Recent years have seen a growing consensus in the philosophical
community that the grandfather paradox and similar logical
puzzles do not preclude the possibility of time travel scenarios
that utilize spacetimes containing closed timelike curves. At
the same time, physicists, who for half a century acknowledged
that the general theory of relativity is compatible with such
spacetimes, have intensely studied the question whether the
operation of a time machine would be admissible in the context
of general relativity theory or theories that attempt to combine
general relativity and quantum mechanics. A time machine is a
device which brings about closed timelike curves—and thus
enables time travel—where none would have existed otherwise. The
physics literature contains various nogo theorems for time
machines, i.e., theorems which purport to establish that, under
physically plausible assumptions, the operation of a time
machine is impossible. We conclude that for the time being there
exists no conclusive nogo theorem against time machines. The
character of the material covered in this article makes it
inevitable that its content is of a rather technical nature. We
contend, however, that philosophers should nevertheless be
interested in this literature for at least two reasons. First,
the topic of time machines leads to a number of interesting
foundations issues in classical and quantum theories of gravity;
and second, philosophers can contribute to the topic by
clarifying what it means for a device to count as a time
machine, by relating the debate to other concerns such as
Penrose's cosmic censorship conjecture and the fate of
determinism in general relativity theory, and by eliminating a
number of confusions regarding the status of the paradoxes of
time travel. The present article addresses these ambitions in as
nontechnical a manner as possible, and the reader is referred
to the relevant physics literature for details
1. Introduction: time travel vs. time machine
The topic of time machines is the subject of a sizable and
growing physics literature, some of which has filtered down to
popular and semipopular presentations. The issues raised by
this topic are largely oblique, if not orthogonal, to those
treated in the philosophical literature on time travel. Most
significantly, the socalled paradoxes of time travel do not
play a substantial role in the physics literature on time
machines. This literature equates the possibility of time travel
with the existence of closed timelike curves (CTCs) or
worldlines for material particles that are smooth,
futuredirected timelike curves with selfintersections. Since
time machines designate devices which bring about the existence
of CTCs and thus enable time travel, the paradoxes of time
travel are irrelevant for attempted “nogo” results for time
machines because these results concern what happens before the
emergence of CTCs. This, in our opinion, is fortunate since the
paradoxes of time travel are nothing more than a crude way of
bringing out the fact that the application of familiar local
laws of relativistic physics to a spacetime background which
contains CTCs typically requires that consistency constraints on
initial data must be met in order for a local solution of the
laws to be extendable to a global solution. The nature and
status of these constraints is the subject of ongoing
discussion. We will not try to advance the discussion of this
issue here; rather, our aim is to acquaint the reader with the
issues addressed in the physics literature on time machines and
to connect them with issues in the philosophy of space and time
and, more generally, with issues in the foundations of physics.
Paradox mongers can be reassured in that if a paradox is lost in
shifting the focus from time travel itself to time machines,
then a paradox is also gained: if it is possible to operate a
time machine device that produces CTCs, then it is possible to
alter the structure of spacetime such that determinism fails;
but by undercutting determinism, the time machine undercuts the
claim that it is responsible for producing CTCs. But just as the
grandfather paradox is a crude way of making a point, so this
new paradox is a crude way of indicating that it is going to be
difficult to specify what it means to be a time machine. This is
a task that calls not for paradox mongering but for
scientifically informed philosophizing. The present article will
provide the initial steps of this task and will indicate what
remains to be done. But aside from paradoxes, the main payoff of
the topic of time machines is that it provides a quick route to
the heart of a number of foundations problems in classical
general relativity theory and in attempts to produce a quantum
theory of gravity by combining general relativity and quantum
mechanics. We will indicate the shape of some of these problems
here, but will refer the interested reader elsewhere for
technical details.
There are at least two distinct general notions of time
machines, which we will call Wellsian and Thornian
for short. In The Time Machine, H. G. Wells (1931)
described what has become science fiction's paradigmatic
conception of a time machine: the intrepid operator fastens her
seat belt, dials the target date—past or future—into the
counter, throws a lever, and sits back while time rewinds or
fast forwards until the target date is reached. We will not
broach the issue of whether or not a Wellsian time machine can
be implemented within a relativistic spacetime framework. For,
as will soon become clear, the time machines which have recently
come into prominence in the physics literature are of an utterly
different kind. This second kind of time machine was originally
proposed by Kip Thorne and his collaborators (see Morris and
Thorne 1988; Morris, Thorne, and Yurtsever 1988). These articles
considered the possibility that, without violating the laws of
general relativistic physics, an advanced civilization might
manipulate concentrations of matterenergy so as to produce CTCs
where none would have existed otherwise. In their example, the
production of “wormholes” was used to generate the required
spacetime structure. But this is only one of the ways in which a
time machine might operate, and in what follows any device which
affects the spacetime structure in such a way that CTCs result
will be dubbed a Thornian time machine. We will only be
concerned with this variety of time machine, leaving the
Wellsian variety to science fiction writers. This will
disappoint the aficionados of science fiction since Thornian
time machines do not have the magical ability to transport the
wouldbe time traveler to the past of the events that constitute
the operation of the time machine. For those more interested in
science than in science fiction, this loss is balanced by the
gain in realism and the connection to contemporary research in
physics.
In Sections 2 and 3 we investigate the circumstances under which
it is plausible to see a Thornian time machine at work. The main
difficulty lies in specifying the conditions needed to make
sense of the notion that the time machine “produces” or is
“responsible for” the appearances of CTCs. We argue that at
present there is no satisfactory resolution of this difficulty
and, thus, that the topic of time machines in a general
relativistic setting is somewhat illdefined. This fact does not
prevent progress from being made on the topic; for if one's aim
is to establish nogo results for time machines it suffices to
identify necessary conditions for the operation of a time
machine and then to prove, under suitable hypotheses about what
is physically possible, that it is not physically possible to
satisfy said necessary conditions. In Section 4 we review
various nogo results which depend only on classical general
relativity theory. Section 5 surveys results that appeal to
quantum effects. Conclusions are presented in Section 6.
2. What is a (Thornian) time machine? Preliminaries
The setting for the discussion is a general relativistic
spacetime (M, g_{ab}) where M
is a differentiable manifold and g_{ab} is a
Lorentz signature metric defined on all of M. The central
issue addressed in the physics literature on time machines is
whether in this general setting it is physically possible to
operate a Thornian time machine. This issue is to be settled by
proving theorems about the solutions to the equations that
represent what are taken to be physical laws operating in the
general relativistic setting—or at least this is so once the
notion of a Thornian time machine has been explicated.
Unfortunately, no adequate and generally accepted explication
that lends itself to the required mathematical proofs is to be
found in the literature. This is neither surprising nor
deplorable. Mathematical physicists do no wait until some
concept has received its final explication before trying to
prove theorems about it; indeed, the process of theorem proving
is often an essential part of conceptual clarification. The
moral is well illustrated by the history of the concept of a
spacetime singularity in general relativity where this concept
received its now canonical definition only in the process of
proving the PenroseHawkingGeroch singularity theorems, which
came at the end of a decades long dispute over the issue of
whether spacetime singularities are a generic feature of
solutions to Einstein's gravitational field equations. However,
this is not to say that philosophers interested in time machines
should simply wait until the dust has settled in the physics
literature; indeed, the physics literature could benefit from
deployment of the analytical skills that are the stock in trade
of philosophy. For example, the paradoxes of time travel and the
fate of time machines are not infrequently confused in the
physics literature, and as will become evident below, subtler
confusions abound as well.
The question of whether a Thornian time machine—a device that
produces CTCs—can be seen to be at work only makes sense if the
spacetime has at least three features: temporal orientability, a
definite time orientation, and a causally innocuous past. In
order to make the notion of a CTC meaningful, the spacetime must
be temporally orientable (i.e., must admit a consistent
time directionality), and one of the two possible time
orientations has to be singled out as giving the
direction of time. Nontemporal orientability is not really an
obstacle since if a given general relativistic spacetime is not
temporally orientable, a spacetime that is everywhere locally
the same as the given spacetime and is itself temporally
orientable can be obtained by passing to a covering spacetime.
How to justify the singling out of one of the two possible
orientations as future pointing requires a solution to the
problem of the direction of time, a problem which is still
subject to lively debate (see Calendar 2001). But for present
purposes we simply assume that a temporal orientation has been
provided. A CTC is then (by definition) a parameterized closed
spacetime curve whose tangent is everywhere a futurepointing
timelike vector. A CTC can be thought of as the world line of
some possible observer whose life history is linearly ordered in
the small but not in the large: the observer has a consistent
experience of the “next moment,” and the “next,” etc., but
eventually the “next moment” brings her back to whatever event
she regards as the starting point.
As for the third condition—a causally innocuous past—the
question of the possibility of operating a device that produces
CTCs presupposes that there is a time before which no CTCs
existed. Thus, Gödel spacetime, so beloved of the time travel
literature, is not a candidate for hosting a Thornian time
machine since through every point in this spacetime there is a
CTC. We make this third condition precise by requiring that the
spacetime admits a global time slice Σ (i.e., a spacelike
hypersurface without edges); that Σ is twosided and partitions
M into three parts—Σ itself, the part of M on the
past side of Σ and the part of M on the future side of
Σ—and that there are no CTCs that lie on the past side of Σ. The
first two clauses of this requirement together entail that the
time slice Σ is a partial Cauchy surface, i.e., Σ is a
time slice that is not intersected more than once by any
futuredirected timelike curve.
Now suppose that the state on a partial Cauchy surface Σ_{0}
with no CTCs to its past is to be thought of as giving a
snapshot of the universe at a moment before the machine is
turned on. The subsequent realization of a Thornian time machine
scenario requires that the chronology violating region
V
⊆
M,
the region of spacetime traced out by CTCs, is nonnull and lies
to the future of Σ_{0}. The fact that V ≠
∅
does not lead to any consistency constraints on initial data on
Σ_{0} since, by hypothesis, Σ_{0} is not
intersected more than once by any timelike curve, and thus,
insofar as the socalled paradoxes of time travel are concerned
with such constraints, the paradoxes do not arise with respect
to Σ_{0}. But by the same token, the option of traveling
back into the past of Σ_{0} is ruled out by the set up
as it has been sketched so far, since otherwise Σ_{0}
would not be a partial Cauchy surface. This just goes to
underscore the point made above that the fans of science fiction
stories of time machines will not find the present context of
discussion broad enough to encompass their vision of how time
machines should operate; they may now stop reading this article
and return to their novels.
As a concrete example of these concepts, consider the (1 +
1)dimensional Misner spacetime (see
Figure 1)
which exhibits some of the causal features of TaubNUT
spacetime, a vacuum solution to Einstein's gravitational field
equations. It satisfies all three of the conditions discussed
above. It is temporally orientable, and a time orientation has
been singled out—the shading in the figure indicates the future
lobes of the light cones. To the past of the partial Cauchy
surface Σ_{0} lies the Taub region where the causal
structure of spacetime is as bland as can be desired. But to the
future of Σ_{0} the light cones begin to “tip over,” and
eventually the tipping results in CTCs in the NUT region.
The issue that must be faced now is what further conditions must
be imposed in order that the appearance of CTCs to the future of
Σ_{0} can be attributed to the operation of a time
machine. Not surprisingly, the answer depends not just on the
structure of the spacetime at issue but also on the physical
laws that govern the evolution of the spacetime structure. If
one adopts the attitude that the label “time machine” is to be
reserved for devices that operate within a finite spatial range
for a finite stretch of time, then one will want to impose
requirements to assure that what happens in a compact region of
spacetime lying on or to the future of Σ_{0} is
responsible for the CTCs. Or one could be more liberal and allow
the wouldbe time machine to be spread over an infinite space.
We will adopt the more liberal stance since it avoids various
complications while still sufficing to elicit key points. Again,
one could reserve the label “time machine” for devices that
manipulate concentrations of massenergy in some specified ways.
For example, based on Gödel spacetime—where matter is everywhere
rotating and a CTC passes through every spacetime point—one
might conjecture that setting into sufficiently rapid rotation a
finite mass concentration of appropriate shape will eventuate in
CTCs. But with the goal in mind of proving negative general
results, it is better to proceed in a more abstract fashion.
Think of the conditions on the partial Cauchy surface Σ_{0}
as encoding the instructions for the operation of the time
machine. The details of the operation of the device—whether it
operates in a finite region of spacetime, whether it operates by
setting matter into rotation, etc.—can be left to the side. What
must be addressed, however, is whether the processes that evolve
from the state on Σ_{0} can be deemed to be responsible
for the subsequent appearance of CTCs.
3. When can a wouldbe time machine be held responsible for
the emergence of CTCs?
The most obvious move is to construe “responsible for” in the
sense of causal determinism. But in the present setting this
move quickly runs into a dead end. For if CTCs exist to the
future of Σ_{0} they are not causally determined by the
state on Σ_{0} since the time travel region V, if
it is nonnull, lies outside the future domain of dependence
D^{+}(Σ_{0}) of Σ_{0}, the
portion of spacetime where the field equations of relativistic
physics uniquely determine the state of things from the state on
Σ_{0}. The point is illustrated by the toy model of
Figure 1.
The surface labeled H^{+}(Σ_{0}) is
called the future Cauchy horizon of Σ_{0}. It is
the future boundary of D^{+}(Σ_{0}), and
it separates the portion of spacetime where conditions are
causally determined by the state on Σ_{0} from the
portion where conditions are not so determined. And, as
advertised, the CTCs in the model of
Figure 1
lie beyond H^{+}(Σ_{0}).
Thus, if the operation of a Thornian time machine is to be a
live possibility, some condition weaker than causal determinism
must be used to capture the sense in which the state on Σ_{0}
can be deemed to be responsible for the subsequent development
of CTCs. Given the failure of causal determinism, it seems the
next best thing to demand that the region V is “adjacent”
to the future domain of dependence D^{+}(Σ_{0}).
Here is an initial stab at such an adjacency condition. Consider
causal curves which have a future endpoint in the time travel
region V and no past endpoint. Such a curve may never
leave V; but if it does, require that it intersects Σ_{0}.
But this requirement is too strong because it rules out Thornian
time machines altogether. For a curve of the type in question to
reach Σ_{0} it must intersect H^{+}(Σ_{0}),
but once it reaches H^{+}(Σ_{0}) it can
be continued endlessly into the past without meeting Σ_{0}
because the generators of H^{+}(Σ_{0})
are past endless null geodesics that never meet Σ_{0}.
This difficulty can be overcome by weakening the requirement at
issue by rephrasing it in terms of timelike curves rather than
causal curves. Now the set of candidate time machine spacetimes
satisfying the weakened requirement is nonempty—as illustrated,
once again, by the spacetime of
Figure 1.
But the weakened requirement is too weak, as illustrated by the
(1 + 1)dimensional version of DeutschPolitzer spacetime (see
Figure 2),
which is constructed from twodimensional Minkowski spacetime by
deleting the points p_{1}–p_{4}
and then gluing together the strips as shown. Every past endless
timelike curve that emerges from the time travel region V
of DeutschPolitzer spacetime does meet Σ_{0}. But this
spacetime is not a plausible candidate for a time machine
spacetime. Up to and including the time Σ_{0} (which can
be placed as close to V as desired) this spacetime is
identical with empty Minkowski spacetime. If the state of the
corresponding portion of Minkowski spacetime is not responsible
for the development of CTCs—and it certainly is not since there
are no CTCs in Minkowski spacetime—how can the state on the
portion of DeutschPolitzer spacetime up to and including the
time Σ_{0} be held responsible for the CTCs that appear
in the future?
The deletion of the points p_{1}p_{4}
means that the DeutschPolitzer spacetime is singular in the
sense that it is geodesically incomplete. It would be too
drastic to require of a timemachine hosting spacetime that it
be geodesically complete. And in any case the offending feature
of DeutschPolitzer can be gotten rid of by the following trick.
Multiplying the flat Lorentzian metric η_{ab} of
DeutschPolitzer spacetime by a scalar function f(x,
t) > 0 produces a new metric η′_{ab} :=
f η_{ab} which is conformal to the original metric
and, thus, has exactly the same causal features as the original
metric. But if the conformal factor f is chosen to “blow
up” as the missing points p_{1}–p_{4}
are approached, the resulting spacetime is geodesically
complete—intuitively, the singularities have been pushed off to
infinity.
A more subtle way to exclude DeutschPolitzer spacetime focuses
on the generators of H^{+}(Σ_{0}). The
stipulations laid down so far for Thornian time machines imply
that the generators of H^{+}(Σ_{0})
cannot intersect Σ_{0}. But in addition it can be
required that these generators do not “emerge from a
singularity” and do not “come from infinity,” and this would
suffice to rule out DeutschPolitzer spacetime and its conformal
cousins as legitimate candidates for time machine spacetimes.
More precisely, we can impose what Stephen Hawking (1992a,b)
calls the requirement that H^{+}(Σ_{0})
be compactly generated; namely, the past endless null
geodesics that generate H^{+}(Σ_{0})
must, if extended far enough into past, fall into and remain in
a compact subset of spacetime. Obviously the spacetime of
Figure 1
fulfills Hawking's requirement—since in this case H^{+}(Σ_{0})
is itself compact—but just as obviously the spacetime of
Figure 2
(conformally doctored or not) does not.
Imposing the requirement of a compactly generated future Cauchy
horizon has not only the negative virtue of excluding some
unsuited candidate time machine spacetimes but a positive virtue
as well. It is easily proved that if H^{+}(Σ_{0})
is compactly generated then the condition of strong causality
is violated on H^{+}(Σ_{0}), which means,
intuitively, there are almost closed causal curves near H^{+}(Σ_{0}).
This violation can be taken as an indication that the seeds of
CTCs have been planted on Σ_{0} and that by the time
H^{+}(Σ_{0}) is reached they are ready to
bloom.
However, we still have no guarantee that if CTCs do bloom to the
future of Σ_{0}, then the state on Σ_{0} is
responsible for the blooming. Of course, we have already learned
that we cannot have the iron clad guarantee of causal
determinism that the state on Σ_{0} is responsible for
the actual blooming in all of its particularity. But we might
hope for a guarantee that the state on Σ_{0} is
responsible for the blooming of some CTCs—the actual ones
or others. The difference takes a bit of explaining. The failure
of causal determinism is aptly pictured by the image of a future
“branching” of world histories, with the different branches
representing different alternative possible futures of (the
domain of dependence of) Σ_{0} that are compatible with
the actual past and the laws of physics. And so it is in the
present setting: if H^{+}(Σ_{0}) ≠
∅,
then there will generally be different ways to extend the
spacetime model beyond H^{+}(Σ_{0}), all
compatible with the laws of general relativistic physics. But if
CTCs are present in all of these extensions, even through the
details of the CTCs may vary from one extension to another, then
the state on Σ_{0} can rightly be deemed to be
responsible for the fact that subsequently CTCs did develop.
Not surprisingly, the wouldbe time machine operator cannot hope
to set conditions on Σ_{0} such that every
mathematically possible extension of D^{+}(Σ_{0})
contains CTCs. Reverting to the example of
Figure 1,
a CTCfree extension of D^{+}(Σ_{0}) is
obtained by having the light cones “tip back up” starting on
H^{+}(Σ_{0}) so that although a closed null
curve develops, no CTCs appear in the extension. Here the
wouldbe time machine operator might hope to save the day by
showing that the extension in question is not physically
possible because the laws of physics dictate that once the light
cones start to tip over in the way shown in
Figure 1,
they keep tipping until CTCs form. That such an appeal to the
laws of physics (which in the end may rule out the operation of
a time machine) cannot be avoided is also shown by a second way
of creating CTCfree extensions of D^{+}(Σ_{0})
in
Figure 1;
namely, allow the light cones to continue to tip but delete a
vertical strip of spacetime extending from H^{+}(Σ_{0})
into the future, thus preventing a timelike curve from circling
around the cylinder and closing on itself. This and similar
examples can be bypassed by the additional reasonable
requirement that the relevant extensions to consider are
maximal, i.e., cannot be further extended in the sense of
being isometrically embeddable as a proper subset of a larger
spacetime. But (as the alert reader will have perceived) this
additional requirement can be finessed by means of the stratagem
used above; namely, multiplying the metric by a conformal factor
does not change the causal features of the model, but by
choosing this factor to approach 0 or ∞ as the deleted strip is
approached the maximality of the extension can be secured.
At this juncture the natural move for the wouldbe time machine
operator is to try to show that the maximal CTCfree extensions
created by the conformal doctoring exceed the bounds of physical
possibility. In general relativity theory these bounds are set
using the stressenergy tensor T_{ab},
which specifies the distribution of matterenergy in the
spacetime (M, g_{ab}). The basic
requirement is that together T_{ab} and g_{ab}
satisfy the Einstein gravitational field equations for all of
M. But without further specificity this requirement does
little to limit the possible metrics of spacetime. For given an
arbitrary metric g_{ab} defined on M, the
Einstein tensor G_{ab}[g_{ab}]—a
functional constructed from the metric g_{ab} and
its derivatives—can be computed and the result taken to define a
stressenergy tensor T_{ab}; the resulting pair
T_{ab} and g_{ab} automatically
satisfies the Einstein gravitational field equations, which have
the form G_{ab}[g_{ab}]
∝
T_{ab}.
The requirement that the Einstein equations be satisfied gains
bite in one of two ways: either by imposing energy conditions on
T_{ab}, such as the weak energy condition
that says that the energy density is nonnegative and/or the
dominant energy condition which says that the flow of
energymomentum is not spacelike; or by requiring that T_{ab}
arises from known matter fields and that together T_{ab}
and g_{ab} satisfy the coupled Einsteinmatter
field equations.
Alas, it seems that the wouldbe time machine operator is not
saved by appeal to these or other requirements that can be
stated in terms of local conditions on T_{ab} and
g_{ab}. A theorem by Krasnikov (2002, 2003)
establishes that for every timeoriented spacetime (M, g_{ab}),
there exists a maximal extension that does not contain any CTCs,
except, perhaps, in the chronological past of the image of M
in the extension. Furthermore, Krasnikov shows how these
extensions can be constructed such that local conditions, such
as satisfying the Einstein field equations or energy conditions,
can be carried over from the initial spacetime to the maximal
CTCfree extension. Thus, if a time machine were defined as a
device which leads to CTCs in all possible extensions of
H^{+}(Σ_{0}) satisfying the conditions
laid down thus far, it appears as if Krasnikov's theorem
effectively prohibits time machines.
The wouldbe time machine operator need not capitulate in the
face of Krasnikov's theorem. Recall that the main difficulty in
specifying the conditions for the successful operation of
Thornian time machines traces to the fact that the standard form
of causal determinism does not apply to the production of CTCs.
But causal determinism can fail for reasons that have nothing to
do with CTCs or other acausal features of relativistic
spacetimes, and it seems only fair to assure that these modes of
failure have been removed before proceeding to discuss the
prospects for time machines. To zero in on the modes of failure
at issue, consider vacuum solutions (T_{ab} ≡ 0)
to Einstein's field equations. Let (M, g_{ab})
and (M′, g′_{ab}) be two such
solutions, and let Σ
⊂
M
and Σ′
⊂
M′
be spacelike hypersurfaces of their respective spacetimes.
Suppose that there is an isometry Ψ from some neighborhood N(Σ)
of Σ onto a neighborhood N′(Σ′) of Σ′. Does it follow, as
we would want determinism to guarantee, that Ψ is extendible to
an isometry from D^{+}(Σ) onto D^{+}(Σ′)?
To see why the answer is negative, start with any solution (M, g_{ab})
of the vacuum Einstein equations, and cut out a closed set of
points lying to the future of N(Σ) and in D^{+}(Σ).
Denote the surgically altered manifold by M* and the
restriction of g_{ab} to M* by g*_{ab}.
Then (M*, g*_{ab}) is also a
solution of the vacuum Einstein equations. But obviously the
pair of solutions (M, g_{ab}) and (M*, g*_{ab})
violates the condition that determinism was supposed to
guarantee as Ψ is not extendible to an isometry from D^{+}(Σ)
onto D^{+}(Σ*). It might seem that the
requirement, contemplated above, that the spacetimes under
consideration be maximal, already rules out spacetimes that have
“holes” in them. But while maximality does rule out the
surgically mutilated spacetime just constructed, it does not
guarantee hole freeness in the sense needed to make sure that
determinism does not stumble before it gets to the starting
gate. That (M, g_{ab}) is hole free in the
relevant sense requires that if Σ
⊂
M
is a spacelike hypersurface, there does not exist a spacetime (M′, g′_{ab})
and an isometric embedding Φ of D^{+}(Σ) into
M′ such that Φ(D^{+}(Σ)) is a proper subset
of D^{+}(Φ(Σ)). A theorem due to Robert Geroch
(1977, 87), who is responsible for this definition, asserts that
if Σ
⊂
M
and Σ′
⊂
M′
are spacelike hypersurfaces in holefree spacetimes (M, g_{ab})
and (M′, g′_{ab}), respectively,
and if there exists an isometry Ψ: M → M′, then Ψ
is indeed extendible to an isometry between D^{+}(Σ)
and D^{+}(Σ′). Thus, hole freeness precludes an
important mode of failure of determinism which we wish to
exclude in our discussion of time machines. It can be shown that
hole freeness entails, but is not entailed by, maximality. And
it is just this gap that gives the wouldbe time machine
operator some hope, for it is not obvious that Krasnikov's
construction, which produces maximal CTCfree extensions, also
produces holefree extensions.
Thus, we propose that one clear sense of what it would mean for
a Thornian time machine to operate in the setting of general
relativity theory is given by the following assertion: the laws
of general relativistic physics allow solutions containing a
partial Cauchy surface Σ_{0} such that no CTCs lie to
the past of Σ_{0} but every extension of D^{+}(Σ_{0})
as a hole free solution of the laws contains CTCs.
Correspondingly, a proof of the physical impossibility of time
machines would take the form of showing that this assertion is
false for the actual laws of physics, consisting, presumably, of
Einstein's field equations plus energy conditions and, perhaps,
some additional restrictions as well. And a proof of the
emptiness of the associated concept of a Thornian time machine
would take the form of showing that the assertion is false
independently of the details of the laws of physics, as long as
they take the form of local conditions on T_{ab}
and g_{ab}.
If such an emptiness proof should be forthcoming, the fan of
time machines can retreat to a weaker concept of Thornian time
machine by taking a page from probabilistic accounts of
causation, the idea being that a time machine can be seen to be
at work if its operation increases the probability of the
appearance of CTCs. Since general relativity theory itself is
innocent of probabilities, they have to be introduced by hand,
either by inserting them into the models of the theory, i.e., by
modifying the theory at the level of the objectlanguage, or by
defining measures on sets of models, i.e., by modifying the
theory at the level of the metalanguage. Since the former would
change the character of the theory, only the latter will be
considered. The project for making sense of the notion that a
time machine as a probabilistic cause of the appearance of CTCs
would then take the following form. First define a normalized
measure on the set of models having a partial Cauchy surface to
the past of which there are no CTCs. Then show that the subset
of models that have CTCs to the future of the partial Cauchy
surface has nonzero measure. Next, identify a range of
conditions on or near the partial Cauchy surface that are
naturally construed as settings of a device that is a wouldbe
probabilistic cause of CTCs, and show that the subset of models
satisfying these conditions has nonzero measure. Finally, show
that conditionalizing on the latter subset increases the measure
of the former subset. Assuming that this formal exercise can be
successfully carried out, there remains the task of justifying
the measure constructed as a measure of objective chance. This
task is especially daunting in the cosmological setting since
neither of the leading interpretations of objective chance seems
applicable. The frequency interpretation is strained since the
development of CTCs may be a nonrepeated phenomenon; and the
propensity interpretation is equally strained since, barring
justso stories about the Creator throwing darts at the Cosmic
Dart Board, there is no chance mechanism for producing
cosmological models.
We conclude that, even apart from general doubts about a
probabilistic account of causation, the resort to a
probabilistic conception of time machines is a desperate
stretch, at least in the context of classical general relativity
theory. In a quantum theory of gravity, a probabilistic
conception of time machines may be appropriate if the theory
itself provides the transition probabilities between the
relevant states. But an evaluation of this prospect must wait
until the theory of quantum gravity is available.
4. Nogo results for (Thornian) time machines in classical
general relativity theory
In order to appreciate the physics literature aimed at proving
nogo results for time machines it is helpful to view these
efforts as part of the broader project of proving chronology
protections theorems, which in turn is part of a still
larger project of proving cosmic censorship theorems. To
explain, we start with cosmic censorship and work backwards.
For sake of simplicity, concentrate on the initial value problem
for vacuum solutions (T_{ab} ≡ 0) to Einstein's
field equations. Start with a threemanifold Σ equipped with
quantities which, when Σ is embedded as a spacelike submanifold
of spacetime, become initial data for the vacuum field
equations. Corresponding to the initial data there exists a
unique maximal development (M, g_{ab}) for
which (the image of the embedded) Σ is a Cauchy surface. This
solution, however, may not be maximal simpliciter, i.e., it may
be possible to isometrically embed it as a proper part of a
larger spacetime, which itself may be a vacuum solution to the
field equations; if so Σ will not be a Cauchy surface for the
extended spacetime, which fails to be a globally hyperbolic
spacetime. This situation can arise because of a poor choice of
initial value hypersurface, as illustrated in
Figure 3
by taking Σ to be the indicated spacelike hyperboloid of (1 +
1)dimensional Minkowski spacetime. But, more interestingly, the
situation can arise because the Einstein equations allow various
pathologies, collectively referred to as “naked singularities,”
to develop from regular initial data. The strong form of
Penrose's celebrated cosmic censorship conjecture
proposes that, consistent with Einstein's field equations, such
pathologies do not arise under physically reasonable conditions
or else that the conditions leading to the pathologies are
highly nongeneric within the space of all solutions to the
field equations. A small amount of progress has been made on
stating and proving precise versions of this conjecture.
One way in which strong cosmic censorship can be violated is
through the emergence of acausal features. Returning to the
example of Misner spacetime (Figure
1),
the spacetime up to H^{+}(Σ_{0}) is the
unique maximal development of the vacuum Einstein equations for
which Σ_{0} is a Cauchy surface. But this development is
extendible, and in the extension illustrated in
Figure 1
global hyperbolicity of the development is lost because of the
presence of CTCs. The chronology protection conjecture
then can be construed as a subconjecture of the cosmic
censorship conjecture, saying, roughly, that consistent with
Einstein field equations, CTCs do not arise under physically
reasonable conditions or else that the conditions are highly
nongeneric within the space of all solutions to the field
equations. Nogo results for time machines are then special
forms of chronology protection theorems that deal with cases
where the CTCs are manufactured by time machines. In the other
direction, a very general chronology protection theorem will
automatically provide a nogo result for time machines, however
that notion is understood, and a theorem establishing strong
cosmic censorship will automatically impose chronology
protection.
The most widely discussed chronology protection theorem/nogo
result for time machines in the context of classical general
relativity theory is due to Hawking (1992a). Before stating the
result, note first that, independently of the Einstein field
equations and energy conditions, a partial Cauchy surface Σ must
be compact if its future Cauchy horizon H^{+}(Σ)
is compact (see Hawking 1992a and Chrusciel and Isenberg 1993).
However, it is geometrically allowed that Σ is noncompact if
H^{+}(Σ) is required only to be compactly generated
rather than compact. But what Hawking showed is that this
geometrical possibility is ruled out by imposing Einstein's
field equations and the weak energy condition. Thus, if Σ_{0}
is a partial Cauchy surface representing the situation just
before or just as the wouldbe Thornian time machine is switched
on, and if a necessary condition for seeing a Thornian time
machine at work is that H^{+}(Σ_{0}) is
compactly generated, then consistently with Einstein's field
equations and the weak energy condition, a Thornian time machine
cannot operate in a spatially open universe since Σ_{0}
must be compact.
This nogo result does not touch the situation illustrated in
Figure 1.
TaubNUT spacetime is a vacuum solution to Einstein's field
equations so the weak energy condition is automatically
satisfied, and H^{+}(Σ_{0}) is compact
and, a fortiori, compactly generated. Hawking's theorem is not
contradicted since Σ_{0} is compact. By the same token
the theorem does not speak to the possibility of operating a
Thornian time machine in a spatially closed universe. To help
fill the gap, Hawking proved that when Σ_{0} is compact
and H^{+}(Σ_{0}) is compactly generated,
the Einstein field equations and the weak energy condition
together guarantee that both the convergence and shear of the
null geodesic generators of H^{+}(Σ_{0})
must vanish, which he interpreted to imply that no observers can
cross over H^{+}(Σ_{0}) to enter the
chronology violating region V. But rather than showing
that it is physically impossible to operate a Thornian time
machine in a closed universe, this result shows only that, given
the correctness of Hawking's interpretation, the observers who
operate the time machine cannot take advantage of the CTCs it
produces.
There are two sources of doubt about the effectiveness of
Hawking's nogo result even for open universes. The first stems
from possible violations of the weak energy condition by
stressenergy tensors arising from classical relativistic matter
fields (see Vollick 1997 and Visser and Barcelo 2000). The
second stems from the fact that Hawking's theorem functions as a
chronology protection theorem only by way of serving as a
potential nogo result for Thornian time machines since the
crucial condition that H^{+}(Σ_{0}) is
compactly generated is supposedly justified by being a necessary
condition for the operation of such machine. But in retrospect,
the motivation for this condition seems frayed. As argued in the
previous section, if the Einstein field equations and energy
conditions entail that all hole free extensions of D^{+}(Σ_{0})
contain CTCs, then it is plausible to see a Thornian time
machine at work, quite regardless of whether or not H^{+}(Σ_{0})
is compactly generated or not. Of course, it remains to
establish the existence of cases where this entailment holds. If
it should turn out that there are no such cases, then the
prospects of Thornian time machines are dealt a severe blow, but
the reasons are independent of Hawking's theorem. On the other
hand, if such cases do exist then our conjecture would be that
they exist even when some of the generators of H^{+}(Σ_{0})
come from singularities or infinity and, thus, H^{+}(Σ_{0})
is not compactly generated.
5. Nogo results in quantum gravity
Three degrees of quantum involvement in gravity can be
distinguished. The first degree, referred to as quantum field
theory on curved spacetimes, simply takes off the shelf a
spacetime provided by general relativity theory and then
proceeds to study the behavior of quantum fields on this
background spacetime. The Unruh effect, which predicts the
thermalization of a free scalar quantum field near the horizon
of a black hole, lies within this ambit. The second degree of
involvement, referred to as semiclassical quantum gravity,
attempts to calculate the backreaction of the quantum fields on
spacetime metric by computing the expectation value <ΨT_{ab}Ψ>
of the stressenergy tensor in some appropriate quantum state
Ψ> and then inserting the value into Einstein's field equations
in place of T_{ab} . Hawking's celebrated
prediction of black hole radiation belongs to this ambit. The
third degree of involvement attempts to produce a genuine
quantum theory of gravity in the sense that the gravitational
degrees of freedom are quantized. Currently loop quantum gravity
and string theory are the main research programs aimed at this
goal.
The first degree of quantum involvement, if not opening the door
to Thornian time machines, at least seemed to remove some
obstacles since quantum fields are known to lead to violations
of the energy conditions used in the setting of classical
general relativity theory to prove chronology protection
theorems and nogo results for time machines. However, the
second degree of quantum involvement seemed, at least initially,
to slam the door shut. The intuitive idea was this. Start with a
general relativistic spacetime where CTCs develop to the future
of H^{+}(Σ) (often referred to as the “chronology
horizon”) for some suitable partial Cauchy surface Σ. Find that
the propagation of a quantum field on this spacetime background
is such that <ΨT_{ab}Ψ> “blows up” as H^{+}(Σ)
is approached from the past. Conclude that the backreaction on
the spacetime metric creates unbounded curvature, which
effectively cuts off the future development that would otherwise
eventuate in CTCs. These intuitions were partly vindicated by
detailed calculations in several models. But eventually a number
of exceptions were found in which the backreaction remains
arbitrarily small near H^{+}(Σ). This seemed to
leave the door ajar for Thornian time machines.
But fortunes were reversed once again by a result of Kay,
Radzikowski, and Wald (1997). The details of their theorem are
too technical to review here, but the structure of the argument
is easy to grasp. The naïve calculation of <ΨT_{ab}Ψ>
results in infinities which have to be subtracted off to produce
a renormalized expectation value <ΨT_{ab}Ψ>_{R}
with a finite value. The standard renormalization procedure uses
a limiting procedure that is mathematically welldefined if, and
only if, a certain condition obtains. The KRW theorem shows that
this condition is violated for points on H^{+}(Σ)
and, thus, that the expectation value of the stressenergy
tensor is not welldefined at the chronology horizon.
While the KRW theorem is undoubtedly of fundamental importance
for semiclassical quantum gravity, it does not serve as an
effective nogo result for Thornian time machines. In the first
place, the theorem assumes, in concert with Hawking's chronology
protection theorem, that H^{+}(Σ) is compactly
generated, and we repeat that it is far from clear that this
assumption is necessary for seeing a Thornian time machine in
operation. A second, and more fundamental, reservation applies
even if a compactly generated H^{+}(Σ) is
accepted as a necessary condition for time machines. The KRW
theorem functions as a nogo result by providing a reductio
ad absurdum with a dubious absurdity: roughly, if you try to
operate a Thornian time machine, you will end up invalidating
semiclassical quantum gravity. But semiclassical quantum
gravity was never viewed as anything more than a stepping stone
to a genuine quantum theory of gravity, and its breakdown is
expected to be manifested when Planckscale physics comes into
play. This worry is underscored by Visser's (1997, 2003)
findings that in chronology violating models transPlanckian
physics can be expected to come into play before H^{+}(Σ)
is reached.
It thus seems that if some quantum mechanism is to serve as the
basis for chronology protection, it must be found in the third
degree of quantum involvement in gravity. Both loop quantum
gravity and string theory have demonstrated the ability to cure
some of the curvature singularities of classical general
relativity theory. But as far as we are aware there are no
demonstrations that either of these approaches to quantum
gravity can get rid of the acausal features exhibited in various
solutions to Einstein's field equations. An alternative approach
to formulate a fullyfledged quantum theory of gravity attempts
to capture the Planckscale structure of spacetime by
constructing it from causal sets. Since these sets must be
acyclic, i.e., no element in a causal set can causally precede
itself, CTCs are ruled out a priori. Actually, a theorem due to
Malament (1977) suggests that any Planckscale approach encoding
only the causal structure of a spacetime cannot permit CTCs
either in the smooth classical spacetimes or a corresponding
phenomenon in their quantum counterparts.
In sum, the existing nogo results that use the first two
degrees of quantum involvement are not very convincing, and the
third degree of involvement is not mature enough to allow useful
pronouncements.
6. Conclusion
Hawking opined that “[i]t seems there is a chronology protection
agency, which prevents the appearance of closed timelike curves
and so makes the universe safe for historians” (1992a, 603). He
may be right, but to date there are no convincing arguments that
such an Agency is housed in either classical general relativity
theory or in semiclassical quantum gravity. And it is too early
to tell whether this Agency is housed in loop quantum gravity or
string theory. But even if it should turn out that Hawking is
wrong in that the laws of physics do not support a Chronology
Protection Agency, it could still be the case that the laws
support an AntiTime Machine Agency. For it could turn out that
while the laws do not prevent the development of CTCs, they also
do not make it possible to attribute the appearance of CTCs to
the workings of any wouldbe time machine. We argued that a
strong presumption in favor of the latter would be created in
classical general relativity theory by the demonstration that
for any model satisfying Einstein's field equations and energy
conditions as well as possessing a partial Cauchy surface Σ_{0}
to the future of which there are CTCs, there are hole free
extensions of D^{+}(Σ_{0}) satisfying
Einstein's field equations and energy conditions but containing
no CTCs to the future of Σ_{0}. There are no doubt
alternative approaches to understanding what it means for a
device to be “responsible for” the development of CTCs.
Exploring these alternatives is one place that philosophers can
hope to make a contribution to an ongoing discussion that, to
date, has been carried mainly by the physics community.
Participating in this discussion means that philosophers have to
forsake the activity of logical gymnastics with the paradoxes of
time travel for the more arduous but (we believe) rewarding
activity of digging into the foundations of physics.
Time machines may never see daylight, and perhaps so for
principled reasons that stem from basic physical laws. But even
if mathematical theorems in the various theories concerned
succeed in establishing the impossibility of time machines,
understanding why time machines cannot be constructed will shed
light on central problems in the foundations of physics. As we
have argued in Section 4, for instance, the hunt for time
machines in general relativity theory should be interpreted as a
core issue in studying the fortunes of Penrose's cosmic
censorship conjecture. This conjecture arguably constitutes the
most important open problem in general relativity theory.
Similarly, as discussed in Section 5, mathematical theorems
related to various aspects of time machines offer results
relevant for the search of a quantum theory of gravity. In sum,
studying the possibilities for operating a time machine turns
out to be not a scientifically peripheral or frivolous weekend
activity but a useful way of probing the foundations of
classical and quantum theories of gravity.


