## 1. Paradoxes Lost?



Modern physics strips away many aspects of the manifest image of time.
Time as it appears in the equations of classical mechanics has no need
for a distinguished present moment, for example. Relativity theory
leads to even sharper contrasts. It replaces absolute simultaneity,
according to which it is possible to unambiguously determine the time
order of distant events, with relative simultaneity: extending an
"instant of time" throughout space is not unique, but
depends on the state of motion of an observer. More dramatically, in
general relativity the mathematical properties of time (or better, of
spacetime)--its topology and geometry--depend upon how
matter is arranged rather than being fixed once and for all. So
physics can be, and indeed has to be, formulated without treating time
as a universal, fixed background structure. Since general relativity
represents gravity through spacetime geometry, the allowed geometries
must be as varied as the ways in which matter can be arranged.
Alongside geometrical models used to describe the solar system, black
holes, and much else, the scope of variation extends to include some
exotic structures unlike anything astrophysicists have observed. In
particular, there are spacetime geometries with curves that loop back
on themselves: closed timelike curves (CTCs), which describe the
possible trajectory of an observer who returns exactly back to their
earlier state--without any funny business, such as going faster
than the speed of light. These geometries satisfy the relevant
physical laws, the equations of general relativity, and in that sense
time travel is physically possible.



Yet circular time generates paradoxes, familiar from science fiction
stories featuring time
 travel:[1]


* *Consistency:* Kurt plans to murder his own grandfather
Adolph, by traveling along a CTC to an appropriate moment in the past.
He is an able marksman, and waits until he has a clear shot at
grandpa. Normally he would not miss. Yet if he succeeds, there is no
way that he will then exist to plan and carry out the mission. Kurt
pulls the trigger: what can happen?
* *Underdetermination:* Suppose that Kurt first travels back
in order to give his earlier self a copy of *How to Build a Time
Machine.* This is the same book that allows him to build a time
machine, which he then carries with him on his journey to the past.
Who wrote the book?
* *Easy Knowledge:* A fan of classical music enhances their
computer with a circuit that exploits a CTC. This machine efficiently
solves problems at a higher level of computational complexity than
conventional computers, leading (among other things) to finding the
smallest circuits that can generate Bach's oeuvre--and to
compose new pieces in the same style. Such easy knowledge is at odds
with our understanding of our epistemic predicament. (This third
paradox has not drawn as much attention.)



The first two paradoxes were once routinely taken to show that
solutions with CTCs should be rejected--with charges varying from
violating logic, to being "physically unreasonable", to
undermining the notion of free will. Closer analysis of the paradoxes
has largely reversed this consensus. Physicists have discovered many
solutions with CTCs and have explored their properties in pursuing
foundational questions, such as whether physics is compatible with the
idea of objective temporal passage (starting with Godel 1949).
Philosophers have also used time travel scenarios to probe questions
about, among other things, causation, modality, free will, and
identity (see, e.g., Earman 1972 and Lewis's seminal 1976
paper).



We begin below with *Consistency*, turning to the other
paradoxes in later sections. A standard, stone-walling response is to
insist that the past cannot be changed, as a matter of logic, even by
a time traveler (e.g., Godel 1949, Clarke 1977, Horwich
1987). Adolph cannot both die and survive, as a matter of logic, so
any scheme to alter the past must fail. In many of the best time
travel fictions, the actions of a time traveler are constrained in
novel and unexpected ways. Attempts to change the past fail, and they
fail, often tragically, in just such a way that they set the stage for
the time traveler's self-defeating journey. The first question
is whether there is an analog of the consistent story when it comes to
physics in the presence of CTCs. As we will see, there is a remarkable
general argument establishing the existence of consistent solutions.
Yet a second question persists: why can't time-traveling Kurt
kill his own grandfather? Doesn't the necessity of failures to
change the past put unusual and unexpected constraints on time
travelers, or objects that move along CTCs? The same argument shows
that there are in fact no constraints imposed by the existence of
CTCs, in some cases. After discussing this line of argument, we will
turn to the palatability and further implications of such constraints
if they are required, and then turn to the implications of quantum
mechanics.


## 2. Topology and Constraints



Wheeler and Feynman (1949) were the first to claim that the fact that
nature is continuous could be used to argue that causal influences
from later events to earlier events, as are made possible by time
travel, will not lead to paradox without the need for any constraints.
Maudlin (1990) showed how to make their argument precise and more
general, and argued that nonetheless it was not completely
general.



Imagine the following set-up. We start off having a camera with a
black and white film ready to take a picture of whatever comes out of
the time machine. An object, in fact a developed film, comes out of
the time machine. We photograph it, and develop the film. The
developed film is subsequently put in the time machine, and set to
come out of the time machine at the time the picture is taken. This
surely will create a paradox: the developed film will have the
opposite distribution of black, white, and shades of gray, from the
object that comes out of the time machine. For developed black and
white films (i.e., negatives) have the opposite shades of gray from
the objects they are pictures of. But since the object that comes out
of the time machine is the developed film itself it we surely have a
paradox.



However, it does not take much thought to realize that there is no
paradox here. What will happen is that a uniformly gray picture will
emerge, which produces a developed film that has exactly the same
uniform shade of gray. No matter what the sensitivity of the film is,
as long as the dependence of the brightness of the developed film
depends in a continuous manner on the brightness of the object being
photographed, there will be a shade of gray that, when photographed,
will produce exactly the same shade of gray on the developed film.
This is the essence of Wheeler and Feynman's idea. Let us first
be a bit more precise and then a bit more general.



For simplicity let us suppose that the film is always a uniform shade
of gray (i.e., at any time the shade of gray does not vary by location
on the film). The possible shades of gray of the film can then be
represented by the (real) numbers from 0, representing pure black, to
1, representing pure white.



Let us now distinguish various stages in the chronological order of
the life of the film. In stage \(S\_1\) the film is young; it has just
been placed in the camera and is ready to be exposed. It is then
exposed to the object that comes out of the time machine. (That object
in fact is a later stage of the film itself). By the time we come to
stage \(S\_2\) of the life of the film, it has been developed and is
about to enter the time machine. Stage \(S\_3\) occurs just after it
exits the time machine and just before it is photographed. Stage
\(S\_4\) occurs after it has been photographed and before it starts
fading away. Let us assume that the film starts out in stage \(S\_1\)
in some uniform shade of gray, and that the only significant change in
the shade of gray of the film occurs between stages \(S\_1\) and
\(S\_2\). During that period it acquires a shade of gray that depends
on the shade of gray of the object that was photographed. In other
words, the shade of gray that the film acquires at stage \(S\_2\)
depends on the shade of gray it has at stage \(S\_3\). The influence of
the shade of gray of the film at stage \(S\_3\), on the shade of gray
of the film at stage \(S\_2\), can be represented as a mapping, or
function, from the real numbers between 0 and 1 (inclusive), to the
real numbers between 0 and 1 (inclusive). Let us suppose that the
process of photography is such that if one imagines varying the shade
of gray of an object in a smooth, continuous manner then the shade of
gray of the developed picture of that object will also vary in a
smooth, continuous manner. This implies that the function in question
will be a continuous function. Now any continuous function from the
real numbers between 0 and 1 (inclusive) to the real numbers between 0
and 1 (inclusive) must map at least one number to itself. One can
quickly convince oneself of this by graphing such functions. For one
will quickly see that any continuous function \(f\) from \([0,1]\) to
\([0,1]\) must intersect the line \(x=y\) somewhere, and thus there
must be at least one point \(x\) such that \(f(x)=x\). Such points are
called fixed points of the function. Now let us think about what such
a fixed point represents. It represents a shade of gray such that,
when photographed, it will produce a developed film with exactly that
same shade of gray. The existence of such a fixed point implies a
solution to the apparent paradox.



Let us now be more general and allow color photography. One can
represent each possible color of an object (of uniform color) by the
proportions of blue, green and red that make up that color. (This is
why television screens can produce all possible colors.) Thus one can
represent all possible colors of an object by three points on three
orthogonal lines \(x, y\) and \(z\), that is to say, by a point in a
three-dimensional cube. This cube is also known as the
"Cartesian product" of the three line segments. Now, one
can also show that any continuous map from such a cube to itself must
have at least one fixed point. So color photography can not be used to
create time travel paradoxes either!



Even more generally, consider some system \(P\) which, as in the above
example, has the following life. It starts in some state \(S\_1\), it
interacts with an object that comes out of a time machine (which
happens to be its older self), it travels back in time, it interacts
with some object (which happens to be its younger self), and finally
it grows old and dies. Let us assume that the set of possible states
of \(P\) can be represented by a Cartesian product of \(n\) closed
intervals of the reals, i.e., let us assume that the topology of the
state-space of \(P\) is isomorphic to a finite Cartesian product of
closed intervals of the reals. Let us further assume that the
development of \(P\) in time, and the dependence of that development
on the state of objects that it interacts with, is continuous. Then,
by a well-known fixed point theorem in topology (see, e.g., Hocking
& Young 1961: 273), no matter what the nature of the interaction
is, and no matter what the initial state of the object is, there will
be at least one state \(S\_3\) of the older system (as it emerges from
the time travel machine) that will influence the initial state \(S\_1\)
of the younger system (when it encounters the older system) so that,
as the younger system becomes older, it develops exactly into state
\(S\_3\). Thus without imposing any constraints on the initial state
\(S\_1\) of the system \(P\), we have shown that there will always be
perfectly ordinary, non-paradoxical, solutions, in which everything
that happens, happens according to the usual laws of development. Of
course, there is looped causation, hence presumably also looped
explanation, but what do you expect if there is looped time?



Unfortunately, for the fan of time travel, a little reflection
suggests that there are systems for which the needed fixed point
theorem does not hold. Imagine, for instance, that we have a dial that
can only rotate in a plane. We are going to put the dial in the time
machine. Indeed we have decided that if we see the later stage of the
dial come out of the time machine set at angle \(x\), then we will set
the dial to \(x+90\), and throw it into the time machine. Now it seems
we have a paradox, since the mapping that consists of a rotation of
all points in a circular state-space by 90 degrees does not have a
fixed point. And why wouldn't some state-spaces have the
topology of a circle?



However, we have so far not used another continuity assumption which
is also a reasonable assumption. So far we have only made the
following demand: the state the dial is in at stage \(S\_2\) must be a
continuous function of the state of the dial at stage \(S\_3\). But,
the state of the dial at stage \(S\_2\) is arrived at by taking the
state of the dial at stage \(S\_1\), and rotating it over some angle.
It is not merely the case that the effect of the interaction, namely
the state of the dial at stage \(S\_2\), should be a continuous
function of the cause, namely the state of the dial at stage \(S\_3\).
It is additionally the case that path taken to get there, the way the
dial is rotated between stages \(S\_1\) and \(S\_2\) must be a
continuous function of the state at stage \(S\_3\). And, rather
surprisingly, it turns out that this can not be done. Let us
illustrate what the problem is before going to a more general
demonstration that there must be a fixed point solution in the dial
case.



Forget time travel for the moment. Suppose that you and I each have a
watch with a single dial neither of which is running. My watch is set
at 12. You are going to announce what your watch is set at. My task is
going to be to adjust my watch to yours no matter what announcement
you make. And my actions should have a continuous (single valued)
dependence on the time that you announce. Surprisingly, this is not
possible! For instance, suppose that if you announce "12",
then I achieve that setting on my watch by doing nothing. Now imagine
slowly and continuously increasing the announced times, starting at
12. By continuity, I must achieve each of those settings by rotating
my dial to the right. If at some point I switch and achieve the
announced goal by a rotation of my dial to the left, I will have
introduced a discontinuity in my actions, a discontinuity in the
actions that I take as a function of the announced angle. So I will be
forced, by continuity, to achieve every announcement by rotating the
dial to the right. But, this rotation to the right will have to be
abruptly discontinued as the announcements grow larger and I
eventually approach 12 again, since I achieved 12 by not rotating the
dial at all. So, there will be a discontinuity at 12 at the latest. In
general, continuity of my actions as a function of announced times can
not be maintained throughout if I am to be able to replicate all
possible settings. Another way to see the problem is that one can
similarly reason that, as one starts with 12, and imagines
continuously making the announced times earlier, one will be forced,
by continuity, to achieve the announced times by rotating the dial to
the left. But the conclusions drawn from the assumption of continuous
increases and the assumption of continuous decreases are inconsistent.
So we have an inconsistency following from the assumption of
continuity and the assumption that I always manage to set my watch to
your watch. So, a dial developing according to a continuous dynamics
from a given initial state, can not be set up so as to react to a
second dial, with which it interacts, in such a way that it is
guaranteed to always end up set at the same angle as the second dial.
Similarly, it can not be set up so that it is guaranteed to always end
up set at 90 degrees to the setting of the second dial. All of this
has nothing to do with time travel. However, the impossibility of such
set ups is what prevents us from enacting the rotation by 90 degrees
that would create paradox in the time travel setting.



Let us now give the positive result that with such dials there will
always be fixed point solutions, as long as the dynamics is
continuous. Let us call the state of the dial before it interacts with
its older self the initial state of the dial. And let us call the
state of the dial after it emerges from the time machine the final
state of the dial. There is also an intermediate state of the dial,
after it interacts with its older self and before it is put into the
time machine. We can represent the initial or intermediate states of
the dial, before it goes into the time machine, as an angle \(x\) in
the horizontal plane and the final state of the dial, after it comes
out of the time machine, as an angle \(y\) in the vertical plane. All
possible \(\langle x,y\rangle\) pairs can thus be visualized as a
torus with each \(x\) value picking out a vertical circular
cross-section and each \(y\) picking out a point on that
cross-section. See
 figure 1.



![a torus diagram: link to extended description below](figure-1.svg)

Figure 1 [An
 extended description of figure 1
 is in the supplement.]




Suppose that the dial starts at angle \(i\) which picks out vertical
circle \(I\) on the torus. The initial angle \(i\) that the dial is at
before it encounters its older self, and the set of all possible final
angles that the dial can have when it emerges from the time machine is
represented by the circle \(I\) on the torus (see
 figure 1).
 Given any possible angle of the emerging dial, the dial initially at
angle \(i\) will develop to some other angle. One can picture this
development by rotating each point on \(I\) in the horizontal
direction by the relevant amount. Since the rotation has to depend
continuously on the angle of the emerging dial, circle \(I\) during
this development will deform into some loop \(L\) on the torus. Loop
\(L\) thus represents all possible intermediate angles \(x\) that the
dial is at when it is thrown into the time machine, given that it
started at angle \(i\) and then encountered a dial (its older self)
which was at angle \(y\) when it emerged from the time machine. We
therefore have consistency if \(x=y\) for some \(x\) and \(y\) on loop
\(L\). Now, let loop \(C\) be the loop which consists of all the
points on the torus for which \(x=y\). Ring \(I\) intersects \(C\) at
point \(\langle i,i\rangle\). Obviously any continuous deformation of
\(I\) must still intersect \(C\) somewhere. So \(L\) must intersect
\(C\) somewhere, say at \(\langle j,j\rangle\). But that means that no
matter how the development of the dial starting at \(I\) depends on
the angle of the emerging dial, there will be some angle for the
emerging dial such that the dial will develop exactly into that angle
(by the time it enters the time machine) under the influence of that
emerging dial. This is so no matter what angle one starts with, and no
matter how the development depends on the angle of the emerging dial.
Thus even for a circular state-space there are no constraints needed
other than continuity.



Unfortunately there are state-spaces that escape even this argument.
Consider for instance a pointer that can be set to all values between
0 and 1, where 0 and 1 are not possible values. That is, suppose that
we have a state-space that is isomorphic to an open set of real
numbers. Now suppose that we have a machine that sets the pointer to
half the value that the pointer is set at when it emerges from the
time machine.



![a diagram: link to extended description below](figure-2.svg)

Figure 2 [An
 extended description of figure 2
 is in the supplement.]




Suppose the pointer starts at value \(I\). As before we can represent
the combination of this initial position and all possible final
positions by the line \(I\). Under the influence of the pointer coming
out of the time machine the pointer value will develop to a value that
equals half the value of the final value that it encountered. We can
represent this development as the continuous deformation of line \(I\)
into line \(L\), which is indicated by the arrows in
 figure 2.
 This development is fully continuous. Points \(\langle x,y\rangle\)
on line \(I\) represent the initial position \(x=I\) of the (young)
pointer, and the position \(y\) of the older pointer as it emerges
from the time machine. Points \(\langle x,y\rangle\) on line \(L\)
represent the position \(x\) that the younger pointer should develop
into, given that it encountered the older pointer emerging from the
time machine set at position \(y\). Since the pointer is designed to
develop to half the value of the pointer that it encounters, the line
\(L\) corresponds to \(x=1/2 y\). We have consistency if there is some
point such that it develops into that point, if it encounters that
point. Thus, we have consistency if there is some point \(\langle
x,y\rangle\) on line \(L\) such that \(x=y\). However, there is no
such point: lines \(L\) and \(C\) do not intersect. Thus there is no
consistent solution, despite the fact that the dynamics is fully
continuous.



Of course if 0 were a possible value, \(L\) and \(C\) would intersect
at 0. This is surprising and strange: adding one point to the set of
possible values of a quantity here makes the difference between
paradox and peace. One might be tempted to just add the extra point to
the state-space in order to avoid problems. After all, one might say,
surely no measurements could ever tell us whether the set of possible
values includes that exact point or not. Unfortunately there can be
good theoretical reasons for supposing that some quantity has a
state-space that is open: the set of all possible speeds of massive
objects in special relativity surely is an open set, since it includes
all speeds up to, but not including, the speed of light. Quantities
that have possible values that are not bounded also lead to counter
examples to the presented fixed point argument. And it is not obvious
to us why one should exclude such possibilities. So the argument that
no constraints are needed is not fully general.



An interesting question of course is: exactly for which state-spaces
must there be such fixed points? The arguments above depend on a
well-known fixed point theorem (due to Schauder) that guarantees the
existence of a fixed point for compact, convex state spaces. We do not
know what subsequent extensions of this result imply regarding fixed
points for a wider variety of systems, or whether there are other
general results along these lines. (See Kutach 2003 for more on this
issue.)



A further interesting question is whether this line of argument is
sufficient to resolve *Consistency* (see also Dowe 2007). When
they apply, these results establish the existence of a solution, such
as the shade of uniform gray in the first example. But physicists
routinely demand more than merely the existence of a solution, namely
that solutions to the equations are stable--such that
"small" changes of the initial state lead to
"small" changes of the resulting trajectory. (Clarifying
the two senses of "small" in this statement requires
further work, specifying the relevant topology.) Stability in this
sense underwrites the possibility of applying equations to real
systems given our inability to fix initial states with indefinite
precision. (See Fletcher 2020 for further discussion.) The fixed point
theorems guarantee that for an initial state \(S\_1\) there is a
solution, but this solution may not be "close" to the
solution for a nearby initial state, \(S'\). We are not aware of any
proofs that the solutions guaranteed to exist by the fixed point
theorems are also stable in this sense.


## 3. The General Possibility of Time Travel in General Relativity



Time travel has recently been discussed quite extensively in the
context of general relativity. General relativity places few
constraints on the global structure of space and time. This
flexibility leads to a possibility first described in print by Hermann
Weyl:



> 
> 
> Every world-point is the origin of the double-cone of the active
> future and the passive past [i.e., the two lobes of the light cone].
> Whereas in the special theory of relativity these two portions are
> separated by an intervening region, it is certainly possible in the
> present case [i.e., general relativity] for the cone of the active
> future to overlap with that of the passive past; so that, in
> principle, it is possible to experience events now that will in part
> be an effect of my future resolves and actions. Moreover, it is not
> impossible for a world-line (in particular, that of my body), although
> it has a timelike direction at every point, to return to the
> neighborhood of a point which it has already once passed through.
> (Weyl 1918/1920 [1952: 274])
> 
> 
> 



A time-like curve is simply a space-time trajectory such that the
speed of light is never equaled or exceeded along this trajectory.
Time-like curves represent possible trajectories of ordinary objects.
In general relativity a curve that is everywhere timelike locally can
nonetheless loop back on itself, forming a CTC. Weyl makes the point
vividly in terms of the light cones: along such a curve, the future
lobe of the light cone (the "active future") intersects
the past lobe of the light cone (the "passive past").
Traveling along such a curve one would never exceed the speed of
light, and yet after a certain amount of (proper) time one would
return to a point in space-time that one previously visited. Or, by
staying close to such a CTC, one could come arbitrarily close to a
point in space-time that one previously visited. General relativity,
in a straightforward sense, allows time travel: there appear to be
many space-times compatible with the fundamental equations of general
relativity in which there are CTC's. Space-time, for instance,
could have a Minkowski metric everywhere, and yet have CTC's
everywhere by having the temporal dimension (topologically) rolled up
as a circle. Or, one can have wormhole connections between different
parts of space-time which allow one to enter "mouth \(A\)"
of such a wormhole connection, travel through the wormhole, exit the
wormhole at "mouth \(B\)" and re-enter "mouth
\(A\)" again. CTCs can even arise when the spacetime is
topologically \(\mathbb{R}^4\), due to the "tilting" of
light cones produced by rotating matter (as in Godel 1949's
spacetime).



General relativity thus appears to provide ample opportunity for time
travel. Note that just because there are CTC's in a space-time,
this does not mean that one can get from any point in the space-time
to any other point by following some future directed timelike
curve--there may be insurmountable practical obstacles. In
Godel's spacetime, it is the case that there are CTCs
passing through every point in the spacetime. Yet these CTCs are not
geodesics, so traversing them requires acceleration. Calculations of
the minimal fuel required to travel along the appropriate curve should
discourage any would-be time travelers (Malament 1984, 1985; Manchak
2011). But more generally CTCs may be confined to smaller regions;
some parts of space-time can have CTC's while other parts do
not. Let us call the part of a space-time that has CTC's the
"time travel region" of that space-time, while calling the
rest of that space-time the "normal region". More
precisely, the "time travel region" consists of all the
space-time points \(p\) such that there exists a (non-zero length)
timelike curve that starts at \(p\) and returns to \(p\). Now let us
turn to examining space-times with CTC's a bit more closely for
potential problems.


## 4. Two Toy Models



In order to get a feeling for the sorts of implications that closed
timelike curves can have, it may be useful to consider two simple
models. In space-times with closed timelike curves the traditional
initial value problem cannot be framed in the usual way. For it
presupposes the existence of Cauchy surfaces, and if there are CTCs
then no Cauchy surface exists. (A Cauchy surface is a spacelike
surface such that every inextendable timelike curve crosses it exactly
once. One normally specifies initial conditions by giving the
conditions on such a surface.) Nonetheless, if the topological
complexities of the manifold are appropriately localized, we can come
quite close. Let us call an edgeless spacelike surface \(S\) a
*quasi-Cauchy* surface if it divides the rest of the manifold
into two parts such that


1. every point in the manifold can be connected by a timelike curve
to \(S\), and
2. any timelike curve which connects a point in one region to a point
in the other region intersects \(S\) exactly once.



It is obvious that a quasi-Cauchy surface must entirely inhabit the
normal region of the space-time; if any point \(p\) of \(S\) is in the
time travel region, then any timelike curve which intersects \(p\) can
be extended to a timelike curve which intersects \(S\) near \(p\)
again. In extreme cases of time travel, a model may have no normal
region at all (e.g., Minkowski space-time rolled up like a cylinder in
a time-like direction), in which case our usual notions of temporal
precedence will not apply. But temporal anomalies like wormholes (and
time machines) can be sufficiently localized to permit the existence
of quasi-Cauchy surfaces.



Given a timelike orientation, a quasi-Cauchy surface unproblematically
divides the manifold into its *past* (i.e., all points that can
be reached by past-directed timelike curves from \(S)\) and its
*future* (ditto *mutatis mutandis*). If the whole past
of \(S\) is in the normal region of the manifold, then \(S\) is a
*partial Cauchy surface*: every inextendable timelike curve
which exists to the past of \(S\) intersects \(S\) exactly once, but
(if there is time travel in the future) not every inextendable
timelike curve which exists to the future of \(S\) intersects \(S\).
Now we can ask a particularly clear question: consider a manifold
which contains a time travel region, but also has a partial Cauchy
surface \(S\), such that all of the temporal funny business is to the
future of \(S\). If all you could see were \(S\) and its past, you
would not know that the space-time had any time travel at all. The
question is: are there any constraints on the sort of data which can
be put on \(S\) and continued to a global solution of the dynamics
which are different from the constraints (if any) on the data which
can be put on a Cauchy surface in a simply connected manifold and
continued to a global solution? If there is time travel to our future,
might we we able to tell this now, because of some implied oddity in
the arrangement of present things?



It is not at all surprising that there might be constraints on the
data which can be put on a locally space-like surface which passes
through the time travel region: after all, we never think we can
freely specify what happens on a space-like surface and on another
such surface to its future, but in this case the surface at issue lies
to its own future. But if there were particular constraints for data
on a partial Cauchy surface then we would apparently need to have to
rule out some sorts of otherwise acceptable states on \(S\) if there
is to be time travel to the future of \(S\). We then might be able to
establish that there will be no time travel in the future by simple
inspection of the present state of the universe. As we will see, there
is reason to suspect that such constraints on the partial Cauchy
surface are non-generic. But we are getting ahead of ourselves: first
let's consider the effect of time travel on a very simple
dynamics.



The simplest possible example is the Newtonian theory of perfectly
elastic collisions among equally massive particles in one spatial
dimension. The space-time is two-dimensional, so we can represent it
initially as the Euclidean plane, and the dynamics is completely
specified by two conditions. When particles are traveling freely,
their world lines are straight lines in the space-time, and when two
particles collide, they exchange momenta, so the collision looks like
an "\(X\)" in space-time, with each particle changing its
momentum at the
 impact.[2]
 The dynamics is purely local, in that one can check that a set of
world-lines constitutes a model of the dynamics by checking that the
dynamics is obeyed in every arbitrarily small region. It is also
trivial to generate solutions from arbitrary initial data if there are
no CTCs: given the initial positions and momenta of a set of
particles, one simply draws a straight line from each particle in the
appropriate direction and continues it indefinitely. Once all the
lines are drawn, the worldline of each particle can be traced from
collision to collision. The boundary value problem for this dynamics
is obviously well-posed: any set of data at an instant yields a unique
global solution, constructed by the method sketched above.



What happens if we change the topology of the space-time by hand to
produce CTCs? The simplest way to do this is depicted in
 figure 3:
 we cut and paste the space-time so it is no longer simply connected
by identifying the line \(L-\) with the line \(L+\). Particles
"going in" to \(L+\) from below "emerge" from
\(L-\) , and particles "going in" to \(L-\) from below
"emerge" from \(L+\).



![a diagram: link to extended description below](figure-3.svg)

Figure 3: Inserting CTCs by Cut and
Paste. [An
 extended description of figure 3
 is in the supplement.]




How is the boundary-value problem changed by this alteration in the
space-time? Before the cut and paste, we can put arbitrary data on the
simultaneity slice \(S\) and continue it to a unique solution. After
the change in topology, \(S\) is no longer a Cauchy surface, since a
CTC will never intersect it, but it is a partial Cauchy surface. So we
can ask two questions. First, can arbitrary data on \(S\) always be
continued to a global solution? Second, is that solution unique? If
the answer to the first question is \(no\), then we have a
backward-temporal constraint: the existence of the region with CTCs
places constraints on what can happen on \(S\) even though that region
lies completely to the future of \(S\). If the answer to the second
question is \(no\), then we have an odd sort of indeterminism,
analogous to the unwritten book: the complete physical state on \(S\)
does not determine the physical state in the future, even though the
local dynamics is perfectly deterministic and even though there is no
other past edge to the space-time region in \(S\)'s future
(i.e., there is nowhere *else* for boundary values to come from
which could influence the state of the region).



In this case the answer to the first question is *yes* and to
the second is *no*: there are no constraints on the data which
can be put on \(S\), but those data are always consistent with an
infinitude of different global solutions. The easy way to see that
there always is a solution is to construct the minimal solution in the
following way. Start drawing straight lines from \(S\) as required by
the initial data. If a line hits \(L-\) from the bottom, just continue
it coming out of the top of \(L+\) in the appropriate place, and if a
line hits \(L+\) from the bottom, continue it emerging from \(L-\) at
the appropriate place.
 Figure 4
 represents the minimal solution for a single particle which enters
the time-travel region from the left:



![a diagram: link to extended description below](figure-4.svg)

Figure 4: The Minimal Solution. [An
 extended description of figure 4
 is in the supplement.]




The particle "travels back in time" three times. It is
obvious that this minimal solution is a global solution, since the
particle always travels inertially.



But the same initial state on \(S\) is also consistent with other
global solutions. The new requirement imposed by the topology is just
that the data going into \(L+\) from the bottom match the data coming
out of \(L-\) from the top, and the data going into \(L-\) from the
bottom match the data coming out of \(L+\) from the top. So we can add
any number of vertical lines connecting \(L-\) and \(L+\) to a
solution and still have a solution. For example, adding a few such
lines to the minimal solution yields:



![a diagram: link to extended description below](figure-5.svg)

Figure 5: A Non-Minimal Solution. [An
 extended description of figure 5
 is in the supplement.]




The particle now collides with itself twice: first before it reaches
\(L+\) for the first time, and again shortly before it exits the CTC
region. From the particle's point of view, it is traveling to
the right at a constant speed until it hits an older version of itself
and comes to rest. It remains at rest until it is hit from the right
by a younger version of itself, and then continues moving off, and the
same process repeats later. It is clear that this is a global model of
the dynamics, and that any number of distinct models could be
generating by varying the number and placement of vertical lines.



Knowing the data on \(S\), then, gives us only incomplete information
about how things will go for the particle. We know that the particle
will enter the CTC region, and will reach \(L+\), we know that it will
be the only particle in the universe, we know exactly where and with
what speed it will exit the CTC region. But we cannot determine how
many collisions the particle will undergo (if any), nor how long (in
proper time) it will stay in the CTC region. If the particle were a
clock, we could not predict what time it would indicate when exiting
the region. Furthermore, the dynamics gives us no handle on what to
think of the various possibilities: there are no probabilities
assigned to the various distinct possible outcomes.



Changing the topology has changed the mathematics of the situation in
two ways, which tend to pull in opposite directions. On the one hand,
\(S\) is no longer a Cauchy surface, so it is perhaps not surprising
that data on \(S\) do not suffice to fix a unique global solution. But
on the other hand, there is an added constraint: data "coming
out" of \(L-\) must exactly match data "going in" to
\(L+\), even though what comes out of \(L-\) helps to determine what
goes into \(L+\). This added consistency constraint tends to cut down
on solutions, although in this case the additional constraint is more
than outweighed by the freedom to consider various sorts of data on
\({L+}/{L-}\).



The fact that the extra freedom outweighs the extra constraint also
points up one unexpected way that the supposed paradoxes of time
travel may be overcome. Let's try to set up a paradoxical
situation using the little closed time loop above. If we send a single
particle into the loop from the left and do nothing else, we know
exactly where it will exit the right side of the time travel region.
Now suppose we station someone at the other side of the region with
the following charge: if the particle should come out on the right
side, the person is to do something to *prevent* the particle
from going in on the left in the first place. In fact, this is quite
easy to do: if we send a particle in from the right, it seems that it
can exit on the left and *deflect* the incoming left-hand
particle.



Carrying on our reflection in this way, we further realize that if the
particle comes out on the right, we might as well send *it*
back in order to deflect itself from entering in the first place. So
all we really need to do is the following: set up a perfectly
reflecting particle mirror on the right-hand side of the time travel
region, and launch the particle from the left so that--*if
nothing interferes with it*--it will just barely hit \(L+\).
Our paradox is now apparently complete. If, on the one hand, nothing
interferes with the particle it will enter the time-travel region on
the left, exit on the right, be reflected from the mirror, re-enter
from the right, and come out on the left to prevent itself from ever
entering. So if it enters, it gets deflected and never enters. On the
other hand, if it never enters then nothing goes in on the left, so
nothing comes out on the right, so nothing is reflected back, and
there is nothing to deflect it from entering. So if it doesn't
enter, then there is nothing to deflect it and it enters. If it
enters, then it is deflected and doesn't enter; if it
doesn't enter then there is nothing to deflect it and it enters:
paradox complete.



But at least one solution to the supposed paradox is easy to
construct: just follow the recipe for constructing the minimal
solution, continuing the initial trajectory of the particle
(reflecting it the mirror in the obvious way) and then read of the
number and trajectories of the particles from the resulting diagram.
We get the result of
 figure 6:



![a diagram: link to extended description below](figure-6.svg)

Figure 6: Resolving the
"Paradox". [An
 extended description of figure 6
 is in the supplement.]




As we can see, the particle approaching from the left never reaches
\(L+\): it is deflected first by a particle which emerges from \(L-\).
But it is not deflected *by itself*, as the paradox suggests,
it is deflected by another particle. Indeed, there are now
*four* particles in the diagram: the original particle and
three particles which are confined to closed time-like curves. It is
not the leftmost particle which is reflected by the mirror, nor even
the particle which deflects the leftmost particle; it is another
particle altogether.



The paradox gets it traction from an incorrect presupposition. If
there is only one particle in the world at \(S\) then there is only
one particle which could participate in an interaction in the time
travel region: the single particle would have to interact with its
earlier (or later) self. But there is no telling what might come out
of \(L-\): the only requirement is that whatever comes out must match
what goes in at \(L+\). So if you go to the trouble of constructing a
working time machine, you should be prepared for a different kind of
disappointment when you attempt to go back and kill yourself: you may
be prevented from entering the machine in the first place by some
completely unpredictable entity which emerges from it. And once again
a peculiar sort of indeterminism appears: if there are many
self-consistent things which could prevent you from entering, there is
no telling which is even likely to materialize. This is just like the
case of the unwritten book: the book is never written, so nothing
determines what fills its pages.



So when the freedom to put data on \(L-\) outweighs the constraint
that the same data go into \(L+\), instead of paradox we get an
embarrassment of riches: many solution consistent with the data on
\(S\), or many possible books. To see a case where the constraint
"outweighs" the freedom, we need to construct a very
particular, and frankly artificial, dynamics and topology. Consider
the space of all linear dynamics for a scalar field on a lattice. (The
lattice can be though of as a simple discrete space-time.) We will
depict the space-time lattice as a directed graph. There is to be a
scalar field defined at every node of the graph, whose value at a
given node depends linearly on the values of the field at nodes which
have arrows which lead to it. Each edge of the graph can be assigned a
weighting factor which determines how much the field at the input node
contributes to the field at the output node. If we name the nodes by
the letters *a*, *b*, *c*, etc., and the edges by
their endpoints in the obvious way, then we can label the weighting
factors by the edges they are associated with in an equally obvious
way.



Suppose that the graph of the space-time lattice is *acyclic*,
as in
 figure 7.
 (A graph is *Acyclic* if one can not travel in the direction
of the arrows and go in a loop.)



![an acyclic graph: link to extended description below](figure-7.svg)

Figure 7: An Acyclic Lattice. [An
 extended description of figure 7
 is in the supplement.]




It is easy to regard a set of nodes as the analog of a Cauchy surface,
e.g., the set \(\{a, b, c\}\), and it is obvious if arbitrary data are
put on those nodes the data will generate a unique solution in the
 future.[3]
 If the value of the field at node \(a\) is 3 and at node \(b\) is 7,
then its value at node \(d\) will be \(3W\_{ad}\) and its value at node
\(e\) will be \(3W\_{ae} + 7W\_{be}\). By varying the weighting factors
we can adjust the dynamics, but in an acyclic graph the future
evolution of the field will always be unique.



Let us now again artificially alter the topology of the lattice to
admit CTCs, so that the graph now is cyclic. One of the simplest such
graphs is depicted in
 figure 8:
 there are now paths which lead from \(z\) back to itself, e.g., \(z\)
to \(y\) to \(z\).



![cyclic graph: link to extended description below](figure-8.svg)

Figure 8: Time Travel on a Lattice. [An
 extended description of figure 8
 is in the supplement.]




Can we now put arbitrary data on \(v\) and \(w\), and continue that
data to a global solution? Will the solution be unique?



In the generic case, there will be a solution and the solution will be
unique. The equations for the value of the field at \(x, y\), and
\(z\) are:

 
\[\begin{align}
x &= vW\_{vx} + zW\_{zx} \\
 y &= wW\_{wy} + zW\_{zy} \\
 z &= xW\_{xz} + yW\_{yz}. 
\end{align}\]


Solving these equations for \(z\) yields

 
\[ z = (vW\_{vx} + zW\_{zx})W\_{xz} + (wW\_{wy} + zW\_{zy})W\_{yz}, \]


or

 
\[ z = \frac{(vW\_{vx}W\_{xz} + wW\_{wy}W\_{yz})} {(1 - W\_{zx}W\_{xz} - W\_{zy}W\_{yz})}, \]


which gives a unique value for \(z\) in the generic case. But looking
at the space of all possible dynamics for this lattice (i.e., the
space of all possible weighting factors), we find a singularity in the
case where \(1-W\_{zx}W\_{xz} - W\_{zy}W\_{yz} = 0\). If we choose
weighting factors in just this way, then arbitrary data at \(v\) and
\(w\) cannot be continued to a global solution. Indeed, if the scalar
field is everywhere non-negative, then this particular choice of
dynamics puts ironclad constraints on the value of the field at \(v\)
and \(w\): the field there must be zero (assuming \(W\_{vx}\) and
\(W\_{wy}\) to be non-zero), and similarly all nodes in their past must
have field value zero. If the field can take negative values, then the
values at \(v\) and \(w\) must be so chosen that \(vW\_{vx}W\_{xz} =
-wW\_{wy}W\_{yz}\). In either case, the field values at \(v\) and \(w\)
are severely constrained by the existence of the CTC region even
though these nodes lie completely to the past of that region. It is
this sort of constraint which we find to be unlike anything which
appears in standard physics.



Our toy models suggest three things. The first is that it may be
impossible to prove in complete generality that arbitrary data on a
partial Cauchy surface can *always* be continued to a global
solution: our artificial case provides an example where it cannot. The
second is that such odd constraints are not likely to be generic: we
had to delicately fine-tune the dynamics to get a problem. The third
is that the opposite problem, namely data on a partial Cauchy surface
being consistent with *many* different global solutions, is
likely to be generic: we did not have to do any fine-tuning to get
this result.



This third point leads to a peculiar sort of indeterminism,
illustrated by the case of the unwritten book: the entire state on
\(S\) does not determine what will happen in the future even though
the local dynamics is deterministic and there are no other
"edges" to space-time from which data could influence the
result. What happens in the time travel region is constrained but not
determined by what happens on \(S\), and the dynamics does not even
supply any *probabilities* for the various possibilities. The
example of the photographic negative discussed in section 2, then,
seems likely to be unusual, for in that case there is a
*unique* fixed point for the dynamics, and the set-up plus the
dynamical laws *determine* the outcome. In the generic case one
would rather expect *multiple* fixed points, with no room for
anything to influence, even probabilistically, *which* would be
realized. (See the supplement on



Remarks and Limitations on the Toy Models.



It is ironic that time travel should lead generically not to
contradictions or to constraints (in the normal region) but to
*underdetermination* of what happens in the time travel region
by what happens everywhere else (an underdetermination tied neither to
a probabilistic dynamics nor to a free edge to space-time). The
traditional objection to time travel is that it leads to
contradictions: there is no consistent way to complete an arbitrarily
constructed story about how the time traveler intends to act. Instead,
though, it appears that the more significant problem is
underdetermination: the story can be consistently completed in many
different ways.


## 5. Slightly More Realistic Models of Time Travel



Echeverria, Klinkhammer, and Thorne (1991) considered the case of
3-dimensional single hard spherical ball that can go through a single
time travel wormhole so as to collide with its younger self.



![a diagram: link to extended description below](figure-9.svg)

Figure 9 [An
 extended description of figure 9
 is in the supplement.]




The threat of paradox in this case arises in the following form.
Consider the initial trajectory of a ball as it approaches the time
travel region. For some initial trajectories, the ball does not
undergo a collision before reaching mouth 1, but upon exiting mouth 2
it will collide with its earlier self. This leads to a contradiction
if the collision is strong enough to knock the ball off its trajectory
and deflect it from entering mouth 1. Of course, the Wheeler-Feynman
strategy is to look for a "glancing blow" solution: a
collision which will produce exactly the (small) deviation in
trajectory of the earlier ball that produces exactly that collision.
Are there always such
 solutions?[4]



Echeverria, Klinkhammer & Thorne found a large class of initial
trajectories that have consistent "glancing blow"
continuations, and found none that do not (but their search was not
completely general). They did not produce a rigorous proof that every
initial trajectory has a consistent continuation, but suggested that
it is very plausible that every initial trajectory has a consistent
continuation. That is to say, they have made it very plausible that,
in the billiard ball wormhole case, the time travel structure of such
a wormhole space-time does not result in constraints on states on
spacelike surfaces in the non-time travel region.



In fact, as one might expect from our discussion in the previous
section, they found the opposite problem from that of inconsistency:
they found underdetermination. For a large class of initial
trajectories there are multiple different consistent "glancing
blow" continuations of that trajectory (many of which involve
multiple wormhole traversals). For example, if one initially has a
ball that is traveling on a trajectory aimed straight between the two
mouths, then one obvious solution is that the ball passes between the
two mouths and never time travels. But another solution is that the
younger ball gets knocked into mouth 1 exactly so as to come out of
mouth 2 and produce that collision. Echeverria et al. do not note the
possibility (which we pointed out in the previous section) of the
existence of additional balls in the time travel region. We conjecture
(but have no proof) that for every initial trajectory of \(A\) there
are some, and generically many, multiple-ball continuations.



Friedman, Morris, et al. (1990) examined the case of source-free
non-self-interacting scalar fields traveling through such a time
travel wormhole and found that no constraints on initial conditions in
the non-time travel region are imposed by the existence of such time
travel wormholes. In general there appear to be no known counter
examples to the claim that in "somewhat realistic"
time-travel space-times with a partial Cauchy surface there are no
constraints imposed on the state on such a partial Cauchy surface by
the existence of CTC's. (See, e.g., Friedman & Morris 1991;
Thorne 1994; Earman 1995; Earman, Smeenk, & Wuthrich 2009;
and Dowe 2007.)



How about the issue of constraints in the time travel region \(T\)?
*Prima facie*, constraints in such a region would not appear to
be surprising. But one might still expect that there should be no
constraints on states on a spacelike surface, provided one keeps the
surface "small enough". In the physics literature the
following question has been asked: for any point \(p\) in \(T\), and
any space-like surface \(S\) that includes \(p\) is there a
neighborhood \(E\) of \(p\) in \(S\) such that any solution on \(E\)
can be extended to a solution on the whole space-time? With respect to
this question, there are some simple models in which one has this kind
of extendability of local solutions to global ones, and some simple
models in which one does not have such extendability, with no clear
general pattern. The technical mathematical problems are amplified by
the more conceptual problem of what it might mean to say that one
could create a situation which forces the creation of closed timelike
curves. (See, e.g., Yurtsever 1990; Friedman, Morris, et al. 1990;
Novikov 1992; Earman 1995; and Earman, Smeenk, & Wuthrich
2009). What are we to think of all of this?


## 6. The Possibility of Time Travel Redux



The toy models above all treat billiard balls, fields, and other
objects propagating through a background spacetime with CTCs. Even if
we can show that a consistent solution exists, there is a further
question: what kind of matter and dynamics could generate CTCs to
begin with? There are various solutions of Einstein's equations
with CTCs, but how do these exotic spacetimes relate to the models
actually used in describing the world? In other words, what positive
reasons might we have to take CTCs seriously as a feature of the
actual universe, rather than an exotic possibility of primarily
mathematical interest?



We should distinguish two different kinds of "possibility"
that we might have in mind in posing such questions (following Stein
1970). First, we can consider a solution as a candidate cosmological
model, describing the (large-scale gravitational degrees of freedom of
the) entire universe. The case for ruling out spacetimes with CTCs as
potential cosmological models strikes us as, surprisingly, fairly
weak. Physicists used to simply rule out solutions with CTCs as
unreasonable by fiat, due to the threat of paradoxes, which we have
dismantled above. But it is also challenging to make an observational
case. Observations tell us very little about global features, such as
the existence of CTCs, because signals can only reach an observer from
a limited region of spacetime, called the past light cone. Our past
light cone--and indeed the collection of all the past light cones
for possible observers in a given spacetime--can be embedded in
spacetimes with quite different global features (Malament 1977,
Manchak 2009). This undercuts the possibility of using observations to
constrain global topology, including (among other things) ruling out
the existence of CTCs.



Yet the case in favor of taking cosmological models with CTCs
seriously is also not particularly strong. Some solutions used to
describe black holes, which are clearly relevant in a variety of
astrophysical contexts, include CTCs. But the question of whether the
CTCs themselves play an essential representational role is subtle: the
CTCs arise in the maximal extensions of these solutions, and can
plausibly be regarded as extraneous to successful applications.
Furthermore, many of the known solutions with CTCs have symmetries,
raising the possibility that CTCs are not a stable or robust feature.
Slight departures from symmetry may lead to a solution without CTCs,
suggesting that the CTCs may be an artifact of an idealized model.



The second sense of possibility regards whether
"reasonable" initial conditions can be shown to lead to,
or not to lead to, the formation of CTCs. As with the toy models
above, suppose that we have a partial Cauchy surface \(S\), such that
all the temporal funny business lies to the future. Rather than simply
assuming that there is a region with CTCs to the future, we can ask
instead whether it is possible to *create* CTCs by manipulating
matter in the initial, well-behaved region--that is, whether it
is possible to build a time machine. Several physicists have pursued
"chronology protection theorems" aiming to show that the
dynamics of general relativity (or some other aspects of physics)
rules this out, and to clarify why this is the case. The proof of such
a theorem would justify neglecting solutions with CTCs as a source of
insight into the nature of time in the actual world. But as of yet
there are several partial results that do not fully settle the
question. One further intriguing possibility is that even if general
relativity by itself does protect chronology, it may not be possible
to formulate a sensible theory describing matter and fields in
solutions with CTCs. (See SEP entry on Time Machines; Smeenk and
Wuthrich 2011 for more.)



There is a different question regarding the limitations of these toy
models. The toy models and related examples show that there are
consistent solutions for simple systems in the presence of CTCs. As
usual we have made the analysis tractable by building toy models,
selecting only a few dynamical degrees of freedom and tracking their
evolution. But there is a large gap between the systems we have
described and the time travel stories they evoke, with Kurt traveling
along a CTC with murderous intentions. In particular, many features of
the manifest image of time are tied to the thermodynamical properties
of macroscopic systems. Rovelli (unpublished) considers a extremely
simple system to illustrate the problem: can a clock move along a CTC?
A clock consists of something in periodic motion, such as a pendulum
bob, and something that counts the oscillations, such as an escapement
mechanism. The escapement mechanism cannot work without friction; this
requires dissipation and increasing entropy. For a clock that counts
oscillations as it moves along a time-like trajectory, the entropy
must be a monotonically increasing function. But that is obviously
incompatible with the clock returning to precisely the same state at
some future time as it completes a loop. The point generalizes,
obviously, to imply that anything like a human, with memory and
agency, cannot move along a CTC.


## 7. Even If There are Constraints, So What?



Since it is not obvious that one can rid oneself of all constraints in
realistic models, let us examine the argument that time travel is
implausible, and we should think it unlikely to exist in our world, in
so far as it implies such constraints. The argument goes something
like the following. In order to satisfy such constraints one needs
some pre-established divine harmony between the global (time travel)
structure of space-time and the distribution of particles and fields
on space-like surfaces in it. But it is not plausible that the actual
world, or any world even remotely like ours, is constructed with
divine harmony as part of the plan. In fact, one might argue, we have
empirical evidence that conditions in any spatial region can vary
quite arbitrarily. So we have evidence that such constraints, whatever
they are, do not in fact exist in our world. So we have evidence that
there are no closed time-like lines in our world or one remotely like
it. We will now examine this argument in more detail by presenting
four possible responses, with counterresponses, to this argument.



**Response 1.** There is nothing implausible or new about
such constraints. For instance, if the universe is spatially closed,
there has to be enough matter to produce the needed curvature, and
this puts constraints on the matter distribution on a space-like
hypersurface. Thus global space-time structure can quite
unproblematically constrain matter distributions on space-like
hypersurfaces in it. Moreover we have no realistic idea what these
constraints look like, so we hardly can be said to have evidence that
they do not obtain.



**Counterresponse 1.** Of course there are constraining
relations between the global structure of space-time and the matter in
it. The Einstein equations relate curvature of the manifold to the
matter distribution in it. But what is so strange and implausible
about the constraints imposed by the existence of closed time-like
curves is that these constraints in essence have nothing to do with
the Einstein equations. When investigating such constraints one
typically treats the particles and/or field in question as test
particles and/or fields in a given space-time, i.e., they are assumed
not to affect the metric of space-time in any way. In typical
space-times without closed time-like curves this means that one has,
in essence, complete freedom of matter distribution on a space-like
hypersurface. (See
 response 2
 for some more discussion of this issue). The constraints imposed by
the possibility of time travel have a quite different origin and are
implausible. In the ordinary case there is a causal interaction
between matter and space-time that results in relations between global
structure of space-time and the matter distribution in it. In the time
travel case there is no such causal story to be told: there simply has
to be some pre-established harmony between the global space-time
structure and the matter distribution on some space-like surfaces.
This is implausible.



**Response 2.** Constraints upon matter distributions are
nothing new. For instance, Maxwell's equations constrain
electric fields \(\boldsymbol{E}\) on an initial surface to be related
to the (simultaneous) charge density distribution \(\varrho\) by the
equation \(\varrho = \text{div}(\boldsymbol{E})\). (If we assume that
the \(E\) field is generated solely by the charge distribution, this
conditions amounts to requiring that the \(E\) field at any point in
space simply be the one generated by the charge distribution according
to Coulomb's inverse square law of electrostatics.) This is not
implausible divine harmony. Such constraints can hold as a matter of
physical law. Moreover, if we had inferred from the apparent free
variation of conditions on spatial regions that there could be no such
constraints we would have mistakenly inferred that \(\varrho =
\text{div}(\boldsymbol{E})\) could not be a law of nature.



**Counterresponse 2.** The constraints imposed by the
existence of closed time-like lines are of quite a different character
from the constraint imposed by \(\varrho =
\text{div}(\boldsymbol{E})\). The constraints imposed by \(\varrho =
\text{div}(\boldsymbol{E})\) on the state on a space-like hypersurface
are:


1. local constraints (i.e., to check whether the constraint holds in
a region you just need to see whether it holds at each point in the
region),
2. quite independent of the global space-time structure,
3. quite independent of how the space-like surface in question is
embedded in a given space-time, and
4. very simply and generally stateable.



On the other hand, the consistency constraints imposed by the
existence of closed time-like curves (i) are not local, (ii) are
dependent on the global structure of space-time, (iii) depend on the
location of the space-like surface in question in a given space-time,
and (iv) appear not to be simply stateable other than as the demand
that the state on that space-like surface embedded in such and such a
way in a given space-time, do not lead to inconsistency. On some views
of laws (e.g., David Lewis' view) this plausibly implies that
such constraints, even if they hold, could not possibly be laws. But
even if one does not accept such a view of laws, one could claim that
the bizarre features of such constraints imply that it is implausible
that such constraints hold in our world or in any world remotely like
ours.



**Response 3.** It would be strange if there are
constraints in the non-time travel region. It is not strange if there
are constraints in the time travel region. They should be explained in
terms of the strange, self-interactive, character of time travel
regions. In this region there are time-like trajectories from points
to themselves. Thus the state at such a point, in such a region, will,
in a sense, interact with itself. It is a well-known fact that systems
that interact with themselves will develop into an equilibrium state,
if there is such an equilibrium state, or else will develop towards
some singularity. Normally, of course, self-interaction isn't
true instantaneous self-interaction, but consists of a feed-back
mechanism that takes time. But in time travel regions something like
true instantaneous self-interaction occurs. This explains why
constraints on states occur in such time travel regions: the states
"*ab initio*" have to be "equilibrium
states". Indeed in a way this also provides some picture of why
indeterminism occurs in time travel regions: at the onset of
self-interaction states can fork into different equi-possible
equilibrium states.



**Counterresponse 3.** This is explanation by woolly
analogy. It all goes to show that time travel leads to such bizarre
consequences that it is unlikely that it occurs in a world remotely
like ours.




**Response 4.** All of the previous discussion completely
misses the point. So far we have been taking the space-time structure
as given, and asked the question whether a given time travel
space-time structure imposes constraints on states on (parts of)
space-like surfaces. However, space-time and matter interact. Suppose
that one is in a space-time with closed time-like lines, such that
certain counterfactual distributions of matter on some neighborhood of
a point \(p\) are ruled out if one holds that space-time structure
fixed. One might then ask



> 
> 
> Why does the actual state near \(p\) in fact satisfy these
> constraints? By what divine luck or plan is this local state
> compatible with the global space-time structure? What if conditions
> near \(p\) had been slightly different?
> 
> 
> 



And one might take it that the lack of normal answers to these
questions indicates that it is very implausible that our world, or any
remotely like it, is such a time travel universe. However the proper
response to these question is the following. There are no constraints
in any significant sense. If they hold they hold as a matter of
accidental fact, not of law. There is no more explanation of them
possible than there is of any contingent fact. Had conditions in a
neighborhood of \(p\) been otherwise, the global structure of
space-time would have been different. So what? The only question
relevant to the issue of constraints is whether an arbitrary state on
an arbitrary spatial surface \(S\) can always be embedded into a
space-time such that that state on \(S\) consistently extends to a
solution on the entire space-time.



But we know the answer to that question. A well-known theorem in
general relativity says the following: any initial data set on a three
dimensional manifold \(S\) with positive definite metric has a unique
embedding into a maximal space-time in which \(S\) is a Cauchy surface
(see, e.g., Geroch & Horowitz 1979: 284 for more detail), i.e.,
there is a unique largest space-time which has \(S\) as a Cauchy
surface and contains a consistent evolution of the initial value data
on \(S\). Now since \(S\) is a Cauchy surface this space-time does not
have closed time like curves. But it may have extensions (in which
\(S\) is not a Cauchy surface) which include closed timelike curves,
indeed it may be that any maximal extension of it would include closed
timelike curves. (This appears to be the case for extensions of states
on certain surfaces of Taub-NUT space-times. See Earman, Smeenk, &
Wuthrich 2009). But these extensions, of course, will be
consistent. So properly speaking, there are no constraints on states
on space-like surfaces. Nonetheless the space-time in which these are
embedded may or may not include closed time-like curves.




**Counterresponse 4.** This, in essence, is the
stonewalling answer which we indicated in section 1. However, whether
or not you call the constraints imposed by a given space-time on
distributions of matter on certain space-like surfaces "genuine
constraints", whether or not they can be considered lawlike, and
whether or not they need to be explained, the existence of such
constraints can still be used to argue that time travel worlds are so
bizarre that it is implausible that our world or any world remotely
like ours is a time travel world.



Suppose that one is in a time travel world. Suppose that given the
global space-time structure of this world, there are constraints
imposed upon, say, the state of motion of a ball on some space-like
surface when it is treated as a test particle, i.e., when it is
assumed that the ball does not affect the metric properties of the
space-time it is in. (There is lots of other matter that, via the
Einstein equation, corresponds exactly to the curvature that there is
everywhere in this time travel worlds.) Now a real ball of course does
have some effect on the metric of the space-time it is in. But let us
consider a ball that is so small that its effect on the metric is
negligible. Presumably it will still be the case that certain states
of this ball on that space-like surface are not compatible with the
global time travel structure of this universe.



This means that the actual distribution of matter on such a space-like
surface can be extended into a space-time with closed time-like lines,
but that certain counterfactual distributions of matter on this
space-like surface can not be extended into the same space-time.
*But note that the changes made in the matter distribution (when
going from the actual to the counterfactual distribution) do not in
any non-negligible way affect the metric properties of the
space-time.* (Recall that the changes only effect test particles.)
Thus the reason why the global time travel properties of the
counterfactual space-time have to be significantly different from the
actual space-time is not that there are problems with metric
singularities or alterations in the metric that force significant
global changes when we go to the counterfactual matter distribution.
The reason that the counterfactual space-time has to be different is
that in the counterfactual world the ball's initial state of
motion starting on the space-like surface, could not "meet
up" in a consistent way with its earlier self (could not be
consistently extended) if we were to let the global structure of the
counterfactual space-time be the same as that of the actual
space-time. Now, it is not bizarre or implausible that there is a
counterfactual dependence of manifold structure, even of its topology,
on matter distributions on spacelike surfaces. For instance, certain
matter distributions may lead to singularities, others may not. We may
indeed in some sense have causal power over the topology of the
space-time we live in. But this power normally comes via the Einstein
equations. But it is bizarre to think that there could be a
counterfactual dependence of global space-time structure on the
arrangement of certain tiny bits of matter on some space-like surface,
where changes in that arrangement by assumption do not affect the
metric *anywhere in space-time in any significant way*. It is
implausible that we live in such a world, or that a world even
remotely like ours is like that.



Let us illustrate this argument in a different way by assuming that
wormhole time travel imposes constraints upon the states of people
prior to such time travel, where the people have so little mass/energy
that they have negligible effect, via the Einstein equation, on the
local metric properties of space-time. Do you think it more plausible
that we live in a world where wormhole time travel occurs but it only
occurs when people's states are such that these local states
happen to combine with time travel in such a way that nobody ever
succeeds in killing their younger self, or do you think it more
plausible that we are not in a wormhole time travel
 world?[5]


## 8. Computational Models



An alternative approach to time travel (initiated by Deutsch 1991)
abstracts away from the idealized toy models described
 above.[6]
 This computational approach considers instead the evolution of bits
(simple physical systems with two discrete states) through a network
of interactions, which can be represented by a circuit diagram with
gates corresponding to the interactions. Motivated by the possibility
of CTCs, Deutsch proposed adding a new kind of channel that connects
the *output* of a given gate back to its
*input*--in essence, a backwards-time step. More
concretely, given a gate that takes \(n\) bits as input, we can
imagine taking some number \(i \lt n\) of these bits through a channel
that loops back and then do double-duty as inputs. Consistency
requires that the state of these \(i\) bits is the same for output and
input. (We will consider an illustration of this kind of system in the
next section.) Working through examples of circuit diagrams with a CTC
channel leads to similar treatments of *Consistency* and
*Underdetermination* as the discussion above (see, e.g.,
Wallace 2012: SS 10.6). But the approach offers two new insights
(both originally due to Deutsch): the *Easy Knowledge* paradox,
and a particularly clear extension to time travel in quantum
mechanics.



A computer equipped with a CTC channel can exploit the need to find
consistent evolution to solve remarkably hard problems. (This is quite
different than the first idea that comes to mind to enhance
computational power: namely to just devote more time to a computation,
and then send the result back on the CTC to an earlier state.) The
gate in a circuit incorporating a CTC implements a function from the
input bits to the output bits, under the constraint that the output
and input match the *i* bits going through the CTC channel.
This requires, in effect, finding the fixed point of the relevant
function. Given the generality of the model, there are few limits on
the functions that could be implemented on the CTC circuit. Nature has
to solve a hard computational problem just to ensure consistent
evolution. This can then be extended to other complex computational
problems--leading, more precisely, to solutions of
**NP**-complete problems in polynomial time (see Aaronson
2013: Chapter 20 for an overview and further references). The limits
imposed by computational complexity are an essential part of our
epistemic situation, and computers with CTCs would radically change
this.


## 9. Quantum Mechanics to the Rescue?



We now turn to the application of the computational approach to the
quantum physics of time travel (see Deutsch 1991; Deutsch &
Lockwood 1994). By contrast with the earlier discussions of
constraints in classical systems, they claim to show that time travel
never imposes any constraints on the pre-time travel state of quantum
systems. The essence of this account is as
 follows.[7]



A quantum system starts in state \(S\_1\), interacts with its older
self, after the interaction is in state \(S\_2\), time travels while
developing into state \(S\_3\), then interacts with its younger self,
and ends in state \(S\_4\) (see
 figure 10).



![a diagram: link to extended description below](figure-10.svg)

Figure 10 [An
 extended description of figure 10
 is in the supplement.]




Deutsch assumes that the set of possible states of this system are the
mixed states, i.e., are represented by the density matrices over the
Hilbert space of that system. Deutsch then shows that for any initial
state \(S\_1\), any unitary interaction between the older and younger
self, and any unitary development during time travel, there is a
consistent solution, i.e., there is at least one pair of states
\(S\_2\) and \(S\_3\) such that when \(S\_1\) interacts with \(S\_3\) it
will change to state \(S\_2\) and \(S\_2\) will then develop into
\(S\_3\). The states \(S\_2, S\_3\) and \(S\_4\) will typically be not be
pure states, i.e., will be non-trivial mixed states, even if \(S\_1\)
is pure. In order to understand how this leads to interpretational
problems let us give an example. Consider a system that has a two
dimensional Hilbert space with as a basis the states \(\vc{+}\) and
\(\vc{-}\). Let us suppose that when state \(\vc{+}\) of the young
system encounters state \(\vc{+}\) of the older system, they interact
and the young system develops into state \(\vc{-}\) and the old system
remains in state \(\vc{+}\). In obvious notation:

 

\[\vc{+}\_1 \vc{+}\_3 \text{ develops into } \vc{-}\_2 \vc{+}\_4.\]


Similarly, suppose that:



\[\begin{align}
&\vc{+}\_1 \vc{-}\_3 \text{ develops into } \vc{+}\_2 \vc{-}\_4, \\
&\vc{-}\_1 \vc{+}\_3 \text{ develops into } \vc{-}\_2 \vc{-}\_4, \text{ and} \\
&\vc{-}\_1 \vc{-}\_3 \text{ develops into } \vc{+}\_2 \vc{+}\_4.
\end{align}\]


Let us furthermore assume that there is no development of the state of
the system during time travel, i.e., that \(\vc{+}\_2\) develops into
\(\vc{+}\_3\), and that \(\vc{-}\_2\) develops into \(\vc{-}\_3\).



Now, if the only possible states of the system were \(\vc{+}\) and
\(\vc{-}\) (i.e., if there were no superpositions or mixtures of these
states), then there is a constraint on initial states: initial state
\(\vc{+}\_1\) is impossible. For if \(\vc{+}\_1\) interacts with
\(\vc{+}\_3\) then it will develop into \(\vc{-}\_2\), which, during
time travel, will develop into \(\vc{-}\_3\), which inconsistent with
the assumed state \(\vc{+}\_3\). Similarly if \(\vc{+}\_1\) interacts
with \(\vc{-}\_3\) it will develop into \(\vc{+}\_2\), which will then
develop into \(\vc{+}\_3\) which is also inconsistent. Thus the system
can not start in state \(\vc{+}\_1\).



But, says Deutsch, in quantum mechanics such a system can also be in
any mixture of the states \(\vc{+}\) and \(\vc{-}\). Suppose that the
older system, prior to the interaction, is in a state \(S\_3\) which is
an equal mixture of 50% \(\vc{+}\_3\) and 50% \(\vc{-}\_3\). Then the
younger system during the interaction will develop into a mixture of
50% \(\vc{+}\_2\) and 50% \(\vc{-}\_2\), which will then develop into a
mixture of 50% \(\vc{+}\_3\) and 50% \(\vc{-}\_3\), which is consistent!
More generally Deutsch uses a fixed point theorem to show that no
matter what the unitary development during interaction is, and no
matter what the unitary development during time travel is, for any
state \(S\_1\) there is always a state \(S\_3\) (which typically is not
a pure state) which causes \(S\_1\) to develop into a state \(S\_2\)
which develops into that state \(S\_3\). Thus quantum mechanics comes
to the rescue: it shows in all generality that no constraints on
initial states are needed!



One might wonder why Deutsch appeals to mixed states: will
superpositions of states \(\vc{+}\) and \(\vc{-}\) not suffice?
Unfortunately such an idea does not work. Suppose again that the
initial state is \(\vc{+}\_1\). One might suggest that that if state
\(S\_3\) is

 
\[\sqt \vc{+}\_3 + \sqt \vc{-}\_3\]


one will obtain a consistent development. For one might think that
when initial state \(\vc{+}\_1\) encounters the superposition



\[\sqt\vc{+}\_3 + \sqt\vc{-}\_3,\]


it will develop into superposition

 
\[\sqt\vc{+}\_2 + \sqt\vc{-}\_2,\]


and that this in turn will develop into

 
\[\sqt\vc{+}\_3 + \sqt\vc{-}\_3,\]


as desired. However this is not correct. For initial state
\(\vc{+}\_1\) when it encounters

 
\[\sqt\vc{+}\_3 + \sqt\vc{-}\_3,\]


will develop into the entangled state

 
\[\sqt\vc{-}\_2\vc{+}\_4 + \sqt\vc{+}\_2\vc{-}\_4.\]


In so far as one can speak of the state of the young system after this
interaction, it is in the mixture of 50% \(\vc{+}\_2\) and 50%
\(\vc{-}\_2\), not in the superposition

 
\[\sqt\vc{+}\_2 + \sqt\vc{-}\_2.\]


So Deutsch does need his recourse to mixed states.



This clarification of why Deutsch needs his mixtures does however
indicate a serious worry about the simplifications that are part of
Deutsch's account. After the interaction the old and young
system will (typically) be in an entangled state. Although for
purposes of a measurement on one of the two systems one can say that
this system is in a mixed state, one can not represent the full state
of the two systems by specifying the mixed state of each separate
part, as there are correlations between observables of the two systems
that are not represented by these two mixed states, but are
represented in the joint entangled state. But if there really is an
entangled state of the old and young systems directly after the
interaction, how is one to represent the subsequent development of
this entangled state? Will the state of the younger system remain
entangled with the state of the older system as the younger system
time travels and the older system moves on into the future? On what
space-like surfaces are we to imagine this total entangled state to
be? At this point it becomes clear that there is no obvious and simple
way to extend elementary non-relativistic quantum mechanics to
space-times with closed time-like curves: we apparently need to
characterize not just the entanglement between two systems, but
entanglement relative to specific spacetime descriptions.



How does Deutsch avoid these complications? Deutsch assumes a mixed
state \(S\_3\) of the older system prior to the interaction with the
younger system. He lets it interact with an arbitrary pure state
\(S\_1\) younger system. After this interaction there is an entangled
state \(S'\) of the two systems. Deutsch computes the mixed state
\(S\_2\) of the younger system which is implied by this entangled state
\(S'\). His demand for consistency then is just that this mixed state
\(S\_2\) develops into the mixed state \(S\_3\). Now it is not at all
clear that this is a legitimate way to simplify the problem of time
travel in quantum mechanics. But even if we grant him this
simplification there is a problem: how are we to understand these
mixtures?



If we take an ignorance interpretation of mixtures we run into
trouble. For suppose that we assume that in each individual case each
older system is either in state \(\vc{+}\_3\) or in state \(\vc{-}\_3\)
prior to the interaction. Then we regain our paradox. Deutsch instead
recommends the following, many worlds, picture of mixtures. Suppose we
start with state \(\vc{+}\_1\) in all worlds. In some of the many
worlds the older system will be in the \(\vc{+}\_3\) state, let us call
them *A*-worlds, and in some worlds, *B*-worlds, it will
be in the \(\vc{-}\_3\) state. Thus in *A*-worlds after
interaction we will have state \(\vc{-}\_2\) , and in *B*-worlds
we will have state \(\vc{+}\_2\). During time travel the \(\vc{-}\_2\)
state will remain the same, i.e., turn into state \(\vc{-}\_3\), but
the systems in question will travel from *A*-worlds to
*B*-worlds. Similarly the \(\vc{+}\) \(\_2\) states will travel
from the *B*-worlds to the *A*-worlds, thus preserving
consistency.



Now whatever one thinks of the merits of many worlds interpretations,
and of this understanding of it applied to mixtures, in the end one
does not obtain genuine time travel in Deutsch's account. The
systems in question travel from one time in one world to another time
in another world, but no system travels to an earlier time in the same
world. (This is so at least in the normal sense of the word
"world", the sense that one means when, for instance, one
says "there was, and will be, only one Elvis Presley in this
world.") Thus, even if it were a reasonable view, it is not
quite as interesting as it may have initially seemed. (See Wallace
2012 for a more sympathetic treatment, that explores several further
implications of accepting time travel in conjunction with the many
worlds interpretation.)



We close by acknowledging that Deutsch's starting
point--the claim that this computational model captures the
essential features of quantum systems in a spacetime with
CTCs--has been the subject of some debate. Several physicists
have pursued a quite different treatment of evolution of quantum
systems through CTC's, based on considering the
"post-selected" state (see Lloyd et al. 2011). Their
motivations for implementing the consistency condition in terms of the
post-selected state reflects a different stance towards quantum
foundations. A different line of argument aims to determine whether
Deutsch's treatment holds as an appropriate limiting case of a
more rigorous treatment, such as quantum field theory in curved
spacetimes. For example, Verch (2020) establishes several results
challenging the assumption that Deutsch's treatment is tied to
the presence of CTC's, or that it is compatible with the
entanglement structure of quantum fields.


## 10. Conclusions



What remains of the grandfather paradox in general relativistic time
travel worlds is the fact that in some cases the states on edgeless
spacelike surfaces are "overconstrained", so that one has
less than the usual freedom in specifying conditions on such a
surface, given the time-travel structure, and in some cases such
states are "underconstrained", so that states on edgeless
space-like surfaces do not determine what happens elsewhere in the way
that they usually do, given the time travel structure. There can also
be mixtures of those two types of cases. The extent to which states
are overconstrained and/or underconstrained in realistic models is as
yet unclear, though it would be very surprising if neither obtained.
The extant literature has primarily focused on the problem of
overconstraint, since that, often, either is regarded as a
metaphysical obstacle to the possibility time travel, or as an
epistemological obstacle to the plausibility of time travel in our
world. While it is true that our world would be quite different from
the way we normally think it is if states were overconstrained,
underconstraint seems at least as bizarre as overconstraint.
Nonetheless, neither directly rules out the possibility of time
travel.



If time travel entailed contradictions then the issue would be
settled. And indeed, most of the stories employing time travel in
popular culture are logically incoherent: one cannot
"change" the past to be different from what it was, since
the past (like the present and the future) only occurs once. But if
the only requirement demanded is logical coherence, then it seems all
too easy. A clever author can devise a coherent time-travel scenario
in which everything happens just once and in a consistent way. This is
just too cheap: logical coherence is a very weak condition, and many
things we take to be metaphysically impossible are logically coherent.
For example, it involves no logical contradiction to suppose that
water is not molecular, but if both chemistry and Kripke are right it
is a metaphysical impossibility. We have been interested not in
logical possibility but in physical possibility. But even so, our
conditions have been relatively weak: we have asked only whether
time-travel is consistent with the universal validity of certain
fundamental physical laws and with the notion that the physical state
on a surface prior to the time travel region be unconstrained. It is
perfectly possible that the physical laws obey this condition, but
still that time travel is not metaphysically possible because of the
nature of time itself. Consider an analogy. Aristotle believed that
water is homoiomerous and infinitely divisible: any bit of water could
be subdivided, in principle, into smaller bits of water.
Aristotle's view contains no logical contradiction. It was
certainly consistent with Aristotle's conception of water that
it be homoiomerous, so this was, for him, a conceptual possibility.
But if chemistry is right, Aristotle was wrong both about what water
is like and what is possible for it. It can't be infinitely
divided, even though no logical or conceptual analysis would reveal
that.



Similarly, even if all of our consistency conditions can be met, it
does not follow that time travel is physically possible, only that
some specific physical considerations cannot rule it out. The only
serious proof of the possibility of time travel would be a
demonstration of its actuality. For if we agree that there is no
actual time travel in our universe, the supposition that there might
have been involves postulating a substantial difference from
actuality, a difference unlike in kind from anything we could know if
firsthand. It is unclear to us exactly what the content of possible
would be if one were to either maintain or deny the possibility of
time travel in these circumstances, unless one merely meant that the
possibility is not ruled out by some delineated set of constraints. As
the example of Aristotle's theory of water shows, conceptual and
logical "possibility" do not entail possibility in a
full-blooded sense. What exactly such a full-blooded sense would be in
case of time travel, and whether one could have reason to believe it
to obtain, remain to us obscure.