240 Appendix B
outward at E
+
, but now with a nonzero index; that is to say, n will
now have zeros at some set of points x,- G M. But we can 'kill' these
zeros by simply cutting out the points x,-, giving us a new noncompact
manifold M
r
= M\{xt} with a Lorentzian metric. Of course, the missing
points make M' rather physically uninteresting, but we can push these
points off to infinity by means of a conformal transformation to obtain a
somewhat more reasonable spacetime. Alternatively, Sorkin has pointed
out that if we drop the requirement of time-orientability, the zeros of n
can be eliminated by cutting out a small ball around each point x,- and
identifying the antipodal points of the resulting spherical boundary. This
process will give us a new compact manifold M" that is topologically the
connected sum of M with copies of real projective space RP
3
. The vector
field n will not be well-behaved under such identifications, of course,
but the line element field
(n,
— n) will be, so we will be able to find a
non-time-orientable Lorentzian metric on M".
B.3 Closed timelike curves and causal structure
The topology changes we have found come at a cost, however: topology-
changing universes typically involve causality violations. We have seen
that if x(E
+
)
=£
#(£~), a topology-changing spacetime must either have
some 'missing points' or else be non-time-orientable. Geroch has found an
even stronger result: if compact initial and final surfaces Z~ and E
+
are
not homeomorphic, then any compact time-oriented Lorentz cobordism
between them must contain closed timelike curves
[122].
The proof of this result is reasonably straightforward. We have seen
that a time-oriented Lorentzian metric on M determines a timelike vector
field
n.
If M is compact and contains no closed timelike curves, an integral
curve of n starting at a point xel" must end at some point x' G Z
4
".
The mapping
x»—•
x
f
is then the required homeomorphism between Z~
and E
+
. Moreover, by following the integral curves forward from Z~, we
can obtain a complete time-slicing of M, thus proving that M must have
the product topology M « [0,1] x Z.
Closed timelike curves signal an obvious violation of causality. To
further describe the causal structure of spacetime, it is useful to define
several new mathematical objects. (For more detail, see [123] or
[145];
note,
though, that some definitions differ slightly from source to source.)
In particular, we need some notion of 'space at a given time' in order to
ask whether the future can be completely determined from data 'now'.
A slice S of an n-dimensional spacetime M is an embedded spacelike
(n
—
l)-dimensional submanifold that is closed as a subset of M. The
condition of closure is required to eliminate 'small' submanifolds such as
the disk x
2
+ y
2
< 1 in R
3
. (The closed disk x
2
+ y
2
< 1 is excluded
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