Abstract
H. G. Wells’ Time Traveller inhabits uniform Newtonian time. Where relativistic/quantum travelers into the past follow spacetime curvatures, pastbound Wellsians must reverse their direction of travel relative to absolute time. William Grey and Robin Le Poidevin claim reversing Wellsians must overlap with themselves or fade away piecemeal like the Cheshire Cat. Selfoverlap is physically impossible but ‘Cheshire Cat’ fades destroy Wellsians’ causal continuity and breed bizarre fusions of travelerstages with opposed timedirections. However, Wellsians who rotate in higherdimensional space can reverse temporal direction without selfoverlap, Cheshire Cats or mereological monstrosities. Alas, hyperrotation in Newtonian space poses dynamic and biological problems, (e.g. gravitational/electrostatic singularities and catastrophic bloodloss). Controllable and survivable Wellsian travel needs topologicallyvariable spaces. Newtonian space, not Newtonian time, is Wellsians’ real enemy.
Introduction
Physical travelers into the past must be Gödelians or Wellsians. The former follow spacetime curvatures; the latter reverse temporal direction relative to absolute time.^{Footnote 1} Newtonian and relativistic physics seemingly forbid Gödelians and Wellsians respectively. Can absolute time admit backward time travel or is curved spacetime required? Grey (1999), Le Poidevin (2005) claim reversing extended Wellsians must overlap with themselves or fade away piecemeal like the Cheshire Cat. Either way, travelercontinuity fails and Wellsian reversals are physically impossible.
However, Wellsians have options besides selfoverlap or fades, which reveal surprising links between time travel, spatial topology and biology. Pace Grey (1999), Le Poidevin (2005), absolutetime travel is possible if space is sufficiently accommodating. Wellsians can reverse without discontinuity or overlap if they rotate in higherdimensional space. However, rotation in Newtonian hyperspace is neither (humanly) controllable nor survivable. Controllable, survivable Wellsian travel requires dynamic, variabletopology spaces. Wellsian time can be Newtonian, absolute and uniform; Wellsian space cannot. Wellsian travel requires that space itself be plastic.
Lewisian time travel
Time travel has been variously defined but David Lewis (1976: p. 145) defined it best:
Inevitably, it involves discrepancy between time and time. Any traveler departs and then arrives at his destination; the time elapsed from departure to arrival (positive, or perhaps zero) is the duration of the journey. But if he is a time traveler, the separation in time between departure and arrival does not equal the duration of the journey.
Discrepancies separate external and personal time. External time is time in a suitablyinclusive reference frame. (E.g. a galaxy’s center of mass.) Personal time is travelertime, not a “further temporal dimension, but rather the way in which time is registered by a given object: a heart beating, hair growing, a minute hand moving, a candle burning”, (Le Poidevin 2005: p. 339). Travelers’ memories, watches, digestion, (etc.) register personal time but none individually constitutes it. (Watch malfunctions or hibernation are not time travel.) Time travel affects everything in travelers’ reference frames.
Following Lewis (1976), traveler identity and personal time require correct causal continuity between travelerstages, and dissolve if interstage continuity fails. Whatever their relations to external time, genuine travelers are personal continuants:
For time travel requires personal identity—he who arrives must be the same person who departed. That requires causal continuity, in which causation runs from earlier to later stages in the order of personal time (Lewis 1976: p. 148).
If the usual causal connections underlying traveleridentity and personal time fail en route, the process isn’t travel. Stages of genuine travelers are not joined merely by (e.g.) coincidental likenesses, spatiotemporal proximity or accidentallyshared purposes, howsoever close.^{Footnote 2} Interstage qualitative similarity alone cannot make just any stageaggregate into a traveler if correct interstage causal links are lacking:
... What unites the stages (or segments) of a time traveler is the same sort of mental, or mostly mental, continuity and connectedness that unites anyone else. The only difference is that whereas a common person is connected and continuous with respect to external time, the time traveler is connected and continuous only with respect to his own personal time (Ibid.).
Furthermore, if travelers are normal concrete things, their constitutive links should be spatiotemporally local. Any process that requires spatiotemporally discontinuous interstage links arguably does not involve bona fide travel. This papers argues that extended Wellsians cannot maintain correct interstage causal links (and hence personal time) unless space has extra dimensions or (preferably) variable structure.
Gödelians and Wellsians
Wells (1895) has a thoroughly Newtonian background. Newtonian space and time are mutually independent, eternal substances. Newtonian time is absolute and uniform—passing everywhere in the same direction, at the same rate. Newtonian worlds as a whole possess unambiguous time functions and absolute simultaneity, i.e. have a universal ‘now’. In relativity, space and time are not independent existents but twin aspects of spacetime. General relativity predicts matter’s presence affects spacetime itself and allows time (in effect) to ‘curve’. Gödel (1949) describes spacetimes so curved as to allow ‘closed timelike curves’ (CTCs): journeys that are always (locally) futuredirected yet eventually rendezvous with their own spatiotemporal starting points. Parts of Gödel universes (e.g. galaxies) can have welldefined timefunctions but Gödel universes in toto cannot: they have no universal ‘now’. (Simultaneity relations cannot be defined for their entirety.) Gödel universes neither begin nor end but simply are: infinite, fourdimensional blocks with strange geometrical ‘twists’ that let travelers visit any (externally) earlier or later times. Wellsians take peculiar journeys in otherwise conventional universes; Gödelians are otherwise conventional travelers in strangelystructured spacetimes. Wellsians actively go against time’s local flow; Gödelians passively follow time’s local direction. (Wellsians resemble helicopters in a uniform breeze; Gödelians, balloons in a cyclone.) In effect, Gödelians merely persist in idiosyncratic directions. Gödelians thus don’t much resemble time travelers in fiction, who can generally go anywhere in history. Indeed, some (notably Le Poidevin 2005) think Gödelians’ relative lack of room for maneuver means they cannot genuinely timetravel. If so, problems for Wellsians doom physical time travel.
Newtonian worlds are temporally wellbehaved. Gödel (1949) universes have CTCs through every point. CTCs have welldefined past and future directions locally (in the traveler’s vicinity) but not globally (viewed as a whole). Gödelians can reencounter their own pasts by traveling into their local future. Past and future are only relatively distinct on CTCs, rather as ‘up’ and ‘down’ are not absolute distinctions but relative to the Earth’s center. (Similarly, clockfaces have welldefined clockwise and anticlockwise directions at every point yet a sufficiently prolonged clockwise journey revisits its starting point.) Reversing direction against external time gives Wellsian travel clear start and endpoints. Such reversals present the problems which are the main focus here. Starts and finishes for Gödelian travel are less clear, as temporal oddities show up only for the whole journey: “In Gödel’s spacetime the local temporal and causal order will not differ from the one we are familiar with in our world; deviations can only occur for global distances”, (Pfarr 1981: p. 1090, emphasis original).
Wells (1895) features no processes recognized by relativity. Relativistic time travel (time dilation or CTCs) requires movement and/or gravitational differences between personal and external reference frames. However, Wells’ (1895) Traveller sits at rest (relative to the Earth) with his personal time varying in direction and/or rate of travel relative to all surrounding objects. Everything outside Wells’ (1895) machine registers one external time, while the machine enjoys a unique personal time without motion or gravitational differences relative to its surroundings. During backward timetravel, Wells’ (1895) Traveller sees everything outside the Machine seemingly go into reverse. Unlike Wellsians, Gödelians see no apparent failures of entropy (or other temporal anomalies) in their immediate extravehicular surroundings. Wellsian travel requires locally and globally backward causation; Gödel travel requires only the latter.
Below are depicted one Gödelian and seven Wellsians in various onedimensional and twodimensional spaces, (‘Linelands’ and ‘Flatlands’ respectively).^{Footnote 3} All illustrations have one timedimension.^{Footnote 4}\(\hbox {Wellsian}_\mathrm{(a)}\) is an unextended (pointlike) object which reverses temporal direction twice (first at \(\upphi \) and then at \(\upomega \)) but otherwise persists normally (Fig. 1).
External time increases and personal time decreases \(\upomega \)to\(\upphi \). Each personal moment holds one Wellsian. Each external moment \(\upomega \)to\(\upphi \) (e.g. m1) holds three Wellsians, the midmost growing younger as external time increases. Viewed externally, two Wellsians seem to converge on \(\upphi \) and disappear, while two seemingly appear ex nihilo at \(\upomega \) and diverge. If these phases form parts of one history, physical continuity (and personal time) must survive \(\upphi \) and \(\upomega \)—otherwise, they are three separate objects and not one continuant. Wellsian roundtrips require both \(\upphi \)style and \(\upomega \)style reversals. Wells’ (1895) Traveller departs from/returns to external times around 10 am and 7.30 pm respectively on one day early in 1894. Meantime, he spends eight personal days visiting 802,701 AD and then c. thirty million AD. He presumably reverses \(\upphi \)style leaving the future and then \(\upomega \)style reentering 1894.^{Footnote 5}
Wellsian reversals seem temporally peculiar but spatially benign—at least for unextended objects. However, extended Wellsians risk overlapping later and earlier selves. What Grey (1999: pp. 60–61) baptizes double occupancy looms pre\(\upphi \) and post\(\upomega \), with “not one but two machines—one going backwards and the other forwards—each apparently occupying (or attempting to occupy) the same location”. Wellsians face identical difficulties at \(\upphi \)style ‘apexinstants’ or \(\upomega \)style ‘nadirinstants’—it’s changing direction that poses problems. Can extended concrete objects make such transitions?
Dowe (2000: p. 446) suggests reversing Wellsians avoid selfoverlap if they also move in space: “To do this sort of time travel you have to take a run up”. Passersby might see a departing Wellsian machine “moving across a field, and ... its reversed later self moving towards it (perhaps an antimatter time machine, perhaps not)”, until “the two collide, apparently annihilating both”, (ibid., emphasis added). However, as Le Poidevin (2005: p. 344) shows and Fig. 2 illustrates, movement during reversal still leaves some overlap inevitable. (At least movement at finite velocity. Infinite velocity, even if coherent, threatens discontinuous existence or multiple overlapping stages.) Moving is not enough.
Stationary \(\hbox {Wellsian}_\mathrm{(b)}\) reverses once at \(t_{2}\) so two copies overlap completely at every moment shown. Cooccupancy is more localized and transitory for moving \(\hbox {Wellsian}_\mathrm{(c)}\). The leading and trailing edges of \(\hbox {Wellsian}_\mathrm{(c)}\) are ‘B’ and ‘A’ respectively. In external time, two of \(\hbox {Wellsian}_\mathrm{(c)}\) first meet at \(t_{1}\), when the forward phase’s edge B meets the backward phase’s edge A. These two \(\hbox {Wellsians}_\mathrm{(c)}\) then progressively overlap until they coincide at \(t_{2}\) and vanish thereafter. \(\hbox {Wellsians}_\mathrm{(b, c)}\) are extended objects which multiply occupy the same spaces for extended periods. If one Wellsian can’t exclude another from its space \(t_{1}t_{2}\), is it still concrete at such times? Genuine solutions to double occupancy must at least let Wellsians behave like concrete objects that have continuous histories.
Time travelers who are Gödelian, unextended or spatiotemporally discontinuous all evade double occupancy. Likewise, diverse tropes, universals, fields or sortals^{Footnote 6} can cooccupy. However, “two [concrete] objects of the same kind (persons, chairs, iron spheres)”, (Le Poidevin 2005: pp. 336–337) cannot cooccupy. Napoleon and his dresssense can cooccupy but Napoleon and Wellington cannot, and nor can two Napoleons.
Interestingly, Le Poidevin (2005: p. 350) thinks backward time travel should be something travelers do (or initiate), and should proceed against time’s global direction:
It is, I think, a moot point whether simply following a closed timelike curve in worlds where there is no global earlierlater direction constitutes genuine timetravel. Arguably, time travel is something that, as we might put it, goes against the grain of spacetime, rather than simply following it.
If Gödelians are mere slaves of curved spacetimes (with no global time direction to buck), only Wellsians could truly timetravel. (For similar sentiments, see Torretti 1999: p. 79, n. 20.) Even Gödel thought Gödelians weren’t strictly time travelers. He thought the possibility of CTCs proved that dynamic time has no objective analogue, therefore our temporal experience tracks only an ‘ideal’ (apparent or nonobjective) time and relativity’s ‘t’ coordinate is not truly timelike. (See Yourgrau 1999). Hence a dilemma: without global time, Gödelians can’t timetravel; Wellsians could (per impossibile) timetravel but spatial problems (e.g. cooccupancy) foil them.
But must time travel be Wellsian? Lewis’s (1976) necessary and sufficient condition for time travel, i.e. discrepancy between external and personal timeregisters, covers Wellsian and Gödelian alike. Lewis (1976) defines time travel by outcomes, not topologies or methods, and leaves unspecified how discrepancies arise. Pace Gödel, suppose a wormhole takes your (apparent) personal history from 2045 to execution for witchcraft in 1645. If so, being told you hadn’t really timetraveled (and/or time is ideal) seems cold comfort. However, Le Poidevin (2005) raises relevant issues even if Gödelians truly timetravel. Gödel universes may lack universal times but other relativistic models permit them.^{Footnote 7} If future physics restores global times, only Wellsian travel remains. Sections 1–3 surveyed Lewisian time travel, Gödelian/Wellsian differences and double occupancy. Next, Sect. 4 differentiates double occupancy from bilocation and persistence problems. Sections 5–11 outline and refine candidate answers to double occupancy.
Two things double occupancy is not
(1) Bilocation bilocated timetravelers occupy two distinct places at the same external time and look like two different people, with different personal ages. Bilocation arises if (e.g.) a traveler meets her earlier self. However, doubleoccupying objects multiply occupy the same place at the same external time. Wellsians or Gödelians can bilocate but only Wellsians doubleoccupy. Spatially stationary Wellsians cooccupy without bilocation. Time travelers who meet but don’t overlap with themselves bilocate without cooccupancy. \(\hbox {Wellsian}_\mathrm{(b)}\) only cooccupies; \(\hbox {Wellsian}_\mathrm{(c)}\) cooccupies and bilocates.
(2) Apersistence problem the two chief philosophies of persistence, endurantism and perdurantism, characterize cooccupancy differently but neither evades it, (cf. Le Poidevin 2005: pp. 337–338). For endurantists, persisting objects are wholly present at a time; for perdurantists, objects extend across time. Endurantism implies Wellsians must wholly and simultaneously occupy the same space more than once. Perdurantism implies distinct temporal parts of Wellsians must cooccupy. Either way, each temporal part (or version) is a spatiallyextended concretum with its own associated mass and shape.
Note Wellsians’ spatial problems are not truly solved by letting Wellsians (e.g.):

exist discontinuously in space or time,

become impalpable, abstract or ghostly,

have identityconditions supervening on noncausal facts,

utilize some form of (quasiNewtonian or quantum) nonlocality,

have different haecceities for forward and backwardtravelling phases,

lack welldefined spatial relations with other simultaneouslyexisting objects,

bear different (temporal parts of) instantiation relations to identical regions etc.
All such exotica seem ad hoc. Worse, they simply grant (Grey 1999; Le Poidevin 2005) that reversing Wellsians can’t behave like classical concrete objects. Wellsians’ problems are spatial and only spatial solutions appear herein. To summarize the problem:

At most one (stage or version of a) concrete object of a given kind can occupy an extended spatial region at a time.

Reversing Wellsians change temporal direction relative to their surroundings.

Wellsians are spatiotemporallycontinuous extended concrete objects.
Following sections aim to show that these attributes are jointly achievable but that no proper solution to Wellsian problems has appeared hitherto. Sections 5–6 criticize previous solutions. Sections 7–9 explore new, if unsuccessful, solutions. Adapting Sect. 8 ‘hyperspace’ model, Sects. 10, 11 propose a true solution involving altering space itself.
Wells on double occupancy
Wells himself variously considered cooccupancy—albeit between travelers and other objects, and not between different stages/versions of travelers. Wells (1895) compares time travelers whizzing through history at different ‘speeds’ to objects crossing space at differing speeds (see Le Poidevin 2005: pp. 338–340). Wellsians who traverse (e.g.) 50 external seconds per personal second travel 50 times ‘faster’ and make 1/50th of their ‘stationary’ impression compared to normallypersisting objects, just as bullets make attenuated visual impressions to observers they pass. However, stationary observers see past speeding bullets and not through them, i.e. derive visual impressions primarily from whichever background the passing bullets briefly obscure. An observer who comoves with a bullet should see it obscure its background just like any other object. Similarly, anybody shot will find high velocity exacerbates bullets’ impact rather than diminishes it. Why should perceptual attenuation make objects less present at places they occupy? Physical presence is the problem, not perception of it.
Wells’ (1894) serial version of The Time Machine at least treats cooccupancy physically as well as perceptually. Once Wells’ (1894) ‘Philosophical Inventor’ has described traveling to and from A.D. 12,203, one of his auditors asks why no future people noticed him while he occupied the same (relative) place for millennia. At first, the Inventor waxes analogical in reply, invoking perception like his (1895) successor:
Suppose, for instance, you put some red pigment on a sheet of paper, it excites a certain visual sensation, does it not? Now halve the amount of pigment, the sensation diminishes. Halve it again, the impression of red is still weaker. Continue the process. Clearly there will always be some pigment left, but a time will speedily arrive when the eye will refuse to follow the dilution, when the stimulus will be insufficient to excite the sensation of red. The presentation of red pigment to the senses is then said to be “below the threshold.” Similarly my rapid passage through time, traversing a day in a minute fraction of a second, diluted the stimulus I offered to the perception of these excellent people of futurity ... (Geduld 1987: p. 159)
Note Wells’ (1894) Inventor glosses machine ‘speed’ using personal/external time: in ‘rapid passage’, fractional personal seconds encompass external days. Conceding the perceptual point (maybe prematurely), the questioner then raises physical cooccupancy:
I suppose while you were slipping thus invisibly through the ages, people walked about in the space you occupied. They may have pulled down your house about your head and built a brick wall in your substance. And yet, you know, it is generally believed that two bodies cannot occupy the same space (Geduld 1987: p. 159).
Here, the Philosophical Inventor replies with a non sequitur about molecular diffusion:
Don’t you know that every body, solid, liquid, or gaseous, is made up of molecules with empty spaces between them? That leaves plenty of room to slip through a brick wall, if you only have momentum enough. A slight rise of temperature would be all one would notice and of course if the wall lasted too long and the warmth became uncomfortable one could shift the apparatus a little in space and get out of the inconvenience (Ibid.).
But varying personal/external time shouldn’t mitigate spatial cooccupancy. Why can’t any solids interpenetrate given sufficient relative momentum? Anyway, gasmolecules slipping through membranes move relative to whatever they permeate; Wells’ machines hold stationary relative to any cooccupied objects. Interpenetration is nothing to the purpose: gaseous diffusion never involves multiple extended concreta simultaneously occupying numerically the same space. (Anyway, as Wells’ 1894/95 travelers can see external events, neither is really impalpable while traveling.) While Wells’ attempts fail, other options have been proposed. Section 6 surveys Le Poidevin’s (2005) suggestions, whereby reversing Wellsians perforce disintegrate (and reintegrate) en route. While Le Poidevin (2005) is clear that such solutions themselves fail, his critique is instructive.
Cheshire Cats and chimeras
Le Poidevin (2005: p. 345 ff.) suggests Wellsians can avoid selfoverlap by departing ‘Cheshire Cat’ fashion, each part having a separate departuretime. \(\hbox {Wellsian}_\mathrm{(d)}\) departs piecemeal: edge B leaves at \(t_{1}\) and edge A at \(t_{2}\). Selfoverlap is avoided because each part reverses at a different apexinstant. However, the need to maintain continuity makes travelers’ forward and backward phases peculiarly contiguous. Henceforth fusions of contiguous temporallyopposed masses are ‘chimeras’ and twoheaded chimeras ‘PushmiPullyus’.^{Footnote 8} Above right, a reversing astronaut’s two phases meet toetotoe at \(t_{1}\) and fuse without overlap. Nearing \(t_{2}\), our astronaut’s remains have dwindled to two contiguous copies of a scalp. This twinscalped chimera diminishes until vanishing at \(t_{2}\) (Fig. 3).
Chimeras and PushmiPullyus can last arbitrarily long, depending on their parent phases’ size and velocity. Travelerorientation determines chimeras’ shape. (Their phases needn’t be mirrorimages, e.g. if travelers rotate while reversing.) Headfirst travelers vanish feetlast, their headless chimeras “too horrible to contemplate”, (Le Poidevin 2005: p. 346). A fourlimbed, backfirst, traveler yields four chimeras—one per limb. Whither the traveler in this fourway fission? That traveler identity could supervene purely on said traveler’s scalp seems implausible but still more so is a traveler surviving for unspecified periods as four spatiallydisjoint objects, (none of them containing a brain). Can physical travelers who lack brains have psychological states?
\(\hbox {Wellsian}_\mathrm{(c)}\) departs in one piece at \(t_{2}\). \(\hbox {Wellsian}_\mathrm{(d)}\) has no clear departuretime: its front (B) reverses at \(t_{1}\) and its back (A) at \(t_{2}\). \(\hbox {Wellsian}_\mathrm{(d)}\) is all gone by \(t_{2}\) but \(t_{2}\) isn’t the departuretime. Continuity and identity fail if Wellsians and their spatial parts cannot attend all external times in their careers: “How can something continue to exist if there are (external) times when only a spatial part of it does so?”, (Le Poidevin 2005: p. 345). Cheshire Cat fades destroy travelers and are not mere interludes in their careers. ‘Cheshire Cat’ solutions must let all a Wellsian’s spatial parts visit all times in its career and retain normal causal powers. Travel shouldn’t oblige travelers to shed themselves.
Cooccupancy, Cheshire Cats and chimeras all imperil personal time. For the Januslike PushmiPullyu at \(t_{1}\), \(t_{2}\) is in its left head’s personal future but its right head’s personal past—it also registers \(t_{1}\) twice in experience (once per head) and once in memory (right head only). However, if Cheshire Cats destroy traveleridentity, personal time fails for PushmiPullyus and memories don’t survive headtohead.
Chimeras have problematic personal time even if causal links survive Cheshire Cat fades. Figure 4 lefthand traveler is pushed rightward at \(t_{1}\). All its personally subsequent stages are displaced rightward, whether externally later or earlier. Figure 4 righthand traveler is pushed leftward at \(t_{1}\). This push affects only externallyearlier stages with \(t_{1}\) twice in their personal past. Leftward pushing has direct effects externally later and earlier; rightward pushing has direct effects only externally earlier. When the leftmost traveler revisits \(t_{1}\), where it reintersects \(t_{1}\) is largely determined by influences exerted at that very (external) time. Even if causal continuity survives reversal, chimeras allow effectively instantaneous causal transmission along a timeslice and hence their interstage causal connections aren’t linearly ordered even in personal time, (cf. Le Poidevin 2005, p. 348). So phases of Cheshire Cats can’t comprise genuine travelers because they lack correct interstage links. However, chimeras’ contiguous masses can’t comprise genuine travelers either because of their surfeit of interstage links. Again, traveleridentity requires that “causation runs from earlier to later stages in the order of personal time,” (Lewis 1976: p. 148).
Mereological universalists^{Footnote 9} think that any sum of temporal parts composes an object but yet insist that genuinely salient continuants only fall under certain sortals (e.g. ‘person’) and embody correct causal links. Mereological universalism alone can’t save Wellsians because chimeras’ saliencecriteria are problematic and their personal time illordered. Fusions of causallydisjoint stages are not persons even if made of persons. The Cheshire Cat diachronic question is ‘When does this thing depart?’; the PushmiPullyu synchronic questions are ‘What is this thing?’ and ‘Is this one thing?’.
Also, electrostatic problems loom as chimeraphases get arbitrarily close together:
How close do the series get to each other? The answer is that they actually make contact, before converging on a single temporal part. This is genuine contact, not the mere close proximity that passes for contact in everyday situations. The subatomic structure of matter imposes limits on how close atoms can get to each other. But [a reversing Wellsian] transcends those limits (Le Poidevin 2005: p. 349).
A reversing hydrogen atom would see its two phases’ nuclei fuse into a single proton while their electronic orbits overlap. Discrete (i.e. only finitelydecomposable) space might alleviate unphysical proximity but Newtonian space is supposed to be continuous. ‘Forward’ charged particles should repel their own ‘backward’ counterparts if they retain constant charge through reversal. Coulomb’s inversesquare law implies arbitrarilysmall separations of like charges require unbounded energy: their mutual repulsion blows up arbitrarily as their separation approaches zero. The closer they get, the harder getting them any closer becomes and the more readily they spring apart. Fusing like charges strictly governed by Coulomb’s law is physically prohibitive.
What if a reversing particle’s phases have opposite charges? Famously, Wheeler–Feynman theory says “positrons can be represented as electrons with proper time reversed relative to true time”.^{Footnote 10} If so, the two phases’ charged particles attract their opposite numbers but their backwardtraveling phases are antimatter. In which case, oppositelydirected phases cannot form chimeras without mutual annihilation. (As Grey (1999: p. 62) notes, Wheeler–Feynman reversals should be highly energetic.) An electrostatic dilemma: reversing constant charges requires unbounded energy but reversing opposed charges threatens mutual annihilation. Physics aside, logic forbids mutual annihilation between an object’s earlier and later phases. An earlier phase can destroy a later without inconsistency. However, a later phase destroying an earlier yields a ‘Grandfather’ (strictly ‘Autoinfanticide’) Paradox. At least Wheeler–Feynman theory suggests how Wellsian reversal might occur: if backwardmatter is antimatter, inverting a particle’s charge should induce Wellsian reversal. However, it remains unclear what could switch charges or reverse uncharged particles. (Section 10 revisits and addresses Wheeler–Feynman issues for Wellsians.) Clearly, appeals to motion, perception, interpenetration or disintegration alone cannot solve double occupancy. Section 7 considers new ‘solutions’, wherein travelers can radically change shape.
Singularity Wellsians
Sections 2–6 suggest some lessons: To avoid cooccupancy and chimeras, Wellsians’ stages must never overlap or form contiguous masses. To avoid Cheshire Cats, all a Wellsian’s spatial parts should be able to visit all external times in its career. Sections 7–11 consider new therapies for cooccupancy, Cheshire Cats and chimeras. Those of Sect. 7 involve radical shapechanges and arbitrarilydense matter. Those of Sections 8–11 allow constant shape and density but require extra spatial dimensions (Sects. 8, 9) or variable topology (Sects. 10, 11). \(\hbox {Wellsians}_\mathrm{(ac)}\) reverse intact while \(\hbox {Wellsian}_\mathrm{(d)}\) decomposes into particles. Spatially twodimensional \(\hbox {Wellsian}_\mathrm{(e)}\) begins circular but drastically changes shape: its xextremities (A, B) get arbitrarily close and its zextremities (C, D) arbitrarily separated at \(t_{1}\). \(\hbox {Wellsian}_\mathrm{(e)}\)’s crosssection is illdefined at \(t_{1}\) but otherwise constant (Figs. 5, 6).
All of \(\hbox {Wellsian}_\mathrm{(e)}\) registers completely at least once on every global timeslice. Its phases also only meet along a line—so no overlapping parts, Cheshire Cat fades or chimeras.
Problem solved? Alas, no: consider what happens at \(t_1\). Completely avoiding overlap while keeping crosssectional area constant drives \(\hbox {Wellsian}_\mathrm{(e)}\) towards nil length and unbounded breadth at \(t_1\): \(\hbox {Wellsian}_\mathrm{(e)}\)’s shrinking towards nil xlength must compensatingly drive it towards unbounded size in the ydirection. So \(\hbox {Wellsian}_\mathrm{(f)}\) must go from finite area and breadth to (effectively) zero area and infinite breadth, and back again, in finite time. Since Newtonian physics sets no theoretical limit on how fast objects can travel, at least Newtonian spacetimes permit the arbitrarilylarge velocities such ‘Singularity’ Wellsians need. However, letting parts of Singularity Wellsians proceed to (and from) arbitrary distances arbitrarily fast also needs remarkable scarcity of other objects around \(t_1\) if catastrophic collisions are to be avoided. If \(\hbox {Wellsian}_\mathrm{(f)}\)’s crosssectional area can vary arbitrarily, then at least its zextremities can be finitely far apart traveling through \(t_1\). Even so, \(\hbox {Wellsian}_\mathrm{(f)}\) still hits a linear singularity of zero area and unbounded density at \(t_1\).
One help for ‘Singularity’ solutions is that Newtonian physics allows arbitrarilydivisible matter. However, can objects survive being compressed to infinitesimal length? Real matter has sharply differentiated structure at subatomic, atomic and molecular levels. ‘Infinitesimal’ stages of Singularity Wellsians must have components arbitrarily smaller than (e.g.) any electron. If no physical structure survives reversal, causal continuity fails and \(\hbox {Wellsian}_\mathrm{(f)}\)’s forward/backward phases are distinct continuants. Also, as Singularity Wellsians become arbitrarily dense around \(t_1\), their resulting runaway gravitation should make them ‘dark stars’: classical bodies whose gravitation is so intense no physical object traveling slower than light can escape from their vicinity.^{Footnote 11} Such Wellsians would be invisible to outside observers when reversing but have large gravitational effects on their surroundings. A dilemma looms: restoring ‘dark star’ travelers to finite density/size after reversal seems difficult even if their causal continuity survives the ‘Singularity’, but double occupancy recurs if any finite xextension survives through \(t_1\). Neither option makes ‘Singularities’ survivable.
Coulomb’s law poses problems here too: arbitrarilycompressing multiple like charges requires unbounded energy. (Although electrostatic repulsion might explain how arbitrarilycompressed travelers regain normal size postreversal.) Preserving Wellsians’ causal continuity requires more than spatiotemporal proximity and continuity for their parts. (Parts of exploding objects retain continuous trajectories yet being exploded destroys objects nonetheless.) Bouncing through \((n1)\)dimensional singularities would destroy ndimensional Wellsians. Cheshire Cat solutions cannot merely let each Wellsianpart have a continuous trajectory; Wellsians (and their parts) must also retain:

(1)
Presence on all global times in their histories,

(2)
Finite size and density,

(3)
Continuous trajectories,

(4)
Constant shape.
Singularity Wellsians can observe (1) while their parts can observe (1) and (3). But neither Singularity Wellsians nor their parts can observe (2) or (4). ‘Singularity’ solutions fail. Section 8 considers a better option: allowing Wellsian space extra dimensions.
Hyperrotation
While unsuccessful, Singularity solutions have instructive features, e.g. phases meeting along a frontier with one fewer dimension than their surrounding space. \(\hbox {Wellsian}_\mathrm{(d)}\)’s leading edge is B in forward travel but A in backward. So \(\hbox {Wellsian}_\mathrm{(d)}\) also changes chirality; its phases incongruent counterparts. Doubleoccupancy and Singularity Wellsians maintain congruency but Cheshire Cats resemble another Wellsian protagonist, that of Wells’ (1896) ‘The Plattner Story’.^{Footnote 12} A mysterious explosive blows schoolmaster Plattner into another realm, where he spends 9 days before dropping back into earthly life. On return, Plattner’s body has seemingly swapped left and right sides. As rotation in \((n+)\)dimensions makes orientable ndimensional shapes nonorientable, Wells’ (1896) narrator wonders if Plattner has been rotated in fourdimensional space.
Abbott’s (1884) twodimensional protagonist, ‘A. Square’, visits onedimensional Lineland in a dream. A. Square imagines trying to persuade Lineland’s King that Flatland exists by speaking to His Linear Majesty from His insides. Lindgren and Banchoff (2010: p. 135) suggest a better demonstration, i.e. A. Square ‘Plattnerizing’ Lineland’s King by rotating His Linear Majesty through \(180^{\circ }\) in Flatland. (Lineland’s King has two voices, bass and tenor, one at each end. Hyperrotating the King should make His tenor voice emanate whence His bass voice came and vice versa.)
If Cheshire Cat phases form a single continuant (and so merely change chirality rather than lose identity), their constituent causal connections must survive. Plattner’s alteration involves only spatial movement, i.e. no time travel occurs and no instant holds multiple Plattners or Plattnerstages. Plattner’s chirality alters but his causal continuity survives. What if Wellsians can also rotate Plattnerfashion? \(\hbox {Wellsian}_\mathrm{(f)}\) from a Newtonian ‘Lineland’ reverses in time while rotating in \((xz)\) Flatland ‘hyperspace’.^{Footnote 13}\(\hbox {Wellsian}_\mathrm{(f)}\) rotates through \(90^{\circ }\,t_{1}\hbox {to}t_{2}\) and (personally later) rotates another \(90^{\circ }\,t_2\hbox {to}t_1\). At \(t_2\), \(\hbox {Wellsian}_{\mathrm{(f)}}\) is instantaneously orthogonal to Lineland. Hyperrotation also ‘Plattnerizes’ \(\hbox {Wellsian}_\mathrm{(f)}\), making forward and backward phases into enantiomorphic counterparts (Figs. 7, 8).
\(\hbox {Wellsian}_\mathrm{(f)}\) starts rotating at \(t_{1}\) but begins backward travel (en bloc) at \(t_{2}\). As rotation begins, \(\hbox {Wellsian}_\mathrm{(f)}\) seems to dwindle to a point, apparently leaving only a dimensionless crosssection of itself in Lineland. (Likewise hyperrotating Flatlanders should present linear crosssections to Flatland observers and hyperrotating threedimensional travelers in turn should leave planar crosssections in Spaceland.) All such hyperrotating travelers would appear to lose a spatial dimension when viewed from within their native spaces.
However, unlike Singularity Wellsians’ genuine dimensionloss, these snapfades and dimensional losses are only apparent. In \((xz)\) Flatland, \(\hbox {Wellsian}_\mathrm{(f)}\) retains constant extension and shape. \(\hbox {Wellsian}_\mathrm{(f)}\) has nil xextension when its phases meet at \(t_2\) so no parts overlap and yet all its spatial parts register (in their correct relative places) on every global timeslice. A rotational Wellsian’s stagesum has one more spatial dimension than any individual stage; ndimensional Wellsians forming \((n+)\)dimensional stageaggregates. Each stage has a unique spatiotemporal location without cooccupancy, Cheshire Cats, chimeras, shapechanges or runaway density. Rotating Wellsians change chirality when reversing but otherwise maintain continuity. Hyperrotation meets all four desiderata for Cheshire Cat solutions while avoiding unphysical proximity (and electron overlap) without finitelysized spatial atoms. It isn’t clear how electromagnetic fields from an ndimensional object would propagate if that object were translated into \((n+)\)dimensional space. However, at least if ndimensional charges generate ndimensional chargefronts and Wellsians’ phases meet obliquely in \((n+)\)space, then their phases don’t overlap ‘fieldtofield’. (Consider how two planes that meet obliquely in three dimensions will overlap only at a line and not at a surface.) Hyperrotating travelers’ phases will only meet at apexinstants and along an axis where they are themselves extensionless. Apexinstants hold only one travelerstage and hence threaten no multiple contiguous or overlapping stages. Hyperrotation allows Wellsians sufficient physical continuity for their personal time to be defined without ambiguity or circularity.
Apex instants have ancestors and (distinct) successors at externallyearlier times. But what temporal directions have travelers at apex instants? Presumably, forward or backward yet there seems no basis to assign either. Does hyperrotation make a single travelerstage (instantaneously) temporally bidirectional and dissolve double occupancy only to yield paradox? Not necessarily: apex instants can be finessed by adapting ‘atat’ theories of motion often invoked to address Zeno’s ‘Arrow’ and ‘Stadium’. ‘Atat’ theories of motion can treat instantaneous velocities as illdefined or as limitvalues; instantaneous temporal directions could be treated likewise.
Wellsians approaching apex instants occupy certain places at certain times. Their personal and external time increase during approach. They then occupy certain places at apex instants. Personally thereafter (but earlier externally), they occupy other places. If travelers lose welldefined timedirections at instants where they also cease bilocation, this isn’t necessarily problematic. If Wellsians exhibit neither bilocation nor chimeras at apex instants, where might a physical basis for attributing opposed timedirections reside? Perhaps reversing Wellsians have welldefined timedirections over any finite period but lack timedirection at an (apex or other) instant. Regardless of what happens at apexinstants, cooccupancy and chimeras yield multiple phases with opposed timedirections which (respectively) overlap or fuse for extended periods. ‘Atat’ views of timedirection alone won’t banish double occupancy. Even if (e.g.) \(\hbox {Wellsian}_\mathrm{(c)}\) has one temporally directionless stage at \(t_2\), its phases protractedly overlap approaching \(t_2\).
Hyperrotation means Wellsians’ forward and backward phases join via a single spatiallycomplete phase which instantaneously lacks timedirection. (Nowhere do oppositelyoriented phases overlap or fuse over extended periods.) If time is continuous, there will be no welldefined last instant where two stages exist but apex instants can still be defined, i.e. as instants whose causal ancestors and (distinct) causal successors occupy numerically the same externally earlier times. Apex instants are junctures between externallyincreasing and externallydecreasing times, visited in that personal order; nadir instants are junctures between externallydecreasing and externallyincreasing times, visited in that personal order. The best joint spatial therapy thus far for cooccupancy, Cheshire Cats and chimeras involves reversing Wellsians simultaneously moving and rotating out of local space. Is Wellsian travel saved? Well, progress has been made but, as Sect. 9 shows, hyperrotation seems little real help if (e.g.) rotation is uncontrollable and/or only available to nonliving things. Only Sects. 10, 11 truly dissolve Wellsians’ problems.
Dynamic and biological hyperrotation problems
Even if hyperrotation avoids cooccupancy and chimeras, new problems arise. How might ndimensional objects rotate/translate into \((n+)\)dimensional space? In Wells (1896), the explosion of a mysterious powder triggers Plattner’s rotation but how a (presumably) 3space explosive could rotate Plattner through one more dimension than the explosive itself enjoys seems unclear. If the explosive changes Plattner’s momentum orthogonally to the direction of any exerted force, it violates Newton’s second law. But if the explosive is spatially fourdimensional, Plattner can only handle an infinitesimal crosssection of its true hyperspatial extent. It’s also unclear how the explosive makes Plattner rotate, since any hyperspatial blastfront centered in, and propagated through, 3space would push Plattner through his center of gravity. Maybe Plattner’s transition is dynamically possible if the explosive impinges on him obliquely from outside 3space. Perhaps something fourdimensional ‘peels’ Plattner from 3space, spins him through \(180^{\circ }\) in 3space about an area (i.e. not an axis) internal to him and then drops him back into 3space as his own earlier self’s enantiomorph. However, Platter (like any 3space creature) seems powerless to initiate or control ‘hyperspatial’ (4space) journeys.
Likewise in Abbott (1884), rather than lowerdimensional spaces pushing their occupants forth into hyperspace, higherdimensional creatures reach down and abstract creatures from lowerdimensional spaces. (A visiting Sphere translates Abbot’s Square into 3dimensional Spaceland.) If hyperrotation requires help from higherdimensional life, controllable travel for ndimensional Wellsians requires not only the existence of, but communications with, amenable \((n+)\)dimensional beings. Spacelanders could assist Flatlanders (or Linelanders), and Flatlanders could assist Linelanders, but no realm can facilitate Wellsian travel for its indigenous life unassisted.
Spaces with \((n > 3)\) are usually thought possible. However, James van Cleve disagrees: “for my part, the intuition that three mutually perpendicular lines leave no room for a fourth, or that a sphere intercepts all paths from inside to outside, is as compelling as the intuition that red excludes blue from the same surface”.^{Footnote 14} If so, the maximallydimensioned possible Wellsians are Flatlanders and humans are too ‘deep’ for hyperrotation. Even allowing \((n >3)\) spaces, maybe only \((n = 3)\) spaces permit life, e.g. if \((n \ne 3)\) spaces forbid functional nervous systems, digestive tracts or stable planetary orbits. (Cf. Whitrow 1955.) If so, no \((n \ne 3)\) life can help 3space Wellsians or receive their help.
How might ndimensional objects interact in \((n+)\)dimensional spaces? Perhaps ndimensional objects seem insubstantial to \((n+)\)dimensional ones: “If the Flatlanders are truly twodimensional, with no thickness at all, then they will be as immaterial as shadows or patches of light”, (Rucker 1986: p. 19). However, if ndimensional objects retain finite mass in \((n+)\)dimensions, ndimensional bodies might strike other objects like perfectly sharp blades or prove insusceptible to manipulation against a fixed \((n+)\)dimensional spatial background. Rather than shadows, finitemass Flatlanders might resemble planar ‘dark stars’ in Spaceland. Likewise, Plattner or A. Square might grievously slice or puncture anything they strike during their hyperspatial sojourns.
Hyperrotating humans poses biological problems too. Enclosed (n)dimensional cavities become open in \((n+)\)dimensions. Hyperrotating any enclosing vessel while it translates whisks away the enclosure. Forces rotating human bodies in 4space must act orthogonally to (e.g.) van der Waals forces, contact forces and surfaceadhesion forces that normally hold those bodies’ fluid contents in. Travelers’ blood will tend to retain its original state of motion and thus escape as all but a crosssection of its containing vessels gets rotated into hyperspace. Hyperrotated travelers would presumably die if they shed most, if not all, of their blood, breath, lymph, cerebrospinal fluid etc. ‘Plattner’ hyperrotation relative to (or out of) local space, i.e. travelers’ immediate surroundings, seems highly injurious to anything like anthropomorphic life. (Assuming intermolecular bonds cannot be extended at will, so each traveler effectively becomes one giant molecule. However, this too would be disastrous for life as we know it: a body formed of one giant molecule would be illequipped to absorb nutrients and lose wastes.)
Could ndimensional Wellsians avoid fluidlosses by traveling shielded by other ndimensional objects? Alas, keeping ndimensional bodies completely fluidtight using ndimensional shielding seems a tall order, since the bodies and shields seem obliged to be perfectly contiguous, i.e. with no finitesized gaps at all. (Imagine fluidfilled Flatlanders rotating while sandwiched between two planar shields.) In any event, shielding necessarily interposes other solid objects between travelerphases, yielding another kind of discontinuity. Hyperrotation might be survivable for ‘solidstate’ Wellsians lacking fluidfilled cavities or Wellsians who upload their conscious functions onto nonbiological vehicles before rotation and retrieve them afterwards. (Maybe Wellsians could rotate while cryogenically frozen—although freezing must completely immobilize all their fluid contents, gaseous and liquid.) Even allowing \((n >3)\) spaces (pace van Cleve 1987), hyperrotated humans seem doomed to die and leave messy residues behind. Section 10. shows how hyperrotation can be controllable and survivable, inspired by William Kingdon Clifford (1886: p. 224 ff.) on spatial structure. Only here are double occupancy and Cheshire Cat problems dissolved, and Wellsian travel vindicated.
Cliffordian and neoCliffordian space
Cavityfilled ndimensional objects open catastrophically if rotated/translated in flat \((n+)\)dimensional spaces. What about more complex topologies? Suitable topologies might let Wellsians take their local ndimensional surroundings with them. Suppose space is not uniform and immutable as Newton assumed but flexible, much as ‘rubber sheet’ models of gravitation depict spacetime.
Newton accepted only separate, flat space and time. General relativity postulates curved spacetime. However, William Kingdon Clifford (1845–1879) imagined locallycurved spaces, intermediate between Newtonian space and Einsteinian spacetime. Specifically, Clifford considered the dynamic implications of spatial curvature varying across places and times. Such spaces can also solve Wellsians’ problems. Inspired by Riemann, Clifford wondered if local spatial curvatures cause apparent force phenomena. Perhaps we treat “merely as physical variations effects which are really due to changes in the curvature of our space; [perhaps] some or all of those causes which we term physical may not be due to the geometrical construction of our space”, (Clifford 1886: p. 224). Cliffordian space and matter relate asymmetrically: topology dictates matterdistribution but not vice versa. Adapting John Wheeler’s summary of general relativity: Cliffordian space tells matter how to move, but matter cannot tell Cliffordian space how to curve.^{Footnote 15}
Clifford also postulated that “degree of curvature may change as a whole with the time [and] change of curvature might produce in space a succession of apparent physical changes” (1886: p. 225). If ndimensional space can bulge like a rubber sheet, portions of it can (effectively) rotate in \((n+)\)dimensional space, carrying traveler phases along with them so they meet without overlapping. Such Wellsians wouldn’t rotate relative to local space but instead travel with local space as it ‘rotates’. \(\hbox {Wellsian}_\mathrm{(g)}\)’s Lineland can form branch loops that smoothly diverge from, and reconnect with, their parent space.
Points \(\hbox {A}_{1}\), \(\hbox {B}_{1}\), \(\hbox {A}_{1}^{\prime }\) and \(\hbox {B}_{1}^{\prime }\) all lie in Lineland. Extremities A and B intersect \(t_{1}\) at \(\hbox {A}_{1}\) and \(\hbox {B}_{1}\) when \(\hbox {Wellsian}_\mathrm{(g)}\) travels forward in time and at \(\hbox {A}_{1}^{\prime }\) and \(\hbox {B}_{1}^{\prime }\) when \(\hbox {Wellsian}_\mathrm{(g)}\) travels backward. \(\hbox {A}_{2}\) and \(\hbox {B}_{2}\) are where extremities A and B lie at \(t_{2}\), i.e. the instant when \(\hbox {Wellsian}_\mathrm{(g)}\) proceeds neither forward nor backward in time. A occupies \(\hbox {A}_{1}\), \(\hbox {A}_{2}\) and \(\hbox {A}_{1}^{\prime }\) in that personal order; B occupies \(\hbox {B}_{1}\), \(\hbox {B}_{2}\) and \(\hbox {B}_{1}^{\prime }\) in that personal order. \(\hbox {Wellsian}_\mathrm{(g)}\) effectively leaves Lineland at \(t_{1}\) then loops back to rejoin Lineland at \(t_{1}\) in another place.
\(\hbox {Wellsian}_\mathrm{(g)}\) escapes both stageoverlap and Cheshire Cat fades but never changes chirality or leaves its local linear space. (Figure 9 shows two space dimensions but \(\hbox {Wellsian}_\mathrm{(g)}\) stays in its local linear space throughout.) \(\hbox {Wellsian}_\mathrm{(g)}\) need not even move, but can merely persist forward \(t_{1}\)to\(t_{2}\), and backward \(t_{2}\hbox {to}t_{1}\). Given correct topology (plus ability to move back in time), \(\hbox {Wellsian}_\mathrm{(g)}\) presents no dynamic, spatial or biological difficulties. Figure 9 loop is only spatial – external time increases \(t_{1}\hbox {to}t_{2}\) along both arms. What happens at \(t_{2}\)? In external time, the loop ceases to exist or runs out of space.
\(\hbox {Wellsian}_\mathrm{(g)}\)’s loop only exists for finite external time \(t_{1}\)to\(t_{2}\), and only twice that interval of personal time (i.e. \(t_{1}\)to\(t_{2}\) then \(t_{2}\)to\(t_{1}\)). \(\hbox {Wellsian}_\mathrm{(g)}\)’s reversal needs precise timing: if reversal begins (externally) before \(t_{2}\), \(\hbox {Wellsian}_\mathrm{(g)}\) doubleoccupies; however, reversal later than \(t_{2}\) is impossible because the loop doesn’t exist (externally) later than \(t_{2}\). If entered by an object unable to reverse temporal direction, the loop is a blind alley.
The need for ‘loops’ confines Wellsian travel to spaces where topologicallysuitable connections exist naturally or can be made. Newtonian space is independent of matter. It affects matter (e.g. dictating matter’s degrees of movement) but cannot be affected by it. Newtonian space is multiply hostile to topologicalrotation Wellsian travel. Tridimensionality is neither necessary nor sufficient for space being Newtonian. However, Newtonian space is essentially time/matterindependent and flat. (See e.g. Maudlin 1993.) Rotating Wellsians require either flat \((n+)\)dimensional space or curved ndimensional space. Varying curvature needs nonuniform space; creating curvature needs mutable space. If space cannot be curved to order, Wellsian travel needs preexisting spatial curvatures. Wells’ (1895) Traveler can go anywhere in history; Cliffordian Wellsians are restricted by available spatial curvatures and so stand to space rather as Gödelians do to spacetime.
Clifford’s timeevolving, locallycurved space remains Newtonian qua being absolute, unaffectable by matter and substantival. Letting space and matterdistribution interact reciprocally comes closer to general relativity. Call matteraffectable, locallycurved, absolute space ‘neoCliffordian’. NeoCliffordian spaces allow the local topological (e.g. ‘loop’) connections requisite for survivable reversals. Such spaces can also remain timeindependent substances even if their structures vary over time. (Neo)Cliffordian time can remain Newtonian, i.e. uniform, independent of space and matter, unaffectable by matter and (crucially) possessing global timeslices.
Controllable Wellsian travel needs plastic neoCliffordian spatial structure. Such permits topological connections that let travelers rotate relative to global space but retain constant orientation relative to their local space. This in turn requires suitablylarge (and enduring) curved regions that join smoothly to otherwise uniform space. Given natural correctlyconfigured regions, Cliffordian universes could permit Wellsian travel even if spatial structure was immune from human control. Admittedly such regions would need to be comparatively common, accommodating and easy to locate if Wellsian travel were to be easy, frequent or regular. Also, even given free ingress/egress to topologicallysuitable regions, some additional mechanism seems required for triggering Wellsians’ temporal reversals. If such ‘looped’ regions temporally reverse their contents automatically, the process risks being involuntary (if not de facto Gödelian) and restricted. In contrast, Wells’ (1895) Traveler can theoretically visit all time and space.
An electrostatic puzzle remains to be addressed. If Wheeler–Feynman backwardmatter is antimatter, what prevents even neoCliffordian Wellsians suffering mutual annihilation when their phases connect? Perhaps any requisite ‘warping’ should also affect local timedirection, so rotating Wellsians hold constant orientation relative to local space and time. But if warping means travelers also share time’s local (i.e. not global) direction, any resulting travel is Gödelian, i.e. not Wellsian at all. However, phases of hyperrotating Wellsians meet only as unextended objects at instants and not as extended objects over extended times. The mutual annihilation problem vanishes if ‘atat’ temporal direction theories mean objects are strictly chargeless at any instant, apex or otherwise. So ‘atat’ temporal directionality means reversing Wellsians need not become antimatter, while neoCliffordian travel dissolves double occupancy.
Conclusions
Although such was perhaps not their aim, Grey (1999), Le Poidevin (2005) show continuous, controllable and survivable backward Wellsian travel can’t occur in strictly Newtonian space. Newtonian spaces threaten time travelers with metaphysical, physical and biological challenges, including discontinuities, cooccupancy, Cheshire Cats, chimeras, singularities and fluidloss. These problems are no respecters of size—Wellsian atoms face discontinuity/cooccupancy dilemmas as much as humans.
Some problems (e.g. discontinuity and cooccupancy) are mutually exclusive but solving some generates others, (e.g. Cheshire Cat fades as solutions to cooccupancy). However, paceGrey (1999), Le Poidevin (2005), extended Wellsians can reverse temporal direction, granted space has extra dimensions or (preferably) variable topology. Wellsians fare best in warped space but do not require Gödelian ‘warped’ time. Hence even if Gödel travel is not time travel, absolute time need not forbid time travel.
So time travel can occur even if Gödel travel is either impossible or not true time travel. Physical objects can reverse in Newtonian time. While Clifford (1886) suggests some spatial resources for Wellsians, Wellsians’ best bet are flexible ‘neoCliffordian’ worlds. In such worlds, time and space remain separate substances yet matter and space can reciprocally affect each other and create local topological identifications. Unlike Newtonian space, Wellsian (neoCliffordian) space must be mutable and curved. Hence Newtonian space poses greater hazards for aspirant Wellsians than Newtonian time. To enjoy unrestricted timetravel, Wellsians must be able to change space itself.
Notes
 1.
 2.
Nor can they be joined solely by copying—which at least is a causal relationship but not of the right kind. Cf. Lewis (1976: p. 148) on ‘counterfeit time travel’ achieved via (demonmanaged) copying.
 3.
 4.
 5.
For probable dates, times and durations of the Traveller’s adventures, see Geduld 1987: p. 40, 48 and 94 n. 14. In discarded 1894 drafts, the Traveller overshoots on his journey home, lands first in a prehistoric swamp and then nearly gets shot as a warlock on New Year’s Eve 1645, (ibid. pp. 187–188).
 6.
Cf. the cooccupying (but sortally distinct) ‘Lumpl’ the clay and ‘Goliath’ the statue in Gibbard (1975).
 7.
 8.
After the antelope with a head at each end in Lofting (1920). Dramatizations often portray the PushmiPullyu as a bicephalous llama.
 9.
E.g. “I claim that mereological composition is unrestricted: any old class of things has a mereological sum”, Lewis (1986: p. 211).
 10.
Feynman (1949): p. 753, n. 7. What Feynman calls ‘proper’ and ‘true’ time are effectively Lewisian personal and external time respectively.
 11.
Dark stars were first discussed in Reverend John Michell’s letter to Henry Cavendish, see Michell (1784).
 12.
 13.
While this paper was in progress, a very interesting, rather different, set of spatial solutions to double occupancy (et al.) were proposed in Bernstein (2015), especially 3.1 (‘Personal Space and External Space’, pp. 165–166). One difference between this paper and Bernstein (2015) is the focus herein on how much of Newtonian space can be preserved in allowing Wellsian reversals; if the arguments herein work, Bernstein’s (2015) ‘personal spaces’ must have certain features (e.g. be dynamic and nonuniform) if Wellsians’ spatial problems are not to resurface in personal space. While nothing herein aims to criticise Bernstein (2015), this paper might help explore how such ‘personal spaces’ could (or should) be structured.
 14.
 15.
Russell rejected variablycurved (e.g. Clifford) space as “logically unsound and impossible to know, and therefore to be condemned à priori”, (1897), (1996 edition: p. 118). Later Russell ruefully admitted “The geometry in Einstein’s General Theory of Relativity is such as I had declared to be impossible”, (Russell 1959), (1997 edition: p. 31). For Russell on Clifford, see (Nickerson and Griffin 2008). If Russell’s (1897) argument worked, it would banish both Wellsian and Gödelian travel a priori but without invoking either the ‘Grandfather Paradox’ or ‘causal loop’ objections definitively defused in Lewis (1976).
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Acknowledgements
Many thanks to two anonymous Synthese referees for helpful comments on earlier drafts of this paper. Many thanks too to my ‘Philosophy of Time Travel’ students over the years for entertaining and informative discussions—in particular Cat McdonaldWade, Emily Paul and Vivek Santayana. A special mention too to Oliver Lunel for the ‘cryogenics’ suggestion. I first encountered the idea of rotating a threedimensional time traveler through fourdimensional space in Grant Morrison’s The Invisibles, (New York: Vertigo/D.C. Comics, 1994–2000)
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Richmond, A. Time travel, hyperspace and Cheshire Cats. Synthese 195, 5037–5058 (2018). https://doi.org/10.1007/s1122901714482
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Keywords
 Time travel
 Dimensionality
 Incongruence