Recent years have seen a growing consensus in the philosophicalcommunity that the grandfather paradox and similar logical puzzles donot preclude the possibility of time travel scenarios that utilizespacetimes containing closed timelike curves. At the same time,physicists, who for half a century acknowledged that the generaltheory of relativity is compatible with such spacetimes, haveintensely studied the question whether the operation of a time machinewould be admissible in the context of the same theory and of itsquantum cousins. A time machine is a device which brings about closedtimelike curves—and thus enables time travel—where nonewould have existed otherwise. The physics literature contains variousno-go theorems for time machines, i.e., theorems which purport toestablish that, under physically plausible assumptions, the operationof a time machine is impossible. We conclude that for the time beingthere exists no conclusive no-go theorem against time machines. Thecharacter of the material covered in this article makes it inevitablethat its content is of a rather technical nature. We contend, however,that philosophers should nevertheless be interested in this literaturefor at least two reasons. First, the topic of time machines leads to anumber of interesting foundations issues in classical and quantumtheories of gravity; and second, philosophers can contribute to thetopic by clarifying what it means for a device to count as a timemachine, by relating the debate to other concerns such asPenrose’s cosmic censorship conjecture and the fate ofdeterminism in general relativity theory, and by eliminating a numberof confusions regarding the status of the paradoxes of time travel.The present article addresses these ambitions in as non-technical amanner as possible, and the reader is referred to the relevant physicsliterature for details.

• 1. Introduction: time travel vs. time machine
• 2. What is a (Thornian) time machine? Preliminaries
• 3. When can a would-be time machine be held responsible for the emergence of CTCs?
• 4. No-go results for (Thornian) time machines in classical general relativity theory
• 5. No-go results in quantum gravity
• 6. Conclusion
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction: time travel vs. time machine

The topic of time machines is the subject of a sizable and growingphysics literature, some of which has filtered down to popular andsemi-popular presentations.^[1] The issues raised by this topic are largely oblique, if notorthogonal, to those treated in the philosophical literature on time travel.^[2] Most significantly, the so-called paradoxes of time travel do notplay a substantial role in the physics literature on time machines.This literature equates the possibility of time travel with theexistence of closed timelike curves (CTCs) or worldlines for materialparticles that are smooth, future-directed timelike curves with self-intersections.^[3] Since time machines designate devices which bring about the existenceof CTCs and thus enable time travel, the paradoxes of time travel areirrelevant for attempted “no-go” results for time machinesbecause these results concern what happens before the emergence of CTCs.^[4] This, in our opinion, is fortunate since the paradoxes of time travelare nothing more than a crude way of bringing out the fact that theapplication of familiar local laws of relativistic physics to aspacetime background which contains CTCs typically requires thatconsistency constraints on initial data must be met in order for alocal solution of the laws to be extendable to a global solution. Thenature and status of these constraints is the subject of ongoingdiscussion. We will not try to advance the discussion of this issue here;^[5] rather, our aim is to acquaint the reader with the issues addressedin the physics literature on time machines and to connect them withissues in the philosophy of space and time and, more generally, withissues in the foundations of physics.

Paradox mongers can be reassured in that if a paradox is lost inshifting the focus from time travel itself to time machines, then aparadox is also gained: if it is possible to operate a time machinedevice that produces CTCs, then it is possible to alter the structureof spacetime such that determinism fails; but by undercuttingdeterminism, the time machine undercuts the claim that it isresponsible for producing CTCs. But just as the grandfather paradox isa crude way of making a point, so this new paradox is a crude way ofindicating that it is going to be difficult to specify what it meansto be a time machine. This is a task that calls not for paradoxmongering but for scientifically informed philosophizing. The presentarticle will provide the initial steps of this task and will indicatewhat remains to be done. But aside from paradoxes, the main payoff ofthe topic of time machines is that it provides a quick route to theheart of a number of foundations problems in classical generalrelativity theory and in attempts to produce a quantum theory ofgravity by combining general relativity and quantum mechanics. We willindicate the shape of some of these problems here, but will refer theinterested reader elsewhere for technical details.

There are at least two distinct general notions of time machines,which we will callWellsian andThornian for short.InThe Time Machine, H. G. Wells (1931) described what hasbecome science fiction’s paradigmatic conception of a timemachine: the intrepid operator fastens her seat belt, dials the targetdate—past or future—into the counter, throws a lever, andsits back while time rewinds or fast forwards until the target date isreached. We will not broach the issue of whether or not a Wellsiantime machine can be implemented within a relativistic spacetimeframework. For, as will soon become clear, the time machines whichhave recently come into prominence in the physics literature are of anutterly different kind. This second kind of time machine wasoriginally proposed by Kip Thorne and his collaborators (see Morrisand Thorne 1988; Morris, Thorne, and Yurtsever 1988). These articlesconsidered the possibility that, without violating the laws of generalrelativistic physics, an advanced civilization might manipulateconcentrations of matter-energy so as to produce CTCs where none wouldhave existed otherwise. In their example, the production of“wormholes” was used to generate the required spacetimestructure. But this is only one of the ways in which a time machinemight operate, and in what follows any device which affects thespacetime structure in such a way that CTCs result will be dubbed aThornian time machine. We will only be concerned with thisvariety of time machine, leaving the Wellsian variety to sciencefiction writers. This will disappoint the aficionados of sciencefiction since Thornian time machines do not have the magical abilityto transport the would-be time traveler to the past of the events thatconstitute the operation of the time machine. For those moreinterested in science than in science fiction, this loss is balancedby the gain in realism and the connection to contemporary research inphysics.

In Sections 2 and 3 we investigate the circumstances under which it isplausible to see a Thornian time machine at work. The main difficultylies in specifying the conditions needed to make sense of the notionthat the time machine “produces” or is “responsiblefor” the appearances of CTCs. We argue that at present there isno satisfactory resolution of this difficulty and, thus, that thetopic of time machines in a general relativistic setting is somewhatill-defined. This fact does not prevent progress from being made onthe topic; for if one’s aim is to establish no-go results fortime machines it suffices to identify necessary conditions for theoperation of a time machine and then to prove, under suitablehypotheses about what is physically possible, that it is notphysically possible to satisfy said necessary conditions. In Section 4we review various no-go results which depend only on classical generalrelativity theory. Section 5 surveys results that appeal to quantumeffects. Conclusions are presented in Section 6.

2. What is a (Thornian) time machine? Preliminaries

The setting for the discussion is ageneral relativisticspacetime \((\mathcal{M},g_{ab})\) where \(\mathcal{M}\) is adifferentiable manifold and \(g_{ab}\) is a Lorentz signature metricdefined on all of \(\mathcal{M}\). The central issue addressed in thephysics literature on time machines is whether in this general settingit is physically possible to operate a Thornian time machine. Thisissue is to be settled by proving theorems about the solutions to theequations that represent what are taken to be physical laws operatingin the general relativistic setting—or at least this is so oncethe notion of a Thornian time machine has been explicated.Unfortunately, no adequate and generally accepted explication thatlends itself to the required mathematical proofs is to be found in theliterature. This is neither surprising nor deplorable. Mathematicalphysicists do not wait until some concept has received its finalexplication before trying to prove theorems about it; indeed, theprocess of theorem proving is often an essential part of conceptualclarification. The moral is well illustrated by the history of theconcept of a spacetime singularity in general relativity where thisconcept received its now canonical definition only in the process ofproving the Penrose-Hawking-Geroch singularity theorems, which came atthe end of a decades long dispute over the issue of whether spacetimesingularities are a generic feature of solutions to Einstein’sgravitational field equations.^[6] However, this is not to say that philosophers interested in timemachines should simply wait until the dust has settled in the physicsliterature; indeed, the physics literature could benefit fromdeployment of the analytical skills that are the stock in trade ofphilosophy. For example, the paradoxes of time travel and the fate oftime machines are not infrequently confused in the physics literature,and as will become evident below, subtler confusions abound aswell.

The question of whether a Thornian time machine—a device thatproduces CTCs—can be seen to be at work only makes sense if thespacetime has at least three features: temporal orientability, adefinite time orientation, and a causally innocuous past. In order tomake the notion of a CTC meaningful, the spacetime must betemporally orientable (i.e., must admit a consistent timedirectionality), and one of the two possible time orientations has tobe singled out as givingthe direction of time.^[7] Non-temporal orientability is not really an obstacle since if a givengeneral relativistic spacetime is not temporally orientable, aspacetime that is everywhere locally the same as the given spacetimeand is itself temporally orientable can be obtained by passing to acovering spacetime.^[8] How to justify the singling out of one of the two possibleorientations as future pointing requires a solution to the problem ofthe direction of time, a problem which is still subject to livelydebate (see Callender 2001, 2017). But for present purposes we simplyassume that a temporal orientation has been provided. A CTC is then(by definition) a parameterized closed spacetime curve whose tangentis everywhere a future-pointing timelike vector. A CTC can be thoughtof as the world line of some possible observer whose life history islinearly ordered in the small but not in the large: the observer has aconsistent experience of the “next moment,” and the“next,” etc., but eventually the “next moment”brings her back to whatever event she regards as the startingpoint.

As for the third condition—a causally innocuous past—thequestion of the possibility of operating a device that produces CTCspresupposes that there is a time before which no CTCs existed. Thus,Gödel spacetime, so beloved of the time travel literature, is nota candidate for hosting a Thornian time machine since through everypoint in this spacetime there is a CTC. We make this third conditionprecise by requiring that the spacetime admits aglobal timeslice \(\Sigma\) (i.e., a spacelike hypersurface without edges);^[9] that \(\Sigma\) is two-sided and partitions \(\mathcal{M}\) intothree parts—\(\Sigma\) itself, the part of \(\mathcal{M}\) onthe past side of \(\Sigma\) and the part of \(\mathcal{M}\) on thefuture side of \(\Sigma\)—and that there are no CTCs that lie onthe past side of \(\Sigma\). The first two clauses of this requirementtogether entail that the time slice \(\Sigma\) is apartial Cauchysurface, i.e., \(\Sigma\) is a time slice that is not intersectedmore 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 ofthe universe at a moment before the machine is turned on. Thesubsequent realization of a Thornian time machine scenario requiresthat thechronology 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 oninitial data on \(\Sigma_0\) since, by hypothesis, \(\Sigma_0\) is notintersected more than once by any timelike curve, and thus, insofar asthe so-called paradoxes of time travel are concerned with suchconstraints, 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 sofar, since otherwise \(\Sigma_0\) would not be a partial Cauchysurface. This just goes to underscore the point made above that thefans of science fiction stories of time machines will not find thepresent context of discussion broad enough to encompass their visionof how time machines should operate; they may now stop reading thisarticle and return to their novels.

Figure 1. Misner spacetime

As a concrete example of these concepts, consider the \((1 +1)\)-dimensional Misner spacetime (seeFigure 1) which exhibits some of the causal features of Taub-NUT spacetime, avacuum solution to Einstein’s gravitational field equations. Itsatisfies all three of the conditions discussed above. It istemporally orientable, and a time orientation has been singledout—the shading in the figure indicates the future lobes of thelight cones. To the past of the partial Cauchy surface \(\Sigma_0\)lies the Taub region where the causal structure of spacetime is asbland as can be desired. But to the future of \(\Sigma_0\) the lightcones begin to “tip over,” and eventually the tippingresults in CTCs in the NUT region.

The issue that must be faced now is what further conditions must beimposed in order that the appearance of CTCs to the future of\(\Sigma_0\) can be attributed to the operation of a time machine. Notsurprisingly, the answer depends not just on the structure of thespacetime at issue but also on the physical laws that govern theevolution of the spacetime structure. If one adopts the attitude thatthe label “time machine” is to be reserved for devicesthat operate within a finite spatial range for a finite stretch oftime, then one will want to impose requirements to assure that whathappens 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 liberaland allow the would-be time machine to be spread over an infinitespace. We will adopt the more liberal stance since it avoids variouscomplications while still sufficing to elicit key points. Again, onecould reserve the label “time machine” for devices thatmanipulate concentrations of mass-energy in some specified ways. Forexample, based on Gödel spacetime—where matter iseverywhere rotating and a CTC passes through every spacetimepoint—one might conjecture that setting into sufficiently rapidrotation a finite mass concentration of appropriate shape willeventuate in CTCs (Earman 1995, Manchak 2016). A similar possibilitypresents itself in Kerr spacetime (Andréka et al. 2008,Doboszewski 2022). But with the goal in mind of proving negativegeneral results, it is better to proceed in a more abstract fashion.Think of the conditions on the partial Cauchy surface \(\Sigma_0\) asencoding the instructions for the operation of the time machine. Thedetails of the operation of the device—whether it operates in afinite region of spacetime, whether it operates by setting matter intorotation, 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 subsequentappearance 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” inthe sense of causal determinism. But in the present setting this movequickly 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 thefuture domain of dependence \(D^+(\Sigma_0)\) of \(\Sigma_0\), the portion of spacetime where the fieldequations of relativistic physics uniquely determine the state ofthings from the state on \(\Sigma_0\).^[12] The point is illustrated by the toy model ofFigure 1. The surface labeled \(H^+ (\Sigma_0)\) is called thefutureCauchy horizon of \(\Sigma_0\). It is the future boundary of\(D^+ (\Sigma_0)\),^[13] and it separates the portion of spacetime where conditions arecausally determined by the state on \(\Sigma_0\) from the portionwhere conditions are not so determined. And, as advertised, the CTCsin the model ofFigure 1 lie beyond \(H^+ (\Sigma_0)\).

Figure 2. Deutsch-Politzer spacetime

Thus, if the operation of a Thornian time machine is to be a livepossibility, some condition weaker than causal determinism must beused to capture the sense in which the state on \(\Sigma_0\) can bedeemed to be responsible for the subsequent development of CTCs. Giventhe failure of causal determinism, it seems the next best thing todemand that the region \(V\) is “adjacent” to the futuredomain of dependence \(D^+ (\Sigma_0)\). Here is an initial stab atsuch an adjacency condition. Consider causal curves which have afuture endpoint in the time travel region \(V\) and no past endpoint.Such a curve may never leave \(V\); but if it does, require that itintersects \(\Sigma_0\). But this requirement is too strong because itrules out Thornian time machines altogether. For a curve of the typein question to reach \(\Sigma_0\) it must intersect \(H^+(\Sigma_0)\), but once it reaches \(H^+ (\Sigma_0)\) it can becontinued endlessly into the past without meeting \(\Sigma_0\) becausethe generators of \(H^+ (\Sigma_0)\) are past endless null geodesicsthat never meet \(\Sigma_0\).^[14] This difficulty can be overcome by weakening the requirement at issueby rephrasing it in terms of timelike curves rather than causalcurves. Now the set of candidate time machine spacetimes satisfyingthe weakened requirement is non-empty—as illustrated, onceagain, by the spacetime ofFigure 1. But the weakened requirement is too weak, as illustrated by the \((1+ 1)\)-dimensional version of Deutsch-Politzer spacetime^[15] (seeFigure 2), which is constructed from two-dimensional Minkowski spacetime bydeleting the points \(p_1\)–\(p_4\) and then gluing together thestrips as shown. Every past endless timelike curve that emerges fromthe time travel region \(V\) of Deutsch-Politzer spacetime does meet\(\Sigma_0\). But this spacetime is not a plausible candidate for atime machine spacetime. Up to and including the time \(\Sigma_0\)(which can be placed as close to \(V\) as desired) this spacetime isidentical with empty Minkowski spacetime. If the state of thecorresponding portion of Minkowski spacetime is not responsible forthe development of CTCs—and it certainly is not since there areno CTCs in Minkowski spacetime—how can the state on the portionof Deutsch-Politzer spacetime up to and including the time\(\Sigma_0\) be held responsible for the CTCs that appear in thefuture?

The deletion of the points \(p_1\)–\(p_4\) means that theDeutsch-Politzer spacetime is singular in the sense that it isgeodesically incomplete.^[16] It would be too drastic to require of a time-machine hostingspacetime that it be geodesically complete. And in any case theoffending feature of Deutsch-Politzer can be gotten rid of by thefollowing 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 thesame causal features as the original metric. But if the conformalfactor \(j\) is chosen to “blow up” as the missing points\(p_1\)–\(p_4\) are approached, the resulting spacetime isgeodesically complete—intuitively, the singularities have beenpushed off to infinity.

A more subtle way to exclude Deutsch-Politzer spacetime focuses on thegenerators of \(H^+ (\Sigma_0)\). The stipulations laid down so farfor Thornian time machines imply that the generators of \(H^+(\Sigma_0)\) cannot intersect \(\Sigma_0\). But in addition it can berequired that these generators do not “emerge from asingularity” and do not “come from infinity,” andthis would suffice to rule out Deutsch-Politzer spacetime and itsconformal cousins as legitimate candidates for time machinespacetimes. More precisely, we can impose what Stephen Hawking(1992a,b) calls the requirement that \(H^+ (\Sigma_0)\) becompactly generated; namely, the past endless null geodesicsthat generate \(H^+ (\Sigma_0)\) must, if extended far enough intopast, fall into and remain in a compact subset of spacetime. Obviouslythe spacetime ofFigure 1 fulfills Hawking’s requirement—since in this case \(H^+(\Sigma_0)\) is itself compact—but just as obviously thespacetime ofFigure 2 (conformally doctored or not) does not.

Imposing the requirement of a compactly generated future Cauchyhorizon has not only the negative virtue of excluding some unsuitedcandidate time machine spacetimes but a positive virtue as well. It iseasily proved that if \(H^+ (\Sigma_0)\) is compactly generated thenthe condition ofstrong causality is violated on \(H^+(\Sigma_0)\), which means, intuitively, there are almost closed causalcurves near \(H^+ (\Sigma_0)\).^[17] This violation can be taken as an indication that the seeds of CTCshave 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 thefuture of \(\Sigma_0\), then the state on \(\Sigma_0\) is responsiblefor the blooming. Of course, we have already learned that we cannothave the iron clad guarantee of causal determinism that the state on\(\Sigma_0\) is responsible for the actual blooming in all of itsparticularity. But we might hope for a guarantee that the state on\(\Sigma_0\) is responsible for the blooming ofsomeCTCs—the actual ones or others. The difference takes a bit ofexplaining. The failure of causal determinism is aptly pictured by theimage of a future “branching” of world histories, with thedifferent branches representing different alternative possible futuresof (the domain of dependence of) \(\Sigma_0\) that are compatible withthe actual past and the laws of physics. And so it is in the presentsetting: if \(H^+ (\Sigma_0) \ne \varnothing\), then there willgenerally be different ways to extend \(D^+ (\Sigma_0)\), allcompatible with the laws of general relativistic physics. But if CTCsare present in all of these extensions, even through the details ofthe CTCs may vary from one extension to another, then the state on\(\Sigma_0\) can rightly be deemed to be responsible for the fact thatsubsequently CTCs did develop.

A theorem due to Krasnikov (2002, 2003 [Other Internet Resources],2014a, 2018) might seem to demonstrate that no relativistic spacetimecan count as embodying a Thornian time machine so understood.Following Krasnikov, let us say that a spacetime condition \(C\) islocal 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 localconditions one might want to impose on physically reasonablespacetimes are Einstein’s gravitational field equations andso-called energy conditions that restrict the form of thestress-energy tensor \(T_{ab}\). An example of the latter that willcome into play below is theweak energy condition that saysthat the energy density is non-negative.^[18] Einstein’s field equations (sans cosmological constant) requirethat \(T_{ab}\) is proportional to the Einstein tensor which is afunctional 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 isisometric 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 showsthat every \(C\)-spacetime \((\mathcal{M}, g_{ab})\) admits a\(C\)-maximal extension \((\mathcal{M}^{max}, g^{max}_{ab})\) suchthat all CTCs in \((\mathcal{M}^{max}, g^{max}_{ab})\) are to thechronological past of the image of \(\mathcal{M}\) in\((\mathcal{M}^{max}, g^{max}_{ab})\). So start with some candidatespacetime \((\mathcal{M}, g_{ab})\) for a Thornian time machine, andapply the theorem to \((D^+ (\Sigma_0), g_{ab}|_{D^+ (\Sigma_0)})\).Conclude that no matter what local conditions the candidate spacetimeis required to satisfy, \(D^+ (\Sigma_0)\) has extensions that alsosatisfies said local conditions but does not contain CTCs to thefuture of \(\Sigma_0\). Thus, the candidate spacetime fails to exhibitthe crucial feature identified above necessary for underwriting thecontention that the conditions on \(\Sigma_0\) are responsible for thedevelopment of CTCs. Hence, it appears as if Krasnikov’s theoremeffectively prohibits time machines.

The would-be time machine operator need not capitulate in the face ofKrasnikov’s theorem. Recall that the main difficulty inspecifying the conditions for the successful operation of Thorniantime machines traces to the fact that the standard form of causaldeterminism does not apply to the production of CTCs. But causaldeterminism can fail for reasons that have nothing to do with CTCs orother acausal features of relativistic spacetimes, and it seems onlyfair to ensure that these modes of failure have been removed beforeproceeding to discuss the prospects for time machines. To zero in onthe 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, andlet \(\Sigma \subset \mathcal{M}\) and \(\Sigma ' \subset\mathcal{M}'\) be spacelike hypersurfaces of their respectivespacetimes. Suppose that there is an isometry \(\Psi\) from someneighborhood \(N(\Sigma)\) of \(\Sigma\) onto a neighborhood\(N'(\Sigma ')\) of \(\Sigma '\). Does it follow, as we would wantdeterminism to guarantee, that \(\Psi\) is extendible to an isometryfrom \(D^+ (\Sigma)\) onto \(D^+ (\Sigma ')\)? To see why the answeris negative, start with any solution \((\mathcal{M}, g_{ab})\) of thevacuum Einstein equations, and cut out a closed set of points lying tothe future of \(N(\Sigma)\) and in \(D^+ (\Sigma)\). Denote thesurgically altered manifold by \(\mathcal{M}^*\) and the restrictionof \(g_{ab}\) to \(\mathcal{M}^*\) by \(g^*_{ab}\). Then\((\mathcal{M}^*, g^*_{ab})\) is also a solution of the vacuumEinstein equations. But obviously the pair of solutions\((\mathcal{M}, g_{ab})\) and \((\mathcal{M}^*, g^*_{ab})\) violatesthe 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 outspacetimes that have “holes” in them. But while maximalitydoes rule out the surgically mutilated spacetime just constructed, itdoes not guarantee hole freeness in the sense needed to make sure thatdeterminism does not stumble before it gets to the starting gate. That\((\mathcal{M}, g_{ab})\) is hole free in the relevant sense requiresthat if \(\Sigma \subset \mathcal{M}\) is a spacelike hypersurface,there does not exist a spacetime \((\mathcal{M}', g'_{ab})\) and anisometric 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 isresponsible for this definition, asserts that if \(\Sigma \subset\mathcal{M}\) and \(\Sigma ' \subset \mathcal{M}'\) are spacelikehypersurfaces in hole-free spacetimes \((\mathcal{M}, g_{ab})\) and\((\mathcal{M}', g'_{ab})\), respectively, and if there exists anisometry \(\Psi : \mathcal{M} \rightarrow \mathcal{M}'\), then\(\Psi\) is indeed extendible to an isometry between \(D^+ (\Sigma)\)and \(D^+ (\Sigma ')\). Thus, hole freeness precludes an importantmode of failure of determinism which we wish to exclude in ourdiscussion of time machines. It can be shown that hole freeness is notentailed by maximality.^[19] And it is just this gap that gives the would-be time machine operatorsome hope, for the maximal CTC-free extensions produced byKrasnikov’s construction are not always hole free (Manchak2009b). But Krasnikov (2009) has shown that the Geroch (1977)definition is too strong: Minkowski spacetime fails to satisfy it! Forthis reason, alternative formulations of the hole-freeness definitionhave been constructed which are more appropriate (Manchak 2009a,Minguzzi 2012).

Thus, we propose that one clear sense of what it would mean for aThornian time machine to operate in the setting of general relativitytheory is given by the following assertion: the laws of generalrelativistic physics allow solutions containing a partial Cauchysurface \(\Sigma_0\) such that no CTCs lie to the past of \(\Sigma_0\)but every extension of \(D^+ (\Sigma_0)\) satisfying ________ containsCTCs (where the blank is filled with some “no hole”condition). Correspondingly, a proof of the physical impossibility oftime machines would take the form of showing that this assertion isfalse for the actual laws of physics, consisting, presumably, ofEinstein’s field equations plus energy conditions and, perhaps,some additional restrictions as well. And a proof of the emptiness ofthe associated concept of a Thornian time machine would take the formof showing that the assertion is false independently of the details ofthe laws of physics, as long as they take the form of local conditionson \(T_{ab}\) and \(g_{ab}\).

Are there “no hole” conditions which show the proposed concept of atime machine is not empty? Let \(J^+(p)\) designate thecausalfuture of \(p\), defined as the set of all points in\(\mathcal{M}\) which can be reached from \(p\) by a future-directedcausal curve in \(\mathcal{M}\). Thecausal past \(J^-(p)\)is defined analogously. Now, we say a spacetime\((\mathcal{M},g_{ab})\) isJ closed if, for each \(p\) in\(\mathcal{M}\), the sets \(J^+(p)\) and \(J^-(p)\) are topologicallyclosed. One can verify that J closedness fails in many artificiallymutilated examples (e.g. Minkowski spacetime with one point removedfrom the manifold). For some time, it was thought that a time machineexisted under this no-hole condition (Manchak 2011a). But this turnsout to be incorrect; indeed a recent result shows that any J closedspacetime \((\mathcal{M},g_{ab})\) of three dimensions or more withchronology violating region \(V \neq \mathcal{M}\) must be stronglycausal and therefore fail to have CTCs (Hounnonkpe and Minguzzi 2019).Stepping back, perhaps there are other no-hole conditions which can beused instead to show that the proposed concept of a time machine isnot empty. But even if such a project were successful, Manchak (2014a,2019) has shown that the time machine existence results can benaturally reinterpreted as “hole machine” existenceresults if one is so inclined. Instead of assuming that spacetime isfree of holes and then showing that certain initial conditions areresponsible for the production of CTCs, one could just as well startwith the assumption of no CTCs and then show that certain initialconditions are responsible for the production of holes. Given theimportance of these no hole assumptions to the time machine advocate,much recent work has focused on whether such assumptions arephysically reasonable in some sense (Manchak 2011b, 2014b). This isstill an open question.

Another open question is whether physically more realistic spacetimesthan Misner also permit the operation of time machines and how generictime-machine spacetimes are in particular spacetime theories, such asgeneral relativity. If time-machine spacetimes turn out to be highlynon-generic, the fan of time machines can retreat to a weaker conceptof Thornian time machine by taking a page from probabilistic accountsof causation, the idea being that a time machine can be seen to be atwork if its operation increases the probability of the appearance ofCTCs. Since general relativity theory itself is innocent ofprobabilities, they have to be introduced by hand, either by insertingthem into the models of the theory, i.e., by modifying the theory atthe level of the object-language, or by defining measures on sets ofmodels, i.e., by modifying the theory at the level of themeta-language. Since the former would change the character of thetheory, only the latter will be considered. The project for makingsense of the notion that a time machine as a probabilistic cause ofthe appearance of CTCs would then take the following form. Firstdefine a normalized measure on the set of models having a partialCauchy surface to the past of which there are no CTCs. Then show thatthe subset of models that have CTCs to the future of the partialCauchy surface has non-zero measure. Next, identify a range ofconditions on or near the partial Cauchy surface that are naturallyconstrued as settings of a device that is a would-be probabilisticcause of CTCs, and show that the subset of models satisfying theseconditions has non-zero measure. Finally, show that conditionalizingon 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 objectivechance. This task is especially daunting in the cosmological settingsince neither of the leading interpretations of objective chance seemsapplicable. The frequency interpretation is strained since thedevelopment of CTCs may be a non-repeated phenomenon; and thepropensity interpretation is equally strained since, barring just-sostories 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 probabilisticaccount of causation, the resort to a probabilistic conception of timemachines is a desperate stretch, at least in the context of classicalgeneral relativity theory. In a quantum theory of gravity, aprobabilistic conception of time machines may be appropriate if thetheory itself provides the transition probabilities between therelevant states. But an evaluation of this prospect must wait untilthe 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-goresults for time machines it is helpful to view these efforts as partof the broader project of provingchronology protectionstheorems, which in turn is part of a still larger project ofprovingcosmic censorship theorems. To explain, we start withcosmic censorship and work backwards.

Figure 3. A bad choice of initial valuesurface

For sake of simplicity, concentrate on the initial value problem forvacuum solutions \((T_{ab} \equiv 0)\) to Einstein’s fieldequations. Start with a three-manifold \(\Sigma\) equipped withquantities which, when \(\Sigma\) is embedded as a spacelikesubmanifold of spacetime, become initial data for the vacuum fieldequations. Corresponding to the initial data there exists a unique^[20] maximal development \((\mathcal{M}, g_{ab})\) for which (the image ofthe embedded) \(\Sigma\) is a Cauchy surface.^[21] This solution, however, may not be maximal simpliciter, i.e., it maybe possible to isometrically embed it as a proper part of a largerspacetime, which itself may be a vacuum solution to the fieldequations; if so \(\Sigma\) will not be a Cauchy surface for theextended spacetime, which fails to be a globally hyperbolic spacetime.^[22] This situation can arise because of a poor choice of initial valuehypersurface, as illustrated inFigure 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 variouspathologies, collectively referred to as “nakedsingularities,” to develop from regular initial data. The strongform of Penrose’s celebratedcosmic censorshipconjecture proposes that, consistent with Einstein’s fieldequations, such pathologies do not arise under physically reasonableconditions or else that the conditions leading to the pathologies arehighly non-generic within the space of all solutions to the fieldequations. A small amount of progress has been made on stating andproving precise versions of this conjecture.^[23]

One way in which strong cosmic censorship can be violated is throughthe emergence of acausal features. Returning to the example of Misnerspacetime (Figure 1), the spacetime up to \(H^+ (\Sigma_0)\) is the unique maximaldevelopment of the vacuum Einstein equations for which \(\Sigma_0\) isa Cauchy surface. But this development is extendible, and in theextension illustrated inFigure 1 global hyperbolicity of the development is lost because of thepresence of CTCs. Thechronology protection conjecture thencan be construed as a subconjecture of the cosmic censorshipconjecture, saying, roughly, that consistent with Einstein fieldequations, CTCs do not arise under physically reasonable conditions orelse that the conditions are highly non-generic within the space ofall solutions to the field equations. No-go results for time machinesare then special forms of chronology protection theorems that dealwith cases where the CTCs are manufactured by time machines. In theother direction, a very general chronology protection theorem willautomatically provide a no-go result for time machines, however thatnotion is understood, and a theorem establishing strong cosmiccensorship will automatically impose chronology protection.

The most widely discussed chronology protection theorem/no-go resultfor time machines in the context of classical general relativitytheory is due to Hawking (1992a). Before stating the result, notefirst that, independently of the Einstein field equations and energyconditions, a partial Cauchy surface \(\Sigma\) must be compact if itsfuture Cauchy horizon \(H^+ (\Sigma)\) is compact (see Hawking 1992aand Chrusciel and Isenberg 1993). However, it is geometrically allowedthat \(\Sigma\) is non-compact if \(H^+ (\Sigma)\) is required only tobe compactly generated rather than compact. But what Hawking showed isthat this geometrical possibility is ruled out by imposingEinstein’s field equations and the weak energy condition. Thus,if \(\Sigma_0\) is a partial Cauchy surface representing the situationjust before or just as the would-be Thornian time machine is switchedon, and if a necessary condition for seeing a Thornian time machine atwork is that \(H^+ (\Sigma_0)\) is compactly generated, thenconsistently with Einstein’s field equations and the weak energycondition, a Thornian time machine cannot operate in a spatially openuniverse since \(\Sigma_0\) must be compact.

This no-go result does not touch the situation illustrated inFigure 1. Taub-NUT spacetime is a vacuum solution to Einstein’s fieldequations 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 \(\Sigma_0\) iscompact. By the same token the theorem does not speak to thepossibility of operating a Thornian time machine in a spatially closeduniverse. To help fill the gap, Hawking proved that when \(\Sigma_0\)is compact and \(H^+ (\Sigma_0)\) is compactly generated, the Einsteinfield equations and the weak energy condition together guarantee thatboth the convergence and shear of the null geodesic generators of\(H^+ (\Sigma_0)\) must vanish, which he interpreted to imply that noobservers can cross over \(H^+ (\Sigma_0)\) to enter the chronologyviolating region \(V\). But rather than showing that it is physicallyimpossible to operate a Thornian time machine in a closed universe,this result shows only that, given the correctness of Hawking’sinterpretation, the observers who operate the time machine cannot takeadvantage of the CTCs it produces.

There are two sources of doubt about the effectiveness ofHawking’s no-go result even for open universes. The first stemsfrom possible violations of the weak energy condition by stress-energytensors arising from classical relativistic matter fields (see Vollick1997 and Visser and Barcelo 2000).^[24] The second stems from the fact that Hawking’s theorem functionsas a chronology protection theorem only by way of serving as apotential no-go result for Thornian time machines since the crucialcondition that \(H^+ (\Sigma_0)\) is compactly generated is supposedlyjustified by being a necessary condition for the operation of suchmachine. But in retrospect, the motivation for this condition seemsfrayed. As argued in the previous section, if the Einstein fieldequations and energy conditions entail that all hole free extensionsof \(D^+ (\Sigma_0)\) contain CTCs, then it is plausible to see aThornian time machine at work, quite regardless ofwhether or not \(H^+(\Sigma_0)\) is compactly generated or not. Of course, it remains toestablish the existence of cases where this entailment holds. If itshould turn out that there are no such cases, then the prospects ofThornian time machines are dealt a severe blow, but the reasons areindependent of Hawking’s theorem. On the other hand, if suchcases do exist then our conjecture would be that they exist even whensome of the generators of \(H^+ (\Sigma_0)\) come from singularitiesor 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 curvedspacetimes, simply takes off the shelf a spacetime provided by generalrelativity theory and then proceeds to study the behavior of quantumfields on this background spacetime. The Unruh effect, which predictsthe thermalization of a free scalar quantum field near the horizon ofa black hole, lies within this ambit. The second degree ofinvolvement, referred to as semi-classical quantum gravity, attemptsto calculate the backreaction of the quantum fields on spacetimemetric by computing the expectation value \(\langle \Psi \mid T_{ab}\mid \Psi \rangle\) of the stress-energy tensor in some appropriatequantum state \(\lvert\Psi\rangle\) and then inserting the value intoEinstein’s field equations in place of \(T_{ab}\).^[26] Hawking’s celebrated prediction of black hole radiation belongsto this ambit.^[27] The third degree of involvement attempts to produce a genuine quantumtheory of gravity in the sense that the gravitational degrees offreedom are quantized. Currently loop quantum gravity and stringtheory are the main research programs aimed at this goal.^[28]

The first degree of quantum involvement, if not opening the door toThornian time machines, at least seemed to remove some obstacles sincequantum fields are known to lead to violations of the energyconditions used in the setting of classical general relativity theoryto prove chronology protection theorems and no-go results for timemachines. However, the second degree of quantum involvement seemed, atleast initially, to slam the door shut. The intuitive idea was this.Start with a general relativistic spacetime where CTCs develop to thefuture of \(H^+ (\Sigma)\) (often referred to as the “chronologyhorizon”) for some suitable partial Cauchy surface \(\Sigma\).Find that the propagation of a quantum field on this spacetimebackground is such that \(\langle \Psi \mid T_{ab} \mid\Psi \rangle\)“blows up” as \(H^+ (\Sigma)\) is approached from thepast. Conclude that the backreaction on the spacetime metric createsunbounded curvature, which effectively cuts off the future developmentthat would otherwise eventuate in CTCs. These intuitions were partlyvindicated by detailed calculations in several models. But eventuallya number of exceptions were found in which the backreaction remainsarbitrarily 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 toreview here, but the structure of the argument is easy to grasp. Thenaïve calculation of \(\langle \Psi \mid T_{ab}\mid\Psi \rangle\)results in infinities which have to be subtracted off to produce arenormalized expectation value \(\langle \Psi \mid T_{ab}\mid\Psi\rangle_R\) with a finite value. The standard renormalizationprocedure uses a limiting procedure that is mathematicallywell-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 thestress-energy tensor is not well-defined at the chronologyhorizon.

While the KRW theorem is undoubtedly of fundamental importance forsemi-classical quantum gravity, it does not serve as an effectiveno-go result for Thornian time machines. In the first place, thetheorem assumes, in concert with Hawking’s chronology protectiontheorem, that \(H^+ (\Sigma)\) is compactly generated, and we repeatthat it is far from clear that this assumption is necessary for seeinga Thornian time machine in operation. A second, and more fundamental,reservation applies even if a compactly generated \(H^+ (\Sigma)\) isaccepted as a necessary condition for time machines. The KRW theoremfunctions as a no-go result by providing areductio adabsurdum with a dubious absurdity: roughly, if you try to operatea Thornian time machine, you will end up invalidating semi-classicalquantum gravity. But semi-classical quantum gravity was never viewedas anything more than a stepping stone to a genuine quantum theory ofgravity, and its breakdown is expected to be manifested whenPlanck-scale physics comes into play. This worry is underscored byVisser’s (1997, 2003) findings that in chronology violatingmodels trans-Planckian physics can be expected to come into playbefore \(H^+ (\Sigma)\) is reached.

It thus seems that if some quantum mechanism is to serve as the basisfor chronology protection, it must be found in the third degree ofquantum involvement in gravity. Both loop quantum gravity and stringtheory have demonstrated the ability to cure some of the curvaturesingularities of classical general relativity theory. But as far as weare aware there are no demonstrations that either of these approachesto quantum gravity can get rid of the acausal features exhibited invarious solutions to Einstein’s field equations. An alternativeapproach to formulate a fully-fledged quantum theory of gravityattempts to capture the Planck-scale structure of spacetime byconstructing it from causal sets.^[31] Since these sets must be acyclic, i.e., no element in a causal setcan causally precede itself, CTCs are ruled out a priori. Actually, atheorem due to Malament (1977) suggests that any Planck-scale approachencoding only the causal structure of a spacetime cannot permit CTCseither in the smooth classical spacetimes or a correspondingphenomenon in their quantum counterparts.^[32]

In sum, the existing no-go results that use the first two degrees ofquantum involvement are not very convincing, and the third degree ofinvolvement is not mature enough to allow useful pronouncements. Thereis, however, a rapidly growing literature on the possibility of timetravel in lower-dimensional supersymmetric cousins of string theory.For a review of these recent results and a discussion of the fate of atime-traveller’s ambition in loop quantum gravity, see Smeenkand Wüthrich (2010).

6. Conclusion

Hawking opined that “[i]t seems there is a chronology protectionagency, which prevents the appearance of closed timelike curves and somakes the universe safe for historians” (1992a, 603). He may beright, but to date there are no convincing arguments that such anAgency is housed in either classical general relativity theory or insemi-classical quantum gravity. And it is too early to tell whetherthis Agency is housed in loop quantum gravity or string theory. Buteven if it should turn out that Hawking is wrong in that the laws ofphysics do not support a Chronology Protection Agency, it could stillbe the case that the laws support an Anti-Time Machine Agency. For itcould turn out that while the laws do not prevent the development ofCTCs, they also do not make it possible to attribute the appearance ofCTCs to the workings of any would-be time machine. We argued that astrong presumption in favor of the latter would be created inclassical general relativity theory by the demonstration that for anymodel satisfying Einstein’s field equations and energyconditions as well as possessing a partial Cauchy surface \(\Sigma_0\)to the future of which there are CTCs, there are hole free extensionsof \(D^+ (\Sigma_0)\) satisfying Einstein’s field equations andenergy conditions but containing no CTCs to the future of\(\Sigma_0\). There are no doubt alternative approaches tounderstanding what it means for a device to be “responsiblefor” the development of CTCs. Exploring these alternatives isone place that philosophers can hope to make a contribution to anongoing discussion that, to date, has been carried mainly by thephysics community. Participating in this discussion means thatphilosophers have to forsake the activity of logical gymnastics withthe 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 principledreasons that stem from basic physical laws. But even if mathematicaltheorems in the various theories concerned succeed in establishing theimpossibility of time machines, understanding why time machines cannotbe constructed will shed light on central problems in the foundationsof physics. As we have argued in Section 4, for instance, the hunt fortime machines in general relativity theory should be interpreted as acore issue in studying the fortunes of Penrose’s cosmiccensorship conjecture. This conjecture arguably constitutes the mostimportant open problem in general relativity theory. Similarly, asdiscussed in Section 5, mathematical theorems related to variousaspects of time machines offer results relevant for the search of aquantum theory of gravity. In sum, studying the possibilities foroperating a time machine turns out to be not a scientificallyperipheral or frivolous weekend activity but a useful way of probingthe foundations of classical and quantum theories of gravity.

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• Rovelli, C., 2008, “Loop Quantum Gravity”, inLiving Reviews in Relativity.

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determinism: causal |space and time: the hole argument |time: thermodynamic asymmetry in |time travel |time travel: and modern physics

Acknowledgments

We thank Carlo Rovelli for discussions and John Norton for comments onan earlier draft. C.W. acknowledges support by the Swiss NationalScience Foundation (grant PBSK1-102693).