featuresPtiy«lca Wofld D«omb«r 1994 2 7

Einstein's equations of relativity do not rule out "closed time-likecurves", bizarre trajectories in space-time that might allow us totravel backwards in time. What are the physical constraints on

such "time machines" and what are the possible repercussions?

The physics oftime travel

JONATHAN Z SIMON

The Queen bawled out, "He's murdering the time! Off with hishead!" Lewis Carroll

TIME travel has traditionally been the domain of sciencefiction, not physics. Fortunately, however, at least withinEinstein's theories of relativity, discussions of time travelare open to physicists as well. Special relativity unifies theconcepts of time and space. General relativity goes beyondunification and allows time and space to warp together inthe presence of matter. General relativity even permitssufficient warping to allow "closed time-like curves".These seemingly perverse trajectories describe pathsthrough space-time that always move forward in localtime (i.e. an observer's watch always runs forward), buteventually end up back where and when they started. Aspace-time that contains closed time-like curves, localizedin one region, can be said to have a "time machine".

Closed time-like curves appear in explicit analyticalsolutions to the Einstein equation of general relativity.Previously such solutions were deemed "unphysical",simply because they contained closed time-like curves.Nevertheless, since these solutions obey the field equa-tions, they should not be rejected out of hand. Indeed,interest in closed time-like curves has increased in the pastdecade. The reasons for this are varied, ranging fromthe practical (if time machines can be built, they wouldhave a lot of potential uses), to the theoretical (perhapsquantum gravity can say something about the existence ofclosed time-like curves, or vice versa) to the philosophical(do the laws of physics allow or prohibit closed time-likecurves?).

Travelling forward in time is easy and does not requiremuch new physics. In Newtonian physics with its absolutebackground time we all travel forward in time at the rate ofone second per second. Special relativity allows us to travelforward in time at faster rates. The so-called twin paradoxis an excellent example. While one twin remains at homein an inertial frame, the other zooms off into the Universe,and then back home, at relativistic speeds with time dilatedby a factor of 1/(1 -v2lcy each way, where v is thetravelling twin's speed and c is the speed of light. Onreturning home the travelling twin, having experienced

much less passage of time than the stationary twin, has"gone into the future" relative to the first twin and allother occupants of Earth.

This form of forward time travel happens quite naturallyall the time. High-energy particle showers formed in theupper atmosphere by cosmic rays contain particles whoselifetimes are much shorter than the time it takes them toreach ground level, but whose speeds are extremely closeto the speed of light. The very fact that we observe theseparticles at ground level means that their rate of timepassage has dilated or, equivalently, that they havetravelled into their future.

Although going into the future is straightforward, wealso need to go into the past - which is difficult - to make aclosed time-like curve. This is where general relativity - or,more precisely, the possibility of non-trivial space-times -is required. Conversely, if closed time-like curves exist,then a traveller can go back into the past. (Note that thereis a distinction between going back into the past, discussedhere, and going "backwards in time", that is having one'swatch tick backwards, which is a separate issue with itsown physics). There is currently no evidence that closedtime-like curves exist. For instance, we do not see futuretourists coming back to visit the present. However, it isnot in the tradition of physics to turn the argument aroundand use this lack of evidence to argue that they cannotexist.

What about the paradoxes of time travel? Can they beused as evidence against the existence of closed time-likecurves? The most serious of these is the "grandparent"paradox. In this scenario a time traveller goes back in timeand kills his or her grandparent before he or she has anychildren. This would be a true paradox. One resolution tothis paradox is to postulate that only self-consistenthistories are allowed. With this postulate the rules of thegame are turned around. We can ask whether a particularphysical theory allows self-consistent evolution that is alsoconsistent with its equation of motion.

We should also be aware that many physical theorieswith different equivalent formulations (such as theSchrodinger and Feynman formulations of quantummechanics) can give different results in the presence of

2 8 Physics World December 1994

1 (a) A billiard table time machine. When a ball enters the right-hand pocket at time ( an earlierversion of the same ball Is released from the left-hand pocket with the same speed at an earlier timet-At. The direction of the earlier version Is the reflection of the incident direction about a lineconnecting the point of entry into the pocket with the centre of the pocket. Although there is only oneball, two or more versions of the same ball can be on the table at certain times. The trajectory shownhere Is not self-consistent because the ball collides with the earlier version of itself at such an anglethat it cannot enter the right-hand pocket, (b) A self-consistent trajectory for the Initial conditionsshown in (a). The ball sets off in the same direction as before but collides with the earlier version atan angle which does not prevent it from entering the right-hand pocket, {c-f) Four (of an infinitenumber of) self-consistent trajectories with the same Initial conditions. The number of passagesthrough the time machine varies from solution to solution

closed time-like curves. Therefore when determining theeffect of closed time-like curves on a particular theory, wehave to define the theory very carefully.

Classical behaviourIn avoiding the grandparent paradox, the self-consistencypostulate at first seems to violate our concept of free will. Itis difficult to imagine oneself as a time traveller when oneis not allowed to make choices which might cause aparadox. But this is a bit of a red herring - this same lack offree will already exists in ordinary Newtonian mechanics,or in any deterministic theory: once the initial values of thefields and derivatives are specified, there is no room forfree will. In this sense free will is not usually addressed inphysics. Still, what might happen if a time traveller didactually meet a grandparent, having decided to kill him orher beforehand? There are many possible outcomes: thetime traveller could feel remorse and decide against it; thegrandparent could convince the traveller not to; or the

time machine might break in antici-pation of the event. The problemhere seems to be that human beingshave too many degrees of freedom tokeep track of. Fortunately the funda-mental physical issues are still pre-sent for a much simpler system - thebilliard ball.

As a simple example of a timemachine, due to Joseph Polchinskiwhile he was at the University ofTexas, we use a specialized billiardtable with two pockets. Any objectthat falls into the right-hand pocketat time t is shot out of the left-handpocket at an earlier time, t-At, withthe same speed but a new direction(figure la). This time machine couldbe implemented using a space-time"wormhole" - a shortcut in space-time (of non-trivial topology) whichconnects two distant points by ashorter path. The points connectedmay be separated in time as well asspace (see figure 2). Wormholeshave not been observed yet but areallowed by the Einstein equations.

However, the trajectory in figurela is not self-consistent - if the ballenters the right-hand pocket at timet, it exits the left-hand pocket at anearlier time, t-At, and this earlierversion of the ball collides with thecurrent version, preventing it fromreaching the right-hand pocket in thefirst place. This captures the bareessence of the grandparent paradox.This system has been analysed ingreat detail by Fernando Echeverria,Gunnar Klinkhammer and KipThorne at Caltech who have foundanswers to these questions, but haveraised new, even more interesting,questions in the process. Theiranalysis is completely classical andnon-relativistic (with the obviousexception of the time machine link-ing the pockets).

The resolution of the paradox in figure la is straightfor-ward - if the ball moves off at the same angle, but onlyreceives a glancing blow from the earlier version of itself, itwill enter the right-hand pocket at a slightly different angleto that in figure la. The direction of the ball leaving theleft-hand pocket will also be slightly different - hence thecollision will be glancing, rather than head on as before.This self-consistent trajectory is shown in figure \b. Itshould be emphasized that, although the initial conditionsin figures la and b are exactly the same, the evolution ofthe latter is self-consistent whilst that of the former is not.

It turns out that every initial condition for this model hasa self-consistent solution. In fact, for many initialconditions there is more than one self-consistent solu-tion, and often an infinite number, which all obeyNewton's laws of classical mechanics. The initial condi-tion shown in figure 1 has an infinite number of self-consistent solutions; most are similar to le and If, wherethe final velocity of the ball is opposite to the initialvelocity, but the number of passages through the time

Physics Work) Oactmbtr 1994 2 9

machine varies from solution to solution.There are also systems of time machines and particles

which have no self-consistent solutions for given initialdata, but these are difficult to construct using Newtonianmodels with simply interacting particles. In any event thequestion of what happens to an initial configuration near atime machine is not answered by Newtonian mechanics,because we lose one of the most cherished simplificationsof ordinary physics - that initial conditions plus theequations of motion should uniquely determine thesolution. Also note that, although non-interacting parti-cles have no such richness (or dearth) of solutions, they donot suffer from the grandparent paradox because theycannot affect their surroundings or each other.

Fields have properties similar to particles. Non-interactingfields, such as electromagnetism in the absence of charges,are free from multiple solutions (and paradoxes). How-ever, fields can also destabilize a time machine bypropagating through it an infinite number of times, andadding their field strengths in each passage. If the timemachine has a focusing effect, this infinite build-up of fieldenergy would back-react on the time machine. Thebilliard-table time machine, however, has a defocusingeffect: incoming spherical waves entering the right-handpocket become outgoing spherical waves from the left-hand pocket. Although some of these waves enter theright-hand pocket again and again, the sum converges andthe field strength remains finite everywhere.

Quantum behaviourThe classical behaviour of particles and fields is sufficientlyrich and confusing that one naturally turns to quantummechanics to (possibly) answer the question of whathappens when something passes through a time machine.It turns out that quantum mechanics gives a more definiteanswer than classical mechanics. However, one must firstpick a definite formulation of quantum mechanics to get adefinite answer, and there are several distinct formulationsof quantum mechanics which are equivalent withoutclosed time-like curves, but which are not equivalent intheir presence.

The most basic formulations of non-relativistic quantummechanics are due to Schrodinger and Heisenberg. Innon-relativistic theories there is a universal time functionwhich all observers agree on. In the Schrodingerformulation, an initial wave function is evolved forwardin (Newtonian) time by the Schrodinger equation, andmeasurements correspond to insertions of time-indepen-dent operators. The Heisenberg formulation is equivalent,but all the time dependence is delegated to the operators.

For theories formulated in the four-dimensional(Minkowski) space-time of special relativity, there aremany "times" to choose from (corresponding to thedifferent proper times of different boosted inertialobservers) and any one time function is as good as anyother. For each time function, at every moment there is athree-dimensional spatial "hypersurface" orthogonal tothe time axis. Evolution along a time axis can be specifiedeither by its time function, or by the progression of thesespace-like "slices". However, the results of all the differentformulations agree irrespective of the way time is definedor evolved.

In general relativity there is generally no preferred timeat all, but space-time, which is now curved or warped, canstill be sliced into space-like slices. This enables the wavefunction to be defined on each slice and the evolution ofthe wave function from one slice to die next to be

described by Schrodinger's equation. Two events on thesame slice can be said to take place at the same time insome generalized notion of labelling, but they are notnecessarily simultaneous in any physical sense.

However, in the presence of closed time-like curves wegenerally lose even our notion of space-like slicing,because nearby points in a space-like slice can be reachedby time-like paths as well as space-like paths (figure 3).This means that the value of the field at one point iscausally related to the value of the field at a neighbouringpoint, which destroys all initial-data formulations ofquantum mechanics, including the Schrodinger andHeisenberg formulations.

A natural alternative to consider is the Feynman pathintegral formulation of quantum mechanics. In this,quantum amplitudes are constructed by summing overall possible paths of a particle (or histories of a fieldconfiguration), with a complex weighting given by e15,where 5 is the action of that path (or history). The

2 (a) A wormhole connecting the two points A and B at the same time.Distances are as shown and not as one would naively expect: thedistance through the wormhole (20 cm) is much less than "directly"across via the rest of space (2 m). (£>) Wormholes can also connect twopoints at different times. If these are separated far enough in externaltime, the wormhole would permit closed time-like curves. In this case aparticle travelling through the wormhole from B to A arrives at A at anearlier external time. (Local time always advances forward for particles,whether moving from A to B or B to A.) Wormholes are not necessary toform closed time-like curves, but they can be useful for visualizing theireffects

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Feynman formulation is relatively insensitive to theexistence of closed time-like curves in the region ofintegration, and in ordinary space-time it is completelyequivalent to the Schrodinger/Heisenberg formulation formost theories. A reasonable supposition is that thehistories summed over should be self-consistent (likefigures \b-f). This summation produces an amplitude togo from an initial state to a final state. The initial and finalstates must be on space-like slices free from any closedtime-like regions, so any closed time-like curves that arecreated must be destroyed eventually.

But even restricting quantum mechanics to path-integralformulations is not enough to determine the effects ofclosed time-like curves, because the "propagator" (whichconnects the initial and final states) does notconverge and various methods must be used toensure convergence. These methods - which involvethe addition of a small imaginary mass, or time,which is set to zero at the end of the calculation - areall equivalent in the absence of closed time-likecurves. (The addition of an imaginary time compo-nent, known as Wick rotation, moves the contour ofintegration slightly off the real axis in the complexplane).

The imaginary mass technique allows a straight-forward quantization of non-interacting particles andfields. However, the spectre of the grandparentparadox does rear its head for interacting fields. It hasbeen shown by John Friedman, Nicholas Papastama-tiou and myself at the University of Wisconsin-Milwaukee mat the scattering amplitude becomesnon-unitary. Unitarity in a quantum theory deter-mines how probabilities are calculated from quantumamplitudes (normally by calculating the modulussquared of the amplitude). The mathematical defini-tion of probability ensures that the sum of the proba-bilities always equals one, but the sum of the squaredamplitudes depends on the physics of the theory.

The usual cause of non-unitarity is that someinformation has been ignored. For example, transi-tion amplitudes for the scattering of free electronsand protons into free electrons and protons will notbe unitary unless bound states, that is hydrogenatoms, are also included. In the case of closed time-likecurves, however, unitarity is violated in an unusual way:the sum of amplitudes squared (the norm) of the statesgenuinely changes.

James Hartle of the University of California at SantaBarbara has developed a framework in which genuine non-unitary evolution is possible with the probability stillalways proportional to the square of the quantumamplitude. If the theory is unitary, the proportionalityfactor is constant (and equal to one). Non-unitarityrequires some additional rule of how to computeprobability. However, the factor of proportionality willno longer be universal. The simplest way to "reconnect"the probability to the squared amplitude is to divide theamplitude by the norm of the initial state evolved in time(i.e. to "renormalize" the amplitude). This norm is notconstant in time and also depends on the initial state.

This method uniquely determines the probability fromthe amplitude, but has two disturbing features. First,quantum mechanics is no longer linear, so the cherishedprinciple of superposition is lost. Secondly, because theprobabilities depend on the future value of the norm of theinitial state, probabilities of events that take place beforethe formation of closed time-like curves depend on whathappens once the closed time-like curves form. This gives

an extra form of causality violation before the closed time-like curves form, independent of any causality violationsoccurring after they form.

A different renormalization technique, proposed byArlen Anderson while at Imperial College, London,effectively sets all the non-unitary sectors of the evolutionto zero, eliminating the disturbing features of Hartle'ssimpler method. This is clearly desirable, but it is not yetknown if this method also wipes out important features inthe process.

There is also a recent claim by Stephen Hawking ofCambridge University that Wick rotation leads to a loss ofquantum coherence (and hence a loss of unitarity differentfrom that just discussed). For quantum fields in

local time

local time local time

local time local time

3 A space-like slice (with one spatial dimension suppressed). There are no closedtime-like curves on the left side of the slice, so an observer (or object with a time-liketrajectory) Intersecting the slice at point C never crosses this slice again. There areclosed time-like curves passing through the slice on the right, Including one passingthrough point A. An observer moving along this curve always moves forward locally Intime but nevertheless re-intersects the space-like slice at A. By distorting the trajectoryslightly, one may start at A and end at B. This means that point B Is In the (causal)future with respect to A even though they are on the same space-like slice. Most Initialdata defined on this slice would be Inconsistent with the time evolution of the system,destroying the usual Initial-data formulation of quantum (and classical) mechanics

equilibrium with a mermal bath (in ordinary flat space-time), the Feynman propagator is periodic in imaginarytime (purely imaginary time corresponds to a Wickrotation of 90°), with the period inversely proportional tothe temperature. Propagation through such a thermal bathresults in a loss of quantum coherence due to theinteraction of pure quantum states with the mixed statesof the bath. In space-times with closed time-like curvessome trajectories are periodic in real time. Wick rotationpicks this up as an interaction with a thermal bath at animaginary temperature. When the Wick rotation isinverted at the end of the calculation, there is still someremnant of the thermal nature of the interaction, whichshould result in a loss of quantum coherence.

There is at least one more version of quantummechanics with yet another set of odd behaviours in thepresence of closed time-like curves. An informationtheoretic formulation of quantum mechanics due toDavid Deutsch of Oxford University, like the Wick-rotated path-integral formulation, routinely destroysquantum coherence, but in a novel way. The elements ofthe density matrix get mixed up in such a way that, in thelanguage of the Everett-Wheeler "many worlds" inter-pretation of quantum mechanics, macroscopic observerscan move and communicate from one "branch" to

Phylka World Oscamtw 1994 31

another by interacting with the closed time-like curveregion (see Deutsch and Lockwood in Further reading).

Classical structureSo how difficult would it be to build a time machine?Probably very difficult, though it is not possible to rulethem out completely. The time machines that are mostdifficult to rule out are microscopic time machines at thePlanck scale (ln = yJ(HG/c3) = 2 x 1CT33 cm, rP, = 7(fiG/c5) = 5 x KT44 s, where h is Planck's constant divided by27i, G is the gravitational constant and c is the speed oflight). At this scale, quantum gravitational effectscompletely dominate classical notions of space-time.However, until we have a firm understanding of quantumgravity, nothing certain can be said (superstring theory, apossible candidate for a quantum theory of gravity, is noteven close to answering such questions). The loss of theclassical notion of a background space-time results in whatJohn Wheeler of Princeton University calls "space-timefoam". In this model, space-time, which is smooth atscales much larger than the Planck scale, breaks up into afoam-like structure at smaller scales, possibly allowingtopology change, causality violation and other unusualfeatures.

What about at larger scales, whether subatomic orgalactic? It is very easy to write down a solution to theEinstein equation of general relativity with closed time-likecurves. First, for any space-time with closed time-likecurves, compute the Einstein curvature tensor on the left-hand (geometric) side of the Einstein equation,Gjjy = SnGT^. Then find matter that satisfies the right-hand (stress-energy) side of the equation. However,according to Frank Tipler of Tulane University in NewOrleans, Stephen Hawking and others, the most interest-ing space-times that allow closed time-like curves requireconditions that are highly unlikely - "naked" curvaturesingularities, non-trivial large distance behaviour, ormatter with negative mass/energy density. Here negativemass/energy does not mean antimatter, but matter that willgravitationally repel other matter, and thus excludes allknown types of classical matter. ("Naked" singularities arepoints of infinite density and stress that are not hiddenbehind a black hole - it is thought that the general theoryof relativity does not allow for naked singularities, with thepossible exception of the Big Bang).

There are still a few open loopholes in these proofs, butspace-times with closed time-like curves that are otherwisewell behaved are quite constrained if they must satisfy theEinstein equation with reasonable matter and curvatures.Although difficult to find, such solutions are notimpossible. As long ago as 1937 van Stockum ofEdinburgh found a solution to the Einstein equationinvolving an infinitely long spinning cylinder (similar torecent solutions containing infinitely long cosmic strings)that contained closed time-like curves. One might scoff ata solution requiring an infinitely long cylinder, but thatalone should not be enough to rule out the solution asunphysical. Indeed infinite cylinders are used routinely inelementary electromagnetism and mechanics because theyare good approximations to non-infinite cylinders (andeasy to solve). Moreover, general relativity is not just alocal theory but also a cosmological theory. One cannotdiscard a solution merely because it is of infinite extent. Asit turns out, von Stockum's theory needs structure atinfinity to overcome the problem of constructing closedtime-like curves theorems without negative mass/energydensities. This means that the solutions are not good

approximations to local solutions, and that closed time-like curves do not appear for long (but finite) rotatingcylinders.

In 1949 Kurt Go'del found a cosmological solution withclosed time-like curves where the Universe is filled withrotating matter. If an observer goes far enough out (at highenough accelerations), he or she can traverse a closedtime-like curve. More recently in 1991 Richard Gott ofPrinceton University found another solution to theEinstein equation with closed time-like curves, this timegenerated by two (non-spinning) cosmic strings passingeach other at high speed. This solution has the interestingproperty that the closed time-like curves do not exist atearly and late times, only at intermediate times near themoment of closest approach. The major problems withthis solution are that the strings must be very massive(their mass and kinetic energy must dominate all of themass/energy in the Universe), and that the regioncontaining the closed time-like curves must be infinitelylarge. All these solutions have closed time-like curvesthreading though the entire Universe, all the way out toinfinity. Although solutions to the Einstein equation, suchclosed time-like curves could not be created in a finite-sized laboratory, be it the solar system or the galaxy.

Time travel and black holesThe best known solution to general relativity containingclosed time-like curves in a bounded region of space is therotating black hole. A black hole is a region of space-timewhere the gravitational field is sufficiently strong, andspace-time is sufficiently warped, that even light cannotescape. The location of the last beams of light, strugglingto get away from the black hole but never able to breakfree, is called the event horizon. (For a non-rotating blackhole this is given by the Schwarzschild radius.)

Inside the black hole the space-time curvature andgravitational forces become so strong that they are infiniteat zero radius (at least according to classical generalrelativity). Anything falling inside the event horizon mustreach this "curvature singularity" in a finite amount of(local observer) time. Once inside the event horizon, theobserver inexorably proceeds from larger to smaller radiusand finally to zero radius. Because the radius exhibits thistime-like, unstoppable, "count-down" behaviour, itbehaves as a time coordinate, not a spatial coordinate.The curvature singularity at zero radius is thus a momentin time, not a position in space, and it is as unavoidable forthe observer trapped inside the black hole as next Tuesdayis for those of us outside the black hole.

Similarly, inside an (ideal) rotating black hole, there is aregion near the curvature singularity where the warpingand rotation is so violent that the rotation angle, (J>,becomes a time-like coordinate. Because (J) is both time-like and periodic (with a period of 2rc) there are closedtime-like curves in this region of the space-time.

The closed time-like curve region of the rotating blackhole solution, however, is not considered physical forreasons unrelated to the closed time-like curves. First, thecurvature singularity lies to the past of the closed time-likecurve region. For observers in that region the singularity isvisible and boundary conditions for what comes out of thesingularity must be imposed by some (unknown) addi-tional physics. In principle anything at all could come outof the singularity and ruin the closed time-like curves.Secondly, and perhaps more importantly, there is anunstable region to the past of the closed time-like curves. Itis widely thought that the end-point of this instability is a

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4 The twin paradox acting on a wormhole can produce a time machine.The arrows connect momenta of the same local time. Closed time-likecurves can form after f = 3. The closed time-like curve chosen In red Isclosed by a wormhole - the difficult part of this experiment

curvature singularity which any observer would meetbefore any closed time-like curves could form.

Solutions with negative mass/energy can also be used tosupport closed time-like curves and evade the theoremsproposed by Hawking, Tipler and others. While there is noknown classical matter with negative mass, it can beproduced from quantum effects widiout too much effort.The simplest example of this is the Casimir force betweentwo uncharged conducting plates (or molecules). Thisforce, which is distinct from and weaker than the van derWaals force, was predicted in 1948 and confirmed in1958, and is caused by vacuum fluctuations of theelectromagnetic field due to the Uncertainty Principle.The fluctuations are smaller in the presence of theconducting plates than they are in free space, which leadsto an attractive force. Furthermore, the energy densitybetween the plates is negative compared to that in freespace. In flat space-time, this negative energy is notenough to maintain closed time-like curves, but in space-time with gravitational curvature (as in our Universe) thereare currently no known obstacles, in principle, to creatingenough negative energy to gravitationally support closedtime-like curves.

Mike Morris and Kip Thome at Caltech and others usethis negative energy to create closed time-like curves usingwormholes to connect one region in space-time to anotherdirectly. Without negative energy to support a wormhole,general relativity predicts that it will collapse in even lesstime than it would take light to cross the wormhole. Therepulsive nature of the gravitational field arising fromnegative energy matter is what supports die wormhole.The two regions connecting the wormhole need not bewidely separated in space, indeed if one can arrange for thetwo ends to be separated narrowly in space, but distandy intime, this is precisely what we need for a time machine.

Conceptually, the easiest way to make a time machineout of a wormhole is via the twin paradox of specialrelativity (figure 4). In this example one twin (Alex) staysat home whilst die other (Brett) zips off at high speed,

rejoining Alex at a later time. Due to time dilation, muchless proper time passes for Brett than for Alex by the timeof their reunion (the arrows in figure 4 connect momentsof the same local time for each twin). If Alex and Brett areclever enough to hold on to die ends of a very shortwormhole while diey are travelling, diey can stay in nearinstantaneous contact for the whole of Brett's journey bycommunicating dirough the wormhole instead of thoughnormal space. In figure 4, r = 3 represents the occurrenceof the first closed light-like curve. Starting at mat time,Alex can send a message to Brett at the speed of lightdiough normal space, and then Brett can send die samemessage back to Alex via die wormhole (almost instanta-neously if the wormhole is very short), where it arrives atthe same moment it was sent. After diat time diere areclosed time-like curves, as demonstrated by the red path,which close via die wormhole. The worst difficulties increating a time machine of this type are finding a wormhole,stabilizing it widi enough negative energy matter andkeeping die ends of die wormhole near the twins.

Amos Ori at die Technion in Israel has recendy found anew solution to die Einstein equation with severalinteresting properties. The closed time-like curves evolvefrom a causally well-behaved past, and die solution istrivially flat and causal at infinity and topologically trivialeverywhere (unlike die wormhole time machine). Further-more, die solution does not require negative energy whendie closed time-like curves first form, but it does havenegative energy after dieir formation. However, it is not yetknown what would happen if die region where negativeenergy is used has positive energy matter put in its place. Itis also not yet known whedier diis solution is stable againstdie classical field (infinite) build-up described earlier.

The future of closed time-like curves?Aside from die classical difficulties described above,quantum field tiieory adds new difficulties to die creationof closed time-like curves. In his "chronology protectionconjecture" Stephen Hawking argues diat physics as awhole conspires to prevent die formation of closed time-like curves. The conjecture is bolstered, but not proven, bya two-tiered support system: die classical properties ofgeneral relativity and die quantum properties of fields incurved space-times. However, die classical dieorems (ofHawking, Tipler and odiers) are not enough to support dieconjecture on their own, especially concerning cosmolog-ical solutions witii closed time-like curves or solutions diatuse negative energy matter.

The second attack is based on die properties of quantumfields in space-times which contain, or are about togenerate, closed time-like curves. Sung-Won Kim andKip Thome at Caltech have shown diat die energy densityof quantum field states grows widiout bound as diemoment of creation of die first closed time-like curvesapproaches. While diis does not rule out closed time-likecurves, it does mean diat die approximation of havingquantum fields evolve in a background space-time widioutgravitationally affecting die space-time itself breaks downas die moment of creation approaches. Unfortunately diecorrect treatment of this strong field back-reaction wouldrequire a full dieory of quantum gravity, which we do nothave. The reason for die divergence of die energy density isdiat die vacuum modes propagate around the closed time-like curves and cause quantum instability, even when diespace-time is stable against classical field build-up.

One possible exception to diis general behaviour mightoccur if the divergence of the energy density is so slow diat

Ptiytlc* World December 1984 3 3

it does not reach the Planck scale (E?\ = ^J (he51G)= 1 x 10iy GeV) until one Planck time (~104J s) beforethe epoch of closed time-like curves. Since this type offluctuation is expected even for ordinary flat space whengravity is quantized (particularly in John Wheeler's pictureof space-time foam), it is not at all obvious that this wouldspell doom for the closed time-like curves. Matt Visser ofWashington University in St Louis has found examples ofspace-times with closed time-like curves formed by twowormholes with this very slow divergence, althoughadmittedly only for very extreme parameters. If the twowormholes are separated by 1 astronomical unit (abouteight light-minutes) the wormholes themselves can only beup to 10~18 cm large, meaning that only subatomicparticles of energies greater than 20 TeV could passthrough, and even then they could only travel eightminutes into the past.

Future thoughtsThe physics of closed time-like curves is a diverse andfascinating subject, and its discussion should not bemonopolized by science fiction aficionados at the expenseof physicists. Moreover, the mere existence of a singleclosed time-like curve anywhere (and at any time) in theUniverse forces us to choose between diflferent versions ofquantum mechanics that, in other circumstances, wouldbe equivalent. •

The question of whether closed time-like curves areforbidden in principle by the laws of physics is still veryopen, though, between general relativity and the generalbehaviour of quantum fields in curved space-times, the

deck is certainly stacked against macroscopic closed time-like curves. The question of microscopic (Planck-scale)closed time-like curves remains open, however, and islikely to remain so until a full theory of quantum gravityhas been found.

It is all one to me where I begin, for I shall come back thereagain in time. Parmenides

Further reading

B Allen and J Z Simon 1992 Time travel on a string Nature 35719D Deutsch and M Lockwood 1994 The quantum physics of timetravel Scientific American 270 68 (March)F Echeverria, G Klinkhammer and K S Thome 1991 Billiardballs in wormhole space-times with closed time-like curves: I.classical theory Phys. Rev. D44 1077J B Hartle 1994 Unitarity and causality in generalized quantummechanics for non-chronal space-times Phys. Rev. D49 6543S W Hawking 1992 Chronology protection conjecture Phys. Rev.D46 603D Politzer 1994 Path integrals, density matrices, and informationflow with closed time-like curves Phys. Rev. D49 3981K S Thome 1991 Do the laws of physics permit closed time-likecurves? Annals of the New York Academy of Sciences 631 182K S Thome 1994 Black Holes and Time Warps (Norton, NewYork)

Z Simon' 1$ tn th» Department ofUniversity of Maryland. GoJtoo* Park, MO £0742, USA

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