay submitted for the Adams Prize, Camb. Univ.

Hawking, S.W., Ellis, F.R. (1973), "The large scale structure of

space-time", Cambr. Univ. Press.

Hicks, N.J. (1965), "Notes on Differential Geometry", D. van Nostrand Co.

Kobayashi, S., Nomizu, K. (1963), "Foundation of Differential Geometry",

Interscience.

Martin, W. (1968), "Space-time from a global viewpoint", King's College.

Penrose, Rv (1965), Phys.Rev.Lett., 14, 57.

Sachs, R.K. (1973), Comm.Math.Phys., 33, 215.

Seifert, H. (1975), "The causal structure of singularities" Lecture

Notes in Mathematics, Vol. 570.

Seifert, Ho (1967), Zat.fur Naturforzchung, A.22, 1356.

Schmidt, B.G., Hajicek, P. (1971), Comm.Math.Phys., 23, 285.

Schmidt, B.G., (1971), G.R.G. Journal, Vol.1, 269. — 113 —

CHAPTER IV

QUANTUM GRAVITY

A. General Discussion and Examples

(a) Classical relativistic singularities remain one of the most

puzzling problems of contemporary physics. As regard to their status, no consensus of opinion exists. Although the general attitude is that the existence of singularities is an undesirable feature for any physical theory, a somewhat compromising way of thought has been adopted by some scientists. This view, motivated perhaps by the reluctancy to abandon such an elegant theory as General Relativity, is best expressed by C.W. Misner (see C.W. Misner, Phys.Rev. 186, 1328,

(1969)): The initial singularity of the big-bang models is essential for the coming into being of the Universe. Let us try to analyze a bit further this view. There are examples in physics where progress has been made from tolerance as well as from intolerance: One could be intolerant of the classical models of the atom because atoms were observed not to suffer catastrophic radiative decay. Conversely,

Einstein (see A. Einstein, Ann.Physik 17, 132 (1905); 22, 180 (1908)) and Bohr (see N. Bohr, Phil.Maga. 26, 1, (1913)) were tolerant of

Plank's theory of radiation (in spite of its singular discontinuities) which violated no observation.

Relativistic cosmology has had reasonable success (the expansion of the Universe, the existence of the 3°K microwave background radiation and the objections to its more dramatic novelty (the initial singularity)) are conceptual, rather than observational. Furthermore, there are no observational indications of, say, an era of contraction preceding the present expansion. On the above basis, one could stretch one's mind, find a more acceptable set of words to describe the mathematical situation — 114 —

identified as "singularity", and then proceed to incorporate this into one's physical thinking until observational difficulties force new revisions. The following examples serve to prove that the acceptance of such a true initial singularity (as opposed to an in- describable early era of extraneous but finite high densities and temperatures) can be a positive element in the theory: The first example refers to the observed 12-h anisotropy of the microwave background radiation. It has been proposed (see C.W. Misner, Phys,Rev.

Lett. 19, 533 (1967), Astrophys.dourn. 151, 431 (1968)) that neutrino viscosity at some stage of the evolution of the universe could lead to the observed finite limit on the above anisotropy. But as the equations governing the problem are regular, well-posed differential equations the continuity of the solutions as functions of the initial conditions forbids the occurrence of arbitrary anisotropy at an instant of the histroy of the Universe, no matter how deep in the past this instant may be. Yet, such an occurrence is necessary if we are to invoke neutrino viscosity, or in fact any other dissipative process, for explaining the present anisotropy of the microwave background radiation. It is here that the concept of a true initial singularity offers a way out. Indeed, since evidently the equations are singular at the initial time (or equivalently are regular, but set the initial conditions at the infinite past), an infinite range of initial conditions could evolve into a finite set of present conditions. In fact all the difficulties which arise from the apparent incompatibility of the early

Universe models with the observed present-day Universe, that is, the fact that at least some features of the present Universe seem to be independent of most parameters specifying the initial conditions,are resolved (at least in principle) in the above context, because initial conditions are specified at a true singularity, or at the infinite past, but not at any finite and regular past era. Hsee footnote (1) on page 116 — 115 —

Another example of one such feature of the present Universe

seems to be its observed homogeneity. We shall see that an under-

standing of this feature can be reached by treating the initial

singularity as an acceptable element of the theory. Indeed, there

is a model called "the mixmaster universe" (see C.W. Misner, Phys.Rev.

Lett. 22, 1071 (1969)) with a very interesting behaviour: the evolution

of this model consists of stable periods during which the model approx-

imates the Taub Universe; h'aving the two spatial axes nearly equal.

The remaining third spatial axis is the symmetry axis of the Taub

Universe.

As we follow the evolution of this Universe we see that each of

these Taub-like periods is eventually destroyed by small non-linear

terms in the metric; a short period of assymetry follows, and then

the solution settles down again to a new stable Taub-like period.

Only this time, the axis of symmetry need not be the same as in the

previous Taub-like period (see Fig.22).

a la

• . . . d e f

a,boc: Taub-like eras. A: Singularity. d,e,f: Asymmetry eras.

FIGURE 22

* See Fig..23 , page 116. -116-

FIGURE 23.''aub-NUT space j (1).Briefly,if the metric is ds2= -dt2+gi (t)dx'dx ,with gid=e2a(e2b)ij,then the Einstein equation for the expansion rate, -4a)/3 for example, is:da/dt =8 R (Ae-3a +Be — 117 —

These prolonged Taub-like eras suggest that the establishment

of homogeneity in the present Universe may be understood in terms of

the mixmaster model. Indeed, it is known that as we approach the

singularity in the Taub model, light rays or more generally causal

interactions can travel around the Universe. The trouble with the

Taub model is that the causal interactions can propagate only in one

direction along the symmetry axis, so any homogenization can occur

only in this direction. The problem is resolved in the mixmaster

model, in which due to the successive change of the symmetry axis

homogenization can occur in all three spatial directions. Thus, one

can anticipate that causal interactions spanning the entire Universe

in all directions are possible near the singularity in the mixmaster

model. Again, the positive role of the singularity becomes obvious,

since if we exclude it as an element of the theory, we would be left

only with a finite number of changes of the symmetry axis which are

inadequate for any homogenization procedure to be effective.

(b) From a careful examination of the two previous examples it

follows that even with this, most optimistic, attitude as regards the status of singularities, one certainly needs a way to avoid

the singularity itself: for in the two examples what is welcomed is

the consequence of the presence of the singularity rather than the

singularity itself. If one is to preserve General Relativity, the

only possible way out seems to be to postulate what is known as "the

cosmic sensorship principle" (see, e.g. "The large-scale structure

of space-time", Hawking and Ellis, Cambridge University Press, 1973).

This principle states that, due to the nature of Einstein's General

theory of Relativity, all the singularities occurring in the theory are "hidden" inside black holes or behind event horizons. More precisely, the cosmic sensorship principle states that General-Relativistic — 118 — space-times contain no "naked" singularities, where "naked" means a singularity which can be linked to a Cauchy surface with at least one causal curve 2w For a period of time this principle was believed to be true, but in 1973-74 H.J. Seifert and his co-workers (see

Comm.Math.Phys..34, 135-148 (1973); 37, 29-40 (1974)) firstly gave examples of specific relativistic space-times containing "naked" singularities and, secondly, showed that the Cauchy problem for General

Relativity allows "naked" singularities to evolve. Furthermore, they showed that both the initial conditions and the equations of state for which a "naked" singularity does evolve, are not of measure zero in the set of all initial conditions and equations of state respectively.

Obviously this disproves the cosmic sensorship principle, at least as stated above. If one is not prepared to accept singularities at all,

"naked" or "hidden", then one has to search for physical mechanisms capable of eliminating the singularities; that is, one is forced to look for refinements or even alternatives of General Relativity. A standard way one might proceed is to try to combine with General

Relativity other, already known, physical theories. Since one success- ful such theory is Quantum Theory, one finally arrives at the concept of Quantum Gravity. Not surprisingly, a number of approaches to quantum gravity, that is to the problem of quantizing the gravitational field, have appeared in the last 40 years This comes naturally if one bears in mind the great complexity of the subject - from both the technical and conceptual points of view - and the very different back- grounds of various workers in this field.

Since the motivation for introducing quantum gravity is to find a way out of singularities, the following question naturally arises in

+ For a review see e.g. Quantum Gravity - An Oxford Symposium, Edited

by C.J. Isham, R. Penrose and D.W. Sciama, Oxford Univ.Press 1975).

See Fig. 24 , page 119. - 119 -

FIGURE 24. A Penrose diagram of a space-time with a fnakedtt singularity.An observer travelling along the curve hits the sin - gularity before he reaches the horizon H}. This is opposite to the case in Fig. l(page 6),where every observer cros- ses the horizon before hitting the singularity. — 120 — one's mind: Does quantum mechanics (or quantum field theory) provide this way out? This question is beyond present power to answer, since the quantization of the gravitational field has not, in its full generality, been achieved. From the existing special results one cannot clearly say what is likely to be the answer to the above question. Indeed, as we shall see, although most results suggest s positive answer, a careful examination of each case reveals that the possibility of non-avoidance of the singularity remains strong. Furthermore, there are also special examples in which the non-avoidance of the singularity is manifest. We shall give such examples later. For the present, let us reformulate in more precise mathematical language the above raised question. We shall do this by transforming it to a proposition, the validity of which is understood to remain in doubt (this is what J,A.Wheeler has called The Unanimity Query", see G.R.G. Vol.8, 713 (1977)): "Given that all solutions of the classical problem (a) are singular (or (b) non-singular), except a set of"measure"zero; then all solutions of the corresponding quantum problem are (a) singular (or (b) non-singular)". In the following two examples the above proposition seems to hold: The first example is motion of a charged particle under the influence of a point center of attraction

(Rutherford problem). As we kinow, a particle headed straight toward the center arrives in a finite time at a condition of infinite kinetic energy, plainly a singularity, but every other motion is non-singular (see Fig.25).

‘If

FIGURE 25 — 121 —

Thus, in this case, the singular solutions form a set of "measure" zero in the set of all solutions. Being this the situation as regards the classical problem, the proposition above predicts that the solutions of the Schrodinger wave equation for the Rutherford problem will be free of singularities, in accordance with the well-known results. The second example is a particle moving along a potential ridge of uniform height that runs to infinite distance in the x-direction

(see Fig.26). The potential has the form

V(x,y,z) = kga2-y2-z2)2 (b.1)

FIGURE: 26

The classical problem then consists of finding the solutions of the equation = m~d2x , that is dt2 2- dx `v= m, (b.2) dt2

The x-component of (b.2) reads 2 0=m.d2 (b.3) dt — 122 — and the solution of (b.3) is x = vot, so that a solution of (b.2) is x = vot, y = 0, z = 0, which evidently is non-singular. The v- and z- components of (b,2) read

2ky d2y , 2kz d2 - m. - . z (b.4) 2 2 3 (a -y -z ) (a2-y2-z2)3 dt2

From the form of equations (b.4) it is obvious that every other solution of (b.2), except the above found, must be singular; for, suppose x=x(t), y=y(t), z=z(t) is a solution of (b.2), different from the above solution x=vot, y=6, z=0, and suppose it is everywhere non-singular: As y(t), z() are everywhere regular and the classical problem must be well posed, d2y/dt2,d2z/dt2 have also to be regular everywhere. On the other hand, precisely because y,z are everywhere regular, there is always a t making the denominator of the left-hand side of (b.4) 0, thus contradicting that y,z, and therefore y , z are everywhere regular. Thus, in this example, we have the opposite situation than the one we had in the previous example: The regular solutions form a set of "measure" zero in the set of all solutions.

The proposition now predicts that all solutions of the corresponding quantum problem will be singular. Indeed, omitting constants, the

Schrodinger equation reads :

v 4 (— ,2 .v) ty (b.5) a

Since V has a singularity along the line (x,, and (t must be interpreted as a density, we see that V has to be singular, in accordance with the proposition. The proposition in page 120 provides no answer when the distinction between one class of solutions and the other is not that of a set of

"measure" sero and the total set diminished by a set of "measure" zero.

The example given below exhibits this behaviour: — 123 —

A particle moves under an inverse cube central attraction. In

the classical case, as we know, there is a critical impact parameter; particles shot in with an impact parameter less than the critical eventually arrive at a condition of infinite kinetic energy, while those shot in with a larger impact parameter eventually escept to infinity.

We thus see that, in the classical case, both the singular and the non-singular solutions form sets of non-zero "measure" in the set of all solutions. Therefore one expects that the same will be true for the corresponding quantum problem. In conformity with this expectation, the solutions of the radial wave equations are free of singularity at the origin for 1-values in excess of a certain critical value, and singular for small 1-values. .

As a common feature of all three examples quoted before we can regard the fact that the ratio of the "measure" of the set of singular solutions to that of the non-singular is kept fixed when going from the classical to the quantum problem. This is exhibited in the following table:

Example 1 Example 2 Example 3

0 00 Classical case 0 = 0 oā 0 Co

m(A) 0 Quantum case 0 namoug_ ` a - 0 m(B) oD 0 woo

where A is the set of the singular solutions, B is the set of the non-singular solutions and m denotes the "measure". In the case of the quoted examples the choice of the "measure" is as follows:

(a) empty sets and sets consisting of a single solution have zero

"measure". (b) sets consisting of a finite number of solutions have finite

"measure"a — 124 —

"measure".

(d) sets consisting of a countably infinite number of solutions have

weakly infinite "measure".

Motivated by the above observation we may propose the following

generalization of the "Unanimity Query" of Wheeler (see page 1 20):

"The transition from the classical to the quantum domain is such that,

given a unanimous (i.e. appropriate for both the classical and the quantum domain) choice of "measure" in the set of all solutions, the ratio of the "measure" of the set of singular solutions to that of the non-singular is kept fixed".

An example where the generalized rule of unanimity might hold

is the following:

As we know, the various singularity theorems predict the occurrence of singularities in solutions of Einstein's field equations under a fairly wide variety of conditions (see e.g. "The large scale structure of space-time" Hawking and Ellis, Cawbr.Univ.Press, 1973). In part-

icular, a big-bang (initial singularity) and a big-stop (final singularity) are predicted for a closed universe with the topology of a 3-sphere.

However, work by Ellis and King (see Ellis, G.F.R. and King, A.R. (1974)

Commun0Math.Phys. 38, 119-156) has shown that there is a class of solutions in which the ending is not a true singularity, but, in their words, a "whimper" (see Fig.27, page l 25). The existence of such solutions seems to reduce the power of the above mentioned singularity

theorems, in the sense of providing an escape from the big-bang or the big-stop or both. Furthermore, one might well suggest that a transition

to the quantum theory would resolve the singularity problem permanently.

Unfortunately, Belinski et al. (see Belinski, V., Khalatnikov, I.,

Collins, B. 24 June 1976 draft preprint entitled "Dealing with the - 125 -

d e h

a b c h FIGURE 27.A "whimper" singularity.Not all observers hit the singularity:observer h escapes and re-emerges to the next expansion era. — 126 —

instability of special solutions of the field equations"), give

indications that the "whimper" solutions form a set of "measure" zero

in the set of all solutions of the field equations for the problem of

a closed universe with a topology of a 3-sphere. Being this the

situation in the classical case, the generalization rule of unanimity

(if valid) predicts that in the quantum case, the best we can expect

is to have a countably infinite number of non-singular solutions and

an imcountably infinite number of singular solutions. Later, we

shall give examples, which indicate that this is indeed the case.

(o) Irrespectively to whether quantum theory does eventually provide

an escape from classical singularities or not, it is interesting and perhaps crucial for our deeper understanding of the whole problem

to analyze in what way this might be done.

In general one can distinguish two ways in which quantum theory may act to eliminate classical relativistic singularities:

via the violation of the energy condition assumed in all the

singularity theorems, or

.• via quantum uncertainty, which introduces new, possibly non-

singular solutions.

This distinction is by no means fundamental but rather reflects our will to study separately the action of quantum theory upon the problem of initial (or final) singularity on the one hand, and the problem of the collapsing star on the other. Indeed, as we shall see, the first way is more appropriate when discussing cosmological singularities, while the second one seems to be more applicable at the final stages of black holes. Of course, one expects that both ways can be invoked in either situation since the way in which one passes from the classical theory to the quantum theory has to be unique (up to equivalence).

Let us now further analyze the above two ways. First, we shall discuss the violation of the energy condition: — 127 —

As is well known, the singularity theorems state that every

general relativistic space-time which (a) satisfies reasonable causality

and generality conditions, (b) possesses a closed trapped surface (or

equivalently a surface to the past or future of which the light cones

start converging), and (c) contains matter which satisfies the energy

condition

(c.l) (Tmn - 2 gmn T)wmwn ),I 0

where Tmn is the stress-energy tensor of the matter, T its trace

and wm any unit timelike vector, must necessarily contain an inexten-

sible timelike curve of bounded acceleration and finite proper length.

In most known cases this is associated with a curvature singularity, although an exception exists (see review in M. Ryan and L. Shepley,

Cosmology, Princ.Univ. Press, Princeton, N.J., 1974).

Requirements (a), (b) above are generally expected to be satisfied. Also, any normal form of matter satisfies the inequality

(c01)0 On the other hand, it is known that once requirement (c) is dropped, singularity-free solutions become available. One example has been provided by Murphy (see G.L. Murphy, Phys.Rev. B8, 4231 (1973)), who postulates matter with sufficiently large second viscosity to violate condition (c.1). The disadvantage of this example is that the postulated macroscopic property does not emerge from any micros- copic model. Another example is the closed Friedmann model of Fulling and Parker (see S. Fulling and L. Parker, Phys.Rev. 07, 2357 (1973)), in which the singularity can be prevented because the energy condition is violated in a natural way by a massive scalar field in certain quantum states. Although there is doubt if the singularity is avoided in every cycle of this model and if the quantum states invoked are realistic, it is worth while to analyze this example a bit more. The massive scalar field appearing above is welcomed, because strong — 128 — interactions in nuclear physics can be regarded as mediated by particles described by such scalar fields. Also, in quantum field theory, in order to resolve renormalization problems, one is forced to accept spontaneous symmetry breaking and the Higgs mechanism (see e.g.

Jeremy Bernstein, Rev.Mod.Phys.,Vol.46, 7 (1974)). In this context, the massive scalar fields are naturally emerging in the process of eliminating the undesired massles Goldstone bosons. An intuitive way of understanding the violation of the energy condition is illustrated in Fig.28, where the vacuum state is understood to be at the upper extreme of the potential, thus allowing negative energy density.

FIGURE 28

Let us now see more rigorously how a massive scalar field can violate condition (c01): As is well known, the stress-energy tensor for a scalar massive field m with mass m has the form

Using this form of Tmn we can calculate the left-hand side of (c.l) and we find

(0 1 2 (c.3) (Tmn - 2 gmnT) wmwn (~ ' mwm) 2 2 m2 .-129—

As we see, the right-hand side of (c.3) can become negative, if m is

large enough, and consequently (c.l) may be violated.

Because violation of the energy condition (c.l) is generally

regarded as physically non-appealing most scientists prefer to assign

the possible avoidance of singularity to the uncertainty principle of

quantum theory. Let us further analyze this notion:

As is well known, Einstein's field equations may be obtained by

variation of the action

1/2d4x (c.4) S 161-1 R(-g) V

where the units have been chosen so that c=G=1, R is the scalar

curvature of space-time, g the determinant of the metric tensor gmn,

and V is the four-volume of space-time under consideration.

One may distinguish the quantum from the classical'domain by

means of the ratio S/h (h is the Planck's constant/2 n ). When

this ratio is large compared with unity we are in the classical region,

and the governing equations are given by SS=O, whereas when the ratio is close to unity quantum effects begin to play an important role.

in the action, S (see (c.4) above) we may add terms corresponding to

matter fields and various interactions among them, but for our discussion

(c.4) is enough. One may think of V in (c.4) above as the region

sandwiched between two chronologically ordered spacelike hypersurfaces

(see Fig.29 , page 130 ). Then, in the classical l , 2 approach, the specification of a 3-geometry 3G1 on E l leads, via

the field equations obtained by (S=O, to the determination of a

3-geometry 3G2 on Z 2 (e.g. Misner, Thorne, Wheeler in Gravitation,

San Francisco (1973)).

For most situations discussed in general relativity, the above

mentioned ratio is large compared with unity, so that the above classical - 130 -

FIGURE 29.0n 1 a 3-geometry 3 G is specified by giving the light-cones at each point of the hypersurface 1• Varying the action (c.4) over the volume V we specify a 3-geometry G2 on E. 2. - 131 -

approach is valid. But on the gravitational collapse of a compact

object S may become small enough to be of the order of h near the

singularity (i.e. in the last stages of the collapse). This means

that the ratio S/h is close to unity and thus we need to examine the

problem via the quantum approach. To do this, we rephrase the above

described classical sandwich problem as follows:

Given two 3-geometries, 3G1 on 1 , 3G2 on what is 2, the probability amplitude for a given system having the action S to make the transition 3G1, 1 --30b 3G2, 2 Appropriate for 2° handling the problem, as formulated above, is the path integral approach

(see e.g. Feynmann, R.P. and Hibbs, A.R. Quantum Mechanics and Path

Integrals, McGraw-Hill, New York 1965). In this context, one uses a generalized notion of the Feynmann rule of summing over all possible histories, that is allows evolution through all possible field equations, even through those which do not follow from $S=O. The result may be expressed through a propagator K(3G1 ,Z1; 3G2, 12). Of course, the execution of this program in its full generality is extremely difficult and has not been carried out. J.V. Narlikar however (see

J.V. Narlikar, Nature, Vol.269, pg® 129 (1977)) has done this for the collapse of spherical dust ball and for geometries only conformal- to the classical solution. The qualitative result is that the dispersion of the wave function diverges as the system approaches the classical singularity. He then interprets this as an avoidance of the singularity by saying that this means that, as we approach the sintularity, more and more geometries are admitted, and thus eventually non-singular solutions are attained. This interpretation, however, seems a little dubious, because the Heisenberg uncertainty principle,o n which it is based, can be applied to a given configuration space, and thus it is not clear how this can be done in the above situation where we have changing geometries and thus changing configuration spaces. — 132 —

B. Quantum Cosmology

(a) One of the most interesting approaches towards quantum gravity

is surely that of so-called quantum models or Quantum Cosmology.

This approach was inaugurated by DeWitt (see B.S . DeWitt, Phys.Rev.

160, 1113 (1967)) and then followed up by Misner (see C.W. Misner,

Phys.Rev. 186, 1319 (1969)) and his collaborators. Since then, an

increasing number of people are working in this field. The basic idea

seems to be to freeze all the degrees of freedom of the system under

consideration except a finite number, and then try to quantize the remaining ones. The benefits of this approach are of double character:

Firstly, by restricting the system to be one of a finite number

of degrees of freedom, one can pay attention to the problems of quantum gravity which are peculiar to the gravitational field rather than to

those which are common to any quantum field theory. In particular,

the gravitational collapse and the influence upon it of quantum effects can be discussed in this framework, as well as a whole range of problems concerning the choice of time, the interpretation of probability, the choice of canonical variables, etc.

Secondly, one can regard this approach as an actual perturbation scheme in which the perturbation is not, as in the usual Feynmann- diagrams-oriented "covariant" quantization schemes, in terms of a coupling constant, but rather in terms of the number of the modes quantized. As regards this point of view one may well ask if the models with a finite number of degrees of freedom can ever describe realistic physical systems, so that the answers obtained from Quantum

Cosmology can be believed to be approximations of quantum gravity.

More theoretically stated, the above question reads as follows:

Does the limit of the answers, as the number of modes quantized approaches infinity, actually exist? — 133 —

Apart from the complexity of the gravitational field, the

answer to the above question is obscured by the usual difficulties of

any quantum field theory, such as ultraviolet divergences and the

confinement problem.

A very important question, which nautrally is raised in one's

mind, is that concerning the definition of a singularity and of the

possible avoidance of it in the context of Quantum Cosmology. Indeed,

it is well known that the definition of a singularity presents serious

problems even in classical relativity theory. In Quantum Cosmology,

because of the probabilistic interpretation of the wave function, the

situation is even more complicated. The problem may be formulated

as follows:

Given that the system under consideration is at a state described

as W(x), what is the meaning of the statement that the system evolves

into a singular state? In order to answer this, we must first discuss

the interpretation of 10(x)12 as a probability density. This is

permissible only if we have a genuine Hilbert space for M. Since

usually x has a continuous spectrum, this is equivalent to the require-

ment of using a Schrodinger-type wave equation. This is also supported by the fact that usually the resulting Hamiltonian is of a square-root

type and explicitly time-dependent, so that a Klein-Gordon type equation

is not appropriate. Let us now return to the above problem. If

indeed 1f(x)12 can be interpreted as a probability density, then

Pe = IW(x)j2dx is the probability that if a measurement of x is made it will lie in the interval t0, e3. Let us suppose that the classical metric has a singularity at x=0. It is commonly asserted

that the condition @(0) = 0 is sufficient for the avoidance of the classical singularity. This could be regarded as a correct interpretation only if 0 were an isolated point in the spectrum of x. If the -134—

spectrum of x is continuous around 0, then presumably the condition

pe=0 for all eE 0, el for some e', could be regarded as implying

the avoidance of the classical singularity at x=0. This definition

is good as long as we have a classical singularity to use it as

a "reference" point. But what about when there isn't such a

"reference" point? How shall we describe a singular situation in

this case?

The answer to the last question can only be intuitive. An

indication that something is going wrong can surely be considered

the fact that 0(x) becomes unboundedly large as x approaches a

point of its spectrum (including infinity, if it is contained in the

spectrum of x). More precisely, one can say that if the transition

rate (i.e. the transition probability per unit time) between one point

of the spectrum of x and all its neighbouring points becomes infinite

then we have a singularity at that point. But for all practical

purposes the infinite character of 0 suffices to indicate a

singularity. In connection with the above,two points need to be

discussed. The first is the problem of the boundary conditions.

Indeed, it is a common practice in quantum mechanics to obtain a

finite wave function by imposing suitable boundary conditions (e.g.

D(x) 0 as x r-- ± 00 ) . This is not always possible in Quantum Cosmology, because usually the imposition of such boundary

conditions leaves us only with the trivial wave function 0(x) = 0.

The second problem, which is a consequence of the first, is. related

with the normalization. Indeed, because of the usually infinite

character of 0 in Quantum Cosmology, one cannot normalize it unless

* These boundary conditions are reasonable for tha Quantum mechanics or Quantum field theory, where the underlying metric is the flat Minkowski metric of Special Relativitje.In Quantum Cosmology however,tue to the general covariance of the underlying General Relativity theory the imposition of such boundary condit.L- ons becomes problematic if not impossible . — 135 —

one uses infinite normalization constants. But this is not entirely

unfamiliar since one is essentially doing the same thing in quantum

field theory when using the various regularization techniques (e.g. a

cutoff constant or dimensional regularization).

Having discussed some of the outstanding difficulties of quantum

cosmology we now briefly mention the two approaches of quantum gravity

immediately applicable to quantum cosmology. These are the Dirac's

Hamiltonian method and the ADM formalism. They are both concerned

with canonical quantization, their main difference being that ADM

assumes the constraints have been solved while Dirac assumes that they

are part of the statement of the theory. In the next section we are

going to give an outline of the ADM formalism.

(b) We are now going to present ADM formalism. Like any other

quantization scheme, one main problem of the ADM formalism is

to investigate the dynamics of General Relativity. However, the

general coordinate invariance of G e.neral Relativity creates basic

problems to the above investigation: Usually, specification of the

field amplitudes and their first time derivatives at a given instant

suffices to determine the time development of the field (viewed as a

dynamical entity). But for General Relativity the metric field g

can be modified at any later time by making a general coordinate

transformation° Furthermore, it is one of the fundamental assumptions

of General Relativity theory that such an operation should not change

the physics. Thus, one is faced with the non-trivial problem of separating the metric into the part carrying the true dynamical inform-

ation and the part characterizing the coordinate system. The above problem is quantitatively similar to that caused by the gage invariance

of the Maxwell field in electromagnetic theory. Indeed, this gage invariance produces difficulties in separating out the dynamical modes in a Lorentz covariant description of the Maxwell field. In general, -- 136 — the qualitative result of invariance properties is to introduce redundant variables in the original formulation of the theory to ensure that the correct transformation properties are maintained. It is this clash with the smaller number of variables needed to describe the dynamics (i.e. the number of independent Cauchy data) which creates the difficulties in the analysis. Thus the main aim is to recast the theory in canonical form, i.e. to express it using the minimal number of variables needed. Once canonical form is reached, the physical interpretation of the quantities involved is immediate and canonical quantization can be directly applied. Two essential aspects of canonical formalism are: (a) that the field equations are of first order in the time derivatives, and (b) that time has been singled out so that the theory is in 3+1 dimensional form. These features are characteris- tic of Hamilton (Poisson bracket) equations of motion.

General Relativity, as we shall see later, can be recasted into canonical form. Possessing also the invariance under arbitrary coordinate transformations, General Relativity is, in this form, precisely analogous to the parametrized form of Hamiltonian mechanics, in which the theory is invariant under an arbitrary reparameterization, just as

General Relativity is invariant under a general coordinate transformation.

The action of General Relativity is thus in "already parameterized" form. The steps with which the "already parameterized" action of

General Relativity is reduced to canonical form are similar and in one- to-one correspondence with those leading from the parameterized form to the canonical form of the Hamiltonian mechanics. It is thus constructive to begin with a brief review of parameterized particle mechanics.

For simplicity, we deal with a system of a finite number M of degree of freedom. As we know, a given state of such a system is described by the general coordinates (qi)i-1,2,..M and their conjugate

-- 137 —

momenta (p;)4.7 m, i.e. by a point (pi,gi) in the configuration

space. The evolution of the system in time is of course governed

by the Hamilton's equations

q. = P H , p. - H i=1,2,..M (b.l) Ap i gi

where El d/dt, and H H(p,q) is the Hamiltonian of the system.

Equations (b.1) follow from the action principle 4'I=O where tl 1 L dt:= (' Epigi -H(P,q))dt (b.2) t2 `'2

It is understood that pi,gi are varied independently, but their

variations vanish at the endpoints tl,t2, i.e. pi (t1 ) = pi(t2), i(t2). Furthermore, if we also vary t, we get the "energy gi(tl) = g conservation" equation dH = )H = 0. The action (b.2) is said to dt 'at

be in canonical form, since every variable occurring in H is also

occurring in the pq term. One may say that the evolution of the system is described by the motion of the point (p,q) in the configurat-

ion space, i.e. by the trajectory (p(t),q(t)). But as t is an

independent (extrinsic) coordinate, i.e. t is not a coordinate of the

configuration space, this trajectory (and thus (b.2)) has to be

invariant under a general reparameterization of this coordinate, i.e.

nder the general transformation t •--. t` = f(t). Evidently the

action I, as given by (b.2), has not that property. One can recast

the action (b.2) in a form which has the above property: Indeed,

regarding t as a function qM+1 of an arbitrary parameter r, we

write (b.2) as ri r1 M+1 I = drL = dr( pig; ) (b.3) r2 r2 .i=1

+H(p,q)=0 is assumed to hold. where ' E d/dr, and the constraint PM+1

- 138 -

One may equally well replace this constraint equation by an additional

term in the action:

1 M+1 . I =dr ( piq! -NR(PM+1, p,q) ) (b.4) r2 i=1

where N(r) is a Lagrange multiplier. Its variation yields the

constraint equation q)=0 which may be any equation with the R(pM+1,p, solution (occurring as a simple root) -H(p,q). Bearing in pM+1= mind that N(r) transforms as dq/dr, one sees that (b,4) has the

desired property i.e. is invariant under the general transformation

r r = r(r). In achieving this general covariance one has paid

a certain price: Firstly, one has been forced to introduce the (M+1)st

degree of freedom with general coordinate q14+1=t and a corresponding conjugate momentum pM+l = -H. Secondly, one has lost the canonical

form of (b.2), due to the appearance of the Lagrange multiplier N in

the"Hamiltonian" H ."' NR of (b.4)(N occurs in H but not in the

pq' term of (b.4)). A further striking feature of H is that

it vanishes by virtue of the constraint R=O. This is again due to

the general covariance of this formulation which allows the recali-

bration r —*r(r). The action I, as given by (b.4) is said to be in parameterized form, and, as we have said (and will see in detail

later), the action of General Relativity may be written in a form

precisely analogous to (b.4). (of course, instead of one Lagrange

multiplier N there will be four, namely Nu , corresponding to

general transformations of all four coordinates x . Thus, the

problem one is faced with is to reduce an action of the type (b.4) to

the canonical form (b.2).

The problem is essentially solved by reversing the steps that

led from (b.2) to (b.4): Indeed, let us suppose that the constraint

has been solved for and let us insert equation R(pM+1,p,q)=0 pM+1

— 139 — the solution pM+1 = -H(p,q) into (b04). We get

rl I = dr( i -H(P,q)gM+1) (b.5) r2 i=1

We have

M M M dql dr(` .Pg -H(p,q) M+1)- Pidgi-H(P>q)dq,=dq.M pdiq H) i=1 i=1 i=1 M+1

Thus (b.5) becomes gM+1 (r1 ) I = (b.6) qM.0(r2)dqM+1( Pigi/dgM+1 -H(i,q)) i=l which, with the notational change qM+l ""' ' t, is identical with (b,2), i.e. is in canonical form. Thus, the reduction has been carried out.

One may note that (bot) could be reached from (b.6) not only by the change qM+1 (r) t but also by the whole series of substitutions qM+1(r) ..... f(r), upon identifying f(r)St. Thus is not gM+1 determined as a function of r by the dynamics. This is better understood by noting that, as we get from (b04), the equation of motion for M+l is : q ōR 1 N(r) (b.7) q M±1 3 PM+1

Indeed, since the Lagrange multiplier N is left undetermined by all dynamical equations of motion, this is also true for qM+1 (by virtue of the above equation). One thus sees that the form (b.2) is reached from (b.6) only after the imposition of the "coordinate condition"

t (or equivalently As we see, from (b.7) this could qM+1 f(r)Et). equivalently be done by specifying N(r) as an explicite function of r.

Summarizing one can say that the general procedure of reducing a parameterized action to canonical form consists of: (I) inserting the solution of the constraints, and (II) imposing coordinate conditions — 140 —

(usually by specifying the Lagrange multiplier N as an explicit

function of r). Having completed the analysis for the particle

mechanics one is now ready to proceed to General Relativity, since,

as we have said before, the steps are precisely analogous.

As is well known, Einstein's field equations follow from an

action principle ōI=O where

I = fd4 x (- I g 1 )1/2R (b.8)

(where from now on Greek indices are 4-dimensional and Latin indices

are 3-dimensional), 'g .I is the determinant of the metric g V R the Ricci scalar computed from g and we consider variations r v ff = . in the metric (i.e. g or the density ji (- ry 1)1/20"Y to, i g Of course, the so obtained Lagrange equations are second-order differ-

ential equations. It is our aim to obtain a canonical form for these

equations, i.e. to put them in the form of Hamilton's equations (b.1).

Equivalently, this means that one has to recast the action (b.8) into

the canonical form (b.2). Thus, the two main features of canonical

form (see the middle of page 136) have to be fulfilled. The first

of them, namely that the resulting equations must be of first order in

their time derivatives, is insured by using a Lagrangian linear in the first derivatives. This is called the Palatini Lagrangian and consists oC in regarding, besides g ry , the Christoffel symbols re, as independent quantities in the variational principle. That is to say,

(b.8) can be reqritten as

v (b.9) I = d4x(- 1 g 1 )1~2 gP R,y (1.- ) r orC est t—t, - e where Rey + l e _ o4 roc, v rot tot,e vs vc,c, Since RH? (1") does not explicitly involve gt , variation of the action (b.9) with respect to g P yields directly the equations -- .141 —

1 Gry R 0 (MO)) ~n►~ - g R

while variation with respect to v yields # P4 g t`y = g "? + g~~ ĪI + g Pgr - g r = 0 which, b~, , Ot, € OC fat as is well known, has the solution:

~roC _ 1 cx ° - (b.11)

Equations (b.10), (b.11) express the full content of Einstein's

General Relativity theory (in the absence of matter).

The second feature of canonical form, i.e. that the equations

must be explicitly solved for the time derivatives, is insured by a

3+1 dimensional breakup of the original four-dimensional quantities

appearing in (b.9).

In order to make this 3+1 dimensional breakup let us define

n., 4g N (_4g00)-1/2 , Ni s -

pij (_4g)1/2 (4 r.0 _ g 4 T-0 grs)q gipgj pq pq rs

(Here and subsequently, if a quantity has not indices to indicate its dinensioriāhty, is marked with a prefix 4 if it is four-dimensional

and is left unmarked if it is three-dimensional).

g13 in (b.12b) is the reciprocal to g13, i.e. g gjk ij = k Three-dimensional indices are raised and lowered byg13, g1 respectively

(e.g. Ni = g13N~). Some useful relations are:

2 - N2,4 g0i = Ni/N2, 4g13 = g13 - NINJ/N b.13) 4g00 = N'N.. In terms of the basic quantities (b.12) the Lagrangian of the action

(b.8) may be written as 4

— 142 —

=(-4g)1/2 L 4R= -g1 aOP 13-NRO-NiRi_2(PiJN.- 1 PNi+N li(g)1/2),i (b.14) J 2 where the stroke means covariant differentiation with respect to g13, and we have defined

1/2 R+ g-1(~ P2 i P= -Pi3P..)) ,R = -2Pij (b.15) Pi igju,R=P1J - -(g) i ~

At this stage, Einstein's field equations are obtained by varying the action

(b.16)

(L is given by (b.14)) with respect to the (now considered independent) quantities gi ,Pij,N,Ni. The variation with respect to N,Ni yields the four constraint equations

R (g.. P1d) =0 (b.17) where RP' given by (b.15). Since, as we will show below, the action

(b.16) is in parameterized form, the reduction to canonical form can now be made by following the general procedure described on page 139:

That is, we impose "coordinage conditions" by specifying NIL as explicit functions of the coordinates x r , and we also insert the solutions of the four constraint equations (b.17) into (b.16). But first we have to show that (b.16) can be written in a form precisely analogous to (b.4) (i.e. in parameterized form). Indeed, the first term on the right hand side of (b.14) is -gi j doP13 = k(gijP13) P13 tiO gi j - and thus (b.16) becomes

11i1/2 I= d 4 +N (g) ) (b.18)

Considering variations of gij,Pij,N and Ni which vanish at the endpoints one can discard the second and last term on the right-hand side — 143 — of (b.18), since they area total time derivative and a divergence respectively. Thus the action finally becomes

I = d4x( P13 - NO-NiRi ) (b.19) alOgii which is precisely analogous to (b.4), i.e. (b.19) is a parameterized theory's action. This form correctly expresses the invariance of the theory under a general transformation of the x14 , since in this formalism x 14 are parameters in exactly the same sense that r was in the particle mechanics case (see page 138 ). That the N and Ni are truly Lagrange multipliers follows from the fact that they do not appear in the pq' (i.e. P1' Gg..) term of L. ij Upon inserting the constraints (b.17) in (b.19) and imposing coordinate conditions by specifying N ,, as explicit functions of x#` , one arrives at an action which is analogous to (b.2), i.e. in canonical form, and thus canonical quantization can then be directly applied. To conclude this section, we briefly mention the generalization of the above formalism to include matter, which is straightforward: If the matter system is described by a Lagrangian LM, one can always put it in a4 parameterized form LM = - e R . Using the action i

I = d x (LG LM) (b.20) 0 (LG is given by (b.19)), we obtain the Einstein field equation

Gtv = 8 yl G T y , where Tr.., is the energy-momentum tensor of the matter system and G Newton's gravitational constant, if we set 1/2(T0i R 4 _ (-2(g)1/2GN2TOO, 2GN(g) + N'T00)). That is to say, Einstein's field equations follow by varying the action

1/2 OO i 1/2(TOi+NiT00) +P N(R0 GN2T )-Ni(R 2GN(g) I = d4x PDOgij. q~ - +2(9) - (b.21)

Variation with respect to N,N1 yields the extended constraint equations — 144 —

00 -2(g)1/2GN2T , Ri 2(g)1/2GN(T0'+ N'TGG) (b.22) where 11 14 are given by (b.15). These constraints replace the constraints (b.17). Thus, summarizing, one may say that Einstein's field equations G y = 8 ii GTS follow by varying the action

I = d4 x (Pi3 P1 *eGgl) (b.23) where PI q are appropriate functions of T13. 1 Upon inserting in (b.23) the constraints (b.22) and imposing coordinate conditions by specifying N,Ni as explicit functions of x ' , we obtain the action (b.23) in canonical form and thus we proceed with canonical quantization as before.

Bertotti-Robinson universe. As a preliminary step we are going to put the Langarian of the electromagnetic field in canonical form: v If F is the antisymmetric tensor describing the electromagnetic field, Maxwell's equations follow, as it is well known, by varying the action I = d4 x LM (c.l) where

4 1/2 i = 2(A + g) (c.2) LM . -A )f f flv

LM is the straightforward covariant generalization of the usual

Lagrangian of the electromagnetic field in the absence of sources t4V 1 Lem = A y , F + 1,v F t . Thus f frIt in (c.2) is the tensor density

f) = ( F Ft/ ) 1/2 F" (c.3)

The variation of the action (c,1) is taken with respect to A sand ff independently. In order to reduce this action into canonical form we first perform the usual 3+1 dimensional breakup: We define — 145 — f Ui _ 1 i j k A E i = Bi -A . (c.4) ' 2 ( k,j J,k)

f'3k where is the well-known totally antisymmetric tensor. Using the gravitational variables defined in (b.12) the lagrangian (c.2) becomes

L 1 OAi-A0Ei,i {V(g)-7'2gi~ ' J+B1BJ)) +N' i.k EJEk (c.5) (E `E (f J )

At this stage, Maxwell's equations are obtained by varying the action

(c.l) (with LM now given by (c.5)) with respect to the independent quantities Ei,Ai and A0. The variation with respect to A0 yields the constraint equation

Ej = 0 ,i (c.6)

Performing the usual orthogonal decomposition of the three-vectors

E',,Al i.e. putting

EOL+EST, Ai= AiL+AiT, trXEiL° 0 = V EiT, VxAiL= o = %•*A.T (c.7) we see that the solution of the constraint equation (c.6) is

El = EiT (L and T refer to the longitudinal and transversal part of the vector). Inserting this solution into (c.5) and noting that

E1T OE L = EILOA.T = E1T.A.L = A.T.A.L = 0 by construction, we finally 7 i 1 1 arrive at the action

OAiT 13 I -5'dX -EST - 2 (g)-1/2g (EiTEjT+BiB3)+Ni(fijkE3TBk) 8) which is in canonical form with respect to the electromagnetic field.

Next we have to find the extended constraints for the coupled Einstein-

Maxwell system. These are found from the form of the action (b.20) where LG is given by (b.19) and LM by (c.8) above. Varying with respect to N,Ni yields the extended constraint equations — 146 —

RO = - (g) g..(E1TE3T + B1B3 (c.9a) -1/2 ~J )

m im k R = g ~i~kEjT B (c.9b)

(where R 1'4 are given by (b,15)).

Thus, summarizing, one may say that the canonical action of the coupled

Einstein-Maxwell system is obtained from the action

I =fdx (L0 + LM) (c.10)

(where LG is given by (b.14) and LM by (c.8)) upon imposing coordinate conditions by specifying N as explicit functions of t x 1 and inserting the constraints (c.9). We are now ready to discuss the Bertotti-Robinson universe. This universe has the metric

2 ds2 =--- + ( x) 2((dx 0)2-(dx1)2-(dx2)2-(dx3) ) (c.11) which satisfies the Einstein-Maxwell equations (in the absence of sources )

FP. R ~ + F + xFt`P xF y = 0, F ?' _ =xF f = 0 (c.12) Y if we put

10 10 b 10 23 F y = 12-e (~ cosB + sinB) = e( r, cosB + ry sinB)• (c.13) Y grV where x means the dual operation and e,B are disposable constants. e2 Also we have identified b - (x1)2+(x2)2+(x3)2•

Thus, one has for the covariant and constravariant components of the metric tensor — 147 —

b 0 0 0 1/b 0 0 0

0 -b 0 0 0 -1/b 0 0 (g )= P` r 0 0 -b 0 (g )= 0 0 -1/b 0 0 0 0 -b 0 0 0 -1/b

One also has

4g = -b4 , g = -b3

We also have for the spatial metric gij

-b 0 0 -1/b 0 0

0 -b 0 0 -1/b 0 (c.16)

0 0 -b 0 0 -1/b

We want to calculate the action (c.10). We thus need to calculate

L and LM. G We start with LO which is given by (b.14):

LO = -9. a0P13-NRO-N1 Ri-2(PiiN 1 1(g)l j 2 /2),i

where the various quantities appearing above are defined in (b1:2),

(b.13) and (b.15), It is convenient to impose the coordinate conditions now, i.e. to specify N,Ni as explicit functions of x : We make

the choice

00)-1/2 = b-1/2, N - -4g 0 N (= 4g (c.17) i 0i -

As we see, our choice is identical with (b.12a) except for a minus sign, so we conclude that the choice (c.17) amounts in adopting the coordinate system to (-x0,x1,x2,x3) instead of (x0,-x1,-x2,-x3).

Consequently, we replace g with -g in all formulas obtained. In order to calculate L0 we first need the Christoffel symbols i jk of metric (c.16). They are : — 1 48 —

= -2x1b/e2 x2b/e2 T-1 1 Ī"121 = Ī 121 i13 = Ī131 = -x3b/e2

1 _ 1 0 T-122. 1- 1 33=x b/e2 r32 123 =

-2 2 _ /e2 _ (c.18) r22= -2x2b/e2 r 12 = 121 = -xib 13 - r 31

2 111 = 1 33 '/ r~323-2 r 2= -x3b/e2

33- -2x3b/e2 3b/e2 12= 1 21 0 r133 = 31= -x

3 3 = _ -xb/2 e2 11 r22 =x3b'2 e r 23= i" 32~

Also, for the metric (c.14) we have.

0 0 (c.19) pa m r-k - Ti ~-k Using (c.18) and the relation R j _ iJ ~ ij,m FLt im,j ,J ~ij mk ~k jm we can calculate the various components R.. the Riemann tensor: IJ

R1l = 4b/e2 -(2(x1 )2+4(x2)2+4(x3)2)b2/e4

R22 = 4b/e2 -(4(x1 )2+2(x2)2+4(x3)2)b2/e4 (c.20)

R33 = 4b/e2 -(4(xl)2+4(x2)2+2(x3)2)b2/e4

Using (c020) we can calculate the Ricci scalar :

11R 2 = 2/e2 R g R33 = 12/e2 -10/e (c.21) 11+g22R22+g33

We are now ready to calculate LG (from (b.14)). From (b.12b) we find (using (c.19))

Pij = 0 (c.22) - 149 -

From (b.15) we find (using (c,15), (c.21), (c.22)) :

RO = 2b3/2/e2, R1 = 0 (c.23)

The last term in (b.14) becomes (using (c.22),(c.17)):

-2(b3/2N11 ) , with N given by (c.17). We thus have

LG = -NR° -2(b 3/2N li)which, by virtue of (c.17),(c.23), becomes ,i

LG = 2b/e2 -2(b3/2(b-1/2} I1),i We have to calculate the second

)'i- -1/2 = term: Al = (b-1/2) t1 = (b-1/2r b - -3/2 (-b22x1 )/e 2 ki z b

l/2x1 +b-1/2(4xib)re2 = b l/2xi/e2 + 4b /e2. The second term of

LG above thus becomes -2(b3/2A1)i = -(2bx1 /e2),i -(4b2x1/e2),i =

1 -6b/e2 + 4((x )2+(x2)2+(x3)2)- - 12b2/e2+16((xl)2+(x2)2+(x3)2)b3/e4 = -2b/e2 -12b2/e2 +16b2/e2 = -2b/e2 + -7 . Thus, LG finally becomes

LG 2b/e2 -2(b3/2A1 ),i = 2b/e2 -2b/e2 +4b2/e2 = 4b2/e2 (c.24)

We now proceed to calculate LM. From (c.13) we have

—0 -be1 0 0

be1 0 0 0 where we have defined 0 be e1 =cosB/e,e2=sinB/e (co25) 0 0 2 ,

_ 0 0 -be2 0 j

Thus we have

!Fry 1 b 4 = e2 e

From (c.4),(c.3) we have

) ii Ei = foi = ( )1 /2F0i = 1/2g00g which, by using !Fry! ( IFftv FOi

(c.14),(c.25) gives El = e2e2b , E2'= 0, E3 = 0. We thus see that -- 150 -

for the orthogonal decomposition of (c.7) we can choose E1T = E1,

E2T = 0, E3T = 0, EL = 0. Using the above and (c.17) LM (which is

given by (c.8)) becomes, on defining ATS = A, :

-1 /2 /2 13 LM = -eel 2 b ~0A b 2 -1 g(EEiT3T + BB)'1

Because of our choice (c.17) of coordinate conditions, the constraints

(c.9b) are identically satisfied, thus we have yet to satisfy (c.9a).

By using (c.23), (c.17) these constraints become

2b3/2/e2 -1 /2g..(E EJ = 2(-g) T + B1B3) and substituting

this to L above we finally have

2 LM = -e1 e2b OA -2b/e2 (c.26)

Using (c.24),(c.26) the action (c.10) is finally obtained in

canonical form:

i = fx(-e22b ?0A -2b/e2 + 4b2/e2) (c.27)

Thus we immediately recognize the canonical variable A, its conjugate momentum PA -e~e2b and the Hamiltonian of the system

C C 2 H - •"'~""'"e PA PA where C1 = 2/e1 e2. Porceeding with canonical ē quantization, i.e. making the substitution A —. A, PA ...y. a b one obtains the Hamiltonian operator 2 h t w H = '2.' 9-2+ Using the Schrodinger equation HD(A,t)=g'a a • e bA e ZA at and putting @ K(A)0L(t) we have for the time dependence of tD the equation dL/L = -jC2dt (C2 a positive constant) with solution

-1C2t L(t) = D2e and for the A dependence of W the equation

- 151 --

2 C dK + ~ - C2K = 0. With the solution K(A) = D1eiEA e dA e dA

(where Bis either of the real roots of the equation C1x2+C1 x-C2e2=0).

1 D2etEAe 2t We thus have ID = D -tC and the probability density

f>. ON = D2D2 is thus constant. Bearing in mind that Bertotti-

Robinson is a universe filled with a constant electromagnetic field

(see P. Dolan, Comm.Math.Phys., 9,161(1968)) the above result is

justified since one should expect that the various values of the

constant electromagnetic field will appear with equal probability.

(d) In this section we are going to quantize Godel`s universe.

Before actually proceeding to the quantization we shall need

a number of properties of this universe which will clarify the various

steps of the quantization procedure. The line-element of Godel`s

universe has the form x ds2 = a2(dx2d - x7e +( 2x1/2)dx2e + 2 ldxodx2 - dx3) (dol)

with a being a constant. If S denotes the four dimensional manifold

with metric (dol) then :

(a) S is homogeneous, i.e. for any two points P,Q of S there is

a transformation of S into itself which carries. P into Q.

This follows from the directly verifiable fact that S admits

the following four transformations:

(I) xo ..,. xō = xo + c, xi ,..,.3 xT = xi for Vo

(II) x2 .._ x2 = x2 + c, xi ... . x.= xi for in

(III)x 3 .,..~,, x3 = x3 + c, xi .... x} = xi for i#3

(IV) xl ~.~ xl = x1 + c, x2 --*- x2 = eCx,xi . . x.= xi for i/1,2

- 152 -

(b) S has rotational symmetry, i.e. for each point P of S there

is a one-parametric group of transformations of S into itself

which carries P into itself.

space x1 = constant.

Properties (b),(c) are seen to hold if one makes the transform-

ation (x0,x1 ,x2,x3) . ►- (t,r,y,O) defined by

exl = cosh(2r) + cososinh(2r), x e = 2 sinosinh(2r), 2

x,-2t xo-2t tar,(2 + - V.) - e-2rtan(O/2) with x3 =2y. 2 VT < 2

Then, after a lengthy calculation one finds that ds2 is given in

the new coordinates in the form

ds2 = 4a2(dt2-dr2-dy2 +(sinh4r-sinh2r)d02 + 2 lsinh2rdodt)

which exhibits directly properties (b) and (c),

(d) S contains closed timelike lines, i.e. it is not possible to

assign a cosmic time t to each event of S in such a way

that t always increases, if one moves in a positive time-like direction.

The metric (d01) leads to the following covariant and contravariant

components:

x -x 1 1 0 e 1 0 -1 0 2e 0

0 -1 0 0 0 -1 0 0

g~a.V =a2 x 2 . g' =a -xl -2x1 e 0 e /2 0 2e 0 -2e 0 (d.2)

0 0 0 -1 _ 0 0 0

The only non-vanishing components of the Ricci tensor are 2x1 x1 = 1,R22 = e ,R 02 = R20 = e (d.3) R00

Thus the Ricci scalar is R = 1/a2 (d.4) - 153 -

We then can see that the metric (d.l) satisfies the Einstein

Field Equations

Gj,,v =8 f GTr (d.5) where G is Newton's constant,

1 1 G = R - 2Rg ~y T Pr =Pu uy - 2g (d.6) rt rv ( t‘ r v

We observe that the matter source is a perfect fluid with constant density and pressure p given by p

r=1/817 Gat ,p=1/2 (d.7) u is the unit four-vectoring the direction of the x0-lines. Since r we shall need them later we list the contravariant and covariant components of the unit four vector along the x0,x1,x2-lines which are correspondingly : -x u =(1 /a,0,0,0) , kft =(0,t/a,0,0) , lw =(0,0, f2e 1/a,0) (d.8) u r =(a,0,aexl ,0) , k =(0,- a,0,0) , ly =( ~2a,0,aex` l/ !~ ,0)

It is apparently not immediately possible to use ADM formalism to quantize the metric (d01), since in view of property (d) there is not a cosmic time t. But, as we shall show, this can be eventually done.

The crucial observation is that it is no loss of generality to restrict attention to the surfaces x3=constant. Since this will prove to be a serious step, we have to justify it. The justification is of double character: Firstly, from a geometrical point of views loss of generality does not occur in view of property (c). Secondly, from a dynamical point of view, again there is no loss of generality in restricting attention to the surfaces x3=constant. This follows from the form if the energy-momentum tensor Tr,v . More precisely, the 1-3v row of Tµ v has the form 1-3v =(0,0,0,a2/2), and by symmetry, this is — 154 —

also the form of T column. Y 3 Thus, we loose no dynamical degree of freedom with the restrict-

ion x3=constant, in the sense that we can arrive at the same wave

equation by only modifying the coordinate conditions fixing the lapse

and shift functions N,Ni. (In fact, only the coordinate condition

fixing N needs to be slightly modified). Thus, restricting all

indices to the values (0,1,2) we obtain from (d.l) the line-element

dw2 = -a2dx1 + a2(e2b)iddxidxj (d.9)

where i,j attain the values 0,2 and b is the 2x2 matrix

I lnl x1/2

(d.10) (bij) = I x /2 xi-ln 2 i mn This metric has the covariant and contravariant components g mn,g which are obtained from (d.2) by restricting to m.n (where r , V now m,n attain the values 0,1,2 only). Similarly, for Gmn and Tmn.

Thus, the metric (d.9) satisfies the Einstein Field Equations

Gmn = 811 GT (d.11) mn

where the quantities involved are given by (d.3),(d.4),(d.6),(d.7)

and (d.8) on making the above restriction on the values attained by

the indices , y r As we now observe, the metric (d.9) possesses a cosmic time t, namely t=x1. That this is so, is evident from the fact that, according

to (d08), kmlm = kmum = 0, that is to say the surfaces xl=constant

are orthogonal to the x1-lines. The "2+1" splitting of metric (d.9)

indicated by its given form, is thus a real one, and hence we are justified using ADM formalism from this point onwards. Of course, we shall use a reduced version of ADM formalism from a "3+1" decomposition — 155 — of a four-space to a "2+1" decomposition of a three-space, but, because of the very nature of the ADM formalism, the reduction is trivially achieved by restricting all three-dimensional indices of Section

(b) to two-dimensional ones. Since, furthermore, as we have explained above, it is the same trivial reduction which leads from the metric

(dol) to the metric (d.9), we conclude that the proposed quantization scheme is technically and conceptually equivalent to the usual applications of the ADM formalism.

The 2-dimensional "spatial" metric of the three-dimensional space-time (d.9) is gij = a2(e2b)ij or

l x1 gij = a2 2x I (from this point onwards all indices attain x 1 e e the values 0,2 only). a4 2x1 The determinant of this metric is g = - `L_ e and evidently, its 2 Ricci scalar is R=0. Also, because of the symmetric nature of gij we are only left with the 3 momenta P00, P02 and P22 conjugate

b02,b22 correspondingly. Of course, in the end we shall to the b00, be left with only one independent momentum, say P02, corresponding to the variable xl, since all bij are linear combinations of x1.

We see that metric (d.9) is homogeneous, i.e. admits the trans- formatiors(see property (a)) (x x.) - 3 (xl,xi+c) where c is a constant. It is therefore permissible to assume that all P13 are, like gij, functions Qf x1 only. Working in units where 8n G/c4=1/2, h=2 fl (c is the velocity of light and h is the Planck constant), we have that, according to ADM, the field equations (d.11) are obtained by varying the action

I = C1 P13dgij (d.12) where C1 is a positive constant. Substituting the form of gij — 156 —

we have dg1 = 2gikdbkj and hence

I = 2C1 ,StPikdbki (d.13)

The action (d.13) is not yet in canonical form, since we have

to satisfy the constraint equations (b.22) :

l ( 1 .,k . 2 , i . ,.,k 2,..2 R + P (d.14) g 2

-2Pki k = 2GN li + N1Tll) (d.15) -g (T

where N,Ni are the lapse and shift functions respectively. These

are to be determined by coordinate conditions (N,Ni are really

Langrange multipliers). Let us first examine the momentum constraints

(d.15): The left-hand side of (d.15) is identically 0 in our case:

Pij are explicit functions of xl only, and the covariant derivative

reduces to a simple derivative (all the Christoffel symbols of the

metric are zero). g1 In order to satisfy (d.15) we thus are led to the choice

From (d.6),(d.7) and (d.8) we obtain T11 = a2/2, Ni = -Turin. Til = 0 for i=0,2. Thus the coordinate conditions fixing N0, N2

are Ni = 0, i=0,2.

We now turn to the Hamiltonian constraint (d.14). First of

all we observe that in the action integral (d.13) P0 does not appear 0 explicitly, owing to the fact that db00 = 0 (b00= lnl). Thus, we

are free to choose P°0 arbitrarily as a function of x1,at the

expense of adding it to the canonical form of the Hamiltonian so that

the total Hamiltonian becomes explicitly xl dependent. We shall

use this freedom to simplify the Hamiltoian constraint. Substituting 21- 2x1 in (d.14) the values R=0, g = - we obtain 2 e

1 0 2 1 2 2 0 2 4 2x1 2 a2 2(P 0) + 2(P 2) + 2(P 2) = a e GN (d.16) 2

— 157 — x On choosing PO0 = a 2 1 and N=1/a \ G (d.16) becomes

0 P22 = : 2t (d.17) P 2

The integrand in (d.13) becomes Pikdbki = 2P02db02+P22db22 = 2P02db02 t 2jP02db22 by virtue of (d.17). On substituting the

values of b02 and b22 from (d.10) this finally becomes P'kdbki =

(1 ± 2t)PO2dx Substituting this last into (d.13) we finally obtain l' the action integral in canonical form:

I = 2(1 ± 214C1 fOdx 1 (d.18)

H We thus see that the canonical form of the Hamiltonian is can = -2(1 ± 2t)C1 02, thus giving the total Hamiltonian

x H = H = -2(1 ± 2i)C1 P02 + a2e l (d.19) can+POO

We now proceed with canonical quantization, making the substitution

xl ,. x1, PO2 --ter and solving the Schrodinger wave equation idx 1 dO He - t--- . This equation, by virtue of the above substitution and dx1

(d,19) becomes: x a2e lm = ((1-2C1 ) )āO or 1

x dm = (TC3+C4t)e lm (d.20) dx1

2 2 2 2 where C4 = a (1-2C1 )/(8C1-4C1 +1), C3 = 2a C14/(8C1- C1+1).

The general solution of (d.20) is

lex1 -C3ex1 C4 @=Ae .e (d.21) — 158 —

Because we have used a Schrodinger wave equation, WX@ can be interpreded as a probability density g and thus 2 C2 exl p(A a 3 which, evidently, blows up at + or - - 159 -

REFERENCES

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Narlikar, J.V. (1977), Nature, 269, 129. Ryan, M., Shepley, L. (1974), "Relativistic Homogeneous Cosmologies", Princeton Univ.Press, Seifert, H.J. et al. (1973), Comm.Math.Phys., 34, 135. Seifert, H.J. et al. (1974), Comm.Math.Phys., 37, 29. Wheeler, J.A. (1977), G.R.G. Journ., 9, 713. — 160 —

CHAPTER V

CONCLUSIONS AND SPECULATIONS

In this last part of the work a presentation will be given of the physical interpretation of the various results contained in it. Also, some speculations will be made about the possible resolution of some of the outstanding problems of the contemporary theories of space-time. Of course, as it is inevitable for a con- clusive section, the various claims will not be stated as mathematical propositions, thus needing rigorous mathematical proof, but rather they will be presented as physically appealing remarks.

To begin with, let us discuss the main result of Chapter I, namely: that the set of all solutions of Einstein's Field Equations, viewed as a subset of the set of all Lorentz metrics, contains no isolated points, when the Whitney fine topology is used to topologize the set of all Lorentz metrics. As we have asserted in Chapter I, this can be viewed as an existence theorem guaranteeing that, given a metric which is a solution of Einstein's field equations, there is always an appropriate coordinate system and a limiting process which yields another metric, also solution of the field equations. The fact of course that the result is proven in a particular topology may, at first sight, seem to restrict its validity. But, as it is proven in

Chapter I, this topology is equivalent to the interval topology, which in turn is the most physical topology one could think of. Another way of looking at this result is to view it as a justification to the various perturbation schemes frequently used to simplify the calculations in General Relativity. Indeed, the above result qualifies the use of perturbative approaches since it guarantees that there are solutions of Einstein's field equations "near" any given solution. In particular, 161 -

the well known weak field approximation is justified by this result.

Furthermore, the result seems to suggest that by repeated application

of the weak field approximation one could arrive at any given solution

of the field equations. In this sense, the above result seems to

substantiate the claim that any region containing an arbitrarily strong

gravitational field can always be reached from a region with an arbitrarily weak gravitational field. In other words, one can start from a flat region and, adding a bit curvature at a time, one could reach any curved region, no matter how strongly it is curved. This last is very reasonable, from a physical point of view, although in practice it may well invoke great technical difficulties.

Another useful result for the various perturbation schemes is the notion of global and local Killing vector fields as developed in page 13 , Indeed, one of the main problems one is faced with when one perturbs a solution of the field equations is the occurrence of linearization instabilities. That is, roughly speaking, the occurrence of solutions to the linearized equations which do not extend to solutions of the full equations. It is known that these linearization instabilities occur (for vacuum spacetimes with or without cosmological constant) whenever the background spacetime admits a global Killing vector and a compact Cauchy surface. This assumption is equivalent to the Global Killing holonomy group being non-trivial. Also, one could say that, whenever the background spacetime admits only a local non-trivial Killing holonomy group, then there will be no linearization instabilities in the perturbation.

Furthermore, reformulating linearization instabilities ideas in the language of Killing holonomy groups has the advantage of being immediately generalizable to quantum gravity because it does not involve the notion of curves of solutions of the field equations which is a — 162 — purely classical one.

Another point which is worthwhile to be mentioned is the example given on pages 5 , 6 . The Schwarzchild solution has been obtained as the canonical "limit" of the Reissner-Norsdstrom solution when the charge parameter tends to zero. In view of the recent ideas about black holes radiating away particles and energy

(see, for example, Gibbons, G., Hawking, S. (1977), Phys.Rev,D.15,

2738) one could give a physical interpretation to the above mentioned example. Indeed, one could regard this example as representing the process in which a "Reissner-Norsdstrom" type black hole goes over to a "Schwarzchild" type one by decharging itself; an idea which is recently widely accepted. As we shall see later, this idea combined with another one produces a mechanism which provides an escape from the General Relativistic singularities.

In Chapter II a differential structure is obtained from a causal one and then a boundary using only the causal structure is constructed.

This boundary is proven to be Hausdorff and,furthermore, its existence is proven to be related to stable causality holding on M. Physically interesting is also the result that, with the induced differential structure, M is made "as smooth as it looks", i.e. M appears continuous to freely falling observers. This is in agreement with the accepted picture of two observers one of which falls into a black hole while the other escapes to infinity: Neither the one who falls into the black hole, nor the one escaping to infinity will notice anything discontinuous happening. The one who escapes to infinity will see the other observer smoothly "slowing down", whereas the one who falls into the black hole will continue to see the manifold structure of space-time holding, at least as long as he does not hit the actual singularity.

Chapter III is concerned with b-boundaries and is thus very — 163 — mathematical by nature. Of some importance, from a physical point of view, is the reformulation of the original definition of the b-boundary in terms of structures of M only. Unfortunately, the structures required are several, not just the causal structure of M, and thus the b-boundary is found not to be equivalent with the causal boundary of Chapter II. This could be directly seen from the properties of the causal boundary (e.g. Hausdorffness) which are not shared in general by the b-boundary.

Chapter IV is very physical by character and thus its physical interpretation needs no further discussion here. In connection with the assertion made in the beginning of this chapter, regarding the status of the cosmic censorship principle, namely that, contrary to the early beliefs of Hawking and Penrose, the principle seems not to hold in General Relativity (see page 117), one could add that very recent results suggest that the assertion is true, i.e. the cosmic censorship principle does not hold. Indeed, it has been shown (see

W.A. Hiscock, Cosmic censorship, black holes and particle orbits,

G.R.G.j o ux;99 (1979)) that, assuming Hawking's area theorem (see

Hawking and Ellis "The large scale structure of space-time", C.U.P.

(1973), ch. 9), one is led to the existence of stable, near-circular orbits arbitrarily close to the horizon (r=2m) of a Schwarzchild black hole. This fact is very odd. Combined with the fact that the area theorem is essentially tied up with the Cosmic Censorship principle, the above result constitutes a strong point against this principle.

As discussed in Chapter IV, part A, section (a), being left with the possibility of occurrence of naked singularities (and thus loss of predictability, occurrence of closed timelike lines, etc. (see

F.J. Tiples (1976), Phys.Rev.Lett. 37, 879)) is disturbing and thus the search for mechanisms capable of eliminating the singularities is — 164 —

now of thebutmost importance. In fact, people have been trying to do

this for several years. As explained in the previous chapter, these

attempts are oriented either towards the use of negative energy or

towards quantum uncertainty. Since both of these concepts are, to

say the least, not overwhelmingly acceptable, it would perhaps be

constructive to conclude this work by a suggestion of a mechanism

capable of eliminating the singularities which certainly does not

involve negative energy, and involves only indirectly the uncertainty

principle. This suggestion is partly motivated by the example on

pages 5, 6, and in particular, by its interpretation given in the

pager 2.The key observation is that the existence of charge (as well

as angular momentum) can prevent the occurrence of singularities (see

Hawking and Ellis "The large scale structure of space-time", C.U.P.,

1973). Guided by this, one is led to seek for mechanisms producing

charge from an initially neutral collapsing star. A physically reasonable and relatively harmless way to do this would be to suggest

that the equivalence between gravitational and inertial mass breaks

down in the presence of strong un-uniform gravitational fields, such

as present in a collapse. Indeed, an hydrogen atom for example, would

finally be separated into protons (and neutrons) and electrons. This

would come because, due to the breakdown of the equivalence between

gravitational and inertial mass, the heavy nucleons would acquire

different acceleration than that of the light electrons. Similarly with other atoms. Now, invoking the picture of a black hole "decharging" itself, and assuming that the rate of emission of the different species of charged particles is not the same (perhaps because of the different location they finally occupy, as explained above), we are left with a charged collapsing star from an initially neutral one. The crucial assumption of the breakdown of the weak equivalence principle — 165 — made above, is not in contradiction with the results of the experiments

establishing the equivalence of the inertial and gravitational mass (to

within 1 part in 10-11, e.g. the Galileo experiments, the Dicke experiments, etc.), as it might seem at first sight: All of these experiments have been carried out in the Earth's weak, almost uniform gravitational field. No doubt, of course, a rigorous mathematical formulation is needed if the above outlined mechanism is to be taken seriously.

Perhaps something along the lines of spontaneously broken symmetries, used in quantum field theory, could constitute the framework of such a mathematical formulation.