## Me at Work

**A man said to the universe:**

**"Sir, I exist!"**

**"However", replied the universe,**

**"The fact has not created in me**

** A sense of obligation".**

-- Stephen Crane

Typically, my work involves walking around the department with bits of paper in my hand, looking very serious. I have also been known to supervise undergraduates, primarily in quantum mechanics and relativity. When I tire of that, I like to sit down and think about problems which arise naturally in theories which attempt to quantize gravity via a path integral prescription. In particular, my research is increasingly centered around superstring theory. Examples of questions which amuse me include:

### M-theory Cosmology

Can we recover realistic cosmological scenarios from string theory?
A key idea in cosmology is the idea of inflation, which asserts
that there was a period of accelerating expansion
in the early universe;
this idea is popular because it provides an elegant solution of several
cosmological problems. In the standard picture, inflationary expansion
is driven by a Higgs

mechanism: One assumes the existence of a scalar
field which has a potential energy capable of generating the inflation.
While this provides a mechanism for inflation which can be recovered
within the framework of the standard model, to date nobody has
shown how inflation can arise directly from compactification
of the low-energy limit of string theory.

Recently, my own research into this problem has developed in various
different directions. First of all, there is the new
proposal
for open inflation put forward by Hawking and Turok.
In the Hawking-Turok model one includes a covariantly constant
four-form $F$, which can contribute to the effective cosmological constant
in four dimensions. From the point of view of M-theory this is a sensible
thing to do, given that the low energy limit of M-theory is known
to be eleven-dimensional supergravity, and the bosonic sector of that
theory contains a four-form. In a recent
paper with
Raphael Bousso,
we point out that once you introduce the four-form $F$ it is natural to
consider the effects of membranes which couple electrically to $F$.
In particular, we consider the effects of such membranes which wrap

an entire bubble of open inflation. We point out that such membranes
can be used, in a very natural way, to eliminate the singularities which
plague the original Hawking-Turok model. We also showed that these membranes
will generically tend to neutralize the effective cosmological constant.

On the other hand, I am increasingly interested in
models where the universe
itself is thought of as a brane moving in some higher dimensional spacetime.
In particular, I am interested in how this picture is explicitly realized
within the context of the strongly coupled heterotic string theory,
which has been identified by Horava and Witten as M-theory compactified on
a circle which has been identified under the action of the Z_2 reflection
symmetry which has two fixed points. Thus, in this Horava-Witten picture
the eleven dimensional circle

of M-theory is replaced by an interval.
At each end of this interval there is a ten-dimensional orbifold fixed plane;
to cancel the resulting anomalies in the usual fashion of Green and
Schwarz, one must include E_8 gauge fields which propagate in each of the
ten-dimensional spacetimes. If you compactify all of this on a Calabi-Yau
space, you obtain a minimal N=1 gauged supergravity in five dimensions
coupled to chiral boundary theories; this effective theory has been
extensively discussed by
Lukas, Ovrut, Stelle,
and Waldram.
Motivated by this rich structure, in a recent
paper
with Harvey Reall, we investigate the motion of a $(D-2)$-brane (or domain
wall) that couples to bulk matter. Usually one would expect the spacetime outside such a wall to be time
dependent however we show that in certain cases it can be static, with consistency of the Israel equations
yielding relationships between the bulk metric and matter that can be used as ans\"atze to solve the
Einstein equations. As a concrete model we study a domain wall coupled to a bulk dilaton with Liouville
potentials for the dilaton both in the bulk and on the wall. Such potentials
are natural in the sense that they will generically arise from compactification
in string theory. The bulk solutions we find are all singular but
some have black hole or cosmological horizons, beyond which our solutions describe domain walls moving
in time dependent bulks. A significant period of world volume inflation occurs if the potential on the wall is
not too steep; in some cases the bulk also inflates (with the wall comoving) while in others the wall moves
relative to a non-inflating bulk. We apply our method to obtain
dynamical cosmological solutions of
Horava-Witten theory compactified on a Calabi-Yau space. For more
information about this work, click
here
to see (and to hear)
a talk which I recently gave out in Santa Barbara.

### Does cosmology require a modification

of M-theory?

As I mentioned above, we would like to recover inflationary cosmology
within the context of M-theory. Indeed, from the observational point
of view there is some evidence that there is a non-vanishing cosmological constant.
However, any inflationary scenario requires a solution to the vacuum
Einstein equations known as de Sitter space (or some close cousin).
From the theoretical point of view de Sitter space has become
somewhat of an anomaly because, of all the things you can get out
of string theory or M-theory, de Sitter space doesn't seem to be one of them.
In a recent paper, written in collaboration with Neil Lambert, we study a theory which may be regarded as a slight extension of eleven-dimensional supergravity that still preserves eleven-dimensional supersymmetry. In particular, this theory (which was introduced by Paul Howe in this paper) is the most general form for eleven dimensional supersymmetry compatible with on-shell superfields. Moreover, as long as the eleven-dimensional space is topologically trivial, this theory can be obtained through a field redefinition and therefore it is completely equivalent to eleven-dimensional supergravity in that case. It is only when the topology of spacetime is non-trivial (such as when we compactify on a circle) that the theory is distinct from ordinary eleven-dimensional supergravity, in the sense that a mass term is introduced which cannot be gauged away.

In our paper, we argue that this extension of eleven-dimensional supergravity reflects an underlying, modified (or massive) M-theory which we call MM-theory. We then prove that de Sitter space is the natural ground state of MM-theory (compactified in a certain way on a circle) contrary to the case in standard M-theory and string theory. For us, this is evidence that we should regard MM-theory as a legitimate extension of the M-theory moduli space.

### Does our proposed extension of M-theory admit an underlying quantum mechanical description?

One objection to MM-theory, which was immediately raised by several people, is that the theory does not admit an action formulation and therefore it is unclear how to make sense of quantizing the theory. In a recent paper, written in collobaration with Neil Lambert, we have attempted to address this problem. In particular, we performed a detailed analysis of the dynamical properties of MM-theory, and we discovered that the theory admits multiple zero-brane solutions. We argue that this must imply the existence of a microscopic quantum mechanical matrix description which yields a massive deformation of the usual M(atrix) formulation of M-theory and type IIA string theory.
We are therefore led to the following intuitive picture: The universe is filled with
a gas

of these zero-branes. On the worldvolume of each brane there resides the quantum mechanical
theory which corresponds to this massive deformation of standard M(atrix) theory. However, this massive
phase of M(atrix) mechanics is unstable, and it wants to decay back to standard M(atrix) theory.
This is why we see such a small cosmological constant now, and also why in principle the cosmological
constant would have been much larger in the distant past.

### Do we live on a Brane?

In any of the modern approaches to quantizing gravity, we are led inevitably to the conclusion that there exist extra dimensions of space. Indeed, as I mentioned above in M-theory there are ten dimensions of space, and one dimension of time. Of course, we need to explain why these extra dimensions are not seen, and to do this we typically invoke the old ideas of Kaluza and Klein: Put simply, we assume that the extra dimensions are curled up on such a small length scale that we are unable to detect them.
Recently, however, there has been an enormous amount of interest
in the possibility that some of the extra dimensions of space may
be quite large. Thus, in this framework the
universe would be a brane moving in some higher dimensional
spacetime, usually referred to as the bulk

. As I mentioned
above, this sort of picture is naturally realized in the
Horava-Witten corner of the M-theory moduli space.

Now, it has long been thought that any attempt to
model the Universe as a single brane
embedded in a higher-dimensional bulk spacetime must inevitably
fail because
the gravitational forces experienced by matter on the brane,
being mediated by gravitons travelling in the bulk, are those
appropriate to the higher dimensional spacetime rather than the
lower dimensional brane. Recently however,
Randall and Sundrum
have argued that there are circumstances under
which this need not be so.
Their model involves a thin distributional

static flat
domain wall or three-brane
separating two regions of five-dimensional anti-de-Sitter spacetime.
They solve for the linearized graviton perturbations and find a
square integrable bound state representing a gravitational
wave confined to the domain wall. They also found the linearized
bulk or Kaluza-Klein

graviton modes. They argue that
the latter decouple from the brane and
make negligible contribution to the force beween two
sources in the brane, so that this force is due primarily to the
bound state. In this way we get an inverse square law
attraction rather than the inverse cube law one might
naively have anticipated.

In a recent paper with Gary Gibbons we provide a *fully non-linear treatment* of the Randall-Sundrum model, which shows that smooth domain wall spacetimes supported by a scalar field separating two anti-de-Sitter like regions admit a single graviton bound state. We do this by deriving solutions which describe a pp-wave propagating in the domain wall background spacetime. We then construct a one-to-one correspondence between geometric objects in the pp-wave metric and linearized objects in the Randall-Sundrum scenario. That is, we explicitly identify which components of the Weyl tensor correspond to the zero mode (the mode trapped on the brane), and which correspond to the Kaluza-Klein modes.

However, somewhat to our surprise, we have found
that generically gravitons propagating in the bulk become singular
on what is a Cauchy horizon in the unperturbed spacetime.
These singularities are somewhat unusual, in that scalar invariants
formed from the curvature tensor do not blow up but rather the
components of the curvature in a parallelly propagated frame along
a timelike geodesic do blow up. Such singularities are called
pp curvature singularities

.

One might worry that these singularities signal a breakdown in our ability to make unitary predictions. However, any statements about unitarity should be restricted to physics *on the brane*. Any pathological effects which may emerge from the singularity will be heavily red-shifted by the time they reach the brane. Consequently, the extent to which these singularities signal a pathology of the theory is at present unclear.

### Black holes in the brane?

If matter trapped on a brane undergoes gravitational collapse then a black hole will form. Such a black hole will have a horizon that extends into the dimensions transverse to the brane: it will be a higher dimensional object.

Within the context of the Randall-Sundrum scenario, we need to make sure that the metric on the domain wall, which is induced by the higher dimensional metric describing the gravitational collapse, is simply the Schwarzschild solution. Otherwise, we will not recover the usual astrophysical properties of black holes and stars (e.g. perihelion precession, light bending etc..).

In a recent paper,
written in collaboration with
Stephen Hawking
and Harvey Reall, we have attempted to address this question. In particular,
we propose that what would appear to be a four-dimensional black hole
from the point of view of an observer in the brane-world, is really
a five-dimensional black cigar

, which extends into the extra fifth
dimension. If this cigar extends all the way down to the adS horizon,
then we recover the metric for a black string in adS. However,
such a black string is unstable near the AdS
horizon. This instability, known as the Gregory-Laflamme

instability,
basically means that the string will want to fragment in the region near
the adS horizon. However, the solution is stable far from the AdS
horizon. Thus, we conclude that the late time solution
describes an object that looks like the black string
far from the AdS horizon (so the metric on the domain wall is
Schwarzschild) but has a horizon that closes off before reaching the
AdS horizon (i.e., this is the tip

of the black cigar).
In fact, we conjecture that this black cigar
solution is the unique stable vacuum solution in five dimensions
which describes the endpoint of gravitational collapse on the
brane. We suspect that the AdS horizon will be non-singular for
the cigar solution, but we cannot prove this because the exact form
of the black cigar metric is at present not known.

### Can black holes trap

matter from the extra dimensions?

If we do live on a brane, a natural question is: What kind of
matter lives in the bulk, i.e., in the extra dimensions? Would matter
in the extra dimensions affect our lives on the brane? Ordinarily,
in the Randall-Sundrum models our brane is repulsive, in the sense that
any massive degrees of freedom in the bulk are driven away from the brane
by the gravitational field. However, if we put a black hole on the brane
we might expect that the attractive gravitational force of the black hole
could overcome this repulsive force, and consequently that we could use
a black hole to capturematter from the extra dimensions.

In a recent paper
I have shown that this intuition is correct, at least for a two-dimensional
brane moving in three spatial dimensions. More precisely, for a toy

Randall-Sundrum model (with one less dimension of space) I have shown
that for any black hole which satisfies a certain bound on the mass
there exists a halo

region surrounding the black hole where particles
from the extra dimension can move in stable orbits. This would imply
that certain brane-world black holes are regions where we can store
bulk degrees of freedom.

### What happens if we throw charge into a brane-world black hole?

In the real world, elementary particles may possess various different types of charge. The gravitational collapse of charged matter trapped on a brane will produce a charged black hole on the brane. We are therefore led to the obvious question: Can charges on the brane affect the geometry of space-time in the extra dimensions? In a recent paper, written with Harvey Reall, Hisa-aki Shinkai and Tetsuya Shiromizu we have answered this question. We showed that a non-rotating charged black hole on the domain wall is *not* described by theblack cigarsolution which corresponds to the collapse of uncharged matter on the brane. Instead, we showed that the presence of a point charge in the brane-world has a pronounced effect on the curvature of the bulk spacetime. In other words,

**putting charge on the brane will affect the gravitational tidal forces in the extra dimensions.**I am currently investigating various consequences of this remarkable result.

### The thermodynamics of brane-worlds

What are the gross thermodynamical properties of brane-worlds? If we do live on a brane, it seems reasonable that thermal effects in the extra dimensions might affect physics on the brane. In a recent paper, written with Andreas Karch and Ali Nayeri (at MIT), we analyzed the thermodynamical properties of brane-worlds, with a focus on the second model of Randall and Sundrum. We pointed out that during an inflationary phase on the brane, black holes will tend to be thermally nucleated in the bulk. This led us to ask the question: Can the black hole - brane-world system evolve towards a configuration of thermal equilibrium? To answer this, we generalized the second Randall-Sundrum scenario to allow for non-static bulk regions on each side of the brane-world. Explicitly, we took the bulk to be a {\it Vaidya-AdS} metric, which describes the gravitational collapse of a spherically symmetric null dust fluid in Anti-de Sitter spacetime. We calculated the late time behaviour of this system, including 1-loop effects, subject to the assumption that the brane-world acts like a perfectly reflecting mirror for the bulk modes. We argued that at late times a sufficiently large black hole will relax to a point of thermal equilibrium with the brane-world environment. We are currently generalizing this work to allow for a brane-world which is partially absorbent, so that bulk fields may (with some small probability) excite brane-world degrees of freedom.

### What are the fundamental degrees of freedom of M-theory?

At present, it is unclear as to what are the fundamental physical degrees of fre
edom
underlying M-theory. Most of what we understand about M-theory
is based on the facts that it has eleven-dimensional supergravity as
its low energy limit and, via compactification on a circle, can be
related to type IIA string theory. While the
string

of the type IIA theory was at one time regarded as the fundamental obje
ct,
it is now clear that this is not the basic degree of freedom
underpinning M-theory as a whole. Of course, one may take the point of view tha
t
there is no truly
fundamental physical degree of freedom but rather that, in different regions of
the M-theory moduli space, different degrees of freedom appear to be fundamental
.

An interesting approach to M-theory, which may resolve at least some
of these conceptual issues, is Matrix theory: a supersymmetric quantum
mechanics of N-by-N matrix degrees of freedom (see, e.g.,
this paper
for a recent review). Matrix theory was originally
conjectured to be equivalent
to M-theory in the infinite momentum frame, in the limit that N is
large. The finite N version has been
further conjectured to be equivalent to the discrete light cone quantization
(DLCQ) of M-theory. The Hamiltonian of Matrix
theory is precisely the low-energy limit of the Hamiltonian describing a system
of N type IIA D0-branes, which are point-like objects where fundamental strings
can end. This makes sense
given that the D0-brane couples to the Ramond-Ramond (R-R) vector of the type II
A
theory, which is itself the Kaluza-Klein vector obtained
by dimensionally reducing M-theory on a circle. In other words,
D0-branes are the partonic

, or fundamental, degrees of freedom
underlying Matrix theory.

### The secret life of a D0-brane

A key fact about D0-branes is that they possess internal

degrees
of freedom, which couple to spacetime fields. More precisely,
the worldvolume theory of a D0-brane contains a multiplet of fermions
which can couple to background spacetime fields. This coupling implies that
a D0-brane may possess multipole moments with respect to the various
type IIA supergravity fields. Different such polarization states of
the D0-brane will thus generate different long-range supergravity
fields, and the corresponding semi-classical supergravity solutions
will have different geometries. In a
recent paper, written with
Dominic Brecher
, we
reconsider such solutions from an eleven-dimensional
perspective. We thus begin by deriving the superpartners

of the
eleven-dimensional graviton. These superpartners are obtained by
acting on the purely bosonic solution with broken supersymmetries and,
in theory, one can obtain the full BPS supermultiplet of
states. When we dimensionally reduce a polarized supergraviton along
its direction of motion, we recover a metric which describes a
polarized D0-brane. On the other hand, if we compactify along the retarded
null direction we obtain the short distance, or near-horizon

,
geometry of a polarized D0-brane,
which is related to finite N Matrix theory. The various dipole
moments in this case can only be defined once the eleven-dimensional metric is
regularized

and, even then, they are formally infinite. We argue,
however, that this is to be expected in such a non-asymptotically flat
spacetime. Moreover, we find that the
superpartners of the D0-brane, in this r --> 0 limit, possess
neither spin nor D2-brane dipole moments.

### Phase transitions in M-theory

What is the role of bubbles of phase transition in massive supergravities?
In a recent paper
with Malcolm Perry and Harvey Reall,
we perform a detailed investigation of the Dirichlet
eight-brane of the Type IIA string theory. We
consider what happens when one breaks the supersymmetry, and find a
new one-parameter family of dynamic

D8-branes which generalize the old
static, supersymmetric D8-brane found by Bergshoeff et al.
Can these dynamic branes teach us anything about the eleven-dimensional
origins of the Romans theory?

### Is it possible to find new impossible objects?

My first research project, back when I was a graduate student at
Christ Church
College (Oxford), was concerned
with impossible objects. This work was motivated
by a paper on *Impossible Figures* written by
Roger Penrose. In particular, Roger studied the cohomology of these figures,
and he was able to define a cohomology element which can be regarded as the obstruction

to possibility; more precisely, this element will tell you whether or not the figure
can be realized as a solid object in three-dimensional space. However, for all of
the figures which Roger considered this obstruction was an element of the
*first* cohomology group of the figure. This corresponds to the fact that most
impossible figures (such as the Penrose tribar)
can be drawn on a non-simply connected, annular region of the plane. A natural
question, therefore, is whether or not it is possible to draw a figure where the
obstruction to possibility is an element of the *second* cohomology group of the
figure. Such an impossible figure would have to be drawn on a closed two-dimensional
surface (such as the surface of a balloon).

Recently, I have returned to this problem. My intuition for how to construct one of these exotic impossible figures is motivated by my studies of the breakdown of spin (and pin) structures. The obstruction to the existence of a (s)pin structure on a two-surface is an element of the second cohomology group. This obstruction is rooted in the fact that there is a sign ambiguity in a spinor field at each point of a manifold (i.e., rotation through 360 degrees gives you -1, whereas rotation through 720 degrees gives you +1, but both of these motions covers the identity element in the tangent bundle). My idea is to replace this spinorial sign ambiguity with some visual ambiguity, such as our inability to distinguish concave and convex. In this sense, such an impossible figure would literally correspond to a visual representation of the breakdown of spin structure.

### Superconducting p-branes

Is it sensible to talk about superconducting

phases of M-theory or string theory?
A first attempt to understand how the phenomena of superconductivity
can occur naturally within the framework of
string theory was made in a
recent
paper with Roberto Emparan and Gary Gibbons.
In our paper, we argue that
extreme p-brane horizons diplay several features which are characteristic
of superconducting media.

Roberto has written up a very nice web site describing this work in more detail.

### Quantum Gravity and the Arrow of Time

When is a spacetime its own antiparticle

? That is, when can one
find a diffeomorphism in the identity connected component of the
diffeomorphism group which reverses time (flips the light-cones
everywhere)? In a
paper
with Gary Gibbons, we showed that
there are topological obstructions to the existence
of such diffeomorphisms - does this tell us something about how the
Arrow of Time

emerges in Quantum Gravity?

### How many dimensions of Time are there?

Is it possible to construct phenomenologically viable
multitemporal Kaluza-Klein models

? Basically, these are models
with more than one dimension of time, only the extra timelike dimensions
are curled up very small so you can't see them. To use technical jargon,
a multitemporal Kaluza-Klein model is any Kaluza-Klein scenario where some of
the internal dimensions are allowed to be timelike. Thus, the internal space
in a multitemporal model is in general some compact manifold $M_i$ admitting
a pseudo-Riemannian metric structure $g_P$ of signature $(p, q)$, which is
acted upon by some isometry group $G$. It is interesting to study these
models in order to determine what pathologies they possess. For example,
in a recent
paper
with Roberto Emparan we showed that
the two-timing Kaluza-Klein vacuum will be
unstable to the runaway pair-production
of massless two-timing monopoles. Since two-timing monopoles are not
spontaneously appearing from nothing around us all of the time, this would
seem to be one more piece of evidence that in fact the multitemporal hypothesis
is not tenable.

### The Holographic Principle and String Theory

One of my favourite current string theory research interests
has been the revival of what is known as the
Holographic Principle

.
Recently, this principle has appeared in the setting of string
theory, where it has been conjectured that the physics of
quantum gravity (in a certain class of spacetime) may be
understood by studying the dynamics of a type of gauge theory
which is defined on the boundary of the spacetime. In some recent
papers (with
Roberto Emparan,
Clifford Johnson
and Rob Myers), we have
shown that this correspondence holds on a broader class of
spacetimes than previously considered. A general feature of all
of these spacetimes is that they are all at least locally asymptotic
to anti-de Sitter (adS) spacetime. This spacetime has closed
timelike curves which can be eliminated by passing to the universal
covering space. However, even in the universal covering space of
adS, all geodesics (which describe the motion of test particles
in the space) which depart from a given event will return to the
same event after some fixed time period. In other words, adS
spacetime describes a universe where everything undergoes the same
repeated cycle. For a deep philosophical discussion of what it
would be like to live in such a spacetime, click
here.

### Counting Einstein Structures

How many Riemannian Einstein metrics (instantons

) are there
(up to diffeomorphism) on a given four-manifold? This is a very hard (and deep)
problem in pure mathematics, but its solution would be of enormous importance
in Euclidean quantum gravity.

### How is the topology of a spacetime related to the causal structure?

A 4-dimensional spacetime (M,g) consists of a topological 4-manifold M together with a Lorentz metric g. If the Lorentz metric is non-singular, it will yield a non-degenerate lightcone at each point in the spacetime. Consequently, given a non-singular Lorentz metric g on M we may select a future-directed vector V at each point of M, so that we obtain a non-vanishing vector field V on M. The converse is also true: Given a non-vanishing vector field we may recover a non-singular Lorentz structure. (Note: For the purposes of this discussion I will assume that (M,g) is space and time orientable).

Now, typically when people think of spacetime topology

,
they think of the topology of the manifold M, or of certain
spacelike submanifolds of M. However, they are ignoring the
topology of the Lorentz structure. Indeed, we may ask the question:
When can a Lorentz metric g' be obtained from a Lorentz metric g
by a continuous deformation? Since the metric g' induces
a vector field V' and the metric g induces a vector field V, we
see that this question is actually equivalent to: When can V'
be obtained from V through a continuous deformation? To see how
to answer this question, first notice that at each point a vector
V points in some direction, which is parametrized by a 3-sphere S(3).
Thus, a non-vanishing vector field V will define a map from
any 3-dimensional submanifold C of M, into the 3-sphere S(3):

V: C ---> S(3)

This map has a degree, which is called the kink number

.
The kink number was first studied in the context of General
Relativity by
Finkelstein and Misner.
Using this information, it is possible to show that the obstruction
to deforming V into V' depends on the degrees of maps from
certain submanifolds of M into S(3). Thus, any attempt to study
the space of Lorentz metrics on a manifold M will automatically
involve the kink number, given that we would at least
need to coarse grain Lorentz metrics into homotopy
equivalence classes.

Now, there are a number of results which relate the topology of the manifold M with the causal structure of (M,g). Perhaps the most celebrated of these results is the Theorem of Geroch, which basically asserts that if the topology of space changes, and the topology change is mediated by a non-singular spacetime (M,g) (here non-singular means that the vector field V induced by g is nowhere vanishing), then (M,g) must contain closed timelike curves (CTCs) - i.e., there must be a time machine somewhere in the spacetime. Thus, to be a bit more technical, if we think of (M,g) as a Lorentz cobordism between some initial spacelike 3-surface C(i) and some final spacelike 3-surface C(f), and if C(i) and C(f) are not homeomorphic, then there must exist CTCs in (M,g).

This result is very pretty, and for some people there was a hope that there might be some connection between the causal structure of (M,g) and the topology of the metric g. Indeed, Gibbons and Hawking conjectured that any compact spacetime (M,g) with a single 3-sphere boundary component S(3) on which the kink number vanishes must contain a time machine. However, in a paper written in collaboration with Roger Penrose, we managed to show that this conjecture is false by proving some theorems which showed that there is *no* connection between the kink number and the existence of closed timelike curves. Thus, we may coarse grain Lorentz metrics into homotopy equivalence classes without worrying about the existence of CTCs.

However, this does not mean that there is *no* relationship between the causal structure of (M,g) and the kink number. In particular, I have made a conjecture which basically states that if an asymptotically flat spacetime (M,g) contains an isolated kink, and if this spacetime is causal and satisfies the strong and generic energy conditions, then (M,g) is non-spacelike geodesically incomplete. At present, this conjecture remains unproven.

#### Update!

Eric Woolgar has performed a detailed analysis of the causal structure of kinky spacetimes, and has managed to prove a limited case of the conjecture. Click here to download a .pdf file of this interesting result.

Of course, the answers to all of these questions (and more) will be revealed when you take a look at my publications and preprints.

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