## 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 
semi-popular 
 presentations.[1]
 The issues raised by this topic are largely oblique, if not 
orthogonal, to those treated in the philosophical literature on time 
 travel.[2]
 Most significantly, the so-called 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, future-directed timelike curves with 
 self-intersections.[3]
 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 "no-go" results for time 
machines because these results concern what happens before the 
emergence of 
 CTCs.[4]
 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;[5]
 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 matter-energy 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 would-be 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 ill-defined. This fact does not prevent progress 
from being made on the topic; for if one's aim is to establish no-go 
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 no-go 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*
 \((\mathcal{M},g\_{ab})\) where
 \(\mathcal{M}\) is a differentiable manifold and
\(g\_{ab}\) is a Lorentz signature metric defined on all
of \(\mathcal{M}\). 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 not 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 Penrose-Hawking-Geroch 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.[6]
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.[7]
 Non-temporal 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.[8]
 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 Callender 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 future-pointing 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, 
Godel 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* \(\Sigma\) (i.e., a spacelike hypersurface without 
 edges);[9]
 that \(\Sigma\) is two-sided and partitions 
 \(\mathcal{M}\) into three parts--\(\Sigma\) itself,
the part of 
 \(\mathcal{M}\) on the past side of \(\Sigma\) and the 
part of 
 \(\mathcal{M}\) on the future side of 
\(\Sigma\)--and that there are no CTCs that lie on the past side of
\(\Sigma\). The first two clauses of this requirement together entail 
that the time slice \(\Sigma\) is a *partial Cauchy surface*, 
i.e., \(\Sigma\) is a time slice that is not intersected more than once 
by any future-directed timelike 
 curve.[10]





Now suppose that the state on a partial Cauchy surface 
\(\Sigma\_0\) 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* 
\(V \subseteq \mathcal{M}\), the region of spacetime traced out by

 CTCs,[11]
 is non-null and lies to the future of \(\Sigma\_0\). The fact 
that \(V \ne \varnothing\) does not lead to any consistency 
constraints on initial data on \(\Sigma\_0\) since, by 
hypothesis, \(\Sigma\_0\) is not intersected more than once by 
any timelike curve, and thus, insofar as the so-called paradoxes of 
time travel are concerned with such constraints, the paradoxes do not
arise with respect to \(\Sigma\_0\). But by the same token, the
option of traveling back into the past of \(\Sigma\_0\) is 
ruled out by the set up as it has been sketched so far, since 
otherwise \(\Sigma\_0\) 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.



![Figure 1: Misner spacetime](figure1.jpg)
Figure 1.
Misner spacetime





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 Taub-NUT 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 
\(\Sigma\_0\) lies the Taub region where the causal structure 
of spacetime is as bland as can be desired. But to the future of 
\(\Sigma\_0\) 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 
\(\Sigma\_0\) 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 \(\Sigma\_0\) is responsible for the CTCs. Or 
one could be more liberal and allow the would-be 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 
mass-energy in some specified ways. For example, based on Godel 
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 \(\Sigma\_0\) 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 \(\Sigma\_0\) can be 
deemed to be responsible for the subsequent appearance of CTCs.


## 3. When can a would-be 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 
\(\Sigma\_0\) they are not causally determined by the state on 
\(\Sigma\_0\) since the time travel region \(V\), if it is
non-null, lies outside the *future domain of dependence* 
\(D^+ (\Sigma\_0)\) of \(\Sigma\_0\), 
the portion of spacetime where the field equations of relativistic 
physics uniquely determine the state of things from the state on 
 \(\Sigma\_0\).[12]
 The point is illustrated by the toy model of 
 Figure 1.
 The surface labeled \(H^+ (\Sigma\_0)\) is 
called the *future Cauchy horizon* of \(\Sigma\_0\). It 
is the future boundary of 
 \(D^+ (\Sigma\_0)\),[13]
 and it separates the portion of spacetime where conditions are 
causally determined by the state on \(\Sigma\_0\) from the 
portion where conditions are not so determined. And, as advertised, 
the CTCs in the model of 
 Figure 1
 lie beyond \(H^+ (\Sigma\_0)\).



![Figure 2: Deutsch-Politzer spacetime](figure2.jpg)
Figure 2.
Deutsch-Politzer spacetime





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 \(\Sigma\_0\) 
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 
\(D^+ (\Sigma\_0)\). 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 \(\Sigma\_0\). But this requirement 
is too strong because it rules out Thornian time machines altogether.
For a curve of the type in question to reach \(\Sigma\_0\) it 
must intersect \(H^+ (\Sigma\_0)\), but once 
it reaches \(H^+ (\Sigma\_0)\) it can be 
continued endlessly into the past without meeting \(\Sigma\_0\)
because the generators of \(H^+ (\Sigma\_0)\)
are past endless null geodesics that never meet 
 \(\Sigma\_0\).[14]
 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 non-empty--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 Deutsch-Politzer 
 spacetime[15]
 (see 
 Figure 2),
 which is constructed from two-dimensional 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 Deutsch-Politzer 
spacetime does meet \(\Sigma\_0\). But this spacetime is not a 
plausible candidate for a time machine spacetime. Up to and including
the time \(\Sigma\_0\) (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 
Deutsch-Politzer spacetime up to and including the time 
\(\Sigma\_0\) be held responsible for the CTCs that appear in 
the future?




The deletion of the points 
\(p\_1\)-\(p\_4\) means that the 
Deutsch-Politzer spacetime is singular in the sense that it is 
*geodesically 
 incomplete*.[16]
 It would be too drastic to require of a time-machine hosting 
spacetime that it be geodesically complete. And in any case the 
offending feature of Deutsch-Politzer can be gotten rid of by the 
following trick. Multiplying the flat Lorentzian metric 
\(\eta\_{ab}\) of Deutsch-Politzer spacetime by a scalar
function \(j(x, t) \gt\) produces a new 
metric \(\eta '\_{ab} :=\) *j 
\(\eta\_{ab}\)* which is conformal to the original metric 
and, thus, has exactly the same causal features as the original 
metric. But if the conformal factor \(j\) 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 Deutsch-Politzer spacetime focuses on 
the generators of \(H^+ (\Sigma\_0)\). The 
stipulations laid down so far for Thornian time machines imply that 
the generators of \(H^+ (\Sigma\_0)\) cannot 
intersect \(\Sigma\_0\). 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 Deutsch-Politzer 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 
\(H^+ (\Sigma\_0)\) be *compactly 
generated*; namely, the past endless null geodesics that generate
\(H^+ (\Sigma\_0)\) 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 
\(H^+ (\Sigma\_0)\) 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 \(H^+ (\Sigma\_0)\) 
is compactly generated then the condition of *strong 
causality* is violated on 
\(H^+ (\Sigma\_0)\), which means, 
intuitively, there are almost closed causal curves near 
 \(H^+ (\Sigma\_0)\).[17]
 This violation can be taken as an indication that the seeds of CTCs 
have been planted on \(\Sigma\_0\) and that by the time 
\(H^+ (\Sigma\_0)\) is reached they are ready
to bloom.




However, we still have no guarantee that if CTCs do bloom to the 
future of \(\Sigma\_0\), then the state on \(\Sigma\_0\) 
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 \(\Sigma\_0\) is responsible for the actual 
blooming in all of its particularity. But we might hope for a 
guarantee that the state on \(\Sigma\_0\) 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) \(\Sigma\_0\) that are compatible with
the actual past and the laws of physics. And so it is in the present 
setting: if \(H^+ (\Sigma\_0) \ne \varnothing\), then there will 
generally be different ways to extend 
\(D^+ (\Sigma\_0)\), 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 \(\Sigma\_0\) 
can rightly be deemed to be responsible for the fact that 
subsequently CTCs did develop.




A theorem due to Krasnikov (2002, 2003 [Other Internet Resources],
2014a) might seem to demonstrate that no relativistic spacetime can
count as embodying a Thornian time machine so understood. Following
Krasnikov, let us say that a spacetime condition \(C\)
is *local* just in case, for any open covering
\(\{V\_{\alpha}\}\) of an arbitrary spacetime \((\mathcal{M}, g\_{ab}),
C\) holds in \((\mathcal{M}, g\_{ab})\) iff it holds in \((V\_{\alpha},
g\_{ab}|\_{V\_{\alpha}})\) for all \(\alpha\). Examples of local
conditions one might want to impose on physically reasonable
spacetimes are Einstein's gravitational field equations and
so-called energy conditions that restrict the form of the
stress-energy tensor \(T\_{ab}\). An example of the latter that will
come into play below is the *weak energy condition* that says
that the energy density is non-negative.[18] Einstein's field equations (sans
cosmological constant) require that \(T\_{ab}\) is proportional to the
Einstein tensor which is a functional of the metric and its
derivatives. Call a \(C\)-spacetime \((\mathcal{M}', g'\_{ab})\) a
\(C\)-*extension* of a \(C\)-spacetime \((\mathcal{M},
g\_{ab})\) spacetime if the latter is isometric to an open proper
subset of the former; and call \((\mathcal{M}, g\_{ab})
C\)-*extensible* if it admits a \(C\)-extension and
\(C\)-*maximal* otherwise. (Of course, \(C\) might be the empty
condition.) Krasnikov's theorem shows that every \(C\)-spacetime
\((\mathcal{M}, g\_{ab})\) admits a \(C\)-maximal extension
\((\mathcal{M}^{max}, g^{max}\_{ab})\) such that all CTCs in
\((\mathcal{M}^{max}, g^{max}\_{ab})\) are to the chronological past of
the image of \(\mathcal{M}\) in \((\mathcal{M}^{max},
g^{max}\_{ab})\). So start with some candidate spacetime
\((\mathcal{M}, g\_{ab})\) for a Thornian time machine, and apply the
theorem to \((D^+ (\Sigma\_0), g\_{ab}|\_{D^+ (\Sigma\_0)})\). Conclude
that no matter what local conditions the candidate spacetime is
required to satisfy, \(D^+ (\Sigma\_0)\) has extensions that also
satisfies said local conditions but does not contain CTCs to the
future of \(\Sigma\_0\). Thus, the candidate spacetime fails to exhibit
the crucial feature identified above necessary for underwriting the
contention that the conditions on \(\Sigma\_0\) are responsible for the
development of CTCs. Hence, it appears as if Krasnikov's
theorem effectively prohibits time machines. 




The would-be 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 ensure 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 \((T\_{ab}
\equiv 0)\) to Einstein's field equations. Let \((\mathcal{M},
g\_{ab})\) and \((\mathcal{M}', g'\_{ab})\) be two such solutions, and
let \(\Sigma \subset \mathcal{M}\) and \(\Sigma ' \subset
\mathcal{M}'\) be spacelike hypersurfaces of their respective
spacetimes. Suppose that there is an isometry \(\Psi\) from some
neighborhood \(N(\Sigma)\) of \(\Sigma\) onto a neighborhood
\(N'(\Sigma ')\) of \(\Sigma '\). Does it follow, as we would want
determinism to guarantee, that \(\Psi\) is extendible to an isometry
from \(D^+ (\Sigma)\) onto \(D^+ (\Sigma ')\)? To see why the answer
is negative, start with any solution \((\mathcal{M}, g\_{ab})\) of the
vacuum Einstein equations, and cut out a closed set of points lying to
the future of \(N(\Sigma)\) and in \(D^+ (\Sigma)\). Denote the
surgically altered manifold by \(\mathcal{M}^\*\) and the restriction
of \(g\_{ab}\) to \(\mathcal{M}^\*\) by \(g^\*\_{ab}\). Then
\((\mathcal{M}^\*, g^\*\_{ab})\) is also a solution of the vacuum
Einstein equations. But obviously the pair of solutions
\((\mathcal{M}, g\_{ab})\) and \((\mathcal{M}^\*, g^\*\_{ab})\) violates
the condition that determinism was supposed to guarantee as \(\Psi\)
is not extendible to an isometry from \(D^+ (\Sigma)\) onto \(D^+
(\Sigma^\*)\). 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 \((\mathcal{M}, g\_{ab})\) is hole free in the relevant sense
requires that if \(\Sigma \subset \mathcal{M}\) is a spacelike
hypersurface, there does not exist a spacetime \((\mathcal{M}',
g'\_{ab})\) and an isometric embedding \(\Phi\) of \(D^+ (\Sigma)\)
into \(\mathcal{M}'\) such that \(\Phi(D^+ (\Sigma))\) is a proper
subset of \(D^+ (\Phi(\Sigma))\). A theorem due to Robert Geroch
(1977, 87), who is responsible for this definition, asserts that if
\(\Sigma \subset \mathcal{M}\) and \(\Sigma ' \subset \mathcal{M}'\)
are spacelike hypersurfaces in hole-free spacetimes \((\mathcal{M},
g\_{ab})\) and \((\mathcal{M}', g'\_{ab})\), respectively, and if there
exists an isometry \(\Psi : \mathcal{M} \rightarrow \mathcal{M}'\),
then \(\Psi\) is indeed extendible to an isometry between \(D^+
(\Sigma)\) and \(D^+ (\Sigma ')\). 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 is
not entailed by maximality.[19] And it is just this gap that gives the
would-be time machine operator some hope, for the maximal CTC-free
extensions produced by Krasnikov's construction are not always
hole free (Manchak 2009b). But Krasnikov (2009) has shown that the
Geroch (1977) definition is too strong: Minkowski spacetime fails to
satisfy it! For this reason, alternative formulations of the
hole-freeness definition have been constructed which are more
appropriate (Manchak 2009a, Minguzzi 2012).





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 \(\Sigma\_0\) such that no CTCs lie to the past of
\(\Sigma\_0\) but every extension of
\(D^+ (\Sigma\_0)\) satisfying \_\_\_\_\_\_\_\_
contains CTCs (where the blank is filled with some "no
hole" condition). 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 \(g\_{ab}\).




Are there "no hole" conditions which show the proposed concept of a time machine is not empty? Let \(J^+(p)\) designate the *causal future* of \(p\), defined as the set of all points in \(\mathcal{M}\) which can be reached from \(p\) by a future-directed causal curve in \(\mathcal{M}\). The *causal past* \(J^-(p)\) is defined analogously. Now, we say a spacetime \((\mathcal{M},g\_{ab})\) is *J closed* if, for each \(p\) in \(\mathcal{M}\), the sets \(J^+(p)\) and \(J^-(p)\) are topologically closed. One can verify that J closedness fails in many artificially mutilated examples (e.g. Minkowski spacetime with one point removed from the manifold). For some time, it was thought that a time machine existed under this no-hole condition (Manchak 2011a). But this turns out to be incorrect; indeed a recent result shows that any J closed spacetime \((\mathcal{M},g\_{ab})\) of three dimensions or more with chronology violating region \(V \neq \mathcal{M}\) must be strongly causal and therefore fail to have CTCs (Hounnonkpe and Minguzzi 2019). Stepping back, perhaps there are other no-hole conditions which can be used instead to show that the proposed concept of a time machine is not empty. But even if such a project were successful, Manchak (2014a, 2019) has shown that the time machine existence results can be
naturally reinterpreted as "hole machine" existence results if one is so inclined. Instead of assuming that spacetime is
free of holes and then showing that certain initial conditions are
responsible for the production of CTCs, one could just as well start
with the assumption of no CTCs and then show that certain initial
conditions are responsible for the production of holes. Given the
importance of these no hole assumptions to the time machine advocate,
much recent work has focused on whether such assumptions are
physically reasonable in some sense (Manchak 2011b,
2014b). This is still an open question.




Another open question is whether physically more realistic spacetimes
than Misner also permit the operation of time machines and how generic
time-machine spacetimes are in particular spacetime theories, such as
general relativity. If time-machine spacetimes turn out to be highly
non-generic, 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 object-language, or by defining measures on sets of
models, i.e., by modifying the theory at the level of the
meta-language. 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 non-zero 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 would-be probabilistic
cause of CTCs, and show that the subset of models satisfying these
conditions has non-zero 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 these as measures 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 non-repeated phenomenon; and
the propensity interpretation is equally strained since, barring
just-so 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. No-go results for (Thornian) time machines in classical general relativity theory




In order to appreciate the physics literature aimed at proving no-go 
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.



![Figure 3: A bad choice of initial value surface](figure3.jpg)
Figure 3.
A bad choice of initial value surface





For sake of simplicity, concentrate on the initial value problem for 
vacuum solutions \((T\_{ab} \equiv 0)\) to Einstein's 
field equations. Start with a three-manifold \(\Sigma\) equipped with 
quantities which, when \(\Sigma\) 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[20]
 maximal development 
 \((\mathcal{M}, g\_{ab})\) for which (the image 
of the embedded) \(\Sigma\) is a Cauchy 
 surface.[21]
 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 \(\Sigma\) will not be a Cauchy surface for the 
extended spacetime, which fails to be a globally hyperbolic 
 spacetime.[22]
 This situation can arise because of a poor choice of initial value 
hypersurface, as illustrated in 
 Figure 3
 by taking \(\Sigma\) 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 non-generic 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.[23]





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 \(H^+ (\Sigma\_0)\) is 
the unique maximal development of the vacuum Einstein equations for 
which \(\Sigma\_0\) 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 non-generic within the space 
of all solutions to the field equations. No-go 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 no-go 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/no-go 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 \(\Sigma\) must be compact if its 
future Cauchy horizon \(H^+ (\Sigma)\) is compact (see
Hawking 1992a and Chrusciel and Isenberg 1993). However, it is 
geometrically allowed that \(\Sigma\) is non-compact if 
\(H^+ (\Sigma)\) 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 \(\Sigma\_0\)
is a partial Cauchy surface representing the situation just before or
just as the would-be Thornian time machine is switched on, and if a 
necessary condition for seeing a Thornian time machine at work is 
that \(H^+ (\Sigma\_0)\) 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 \(\Sigma\_0\) must be compact.




This no-go result does not touch the situation illustrated in 
 Figure 1.
 Taub-NUT spacetime is a vacuum solution to Einstein's field equations
so the weak energy condition is automatically satisfied, and 
\(H^+ (\Sigma\_0)\) is compact and, a 
fortiori, compactly generated. Hawking's theorem is not contradicted 
since \(\Sigma\_0\) 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 \(\Sigma\_0\) is compact and 
\(H^+ (\Sigma\_0)\) 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 \(H^+ (\Sigma\_0)\) must 
vanish, which he interpreted to imply that no observers can cross 
over \(H^+ (\Sigma\_0)\) 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 
no-go result even for open universes. The first stems from possible 
violations of the weak energy condition by stress-energy tensors 
arising from classical relativistic matter fields (see Vollick 1997 
and Visser and Barcelo 
 2000).[24]
 The second stems from the fact that Hawking's theorem functions as a 
chronology protection theorem only by way of serving as a potential 
no-go result for Thornian time machines since the crucial condition 
that \(H^+ (\Sigma\_0)\) 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 \(D^+ (\Sigma\_0)\) 
contain CTCs, then it is plausible to see a Thornian time machine at 
work, quite regardless ofwhether or not 
\(H^+ (\Sigma\_0)\) 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 
\(H^+ (\Sigma\_0)\) come from singularities 
or infinity and, thus, \(H^+ (\Sigma\_0)\) is
not compactly 
 generated.[25]



## 5. No-go 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 semi-classical quantum gravity, attempts 
to calculate the backreaction of the quantum fields on spacetime 
metric by computing the expectation value 
\(\langle \Psi \mid T\_{ab} \mid \Psi \rangle\) of the stress-energy 
tensor in some appropriate quantum state \(\lvert\Psi\rangle\) and then 
inserting the value into Einstein's field equations in place of 
\(T\_{ab}\).[26]
 Hawking's celebrated prediction of black hole radiation belongs to 
this 
 ambit.[27]
 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.[28]





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 no-go 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 \(H^+ (\Sigma)\) (often referred to 
as the "chronology horizon") for some suitable partial 
Cauchy surface \(\Sigma\). Find that the propagation of a quantum field 
on this spacetime background is such that 
\(\langle \Psi \mid T\_{ab} \mid\Psi \rangle\) "blows up" as
\(H^+ (\Sigma)\) 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 
 \(H^+ (\Sigma)\).[29]
 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 naive calculation of 
\(\langle \Psi \mid T\_{ab}\mid\Psi \rangle\) results in infinities 
which have to be subtracted off to produce a renormalized expectation
value 
\(\langle \Psi \mid T\_{ab}\mid\Psi \rangle\_R\) with
a finite value. The standard renormalization procedure uses a 
limiting procedure that is mathematically well-defined if, and only 
if, a certain condition 
 obtains.[30]
 The KRW theorem shows that this condition is violated for points on 
\(H^+ (\Sigma)\) and, thus, that the expectation value
of the stress-energy tensor is not well-defined at the chronology 
horizon.




While the KRW theorem is undoubtedly of fundamental importance for 
semi-classical quantum gravity, it does not serve as an effective 
no-go result for Thornian time machines. In the first place, the 
theorem assumes, in concert with Hawking's chronology protection 
theorem, that \(H^+ (\Sigma)\) 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 \(H^+ (\Sigma)\) is accepted as a necessary 
condition for time machines. The KRW theorem functions as a no-go 
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 semi-classical quantum gravity. But 
semi-classical 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 Planck-scale physics 
comes into play. This worry is underscored by Visser's (1997, 2003) 
findings that in chronology violating models trans-Planckian physics 
can be expected to come into play before 
\(H^+ (\Sigma)\) 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 fully-fledged quantum theory of 
gravity attempts to capture the Planck-scale structure of spacetime 
by constructing it from causal 
 sets.[31]
 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 Planck-scale 
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.[32]





In sum, the existing no-go 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. 
There is, however, a rapidly growing literature on the possibility of
time travel in lower-dimensional supersymmetric cousins of string 
theory. For a review of these recent results and a discussion of the 
fate of a time-traveller's ambition in loop quantum gravity, see 
Smeenk and Wuthrich (2010).


## 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 semi-classical 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 Anti-Time 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 would-be 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 
\(\Sigma\_0\) to the future of which there are CTCs, there are 
hole free extensions of \(D^+ (\Sigma\_0)\) 
satisfying Einstein's field equations and energy conditions but 
containing no CTCs to the future of \(\Sigma\_0\). 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.