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\begin{document}

\begin{center}
{\large\bf Space-Time Structure from a Global Viewpoint\footnotemark
     \footnotetext{
     Based in part on a series of lectures given by the author at Birkbeck 
     College in January, 1968. I wish to thank M. WALKER for making available his 
     notes on those lectures.}}\\
{\sc R. Geroch\footnotemark\footnotetext{
     National Science Foundation postdoctoral Fellow.\\
     Present address: Department of Physics, University of Texas, Austin, Tex.}}\\
{\small Department of Mathematics, Birkbeck College - London}
\end{center}

\vskip .25in\ni{\bf Introduction.}

Global methods have, in the last ten years or so, begun to make a significant
impact on thinking in general relativity.  These methods consist of
a collection of more or less unrelated tools which can provide greater insight
into a given (and often local) problem and which occasionally suggest new
and interesting problems of their own.  Global techniques are not 
1) a highly mathematical alternative to local differential geometry,
2) a fixed body of knowledge, developed to carry out some specific program 
in general relativity
or 3) difficult.

The effective application of global methods often depends to a large extent
on intuition.  It is extremely difficult to understand global arguments 
(much less to discover and prove new theorems) unless one already has a good 
qualitative idea of what is to be done. Unfortunately, it is hard to develop
an intuition, for most papers either ignore global techniques completely or
else assume that the requisite background material has already been covered.
In this paper, we shall present some of these intuitive concepts on which so
much of the global work is based.

What are global techniques good for? Of course, the principal application
is to proving theorems about space-times.  To this end, local and global 
methods are often used together. What one is able to do---even with local 
tensor calculus is determined to some extent by the global properties of 
the particular space-time.  Furthermore, global methods permit one to isolate
and \marginpar{72} 
study in detail certain aspects of a space-time, for example, its causal 
structure
and topology. Since such properties are perhaps more ``fundamental''
than the differentiable structure on which tensor analysis is based, it is 
important to know which results about a space-time involve only its causal
structure, only its topology, etc.  Finally, it is often necessary, in both
local and global arguments, to set bounds on how pathological are the 
space-times being considered.  Practically all pathologies are of a global 
nature. Global techniques, therefore, are used to classify pathological 
situations with a view toward deciding which situations are to be ruled out 
by physical considerations.  This role cannot be dismissed as mere 
hair-splitting. How, after all, is one to identify the space-times which are 
too pathological to be of physical interest unless he has at least examined 
the possibilities which can arise?

Each global technique consists of a collection of constructions, theorems,
and counterexamples.  Fortunately, given a general idea of what constructions
are available and how they are used, it is fairly easy to reconstruct the
proofs of most of the theorems.  Thus, statements of theorems are not 
a very important part of the subject.  It is far more useful to have a good 
qualitative idea of what is available than a long list of results. 
The counterexamples, on the other hand, are extremely important. They provide 
not only a testing ground for new ideas, but also models to serve as a guide 
in checking and discovering theorems.

The purpose of this paper is to indicate in general terms what global
techniques are available and what each one is good for.  While it has been
necessary to omit not only a number of minor theorems, but also many
details in the proofs, we have tried to include a reasonably complete survey
of the constructions and counterexamples used in global work. 
Furthermore, in order to give some idea of how the techniques are used, 
we have given some of the more important applications in each case. 
With a few exceptions, all definitions and theorems are stated precisely.
A number of exercises are scattered throughout the paper.

\vskip .25in\ni{\bf 1. --- Topology and manifolds.}

Very little topology has so far found application in general relativity.
Essentially, all that is needed is the notion of a topological space, 
some definitions (continuous, compact, closed), and a few elementary theorems.
Unfortunately, most text books in topology contain a great deal more material
than is normally required for applications in relativity.  A good introductory
discussion of general topology is contained in the first three chapters of
Wallace's book [1].

In this Section we shall discuss briefly what topology is and where it is 
useful 
\marginpar{73}
in relativity. We then introduce certain global properties---mostly of a
negative character---about manifolds.

Let $X$ denote the Euclidean plane.  Given two points, $(x,y)$ and $(x',y')$
in the plane, we define their separation,
\begin{equation}
    d=[(x-x')^2+(y-y')^2]^{\mbox{$\frac{1}{2}$}}
\end{equation}
The distance function (1.1) leads to the concept of an open set. 
Intuitively speaking, a set $O$ in the plane is open if $O$ contains all 
points, ``sufficiently close'' to each point of $O$.  More precisely, 
$O$ is open if, for every point $p\in O$, there is a disk centered at $p$ 
which lies entirely in $O$ (the disk representing points sufficiently close 
to $p$).  These open sets satisfy the following three conditions:

\begin{itemize}
        \item T1) The union of an arbitrary collection of open sets is open.

        \item T2) The intersection of two open sets is open\footnote{
    Therefore, the intersection of any finite collection of open sets is open.
    Note, however, that the introduction of an infinite collection of open 
    sets need not be open.  For example, the intersection of all disks
    centered at the origin of $X$ consists of the origin alone, and this
    single point is certainly not an open set.}.

        \item T3) The entire set $X$ and the empty set are both open.
\end{itemize}

We may interpret these open sets as defining the notion ``sufficiently close''.
That is, intuitive statements of the form ``All points sufficiently close 
to $p$ have the property $\ldots$'' are to be replaced by introducing
precise statements of the form ``There is an open set containing $p$ 
all of whose points have the property $\ldots$''.
On the other hand, the original distance function (1.1) defines a stronger
notion: ``how close''.

The idea of topology is to retain the concept of ``sufficiently close'',
but in such a form that we do not have the additional information of
``how close''.  More precisely, a topological space [1] is the set $X$ 
along with a collection of sets of $X$, called {\bf open sets}, subject to the 
conditions T1)-T3).  A set $C$ is then {\bf closed} if its complement, $X-C$,
is open, \ie, if all points sufficiently close to any point outside of $C$ 
are outside of $C$. Note that a set need not be either open or closed. 
Two or three other definitions concerning topological spaces will be 
introduced later when they are needed.

\prop{Exercise}{Show that the usual topology on Minkowski space is Lorentz
invariant (\ie, that Lorentz transformations take open sets to open sets).
Show that this topology cannot arise from any Lorentz-invariant 
distance-function.}

\marginpar{74}
There are two situations in which topology makes an appearance in
relativity.  The first involves the topology of the underlying manifold
space-time\footnote{
        By a {\bf space-time} we shall understand a 4-dimensional manifold
        with a $C^\oo$ metric $g\ab$ of Lorentz signature, $(+,-,-,-)$.  (We 
        occasionally use the word space-time to refer to lower-dimensional 
        examples.)  All manifolds will be assumed to be Hausdorff and 
        paracompact.}
Of course, a space-time has a great deal more structure than
merely this notion of ``sufficiently close'', but it turns out to be more
appropriate to express certain arguments purely in terms of the topology.
The reason for introducing topology is primarily one of convenience: an
argument, stated topologically, often becomes more transparent. 
The second place in which topologies are used is in the discussion of 
certain ``large'' sets which arise in relativity, notably the set of all 
Lorentz metrics on a manifold [2,3] and the set of all causal curves joining 
two points [4,5].  Again, it would certainly be possible to avoid all 
terminology associated with topology, but it is more convenient not to 
impose such arbitrary a rule.

Every space-time is based on a manifold, and every manifold has a
topology [6].  Thus, certain very coarse features of the space-time are 
determined by its topology.  We are already familiar with two examples of 
topological spaces.  Euclidean n-space \Re{n} and the n-dimensional sphere 
$S^n$\footnote{
        The topology on \R{n} is given by the distance function analogous 
        to (1.1).  The topology on $S^n$ may be obtained by embedding the 
        sphere in the usual way in Euclidean $(n+1)-$space, and taking for 
        the distance function the Euclidean distance between points of the 
        sphere. We shall use the symbols \R{n} and $S^n$ to mean either the 
        topological spaces or the differentiable manifolds.}.  
We can obtain other manifolds from these by taking {\em products} of 
manifolds [6].  For example, the product of \R{1} (the line) and \R{2}
(the plane), the product of \R{1} and $S^1$ is the cylinder, and the product 
of $S^1$ and $S^1$ is the torus, $S^1 \times S^1$ (not $S^2$).
Many of the manifolds encountered in relativity can be constructed from \R{n}
and $S^n$ by taking products.

Consider, for example, the (extended) Reissner-Nordstr\"om solution [7].
The 2-spheres $r={\rm const}$ are surfaces of spherical symmetry.  It is 
convenient, therefore, to draw a picture of this space-time by assigning
a single point to represent each 2-sphere.  The result is shown in Fig. 1a). 
In order to make the diagram easier to draw, we have brought the asymptotic 
regions in from infinity to a finite distance [8].   Such a transformation is 
permissible because it does not affect the situation from the point of view 
of the underlying manifold.  Therefore, the manifold of the 
Reissner-Nordstr\"om solution is the product of the 2-sphere $S^2$ and the 
plane.   But this Figure is topologically the same as the plane, \R{2}. 
To prove this fact, we have redrawn the plane Figure in Fig. 1b), but now 
with a set of co-ordinate lines.  The correspondence between the figure and 
the plane, obtained by mapping the 
\marginpar{75} 
co-ordinate lines into a rectangular grid on the plane, is continuous\footnote{
        A mapping from one topological space to another is said to be 
        {\bf continuous} if the inverse image of each open set in the 
        second space is an open set in the first.\\
        {\bf Exercise}  Verify that, for functions of a real variable 
        (maps from \R{1} to \R{1}), this definition reduces to the usual
        $\epsilon$-$\delta$ definition of continuity.},
both from the figure to the plane and from the plane to the figure. 
(Such a mapping is called a {\bf homeomorphism}, or {\bf topological 
equivalence}.)  We conclude that the Reissner-Nordstr\"om manifold is 
$S^2\times\R{2}$.

\prop{Exercise}{Verify that the Schwarzschild manifold is also 
        $S^2\times\R{2}$.}

What information does this conclusion reveal about the Reissner-Nordstr\"om
solution?  The answer is, very little indeed!  It does not tell us about the
location or even the existence of singularities.  It does not tell us that 
the Reissner-Nordstrom solution has no closed timelike curves.  It does not 
tell us whether or not the Reissner-Nordstr\"om solution has a Cauchy surface. 
It does not tell us that the Reissner-Nordstr\"om solution has asymptotic 
regions, or a ``wormhole'' representing a charge.  What the statement
\marginpar{76}
``The Reissner-Nordstrom manifold is $S^2\times\R{2}$.'' does say is
precisely what the underlying manifold---the arena for the geometry---is.

This point---that the underlying manifold gives practically no information
about the spacetime---is an important one.   We illustrate it with a few 
examples.

Consider the open disk $D_1$ in the plane (Fig.~2a)). (Topologically, the
disk is equivalent to the entire plane.)  One can see at a glance from 
Fig.~2a) how to put a boundary on our disk.  One might imagine, therefore, 
that if this were to represent a space-time, then the ``singular points''
(if any) would be represented by certain points on the boundary. Other points 
on the boundary might be interpreted as ``points at infinity'' as in Fig.~1a)). 
In fact, placing this particular boundary on the disk we have used more than 
just the manifold structure of the disk.  Consider the second disk $D_2$ 
in Fig.~2b). We map the disk $D_1$ onto $D_2$  by mapping the radial spokes 
as shown (a homeomorphism).  Although $D_1$ and $D_2$ represent exactly the 
same manifold, the ``natural'' boundary for $D_2$ is quite different from that 
for $D_1$.  For example, the boundary points $p$ and $q$ of $D_1$ are
mapped into the single boundary point $r$ of $D_2$.  We see that the
notion of a ``natural boundary'' is simply not defined for a manifold; 
some additional structure, such as a Lorentz metric or a causal structure
is needed.  Although one might, intuitively, feel that certain statements
concerning ``singular surfaces'' etc. for space-time should be meaningful, 
the precise formulation of such statements may turn out to be a difficult 
problem.

Consider the two curves $\gamma_p$ and $\gamma_q$ in $D_1$, consisting of 
spokes radiating from the center to the boundary points $p$ and $q$, 
respectively.  In $D_1$, these curves
\marginpar{77}
do not appear to ``meet at the boundary'', yet the appearance is just the
reverse in $D_2$.  That is to say, the notion of two curves ``meeting at 
infinity'' is also not defined by the manifold itself.

Finally, let $\xi^a$ and $\eta^a$ be two vector fields on $D_1$.  What
do the following mean: ``$\xi^a$ approaches $\eta^a$ at the boundary''
and ``$\xi^a$ approaches zero at the boundary''?
Using mappings similar to that above, it can easily be
that neither of these statements can be given any reasonable meaning.
Two vector fields whose components in one co-ordinate system, approach each
other as we approach the boundary will not in general have this property 
in another co-ordinate system.

\prop{Exercise}{Verify that the following statements have no obvious meaning.\\
         1) The Friedmann models have a point singularity;\\ 
              \tab the Schwarzschild solution has a line singularity.\\
         2) A spacelike surface is asymptotically null.\\
         3) A timelike curve is asymptotically null.\\
         4) The metric approaches the Minkowski metric as $r\to\oo$.\\
         5) Two geodesics approach the singularity from the same direction.}

We next attempt, to construct a space-time whose underlying manifold
is the same as that of the Reissner-Nordstr\"om 
solution---$S^2\times\R{2}$---but which has a very different geometrical 
structure.  There is a general and extremely useful method for constructing
examples to illustrate global properties: ``cutting holes and patching
together''.

The method of cutting holes is based on the following fact: if $M$ is a 
space-time and $C$ is any closed subset of $M$, then $M$ with the region $C$ 
removed is also a space time.  Removing selected regions $C$ from a known 
space-time (usually Minkowski space) provides a powerful technique to find
space-times having certain properties.  Consider Minkowski space with the
$t$-axis (in the usual co-ordinates) removed.  This space-time is topologically
$S^2\times\R{2}$ ({\bf Exercise}{\em\ Prove it.}).  Although this space-time 
has the same topology as the Reissner-Nordstr\"om solution, it is very 
different in most other ways.  It has only a single ``asymptotic infinity''
rather than an infinite number, its ``singularity'' has an entirely different 
structure from that of the Reissner-Nordstr\"om solution, etc.

The second technique is that of ``patching together''. Suppose for example
that we wanted to decide whether or not there is a space-time $M$ which is 
without singularity, satisfies Einstein's equations (source-free),
and has a compact\footnote{
        We may define a manifold $S$ as {\bf compact} if every sequence
        of points of $S$ has a point of accumulation.  Intuitively speaking, 
        compactness means that the manifold has no ``open edges'' or
        ``edges which go off to infinity''.  For example, a sphere, a torus,
        and a closed disk are compact while an open disk, a plane, and a 
        cylinder are noncompact.}
\marginpar{78}
spacelike 3-surface (\ie, $M$ is to represent a nonsingular closed universe).
It is very easy to construct such a model.  Begin with a timelike tube in
Minkowski space whose cross-section is a cube (Fig.~3).  Now identify opposite
faces of each cross-section to form compact cross-sections.  (Of course,
one carry out such identifications only when the metric can be joined 
smoothly.)  The resulting space-time $M$ satisfies all our conditions
and, in fact, one more: it is flat.  (Topologically, $M$ is 
$S^1\times S^1\times S^1\times \R{1}$.)  It is an open question whether
or not there are nonflat model which satisfy our three conditions [2].

One occasionally hears the view that space-times constructed by cutting and
pasting are ``artificial'' and should be ruled out.  The problem here is that
it is not enough to say merely that certain space-times leave one 
uncomfortable unless he can pin down precisely which space-times are to be
accepted and which will not.  No suitable criterion has yet been found. 
(It would not do to say ``space-times constructed by cutting and patching are 
unacceptable'', for any space-time can be represented in this way.)  
It is felt that such space-times, while they do appear rather artificial,
must be accepted, at least until a suitable criterion for their rejection is 
discovered.  Much work on global properties consists of trying to formulate 
just such intuitive ideas.

The space-times obtained by cutting and patching are not normally considered
as serious models for our universe.  However, the mere existence of a 
space-time having certain global features suggests that there are many 
models---some perhaps quite reasonable physically---with similar properties.
For example, Minkowski space with the line removed is very similar in its
global properties to certain Weyl solutions.  The idea of the method is not 
so much to construct realistic models of our universe as to show what is 
possible within the context of space-times.  Its power stems from the fact 
that one is able to mold the global properties of a space-time without having 
to worry about details of the metric.

\prop{Exercise}{Find a flat space-time any two of whose points can be joined by 
        a timelike curve.  Find a flat space-time which is topologically
        $S^1\times\R{3}$.}

\vskip .25in\ni{\bf 2. --- Direction fields, orientation.}
\marginpar{79}

A number of questions in relativity are dealt with most easily by introducing
a timelike direction field. Such fields carry certain of the information
contained in the light-cone structure, but in a more convenient form. In this
Section we shall see how direction fields arise in relativity and what they
can be used for.  In particular, direction fields lead to the notions of
space and time orientation.

Let $M,g\ab$ be a space-time.  Choose an arbitrary positive-definite metric
$h\ab$ on $M$.\footnote{
        It is well-known [33] that such a metric exists on every (Hausdorff,
        paracompact) manifold.}
This $h\ab$ defines a {\bf direction field} (\ie, a vector field which is
nonzero at each point, which is defined at each point only up to an arbitrary
non-zero factor) on $M$ as follows [4].  Let $p$ be a point of $M$.
At the point $p$, choose, from among all vectors whose norm with respect
to $h\ab$ is unity, that vector whose norm with respect to $g\ab$
is maximum.  Since the unit vectors with respect to $h\ab$ define a
sphere while the vectors having a given norm with respect to $g\ab$
define a hyperboloid (Fig.~4), it is clear that there will always
exist a maximum, unique up to sign.  The timelike direction field so defined 
depends, of course, on the choice of $h\ab$. However, certain important 
properties of the field turn out to be independent of this
choice. 

Conversely, let $M$ be a 4-dimensional manifold, and let $\alpha\xi^a$ be a
direction field on $M$.\footnote{
     We shall use the notation ``$\alpha\xi^a$'' for a
     direction field.  However, it cannot be assumed, as we shall see alter,
     that the direction field consists of all nonzero scalar multiples of 
     some fixed vector field.}
We shall construct a Lorentz metric $g\ab$ on $M$ for which $\alpha\xi^a$
\marginpar{80}
is timelike.  Choose an arbitrary positive-definite metric on $h\ab$ on $M$
and set [4]
\begin{equation}
     g\ab= -h\ab + 2h_{ap}\xi^p h_{bq}\xi^q(h_{cd}\xi^c \xi^d)^{-1}
\end{equation}
(Note that (2.1) is independent of the scaling of $\xi^a$.  This $g\ab$ has
Lorentz signature, and $\alpha\xi^a$ is timelike with respect to $g\ab$.  Of
course, the direction field does not define a Lorentz metric uniquely.

As an example of the use of direction fields, let us find the condition for 
a given manifold $M$\ \footnote{
     We assume in the discussion that $M$ is connected.}
to admit a Lorentz metric.  As we have seen, the existence of a Lorentz metric
is equivalent to the existence of a direction field.  It turns out that there
are two separate cases, depending on whether or not $M$ is compact.

Suppose first that $M$ is not compact.  Choose on $M$ an arbitrary direction
field which is regular everywhere except, possibly, at certain isolated
singular points (Fig.~5a)).  (Such a field always exits [9].) The idea is to
``push the singular points off to infinity'' and thus obtain a nonsingular
direction field.  Let $p$ be a singular point.  Choose a curve $\gamma$ from $P$
which ``goes off to infinity''\footnote{
     More precisely, $\gamma$ must eventually leave and remain outside of
     every compact set.},
and let $U$ be a small tubular neighborhood of $\gamma$\ \footnote{
     The {\bf interior} of a set $A$ is the ``largest'' open set contained
     in $A$ or, more precisely, the union of all open sets contained in $A$ 
     (see axiom~T1).  A {\bf neighborhood} of a set $B$ is any set whose 
     interior contains $B$.}
(Fig.~5a)).  The manifold consisting of $M$ with the tube $U$ removed is
equivalent (diffeomorphic) to $M$ (Fig.~5b) and~5c)).  We thus obtain a
direction field on $M$ for which $p$ is no longer a singular point (Fig.~5c)).
Continuing in this way, we obtain a nonsingular direction field, and thus a
Lorentz metric, on $M$.

\prop{Exercise}{Show that the above argument applies even in the case in which
there are an infinite number of isolated singular points.}

We have shown that every noncompact manifold carries at least one (and
therefore an infinite number) of Lorentz metrics.  That is to say, every
noncompact manifold is a possible {\em arena} for space-time physics.
Penrose[10] has obtained a stronger version of this result: the Lorentz metric
can always be so chosen that there are no closed timelike curves.

The situation is quite different in the compact case.  It is known [11] that a
necessary and sufficient condition for a compact manifold $M$ to carry a
Lorentz metric is the vanishing of a certain integer, called the Euler
characteristic, associated with $M$.  However, there is a more interesting and
much simpler result along these lines: every compact space-time has closed
timelike curves [12,13].  Let $M$ be a compact manifold with a Lorentz metric,
and let $\alpha\xi^a$ be a timelike direction field on $M$.  Choose any point
$p\in M$.  Draw the timelike curve $\gamma$ which begins at $p$ and is
everywhere tangent to $\alpha\xi^a$.  Choose a sequence of points $p_i$ on
$\gamma$ such that every point of $\gamma$ lies beyond some p.  Since $M$ is
compact, the $p_i$ have an accumulation point\footnote{
     An {\bf accumulation point} of a sequence is a point every
     neighborhood of which contains an infinite number of points in the 
     sequence.  A {\bf limit point} of a sequence is a point every 
     neighborhood of which contains all points of the sequence beyond a 
     certain member.\\
     {\bf Exercise} {\em Show that a sequence of points on a manifold can 
     have many accumulation points, but no more that one limit point.}},
say at $q$ (Fig.~6).  Choose a cylindrical neighborhood $N$ of $q$ such that
every trajectory of $\alpha\xi^a$ which passes through $N$ can be distorted, 
in $N$, so that it remains timelike and passes through the point $q$.  Since
$\alpha\xi^a$ is timelike, every trajectory of $\alpha\xi^a$ which enters $N$
must re-emerge from $N$.  Since the $p_i$ accumulate at $q$, our trajectory
$\gamma$ must continually re-enter $N$.  By distorting $\gamma$ on two of its
successive passages through $N$, we obtain a closed timelike curve, beginning
and ending at $q$.  That is to say, if one wishes to avoid closed timelike
curves, then {\em no} compact manifold is a suitable arena for space-time
physics. 

\prop{Exercise}{Find an example of a compact space-time containing two points
     which cannot be joined by a timelike curve.}

Using similar arguments, one can prove the following result: 
Let $S$ and $S'$ be two compact 3-manifolds.  Then there exists a space-time
$M$ having $S$ and $S'$ as spacelike sections, and such that the region between
$S$ and $S'$ is compact.  If, however, $S$ and $S'$ have different topologies,
then $M$ will always have closed timelike curves [13].  That is, the topology
of spacelike sections in a space-time can change in an arbitrary way from epoch
to another, but such changes are accompanied by closed timelike curves.

\prop{Exercise}{Show that, given two 3-manifolds $S$ and $S'$, there exists a
     space-time $M$, without closed timelike curves, having $S$ and $S'$ as
     spacelike sections.}

Another property of space-times which can be described in terms of direction
fields is time- and space-orientation.  Let $\alpha\xi^a$ be a timelike
direction field in $M$.  With any closed curve $\gamma$ (not necessarily
timelike) beginning and ending at a point $p$, we associate three integers as
follows.

1) Choose one of the two possibilities for the sign of $\alpha\xi^a$ at the
point $p$.  Carry this choice continuously around $\gamma$.  On returning to
$p$, we will obtain either the same or the opposite sign as that with which
began.  That is, $\gamma$ may either be time-preserving ($+1$) or time-reversing
($-1$).

2) Choose three spacelike vectors at $p$ which are linearly independent, and
such that the 3-plane they span does not contain $\alpha\xi^a$.  Carry these
three vectors around $\gamma$ in an arbitrary continuous way, always keeping
the vectors spacelike and independent of each other and of $\alpha\xi^a$.  On
returning to $p$, our vectors will have either the same or opposite
orientation as that with which they began.  That is, $\gamma$ may be either
space-preserving ($+1$) or space-reversing ($-1$).

3) Choose a sign of charge which is to be called ``positive'' at $p$, and carry
this choice continuously around $\gamma$.  On returning to $p$, our choice
of ``positive'' charge may either agree or disagree with that originally made at
$p$.  That is, $\gamma$ may be either charge-preserving ($+1$) or
charge-reversing ($-1$).

Thus, with every closed curve we associate three integers, each either $+1$ or
$-1$.  The first two integers depend only on the geometry of the space-time (and,
in particular, are independent of the direction field and the mode of
transport), while the third involves also the physical properties of the
space-time.  The integers are clearly invariant under continuous deformations
of the curve $\gamma$.\footnote{
     That is, the integers depend only on the holonomy class of the curve
     (in fact, only on its homology class) [1].  If we combine curves (in the 
     sense of homotopy [1]), the the corresponding integers for the two curves
     are multiplied.  Thus, we obtain a homomorphism from the group $\pi_1(M)$ 
     into the group of all triples of integers, each either $+1$ or $-1$.}

A space-time $M$ is said to be {\bf time-} (respectively {\bf space-,
charge-}) {\bf orientable} if for every closed curve in $M$ is time-
(respectively space-, charge-) preserving [11].  For example, that a space-time
be time-orientable means that we may speak of the ``future'' and ``past'' light
cones at each point.  These three notions are completely independent, \ie, a
space-time can be orientable in any combination of the three senses.  An
example of a non-time-orientable space-time is given in Fig.~7.  Finally, a
space-time is said to be {\bf orientable} if every curve is both space- and
time-preserving or neither.

\prop{Exercise}{Using a M\"obius strip, construct an example which is
     space-orientable but not time-orientable.}

\prop{Exercise}{Prove that whether or not a space-time is orientable depends
     only on the underlying manifold, not on the metric.}

\prop{Exercise}{Prove that a space-time is time-orientable if and only if it
     carries a timelike vector field.}

\prop{Exercise}{Show that every space-time which is topologically \R{4} is
     time-, space-, and charge-orientable.}


\marginpar{84}
How can one decide what type of orientability our own universe satisfies?  It
turns out that this can be accomplished with local experiments, provided we
assume that the results hold ``universally'', \ie, provided we assume the 
strong principle of equivalence [14].  Suppose for example that, in every local
region, one can, by means of statistical mechanics, determine a ``future'' time
direction.  Then our universe must certainly be time-orientable.  By a similar
argument, one can show[15] that if, in each local neighborhood, one has the
following results: 
1) an experiment demonstrating noninvariance under separate charge and
     parity reversal [16,17], 
2) an experiment noninvariance under separate simultaneous charge-parity 
     reversal [18], and
3) the {\em CPT theorem} [19], then every closed curve must reverse either 
     all or none of time, space, and charge.
(In particular, our universe must be orientable.)  If, in addition,
time-orientability is assumed, then our universe must be orientable in
all three senses.


\vskip .25in\ni{\bf 3. --- Causal structure I.  The domain of influence.}

We have seen in Sect. 1 that there is associated a topology with the underlying
manifold of any space-time.  We have also seen that this topology gives very
little information about the physics of the space-time.  In particular,
it does not involve the metric in any way.  Is there any structure on our
space-time which refers to the metric, but which does not involve all the
complications inherent in the metric and differential structure?  One of the
most important of these is causal structure.  (Can event $p$ influence event
$q$ by means of a signal?) It turns out [20] that a surprising number of global
arguments concerning space-times can be formulated in terms of this
particular---and very simple---aspect of the metric structure.  The advantage of
using causal structure is that arguments are thereby simpler and more
transparent.  Furthermore, causal structure appears, physically, to be more
``fundamental'' than, for example, differential structure, and so it is useful
to know which results depend only on that structure.

Let $M,g\ab$ be a space-time.  We assume throughout this and the following
Section that M is time-oriented, \ie, that there is given a continuous choice
future light-cone at each point.  We write $p\ll q$ to mean that there is a
smooth, future-directed timelike curve\footnote{
     Since ``timelike curve'' means that the tangent vector to the curve must
     have positive norm, the ``zero curve'' (which always remains at $p$)
     is not counted as timelike.}
which begins at the point $p$ and ends at the point $q$ [20].

This is the fundamental relation of causal space theory.  We derive two of its
properties.

\marginpar{85}
\prop{CS1)}{If $p\ll q$ and $q\ll r$, then $p\ll r$. (See Fig.~8.)}

\ni The condition that $M$ have no closed timelike curves may be expressed 
in the following form:
\prop{CS2)}{For no point $p$ is $p\ll p$.  The proof is obvious.\footnote{
     See the previous footnote.}}
\ni We shall not automatically assume the absence of closed timelike curves
in what follows.

\prop{Exercise}{Prove that if a space-time has one closed timelike curve,
     then it has an infinite number.}

Let $p\in M$.  We denote by \I-p (respectively, \I+p) the set of all points~$q$
such that $q\ll p$ (respectively, $q\gg p$).  This set is called the 
{\bf past} (respectively, {\bf future}) of $p$.  For example, the past of a
point $p$ in Minkowski space is the interior of the past light cone of $p$.

\prop{Exercise}{Show that, if $p\ll q$, then $\I-q \supset \I-p$.\\
     Show that the converse is false (\eg in Minkowski space).}

There is a relation between the causal structure and the topology of $M$.

\prop{Claim}{For any point $p$, \I-p is open.}

\proof{To prove this result, we must show that arbitrarily small variations 
     of any point~$q\in\I-p$ result in points again in \I-p.  
     Let $\gamma$ be a timelike curve from $p$ to $q$ (Fig.~9), and 
     $r$ be a point on this curve which is near to $q$.  Since $r$ is 
     close to $q$ and timelike-related to $q$, \I-r certainly includes a 
     neighborhood of $q$.  But $r\ll p$, and so $\I-r \subset \I-p$.  
     Since \I-p contains a neighborhood of each of its points, \I-p is open.
     It follows from axiom~T1 for a topological space that, for any set $A$, 
     $I^-[A]$ (defined as the union of the pasts of the points of $A$) is 
     open.}

To illustrate this interplay between topology and causal structure, consider 
the example of Fig.~10.  The sequence $p_i$ approaches $p$ as a limit.  The 
point~$q$ is such that $q\ll p_i$, for each $i$.  However, $q$ cannot be joined 
to $p$ by a timelike (or even null) curve.  Is the converse true?  That is,
is it true that, if the $p_i$ approaches $p$ and $q\ll p$, then $q\ll p_i$
for sufficiently large $i$?  The answer is yes.

\marginpar{86}
\prop{Claim}{If the $p_i$ approaches $p$ and $q\ll p$, then $q\ll p_i$
     for sufficiently large $i$.}

     \proof{\I+q is open and contains $p$.  Therefore, \I+q contains the $p_i$
          for sufficiently large $i$.  That is, $q\ll p_i$ for sufficiently 
          large $i$.}

This relationship between the pasts of the points of a sequence and the past 
of their limit is not completely obvious.  Yet it is easy to rederive the
result when needed using the constructions and examples of causal-space
theory.

Since sets of the form \I-p and \I+q are open, it follows from axiom~T2
that sets of the form $\I-p \cap \I+q$ are open.  Therefore, by axiom~T1,
arbitrary unions of sets of this form are open.  It is an interesting
question whether the converse is also true.  That is, if $U$ is a given open
set in $M$, is it true that $U$ can be written as a union of sets of the form
$\I-p \cap \I+q$?  An affirmative answer would imply that the topological
structure of $M$ follows already from its causal structure.

In general, the answer to our question is ``no''.

\marginpar{87}
\prop{Claim}{Given an open set $U \subset M$, $U$ is not necessarily
     expressible as a unions of sets of the form $\I-p \cap \I+q$.}

     \proof{Let us construct a counterexample.\\
          Consider 2-dimensional Minkowski-space\footnote{
               Two-dimensional examples are frequently used because they
               are easier to draw and to visualize.  Why is this not an 
               unphysical procedure, since our own universe is 4-dimensional?
               It is always necessary to check such examples that the same
               features can appear in a 4-dimensional space-time.}.
          Take a horizontal strip of Minkowski-space lying between the lines 
          $t=0$ and $t=1$ (Fig.~11).  The idea is to violate causality so badly
          that sets of the form $\I-p \cap \I+q$ will be ``too large'' for
          their unions to comprise all open sets.  To achieve this end,
          identify the edge $t=0$ with the edge $t=1$.  The result is a
          cylindrical space-time in which ``time'' goes around the cylinder 
          (Fig.~11).  There are closed timelike curves.  Furthermore,
          {\em any} two points may be joined by a timelike curve 
          (\eg by a spiral on the cylinder whose pitch is less than $1$).
          Thus, any point $p$, \I-p is the entire manifold $M$, and so there
          is only one set of the form $\I-p \cap \I+q$, namely, $M$.
          Clearly, no open set, except $M$ itself, can be written as a
          union of sets of the form $\I-p \cap \I+q$.}

\ni The point here is that, in a space-time as severely acausal as that of 
Fig.~11, the causal structure gives practically no information:
$p\ll q$ for any two points.  It could hardly be expected that such
an uninteresting causal structure would be capable of defining
the topology of $M$.

\prop{Exercise}{Show that any open set in Minkowski space can be written
     as the union of sets of the form $\I-p \cap \I+q$.}

We shall see later that it is possible to infer the topology of $M$ from its
causal structure provided we assume that the latter is ``reasonable'' in
a certain sense.

By far the most important element of the causal structure is the 
relation~``$\ll$''.  However, there is a second causal relation which can
be defined on a space-time.  A smooth curve $\gamma$ will be called a
{\bf causal curve} if its tangent vector is either timelike or null
at each point.  (The tangent vector may be timelike at some points and
null at others\footnote{
     We consider the zero-vector as null and both future- and past-
     directed.  Thus, the ``zero curve'', which remains entirely at a
     point~$p$, counts as a causal curve.}.)
We write $p<q$ ({\em p causally-precedes q}) if there is a future-directed
causal curve from $p$ to $q$.  Finally, \J-p is the set of all points which
causally precede $p$.  For example, if $p$ is a point in Minkowski space,
then \J-p is the closed past light-cone at $p$ (\ie, including \I-p,
the null-cone, and the point~$p$ itself).  Clearly, we have

\prop{CS3)}{If $p\ll q$, then $p < q$.}

\marginpar{88}
\prop{Exercise}{Show that, if $p<q$, then $p$ and $q$ may be joined by a 
     future-directed null curve.}

The fundamental relation determining when points may be joined by a
timelike curve is the following:  

\prop{Claim}{Let $\gamma_1$ and $\gamma_2$ be future-directed causal curves
     joining $p$ to $q$ and $q$ to $r$, respectively.  Suppose $p\not\ll r$.
     Then $\gamma_1$ and $\gamma_2$ are null geodesics, and their tangent
     vectors coincide at $q$.}

     \proof{The proof is essentially a generalization of the argument of
          Fig.~8.  If $\gamma_2$ were not a null geodesic, then $gamma_2$
          would enter the past of $r$.  Hence, $q$, the entire 
          curve~$\gamma_2$, and the point~$p$ would all be in \I-r.
          Similarly for $\gamma_1$. If $\gamma_1$ and $\gamma_2$ had
          different tangents at $q$, then we could round off the corner
          of the combined curve at $q$, and thus obtain a causal curve
          from $p$ to $r$ which is timelike near $q$.  This curve would
          therefore enter \I-r in the ``rounding-off region'', and thereafter
          remain in \I-r.}

\prop{Exercise}{Using th above result, establish the following properties 
     of~``$\ll$'' and~``$<$'':}
\prop{CS4)}{If $p<q<r$, then $p<r$.}
\prop{CS5)}{If $p<q\ll r$, then $p\ll r$.}
\prop{CS6)}{If $p\ll q<r$, then $p\ll r$.}

The condition that there be no closed causal curves cannot be written
in a form analogous to CS2.  In fact, it follows from the definition 
that\footnote{
     See the previous footnote.}
\prop{CS7)}{$p<p$ for every point $p$.}

\ni However, this requirement can be expressed in the following way:
\prop{CS8)}{If $p<q<p$, then $p=q$.}


That is, there are defined on every space-time two causal relations subject
to CS1, CS3, CS4, CS5, CS6, and CS7.  If, in addition, there are no closed
causal curves, then these relations also satisfy CS2 and CS8.  A general
{\bf causal space} [20] is a set $X$ with two relations, subject to CS1-CS8,
but not assumed to come from any manifold or metric signature on $X$.
A number of results normally associated with relativity theory can,
in fact, be proven for general causal spaces.

Having discussed some properties that ``$\ll$'' and ``$<$'' {\em do} satisfy,
let us next examine some properties that are {\bf not} satisfied.

\ni 1) Sets of the form \J-p are closed in Minkowski space, but this is not true
in general (Fig.~12).

\ni 2) If the past of $p$ contains the past of $q$ in Minkowski space, then
$q<p$.  This is not true in general (Fig.~12).

\marginpar{89}
\ni 3) In Minkowski space, two points with the same past and the same future 
coincide.  This, too, is not true in general (Fig.~13).  

The above examples are pathological, but that is precisely the point!  
Should one of these properties be required in the course of a proof,
then some further condition must be imposed on the space-time.

Many of the above examples involve closed timelike curves.
What is wrong with such space-times [21,22]?  One would like to argue that they
are artificial because our own universe does not have closed timelike curves,
or, at least, none which pass through the earth.  Such a strong assertion might 
appear to be very difficult to verify, for if there is a closed timelike curve 
which passes through the earth, it would probably wander a great distance from 
the earth before coming back to itself.  One could, nonetheless, explore such 
possibilities.  If we program a rocket ship to traverse a suspected curve,
we should not have to wait 
\marginpar{90}
millions of years for its return.  The ship would 
return the instant we lit the fuse.  The experiment, then, consists of 
programming the ship to follow such a curve, lighting the fuse, and looking 
over our shoulder for its return.  If the ship simply flies off, then our 
suspected closed timelike curve is not such an object.

As nobody has tried such an experiment, why do we suspect no closed timelike 
curves?  There is a much easier experiment.  We ask the following question:  
how large is the class of solutions of, say, Maxwell's equations in a 
space-time with closed timelike curves?  We would expect, in general, to find 
very few.  Give initial data for Maxwell's equations on a spacelike surface $S$.
If there are closed timelike curves through $S$, we expect radiation to be 
reflected off the ``bumps'' in the curvature and eventually to be re-registered 
on $S$.  Thus, our initial data might, in general, be incompatible with the
space-time geometry.  Since the only restriction on electromagnetic fields
that we observe in our local neighborhood is Maxwell's equations, there is at 
least a reason to expect that causality violation is not too severe in our 
neighborhood.

Let us try to formulate, mathematically, this additional causality condition on 
models of our universe.  The most obvious choice is to require that there be
no closed timelike curves.  But is this the ``right'' condition: could a
space-time have no closed timelike curves but nonetheless display acausal
behavior?  The answer is ``yes'': the space-time of Fig.~13 has no closed
timelike curves, but a single closed null curve $C$.

Even to rule out closed causal curves is not the strongest possible condition.
Remove from Fig.~13 the single point $p$, thus breaking the only closed
causal curve in the space-time.  This space still appears causally anomalous,
for there are ``almost closed'' timelike curves through $q$.  More precisely,
given any sufficiently small neighborhood $U$ of $q$, there is a timelike curve
which begins in $U$, leaves $U$, and then re-enters $U$.  Such a space-time is
said to violate {\bf strong causality} [23].  This is precisely the type of
causality violation that one normally wants to rule out in proving theorems on
the existence of singularities [24,23].

\prop{Exercise}{Prove that in a strongly-causal space-time, the topology 
     follows from the causal structure [20].}

\prop{Exercise}{Prove that in a strongly-causal space-time distinct points
     have distinct pasts [20].}

There are even higher levels of causality violation than this strong-causality
(Fig.~14).  What, then, is the ultimate causality condition?  A space-time
is said to be {\bf stably-causal} [3] if, when the metric is varied by an
arbitrary but sufficiently small amount, there results a space-time with no
closed timelike curves.  The words ``sufficiently small'' suggest that the precise
formulation of stable-causality will require the notion of a topology on the
set of all Lorentz metrics on a manifold.
\marginpar{91}
The appropriate topology is called $F^0$ [2]: a neighborhood of a metric 
$g\ab$ consists of all metrics whose value at each point is within a
certain range of $g\ab$, where this ``range'' varies continuously but
otherwise arbitrarily over $M$.  Most common space-times---Minkowski space,
the Schwarzschild, Reissner-Nordstr\"om, and Friedman solutions---are 
stably-causal.

In fact, it is possible to give a general criterion for when a space-time
is stably causal.  Suppose that a space-time $M$ has a time function, \ie
a scalar field $t$ whose gradient is everywhere timelike.  Then $M$ is certainly 
stably-causal.  (Vary the metric at each point in such a way that $\grad_a t$
remains timelike.  The ``varied'' space-times also admit $t$ as a time function,
and so, since t increases along timelike curves, have no closed timelike 
curves.)  It turns out that the converse is also true:

\prop{Claim}{Every stably-causal space-time has a time function [23].}

Thus stable-causality---the ultimate causality condition---is completely
equivalent to the existence of a cosmic time\footnote{
     The equal-time surfaces need not, however, have all the properties
     associated with spacelike surfaces.  In particular, they will, in
     general be disconnected.}.
(There are, of course, many time-functions, so there is no notion
of simultaneity in a stably-causal space-time.)

\vskip .25in\ni{\bf 4. --- Causal structure II.  The domain of dependence.}

In the previous section we studied one aspect of causal structure:
{\em can event $p$ influence event $q$ by means of a signal?}
We shall now be concerned with a second aspect of causal structure: 
{\em is the situation at an event $p$ completely determined by that on
a set $S$?}  Thus, whereas the domain-of-influence is defined by a
\marginpar{92}
relation between points and points, the domain-of-dependence is defined
by a relation between points and sets.  The assumption of 
Section~3---that all space-times are time-oriented---remains in force in the
present section.

Let $M,g\ab$ be a space-time, and let $S$ be a subset of $M$.
We assume that $S$ is {\bf achronal}, \ie, that no two points of $S$
may be joined by a timelike curve\footnote{
     This assumption has two purposes.  Firstly, it ensures that $S$
     represents the type of region on which we might give initial data,
     \ie, a surface which is locally spacelike or null.  Secondly, it
     automatically rules out causality violations in which data given
     on $S$ may later be re-registered on $S$.  For example, the space-time
     of Fig.~11 has no achronal sets at all.}.
It is convenient to think of $S$ as a spacelike surface on which to give
initial-data for, for example, Maxwell's equations.  Since signals in
relativity travel along causal curves, we expect the situation at a point $p$
to be completely determined by that on $S$ provided every such curve from
$p$ intersects $S$.  More precisely, the {\bf future domain-of-dependence}
[5,10,23,25] of $S$, \D+S, is the collection of all points $p$ such that
every past-directed timelike curve (without past-endpoint) from $p$
intersects $S$.  (It is convenient to include $S$ in its own
domain of dependence.)  The {\bf past domain-of-dependence}, \D-S, is
defined in a similar way by interchanging the roles of past and future.

The typical behavior of the domain of dependence is illustrated in Fig.~15.
The space-time is 3-dimensional Minkowski space with a hole, marked ``A'',
removed.  The achronal set $S$ is a hyperboloid of revolution, also with a
hole removed.
\marginpar{93}
The sets \D{\pm}S are shown in the figure.  We see from Fig.~15 that
holes in either the space-time itself or in the surface $S$ create
``shadows'' in the domain-of-dependence of $S$.  Intuitively, information
can enter or leave the space-time through these ``holes'' without it
being registered on $S$.

    We see from Fig.~15 that a past-directed {\em causal} curve from a point 
(\eg $r$) of \D+S need not necessarily intersect $S$.  However, 

\prop{Claim}{If $p$ lies in the interior of \D+S, then every past-directed 
        causal-curve from $p$ does intersect $S$ [5].}
\ni To show this, suppose that $p$ is in the interior of \D+S and let $\Gamma$
be a past-directed causal curve from $p$ which fails to intersect $S$. 
Choose a point $q\in\D+S$ which lies to the future of $p$.  It is fairly clear 
intuitively that one can always find a past-directed timelike curve $\gamma$
from $q$ which remains always to the future of $\Gamma$.  Therefore, $\gamma$ 
does not intersect $S$ either, which contradicts our assumption that 
$q\in\D+S$.

Another property of the domain-of-dependence is that none of its points can 
lie on a closed timelike curve.  The proof is easy. Suppose $p\in\D+S$,
and let $p$ lie on a closed timelike curve $\gamma$.  If $\gamma$ does not 
intersect $S$, then the curve obtained by continually describing $\gamma$ has 
no past-endpoint and does not intersect $S$, thus contradicting our assumption 
that $p\in\D+S$.  On the other hand, if the closed timelike curve $\gamma$
intersects $S$, then $S$ cannot be achronal.

\prop{Exercise}{Show that $S$ is closed if and only if \D+S is closed.\\
        Find an example for which $\D+S\neq\D+{\bar S}$.}

\prop{Exercise}{Let $p\in\D+S$.  Show that $\I-p\cap\I+S \subset \D+S$.}

The first important concept derived from the domain of dependence is that
of a Cauchy surface [5,10,24].  An achronal set $S$ is said to be a
{\bf Cauchy surface} for the space-time $M$ if every point of $M$ lies in either
\D+S or \D-S.

\prop{Exercise}{Prove that a point lies in both \D+S and \D-S if and only if
        it lies in $S$.}
That S be Cauchy surface for $M$ means, intuitively, that the entire
evolution of $M$, past and future, is completely determined by $S$.

Does our own universe possess a Cauchy surface?  It is very difficult to gather
evidence on this question.  Consider, for example, the Reissner-Nordstr\"om
solution (Fig.~16).  In the immediate neighborhood of the surface $S$,
it looks as though $S$ might be a Cauchy surface.  (The geometry in a 
neighborhood differs by a small amount from that in a neighborhood of a similar
surface in the (extended) Schwarzschild solution, and in that case $S$
is Cauchy surface.)  However, as the geometry evolves in the 
Reissner-Nordstr\"om solution, the ``throat'' of the wormhole first contracts 
and then re-expands, allowing points ($p$ in the figure) which are not in \D+S.  
Thus, $S$ turns out not to be Cauchy surface.  However, it would have been 
difficult to predict this result in advance---\ie, by examining only a 
neighborhood of $S$.

\prop{Exercise}{Show that the Friedmann models have a Cauchy surface.}
It is a very strong condition
\marginpar{94}
on a space-time that it admit a Cauchy surface.  We shall now derive some
implications of this condition.

We have seen above that no point in the domain-of-dependence can lie on
a closed timelike curve.  Therefore, a space-time with a Cauchy surface cannot
have any closed timelike curves.

We can also obtain information about the topology of the space-time.
Choose an arbitrary timelike direction field $\alpha\xi^a$ on $M$.
Since $S$ is a Cauchy surface, every trajectory of $\alpha\xi^a$ intersects
$S$, and, since $S$ is achronal, no trajectory intersects $S$ more than once.
That is to say, we may describe each point $p$ of $M$ by giving
\ni\tab i) the point at which the trajectory through $p$ intersects $S$, and
\ni\tab ii) the distance from $p$ to $S$ along that trajectory.\\
Consequently, the space-time $M$, is topologically, $S\times\R{}$ [5,10].

In fact, the above is true in an even stronger form [5]: not only can
$M$ be written in the form of a product, $S\times\R{}$, of a
3-manifold and the real line, but this product can be so chosen that, for each
real number in \R{}, the corresponding surface $S$ is a spacelike 
({\em E.g.,} a spacelike hyperplane is a Cauchy surface for Minkowski space, 
and Minkowski space can be filled with a one-parameter family of such 
hyperplanes.)  It now follows that every space-time with a Cauchy surface
is stably causal.  (Choose the permitted variations in the metric at each
point to be those under which the $S$-surfaces remain spacelike.  Then
each varied metric is also free of closed timelike curves.)

\prop{Exercise}{Show that any two Cauchy surfaces in a space-time have the
        same topology.}

\prop{Exercise}{Find an example of a space-time which can be written as a 
        product $S\times\R{}$, with each embedded $S$ spacelike, but which
        has no Cauchy surface.}

One of the most useful tools for studying the domain-of-dependence is the
Cauchy horizon [5,10,23].  The {\bf future Cauchy horizon}, \H+S, of an
achronal set $S$ is defined as the ``future boundary'' of the 
domain-of-dependence of $S$, or, more precisely, as the set of all
points \D+S no points to whose future is \D+S.  The future Cauchy
horizon of a surface in the Reissner-Nordstr\"om solution is
indicated in Fig.~16.

\prop{Exercise}{Find the Cauchy horizons in Fig.~5.}

The Cauchy horizon in these examples is a null-space.  This is a general
and a very important property of the Cauchy horizon.  More precisely,
we have the following result [5, 23]: let $p$ be a point of \H+S,
such that $p\not\in S$.  Then there is a past-directed null-geodesic
segment from $p$ which lies in \H+S.  This geodesic either remains entirely
in \H+S, or else emerges from \H+S at point of the closure of $S$.

The proof is illustrated in Fig.~17.  Let $p$ be a point in \H+S.
Draw a small neighborhood $U$ of $p$ which does not intersect $S$, and let
$p_i$ ($i=1,2,\cdots$) be a sequence of points which lie in $\I+p$ and
in $U$, and which lie in \I+p and in $U$, and which approach $p$ in the limit.
By definition of \H+S, each of the $p_i$ lies on a timelike curve $\gamma_i$
which does not intersect $S$.  The points $q_i$ at which the $\gamma_i$ leave
the neighborhood $U$ cannot lie to the past of $p$ (for points in both $U$
and \I-p are in \D+S, while the $\gamma_i$, since they do not intersect
$S$, cannot enter \D+S.)  Therefore, the $q_i$ have an accumulation
point $q$ which is in \H+S, and which is on the null cone of $p$.
Draw the segment $\Gamma$ of a null geodesic from $p$ to $q$.  We now
repeat the same construction beginning from $q$.  The resulting null-geodesic
segment $\Gamma'$ from $q$ must have the same tangent as $\Gamma$ at $q$, 
for otherwise $\Gamma'$ would enter \I-p.  Continuing in this way, we 
construct a past-directed null geodesic from $p$ which lies in \H+S.
This construction can only come to an end if, at some stage, we reach 
a point every neighborhood of which intersects $S$.  That is, the null
geodesic can emerge from \H+S only at a point of the closure of $S$.

\prop{Exercise}{Let $p\in\D+S$.  Show that any future-directed timelike curve
        from $p$ which emerges from \D+S does so at a point of \H+S.}

\prop{Exercise}{Find an example in which \H+S has a closed null curve.\\
        Show that \H+S is always achronal.}

\marginpar{96}
The above property of null geodesics is often the most convenient way
to establish the existence of a Cauchy surface.  One uses some condition 
(such as an energy condition) to show that null geodesics of the type required
above cannot exist.  It follows that $S$ cannot have a Cauchy horizon and
must, therefore, be a Cauchy surface.

\prop{Exercise}{Show that if an achronal set $S$ has no Cauchy horizons, then
        $S$ is Cauchy surface.}

We conclude this section with one further property of Cauchy surfaces.
If $S$ is a Cauchy surface, and $p$ a point of $M$, then there is a timelike 
geodesic from $p$ to $S$ whose length is greater than (or equal to) that of 
every other causal curve from $p$ to $S$.  The proof [5, 24] is rather 
technical, and so we can only give it in outline.  We have seen that every 
causal curve from $p$ intersects $S$.  This set of causal curves\footnote{
        It is necessary for this argument to include all continuous
        causal curves, whether or not they are differentiable.}
is first given a topology.  It is then shown that 1) this set
of curves is compact, and 2) the length of a curve is a continuous
function\footnote{
        In fact, the length is only upper semi-continuous, but this
        suffices for the result.}
on this set.  It then follows that there is a curve of maximal length.
It is clear that the maximal curve must be a geodesic (for otherwise we
could lengthen it by straightening it out), and that it must intersect
$S$ orthogonally (for otherwise we could lengthen it by slightly moving
its point of intersection with $S$.)  The existence of a maximal geodesic 
is frequently used in proving theorems on the existence of singularities
[10, 24].

\prop{Problem}{{\rm The domain-of-dependence is normally derived from the domain 
        of influence.  (This program has been carried out for certain causal
        spaces [26].)}\\Do the reverse.  That is, begin with the 
        domain-of-dependence as the basic structure, subject to certain axioms,
        and derive from it the domain-of-influence.}

\vskip 0.25in\ni{\bf 5.---Asymptotic properties.}

The purposes of this section are to briefly describe Penrose's definition
[8, 27] of asymptotic simplicity and to see what are the global implications
of this definition.  The idea of asymptotic simplicity is to avoid entirely
the use of limits of co-ordinate-dependent quantities ``as $r\to\oo$''.
Instead, one introduces a conformal transformation to bring ``infinity'' in
to a finite boundary surface on the manifold.  Asymptotic properties are then
to be described as local geometrical properties of this surface \Scri{}.

Let $M, g\ab$ be a space-time.  Our first assumption is that there is given a
boundary surface \Scri{} for $M$\footnote{
        The resulting manifold with boundary will {\em not} be compact.
        That is, the ``points'' $i^0, i^\pm$ (in Penrose's notation)
        will not be considered here.}.
This manifold, consisting of $M$ with the boundary \Scri{} attached, will be
denoted by $\tilde{M}$.  It is the conformal geometry we wish to carry over
from $M$ to its boundary.  We assume, therefore, that there is a metric 
$\tilde{g}\ab$ on $\tilde{M}$ such that $\tilde{g}\ab=\Omega^2 g\ab$ on $M$,
where $\Omega$ is a non-negative scalar field defined on $\tilde{M}$.

We must impose some further conditions on this system to ensure that
the definition will reflect our intuitive idea of asymptotic flatness.
In the first place, we want \Scri{} to represent ``infinity'', and not
some finite hypersurface.  We assume, therefore, that the conformal factor 
$\Omega$ approaches zero on \Scri{}, \ie, that there is an infinite
``stretching'' in passing from $\tilde{g}\ab$ to $g\ab$.  Secondly,
we want this stretching factor to behave ``like $r^{-1}$'' as we approach
\Scri{}.  This assumption requires that $\grad_a\Omega$ be nonzero on
\Scri{}.\footnote{
        Note that there is no conflict here with our earlier assertion
        that the limit of a vector field ``at infinity'' is not defined.
        $\grad_a\Omega$ is a well-defined vector-field on the manifold
        $\tilde{M}$, and we simply ask that it be nonzero at certain points
        of $\tilde{M}$, namely, at points of \Scri{}.}
Finally, we need some assumption to ensure that \Scri{} represents the
``entire surface at infinity'', and not merely some portion of it.  We 
therefore assume that every null geodesic in $M$ intersect \Scri{} in
two points, one to the future and one to the past.

To summarize, a space-time $M$ is {\bf asymptotically simple} [27] if $M$
can be given a boundary \Scri{} and $\tilde{M}$, defined as $M$ with \Scri{}
attached, can be given a non-singular metric $\tilde{g}\ab$, such that:
\ni\tab 1) $\tilde{g}\ab=\Omega^2 g\ab$ on $M$, where $\Omega>0$ on $M$.
\ni\tab 2) $\Omega=0$ and $\grad_a\Omega\neq 0$ on \Scri{}.
\ni\tab 3) Every null-geodesic in $M$ intersects \Scri{} in just two points.

It follows from these conditions, and the further assumption that Einstein's
source-free equations (with respect to $g\ab$) hold in a neighborhood of
\Scri{}, that \Scri{} be null [27].  We shall assume hereafter that
\Scri{} is null.

Minkowski space, for example, is asymptotically simple.  The conformal
factor can be chosen to be [8]
$$\Omega=(1+x^2+y^2+z^2)^{-\mbox{$\frac{1}{2}$}}$$
in the usual coordinates.  The resulting manifold $\tilde{M}$ is shown
in Fig.~18.  We see that \Scri{} is divided into two parts, \Scri+
and \Scri-, consisting of points reached by future-directed and past-directed 
null geodesics, respectively.

Asymptotic simplicity, it turns out, implies practically every criterion of
\marginpar{98}
``niceness'' we have discussed so far.  As a review of Sect.~1-4, we shall now
derive some of these results.

\prop{Claim}{The first and most important consequence of asymptotic simplicity 
        is the existence of a Cauchy surface.}
\ni To prove this result, first choose an area element $dA$ on \Scri{} such that
the total area of \Scri+ and \Scri- are both one.  For each point $p$ of $M$
we define
$$A^\pm(p)=\int_{\I{\pm}p \cap \Scri{}} dA$$
(Fig 18).  Along any future-directed causal curve, $A^+$ is strictly
decreasing and $A^-$ is strictly increasing.  Let $S$ denote the surface
given by $A^+=A^-$.  Since $A^+$ is decreasing and $A^-$ is increasing
along every future-directed timelike curve, $S$ is certainly achronal.
Furthermore, $A^+$ (respectively, $A^-$) approaches zero along every 
future-directed (respectively, past-directed) timelike curve without
endpoint\footnote{
        If the curve $\gamma$ reaches \Scri+, then the limit of \I-p 
        ($p\in\gamma$) is a single null-generator of \Scri+.  
        If the curve fails to reach \Scri+ (\ie, goes to ``$I^+$''), 
        then the limit is empty.}.
Therefore, every such curve intersects $S$.  That is to say, $S$ is an achronal 
set intersected by every timelike curve, and hence $S$ is a Cauchy surface.
The location of this Cauchy surface is shown in Fig.~18.

Every spacetime with a Cauchy surface is causally stable.  Therefore,
every asymptotically-simple spacetime is stably-causal and, in particular, 
has no closed timelike-curves.

Since $M$ has a Cauchy surface, $M$ is topologically the product of that
surface and the real line.  To obtain further information about the
topology of $M$, we use a method of counting null-geodesics\footnote{
        The argument outlined below arose in the course of a conversation
        we R. PENROSE.}.
Let $N$ denote the collection of all null-geodesics in $M$.  This set is
a manifold.  The idea is to describe $N$ in terms of $S$ and again in
terms of \Scri+ (since all null geodesics intersect both $S$ and \Scri+).

\marginpar{99}
Let $p$ be a point of $S$.  The set of all null-geodesics emanating from $p$
is just the set of null-directions at $p$, and hence is, topologically, a 
2-sphere $S^2$.  Therefore, the topology of $N$ is $S\times S^2$.\footnote{
        Strictly speaking, we have only shown that $N$ is the direction
        bundle of $S$.  However, the direction bundle of every orientable
        3-manifold is trivial [33].}
Let $q$ be a point of \Scri+.  The set of null-geodesics emanating from $q$ 
is a 2-sphere.  However, one of these null-geodesics lies in \Scri+, and
so does not enter $M$.  Hence, the null-geodesics from $q$ which do
enter $M$ are, topologically, a 2-sphere minus a single point, \ie
a plane \R{2}.  Hence, $N$ is topologically $\Scri{+}\times\R{2}$.
But \Scri+ is the union of its null-generators, and so we have
$\Scri+=\R{}\times K$, where $K$ is some 2-manifold.  Therefore,
$N=K\times\R{3}$.

We have shown that $S\times S^2=K\times \R{3}$.  This equation can only hold
if $K$ is topologically a 2-sphere and $S$ is topologically \R{3}.\footnote{
        To prove this statement, take the second homotopy group [1]
        of each side.  It follows that $\pi_2(K)$ must be nonzero.
        Since $K$ is a 2-manifold, it must be topologically $S^2$.
        It then follows that $N$ is topologically \R{3}.}
Since $K$ is an $S^2$, \Scri+ (and similarly \Scri-) is topologically
$S^2\times\R{}$.  This is a well-known result of PENROSE [27].

Since $S$ is topologically \R{3}, $M$ itself is topologically \R{4}.
That is, every asymptotically-simple spacetime is topologically the same
as Minkowski space.  In particular, $M$ is time-, space-, and 
charge-orientable.

\vskip 0.25in\ni{\bf Appendix}

\vskip 0.25in\ni{\bf Hausdorff, connected and paracompact manifolds.}

In sect.~3 we discussed some of the pathological situations which can occur
in the causal structure of a space-time. It is natural to ask whether any 
similar anomalies can arise in the underlying manifold itself. There are, 
roughly speaking, three classes of pathological manifolds.
A non-Hausdorff manifold contains points which cannot be properly separated 
by open sets.   A disconnected manifold can be divided into two or more 
disjoint manifolds. A nonparacompact manifold is ``too large''.
One normally assumes that the manifolds dealt with in relativity are 
Hausdorff, connected, and paracompact.  Furthermore, it has not so far been 
useful to relax any of these assumptions. Since these assumptions often appear
in the statements of theorems, however, it is of interest to know just what 
situations are thereby being ruled out.

A manifold (or, more generally, a topological space) is said to be
{\bf Hausdorff} if any two distinct points are contained in disjoint open sets.
Although
\marginpar{100}
the property of being Hausdorff does not follow from the usual definition
of a manifold\footnote{
        The situation with regard to the other separation axioms is as follows.
        Every manifold is $T_0$ and $T_1$.  Every Hausdorff ($T_2$) manifold
        is $T_3$ (regular).  To ensure that a Hausdorff manifold be $T_4$
        (normal), it is necessary to impose some further condition,
        for example, the equivalent conditions of the theorem below.},
almost all manifolds encountered in relativity are Hausdorff.


An example of a non-Hausdorff manifold is given in Fig.~19. The points
$p$ and $p'$ have the property that every open set containing $p$ intersects 
every open set containing $p'$.  What is pathological about Fig.~19?
Consider the vector-field indicated in the figure.  We ask for the trajectory 
of this field which passes through the point $r$. This trajectory is unique 
until we reach the edge of the half-plane $A$. However, the curve can then
continue as a trajectory of our vector-field, into either $B$ or $C$.
It is normally in situations in which one wishes to have uniqueness of such 
trajectories (or, similarly, of geodesics) that one must impose
the condition of Hausdorffness.  Some further examples of non-Hausdorff
manifolds are given in [28].  In fact, the most natural extension
of Taub space [29], in which both NUT extension [30, 31] are carried
out simultaneously, results in a non-Hausdorff manifold.

    A manifold is said to be {\bf connected} if any two points on the manifold
can be joined by a continuous curve.  For example, the manifold consisting of
two disjoint planes is not connected. Physically, a disconnected space-time
represents two or more separate universes: no matter what the metric is,
observers on different components of the space can never communicate with each
other.  It is more convenient, therefore, to study each connected
component separately, that is, to impose the condition that models of our 
universe are connected.

The final condition to be placed on manifolds is that they be not ``too large''.
This condition can be stated in a number of equivalent ways.

\prop{Theorem}{Let $M$ be a connected Hausdorff manifold.  The following
        conditions are equivalent\footnote{
                A space is said to {\bf separable} if it possess a
                countably dense subset.  For manifolds, the equivalent
                conditions of the theorem imply separability, but
                the converse is not true.}
        \ni\tab 1) A countable number of co-ordinate patches suffice 
                        to cover $M$.
        \ni\tab 2) $M$ may be given a nonsingular positive-definite 
                        metric $h\ab$ [33].
        \ni\tab 3) $M$ may be given a nonsingular connection [34].
        \ni\tab 4) If $U_\alpha$ is any collection of open sets whose union
                        is $M$, then there is a countable number of
                        $U_\alpha$ whose union is also $M$.
        \ni\tab 5) $M$ may be written as the union of a countable collection 
                        of compact sets.
        \ni\tab 6) $M$ is triangulable [35].
        \ni\tab 7) $M$ has a countable basis for its topology [36].
        \ni\tab 8) $M$ is paracompact [36].}

\ni It is a good exercise in topology to prove that 1), 4), 5), 7), and 8)
are equivalent.  It follows in particular from condition 3) that every 
space-time based on a paracompact manifold, for the metric defines a 
nonsingular connection.

We conclude this appendix with an example [37] of a manifold which is not 
paracompact.  Let $A$ denote the half-plane $z=0, y>0$ in 3-dimensional space.
The manifold $M$ consists of the half-plane $A$ along with the family
of half-planes $z={\rm const}, y\leq 0$ (Fig.~20).  In the interiors of these 
half-planes, the topology is the usual topology of the plane.  It is only 
necessary to state what the neighborhoods of points on the edges of the 
half-planes, \ie, at $y=0$.  
\marginpar{102}
For this purpose draw, for $y>0$, a collection of straight lines which are
parallel to the $xz$-plane, and whose slope depends only on $y$.  The slope is
to approach $1$ as $y\to\oo$, and $0$ as $y\to 0$.  A neighborhood of a point 
$p$ is constructed as follows.  Draw a horizontal open disk $D$ through $p$.
An open neighborhood of $p$ consists of all points in the left half-plane
intersected by $D$ and all points of $A$ which are intersected by lines which 
pass through $D$. 
({\bf Exercise}{\em Verify that Fig.~20 is a connected Hausdorff manifold.})
This manifold $M$ is not paracompact, for there exists an uncountable collection
of points of $M$ (namely, the points $x=y=-1$) without any accumulation point,
which violates, for example, condition 7) above.

\vskip 0.5in\ni{\bf REFERENCES}
{\footnotesize
\ni [1] A.H. WALLACE: {\em Algebraic Topology} (New York, 1963).
\ni [2] R. GEROCH: {\em Singularities}, article in: {\em Relativity}. 
\ni\tab      edited by M. CARMELI, S. FICKLER and L. WITTEN (New York, 1970).
\ni [3] S.W. HAWKING: {\em Proc. Roy. Soc.}, 308 A, 433 (1969).
\ni [4] A. AVEZ: {\em Ann. Inst. Fourier}, 13, 105 (1963).
\ni [5] R. GEROCH: {\em Journ. Math. Phys.}, 11, 437 (1970).
\ni [6] J.R. MUNKRES: {\em Elementary Differentiable Topology}, in 
\ni\tab      {\em Ann. Math. Studies}, No. 54 (Princeton, 1954).
\ni [7] J.C. GRAVES and D.R. BRILL: {\em Phys. Rev.}, 120, 1507 (1960).
\ni [8] R. PENROSE: {\em Conformal treatment of infinity}, article in:
\ni\tab      {\em Relativity, Groups, and Topology}, edited by
\ni\tab      C. DeWITT and B. DeWITT (New York, 1964).
\ni [9] R. ABRAHAM and J. ROBBIN: {\em Transversal Mappings and Flows} 
\ni\tab      (New York, 1967)
\ni [10] R. PENROSE: {\em Structure of space-time}, article in: 
\ni\tab      {\em Battelle Rencontres}, edited by C. DeWITT and J.A. WHEELER 
\ni\tab      (New York, 1968).
\ni [11] L. MARKUS: {\em Ann. Math.}, 62, 411 (1955).
\ni [12] R.W. BASS and L. WITTEN: {\em Rev. Mod. Phys.}, 29, 452 (1957).
\ni [13] R. GEROCH: {\em Journ. Math. Phy.}, 8, 782 (1967).
\ni [14] R.H. DICKE: {\em Science} 129, 621 (1959).
\ni [15] R. GEROCH: {\em Singularities in the space-time of general relativity},
\ni\tab      Ph.D. Thesis, (Department of Physics, Princeton University,
\ni\tab      June 1967).
\ni [16] J.H. CHRISTENSON, J.W. CRONIN, V.L. FITCH and R. TURLEY:
\ni\tab      {\em  Phys. Rev. Lett.}, 13, 138 (1964).
\ni [17] C.S. WU: {\em Phys. Rev.}, 105,1413 (1957).
\ni [18] R.L. GARWIN, L.M. LEDERMAN and M. WEINRICH: {\em Phys. Rev.}, 105, 1415 
\ni\tab      (1957)
\ni [19] R.F. STREATER and A.S. WIGHTMAN: {\em PCT, Spin and Statistics and All That}
\ni\tab      (New York, 1964).
\ni [20] E.H. KRONHEIMER, R. PENROSE: {\em Proc. Camb. Phil. Soc.}, 63, 481 (1967).
\ni [21]  P. HAVAS: {\em Synth\`ese}, 18, 75 (1968).
\ni [22] H. REICHENBACH: {\em The Philosophy of Space and Time} (New York, 1958).
\ni [23] S.W. HAWKING: {\em Proc. Roy. Soc.}, 300 A, 187 (1957).
\ni [24] S.W. HAWKING: {\em Proc. Roy. Soc.}, 249 A, 511 (1966).
\ni [25] R. COURANT and D. HILBERT: {\em  Methods of Mathematical Physics}, Vol. 2
\ni\tab      (New York, 1964).
\ni [26] R. GEROCH, E.H. KRONHEIMER and  R. PENROSE: {\em Ideal points in space-time}
\ni\tab      in {\em Proc. Roy. Soc.} (submitted for publication).
\ni [27] R. PENROSE: {\em Proc. Roy. Soc.}, 284 A, 159 (1965).
\ni [28] R. GEROCH: {\em Journ. Math. Phys.}, 9, 450 (1968).
\ni [29] A. H. TAUB: {\em Ann. of Math}, 53, 472 (1951).
\ni [30] C.W. MISNER: {\em Journ. Math. Phys.}, 4, 924 (1963).
\ni [31] E.T. NEWMAN, L. TAMBURINO and T. UNTI: {\em Journ. Math. Phys.}, 4, 915 (1963).
\ni [32] R. GEROCH: {\em The structure of singularities}, article in 
\ni\tab      {\em Battelle Recontres}, edited by C. DEWITT and J.A. WHEELER 
\ni\tab      (New York, 1968).
\ni [33] N. STEENROD: {\em The Topology of Fibre Bundles} (Princeton, 1951).
\ni [34] R. GEROCH: {\em Journ. Math. Phys.}, 9, 1739 (1968).
\ni [35] J.H.C. WHITEHEAD: {\em Ann. Math}, 41, 809 (1940).
\ni [36] J.L. KELLY: {\em General Topology} (New York, 1955).
\ni [37] G. SPRINGER: {\em Riemann Surfaces} (Reading, Mass., 1957), p. 56.
}
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