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Time Travel and Modern Physics

Time travel has been a staple of science fiction. With the advent of general relativity it has been entertained by serious physicists. But, especially in the philosophy literature, there have been arguments that time travel is inherently paradoxical. The most famous paradox is the grandfather paradox: you travel back in time and kill your grandfather, thereby preventing your own existence. To avoid inconsistency some circumstance will have to occur which makes you fail in this attempt to kill your grandfather. Doesn’t this require some implausible constraint on otherwise unrelated circumstances? We examine such worries in the context of modern physics.

1. Paradoxes Lost?

2. topology and constraints, 3. the general possibility of time travel in general relativity, 4. two toy models, 5. slightly more realistic models of time travel, 6. the possibility of time travel redux, 7. even if there are constraints, so what, 8. computational models, 9. quantum mechanics to the rescue, 10. conclusions, other internet resources, related entries.

• Supplement: Remarks and Limitations on the Toy Models

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 Gödel 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., Gödel 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.

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 .

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.

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.

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 Gödel 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 Gödel’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.

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

• every point in the manifold can be connected by a timelike curve to \(S\), and
• 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 \(L+\).

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:

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:

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 :

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.)

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 \(z\).

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:

Solving these equations for \(z\) yields

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.

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.

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, & Wüthrich 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, & Wüthrich 2009). What are we to think of all of this?

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 Wüthrich 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.

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:

• 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),
• quite independent of the global space-time structure,
• quite independent of how the space-like surface in question is embedded in a given space-time, and
• 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, & Wüthrich 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 ]

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: § 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.

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 ).

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:

Similarly, suppose that:

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

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

it will develop into superposition

and that this in turn will develop into

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

will develop into the entangled state

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

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.

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.

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How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

• Adlam, Emily, unpublished, “ Is There Causation in Fundamental Physics? New Insights from Process Matrices and Quantum Causal Modelling ”, 2022, arXiv: 2208.02721. doi:10.48550/ARXIV.2208.02721
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Can we time travel? A theoretical physicist provides some answers

Emeritus professor, Physics, Carleton University

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Peter Watson received funding from NSERC. He is affiliated with Carleton University and a member of the Canadian Association of Physicists.

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Time travel makes regular appearances in popular culture, with innumerable time travel storylines in movies, television and literature. But it is a surprisingly old idea: one can argue that the Greek tragedy Oedipus Rex , written by Sophocles over 2,500 years ago, is the first time travel story .

But is time travel in fact possible? Given the popularity of the concept, this is a legitimate question. As a theoretical physicist, I find that there are several possible answers to this question, not all of which are contradictory.

The simplest answer is that time travel cannot be possible because if it was, we would already be doing it. One can argue that it is forbidden by the laws of physics, like the second law of thermodynamics or relativity . There are also technical challenges: it might be possible but would involve vast amounts of energy.

There is also the matter of time-travel paradoxes; we can — hypothetically — resolve these if free will is an illusion, if many worlds exist or if the past can only be witnessed but not experienced. Perhaps time travel is impossible simply because time must flow in a linear manner and we have no control over it, or perhaps time is an illusion and time travel is irrelevant.

Laws of physics

Since Albert Einstein’s theory of relativity — which describes the nature of time, space and gravity — is our most profound theory of time, we would like to think that time travel is forbidden by relativity. Unfortunately, one of his colleagues from the Institute for Advanced Study, Kurt Gödel, invented a universe in which time travel was not just possible, but the past and future were inextricably tangled.

We can actually design time machines , but most of these (in principle) successful proposals require negative energy , or negative mass, which does not seem to exist in our universe. If you drop a tennis ball of negative mass, it will fall upwards. This argument is rather unsatisfactory, since it explains why we cannot time travel in practice only by involving another idea — that of negative energy or mass — that we do not really understand.

Mathematical physicist Frank Tipler conceptualized a time machine that does not involve negative mass, but requires more energy than exists in the universe .

Time travel also violates the second law of thermodynamics , which states that entropy or randomness must always increase. Time can only move in one direction — in other words, you cannot unscramble an egg. More specifically, by travelling into the past we are going from now (a high entropy state) into the past, which must have lower entropy.

This argument originated with the English cosmologist Arthur Eddington , and is at best incomplete. Perhaps it stops you travelling into the past, but it says nothing about time travel into the future. In practice, it is just as hard for me to travel to next Thursday as it is to travel to last Thursday.

Resolving paradoxes

There is no doubt that if we could time travel freely, we run into the paradoxes. The best known is the “ grandfather paradox ”: one could hypothetically use a time machine to travel to the past and murder their grandfather before their father’s conception, thereby eliminating the possibility of their own birth. Logically, you cannot both exist and not exist.

Read more: Time travel could be possible, but only with parallel timelines

Kurt Vonnegut’s anti-war novel Slaughterhouse-Five , published in 1969, describes how to evade the grandfather paradox. If free will simply does not exist, it is not possible to kill one’s grandfather in the past, since he was not killed in the past. The novel’s protagonist, Billy Pilgrim, can only travel to other points on his world line (the timeline he exists in), but not to any other point in space-time, so he could not even contemplate killing his grandfather.

The universe in Slaughterhouse-Five is consistent with everything we know. The second law of thermodynamics works perfectly well within it and there is no conflict with relativity. But it is inconsistent with some things we believe in, like free will — you can observe the past, like watching a movie, but you cannot interfere with the actions of people in it.

Could we allow for actual modifications of the past, so that we could go back and murder our grandfather — or Hitler ? There are several multiverse theories that suppose that there are many timelines for different universes. This is also an old idea: in Charles Dickens’ A Christmas Carol , Ebeneezer Scrooge experiences two alternative timelines, one of which leads to a shameful death and the other to happiness.

Time is a river

Roman emperor Marcus Aurelius wrote that:

“ Time is like a river made up of the events which happen , and a violent stream; for as soon as a thing has been seen, it is carried away, and another comes in its place, and this will be carried away too.”

We can imagine that time does flow past every point in the universe, like a river around a rock. But it is difficult to make the idea precise. A flow is a rate of change — the flow of a river is the amount of water that passes a specific length in a given time. Hence if time is a flow, it is at the rate of one second per second, which is not a very useful insight.

Theoretical physicist Stephen Hawking suggested that a “ chronology protection conjecture ” must exist, an as-yet-unknown physical principle that forbids time travel. Hawking’s concept originates from the idea that we cannot know what goes on inside a black hole, because we cannot get information out of it. But this argument is redundant: we cannot time travel because we cannot time travel!

Researchers are investigating a more fundamental theory, where time and space “emerge” from something else. This is referred to as quantum gravity , but unfortunately it does not exist yet.

So is time travel possible? Probably not, but we don’t know for sure!

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Time Travel without Regrets

Time travel is not ruled out by general relativity, but it might well create problems for the laws of common sense. In the 28 January Physical Review Letters , a team proposes a new way of deciding the possibility or impossibility of quantum states that travel forward and backward in time. The new criterion automatically disallows quantum versions of the “grandfather paradox,” in which a person travels back in time and kills her ancestor, thereby ensuring her own demise. The team also performed an experiment that illustrates the paradox-nullifying mechanism.

General relativity, Einstein’s theory of space and time, allows the existence of closed timelike curves (CTCs)–paths that go forward in time, then back again to reconnect and form closed loops. Although it’s unclear whether CTCs can be created, physicists have nevertheless explored their possible consequences, including their influence on quantum mechanics.

An ordinary quantum event might involve two particles moving forward in time, changing each other by interacting at some time, then going their separate ways into the future. However, if one outgoing particle enters a CTC, it can double back and become one of the ingoing particles–thus influencing its own transformation. In 1991, Oxford University physicist David Deutsch proposed a consistency condition to avoid time-travel paradoxes: a particle that loops back in time in this way should be in the same quantum state when in reappears in the immediate past of the interaction as it was when it departed the interaction for the immediate future [1] .

To see how this condition works, imagine a quantum particle having states labeled 0 and 1. It travels around a CTC and, on its return, interacts with an “external” particle in such a way that 0 becomes 1 and 1 becomes 0. Such a particle presents a quantum grandfather paradox: when it comes back around the loop, it flips its former self to the opposite state. However, Deutsch showed that consistency is possible if the particle is in a superposition–a state that is equal parts 0 and 1. The interaction exchanges the 0 and the 1, but the state overall remains unchanged. For this to work, the external particle must also be in a superposition that flips back and forth.

The paradox is avoided, but a difficulty arises if the external particle is measured. Then it cannot remain in a superposition but must become definitely either 0 or 1–which means that the CTC particle cannot remain in a superposition, either. To preserve consistency, Deutsch argued that the CTC particle must exist in two parallel universes–the “1-universe” and the “0-universe”–and continually switch between them, so that no contradiction occurs in either one.

Lorenzo Maccone, of the Massachusetts Institute of Technology and the University of Pavia, Italy, and his colleagues propose a more stringent condition that avoids these difficulties. They require that any measurement of the particle going into the future should yield the same result as measuring it when it returns from the past. So any state that would alter the past when it came around again is disallowed, and no grandfather-type paradoxes can arise.

Perhaps surprisingly, “we can still have CTCs even with this strong condition,” Maccone says. Only states that avoid paradoxes after the interaction are able to exist beforehand, so the team calls their condition “post-selection.”

To demonstrate these ideas, the team performed an experiment with photons showing that the consistency condition indeed picks out specific states and destroys all the rest. Lacking an actual CTC to perform the post-selection, the team created photons in a specific quantum state for the input, a state where the polarization was not known or measured but had a correlation with another property, associated with the photon’s path. As the photon went through the experiment, it experienced changes that mimicked the 0-to-1 flipping that occurs in the imagined time-travel arrangement. The team found that only those photons that wouldn’t lead to paradoxes made it through unscathed. Although the result is in line with expectation, no one has simulated time travel in this way before.

An odd consequence of post-selection is that because the presence of a CTC annuls paradoxical states completely, it can disallow some states that seem innocuous today but have unacceptable consequences later. “In principle, one could detect the future existence of time machines by … looking for deviations now from the predictions of quantum mechanics,” says Todd Brun of the University of Southern California in Los Angeles. Although, he adds, it’s hard to know in advance what to measure.

–David Lindley

David Lindley is a freelance science writer in Alexandria, Virginia.

• D. Deutsch, “Quantum Mechanics near Closed Timelike Lines,” Phys. Rev. D 44, 3197 (1991)

More Information

Time Travel Beats Quantum Mechanics ( Focus story from 2009)

Closed Timelike Curves via Postselection: Theory and Experimental Test of Consistency

Seth Lloyd, Lorenzo Maccone, Raul Garcia-Patron, Vittorio Giovannetti, Yutaka Shikano, Stefano Pirandola, Lee A. Rozema, Ardavan Darabi, Yasaman Soudagar, Lynden K. Shalm, and Aephraim M. Steinberg

Phys. Rev. Lett. 106 , 040403 (2011)

Published January 27, 2011

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Physical Review A

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Treating time travel quantum mechanically

John-mark a. allen, phys. rev. a 90 , 042107 – published 9 october 2014.

• Citing Articles (19)
• INTRODUCTION
• MODELING TIME TRAVEL WITH QUANTUM…
• D-CTCs AND P-CTCs
• ALTERNATE THEORIES
• ACKNOWLEDGMENTS

The fact that closed timelike curves (CTCs) are permitted by general relativity raises the question as to how quantum systems behave when time travel to the past occurs. Research into answering this question by utilizing the quantum circuit formalism has given rise to two theories: Deutschian-CTCs (D-CTCs) and “postselected” CTCs (P-CTCs). In this paper the quantum circuit approach is thoroughly reviewed, and the strengths and shortcomings of D-CTCs and P-CTCs are presented in view of their nonlinearity and time-travel paradoxes. In particular, the “equivalent circuit model”—which aims to make equivalent predictions to D-CTCs, while avoiding some of the difficulties of the original theory—is shown to contain errors. The discussion of D-CTCs and P-CTCs is used to motivate an analysis of the features one might require of a theory of quantum time travel, following which two overlapping classes of alternate theories are identified. One such theory, the theory of “transition probability” CTCs (T-CTCs), is fully developed. The theory of T-CTCs is shown not to have certain undesirable features—such as time-travel paradoxes, the ability to distinguish nonorthogonal states with certainty, and the ability to clone or delete arbitrary pure states—that are present with D-CTCs and P-CTCs. The problems with nonlinear extensions to quantum mechanics are discussed in relation to the interpretation of these theories, and the physical motivations of all three theories are discussed and compared.

• Received 22 January 2014

DOI: https://doi.org/10.1103/PhysRevA.90.042107

©2014 American Physical Society

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• Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom
• * [email protected]

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Schematic diagram for the standard form circuit described in the text. The n CR and m CV qubits are shown entering the gate labeled by unitary U . The double bars represent the time-travel event and may be thought of as two depictions of the same spacelike hypersurface, thus forming a CTC. The dashed lines represent the spacelike boundaries of the region in which time travel takes place; CV qubits are restricted to that region.

Circuit diagram for the unproven theorem circuit of Ref. [ 16 ] in the standard form. The book qubit is labeled B , mathematician labeled M , and time traveler labeled T . The input to the circuit is taken to be the pure state | 0 〉 B | 0 〉 M . The notation used for the gates is the standard notation as defined in Ref. [ 14 ] and describes a unitary U = s w a p M T c n o t B M c n o t T B .

The equivalent circuit model. (a) A circuit equivalent to the standard form of Fig.  1 with V = s w a p U . (b) The same circuit “unwrapped” in the equivalent circuit model. There is an infinite ladder of unwrapped circuits, each with the same input state ρ i . To start the ladder, an initial CV state σ 0 is guessed. The CV state of the N th rung of the ladder is σ N . The output state ρ f is taken after a number N → ∞ of rungs of the ladder have been iterated.

Schematic diagram illustrating the protocol defining the action of P-CTCs as described in the text. When Alice measures the final state of the B and A qubits to be | Φ 〉 , the standard teleportation protocol would have Bob (who holds the other half of the initially entangled state) then holding the state with which Alice measured the A system. There are two unusual things about this situation: first, that Bob's system is the same one that Alice wants to teleport, just at an earlier time, and, second, that Alice can get the outcome | Φ 〉 with certainty and so no classical communication to Bob is required to complete the teleportation.

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Scientists Discovered How to Speed Up Time. Seriously.

We can reverse it, too.

• In the physical world, time marches in one direction, but things aren’t so straight forward in the quantum realm.
• This isn’t exactly time travel , but is instead implementing or reverting to different quantum states from different points in time.

In the subatomic universe of quantum physics, you can achieve things considered impossible in our flesh-and-blood physical world. Things like superposition , entanglement , and even teleportation all seem possible when things go quantum. Now, scientists from the Austrian Academy of Sciences (ÖAW) and University of Vienna are adding a kind of time travel to the list.

In a series of papers published throughout the past few years on preprint servers and in various online journals (including Optica , arXiv , and Quantum ), researchers including ÖAW’s Miguel Navascués and University of Vienna’s Philip Walther explain the possibility of speeding up, slowing down, and even reversing the flow of time within a quantum system.

Navascués compares the phenomenon to different movie-watching experiences. “In a theater [classical physics], a movie is projected from beginning to end, regardless of what the audience wants,” he explains to the Spanish-language newspaper El País . “But at home [the quantum world], we have a remote control to manipulate the movie. We can rewind to a previous scene or skip several scenes ahead.”

The researchers achieved this by “evolving” a single photon as it passes through a crystal . Using an experimental device called a “quantum switch,” the single photon of light returns to its previous state before it ever makes the journey. In a way, this is less Doc Brown-style time travel and more about reverting or otherwise altering the states of quantum particles, or “time translation” as Navascués described in 2020 .

However, this isn’t exactly like a rewind button on your TV because usually, viewers can see how things got from plot point A to B—just sped up and in reverse. In quantum mechanics, however, simply observing a system causes it to change, which makes it impossible to track a system’s progress through time. Crucially, these rewinding protocols still work because they can be performed without knowing what the changes were or its “internal dynamics,” according to the scientists.

And this quantum time machine doesn’t just go one direction—Navascués says they’ve also hit upon a method for going forward in evolutionary time as well. He tells El País :

“To make a system age 10 years in one year, you must get the other nine years from somewhere. In a year-long experiment with 10 systems, you can steal one year from each of the first nine systems and give them all to the tenth. At the end of the year, the tenth system will have aged 10 years; the other nine will remain the same as when the experiment began.”

Sadly, these sci-fi findings in the quantum world can’t be sized up to send humans backward and forward in time, because a single human represents a mind-boggling amount of information to “rejuvenate”—in fact, the scientists estimate it would take millions of years to pull it off for just one second.

But for the teams at ÖAW and the University of Vienna, the point isn’t jetting off to the distant future of 2015 , but the ability to increase the capability of quantum processors by arming them with the possibility of reversing errors in a system. After all, if life had a rewind button, wouldn’t you use it?

Darren lives in Portland, has a cat, and writes/edits about sci-fi and how our world works. You can find his previous stuff at Gizmodo and Paste if you look hard enough. 

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Is Time Travel Possible?

We all travel in time! We travel one year in time between birthdays, for example. And we are all traveling in time at approximately the same speed: 1 second per second.

We typically experience time at one second per second. Credit: NASA/JPL-Caltech

NASA's space telescopes also give us a way to look back in time. Telescopes help us see stars and galaxies that are very far away . It takes a long time for the light from faraway galaxies to reach us. So, when we look into the sky with a telescope, we are seeing what those stars and galaxies looked like a very long time ago.

However, when we think of the phrase "time travel," we are usually thinking of traveling faster than 1 second per second. That kind of time travel sounds like something you'd only see in movies or science fiction books. Could it be real? Science says yes!

This image from the Hubble Space Telescope shows galaxies that are very far away as they existed a very long time ago. Credit: NASA, ESA and R. Thompson (Univ. Arizona)

How do we know that time travel is possible?

More than 100 years ago, a famous scientist named Albert Einstein came up with an idea about how time works. He called it relativity. This theory says that time and space are linked together. Einstein also said our universe has a speed limit: nothing can travel faster than the speed of light (186,000 miles per second).

Einstein's theory of relativity says that space and time are linked together. Credit: NASA/JPL-Caltech

What does this mean for time travel? Well, according to this theory, the faster you travel, the slower you experience time. Scientists have done some experiments to show that this is true.

For example, there was an experiment that used two clocks set to the exact same time. One clock stayed on Earth, while the other flew in an airplane (going in the same direction Earth rotates).

After the airplane flew around the world, scientists compared the two clocks. The clock on the fast-moving airplane was slightly behind the clock on the ground. So, the clock on the airplane was traveling slightly slower in time than 1 second per second.

Credit: NASA/JPL-Caltech

Can we use time travel in everyday life?

We can't use a time machine to travel hundreds of years into the past or future. That kind of time travel only happens in books and movies. But the math of time travel does affect the things we use every day.

For example, we use GPS satellites to help us figure out how to get to new places. (Check out our video about how GPS satellites work .) NASA scientists also use a high-accuracy version of GPS to keep track of where satellites are in space. But did you know that GPS relies on time-travel calculations to help you get around town?

GPS satellites orbit around Earth very quickly at about 8,700 miles (14,000 kilometers) per hour. This slows down GPS satellite clocks by a small fraction of a second (similar to the airplane example above).

GPS satellites orbit around Earth at about 8,700 miles (14,000 kilometers) per hour. Credit: GPS.gov

However, the satellites are also orbiting Earth about 12,550 miles (20,200 km) above the surface. This actually speeds up GPS satellite clocks by a slighter larger fraction of a second.

Here's how: Einstein's theory also says that gravity curves space and time, causing the passage of time to slow down. High up where the satellites orbit, Earth's gravity is much weaker. This causes the clocks on GPS satellites to run faster than clocks on the ground.

The combined result is that the clocks on GPS satellites experience time at a rate slightly faster than 1 second per second. Luckily, scientists can use math to correct these differences in time.

If scientists didn't correct the GPS clocks, there would be big problems. GPS satellites wouldn't be able to correctly calculate their position or yours. The errors would add up to a few miles each day, which is a big deal. GPS maps might think your home is nowhere near where it actually is!

In Summary:

Yes, time travel is indeed a real thing. But it's not quite what you've probably seen in the movies. Under certain conditions, it is possible to experience time passing at a different rate than 1 second per second. And there are important reasons why we need to understand this real-world form of time travel.

If you liked this, you may like:

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Quantum time: Is this where the flow of existence comes from?

Why we perceive the passage of time is one of the biggest mysteries of physics. Now we could have found its source – in our most potent theory of reality

By Michael Brooks

18 April 2018

Henrik Sorensen/Getty

THE philosopher Augustine of Hippo once wrote that he felt he knew what time was, so long as no one asked him. Fast forward 16 centuries and the picture has hardly changed, says physicist Carlo Rovelli . Time is “perhaps the greatest mystery”, he says. “At the most fundamental level we currently know of, there is little that resembles time as we experience it.”

The passage of time – a uniform, universal flow that transports us inexorably from a past we cannot revisit to a future we cannot know – is perhaps the most fundamental experience of our existence. Yet our best theories suggest that it is not real . Time doesn’t flow, and past, present and future cannot sensibly be defined. There isn’t even one single time that governs the order of events.

Rovelli, who works at Aix-Marseille University in France, is just one of many physicists hunting for a better answer. As they do, a new generation of experiments is giving hope that we can probe the nature of time more searchingly than ever before. In all this, a strange realisation is crystallising, one that perhaps brings us a little closer to the heart of the mystery. Maybe we shouldn’t be so worried about our ignorance of time. Perhaps, at some level, time is just that – ignorance.

A century ago, Albert Einstein revolutionised our ideas of time. His theories of relativity gave time a physical identity as part of space-time, a malleable fabric on which reality’s events play out (see “ Why now doesn’t exist, and other strange facts about time ”).

Einstein’s ideas are mathematically sound and rigorously confirmed…

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September 2, 2014

Time Travel Simulation Resolves “Grandfather Paradox”

What would happen to you if you went back in time and killed your grandfather? A model using photons reveals that quantum mechanics can solve the quandary—and even foil quantum cryptography

By Lee Billings

On June 28, 2009, the world-famous physicist Stephen Hawking threw a party at the University of Cambridge, complete with balloons, hors d'oeuvres and iced champagne. Everyone was invited but no one showed up. Hawking had expected as much, because he only sent out invitations after his party had concluded. It was, he said, "a welcome reception for future time travelers," a tongue-in-cheek experiment to reinforce his 1992 conjecture that travel into the past is effectively impossible.

But Hawking may be on the wrong side of history. Recent experiments offer tentative support for time travel's feasibility—at least from a mathematical perspective. The study cuts to the core of our understanding of the universe, and the resolution of the possibility of time travel, far from being a topic worthy only of science fiction, would have profound implications for fundamental physics as well as for practical applications such as quantum cryptography and computing.

Closed timelike curves The source of time travel speculation lies in the fact that our best physical theories seem to contain no prohibitions on traveling backward through time. The feat should be possible based on Einstein's theory of general relativity, which describes gravity as the warping of spacetime by energy and matter. An extremely powerful gravitational field, such as that produced by a spinning black hole, could in principle profoundly warp the fabric of existence so that spacetime bends back on itself. This would create a "closed timelike curve," or CTC, a loop that could be traversed to travel back in time.

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Hawking and many other physicists find CTCs abhorrent, because any macroscopic object traveling through one would inevitably create paradoxes where cause and effect break down. In a model proposed by the theorist David Deutsch in 1991, however, the paradoxes created by CTCs could be avoided at the quantum scale because of the behavior of fundamental particles, which follow only the fuzzy rules of probability rather than strict determinism. "It's intriguing that you've got general relativity predicting these paradoxes, but then you consider them in quantum mechanical terms and the paradoxes go away," says University of Queensland physicist Tim Ralph. "It makes you wonder whether this is important in terms of formulating a theory that unifies general relativity with quantum mechanics."

Experimenting with a curve Recently Ralph and his PhD student Martin Ringbauer led a team that experimentally simulated Deutsch's model of CTCs for the very first time, testing and confirming many aspects of the two-decades-old theory. Their findings are published in Nature Communications. Much of their simulation revolved around investigating how Deutsch's model deals with the “grandfather paradox,” a hypothetical scenario in which someone uses a CTC to travel back through time to murder her own grandfather, thus preventing her own later birth. ( Scientific American is part of Nature Publishing Group.)

Deutsch's quantum solution to the grandfather paradox works something like this:

Instead of a human being traversing a CTC to kill her ancestor, imagine that a fundamental particle goes back in time to flip a switch on the particle-generating machine that created it. If the particle flips the switch, the machine emits a particle— the particle—back into the CTC; if the switch isn't flipped, the machine emits nothing. In this scenario there is no a priori deterministic certainty to the particle's emission, only a distribution of probabilities. Deutsch's insight was to postulate self-consistency in the quantum realm, to insist that any particle entering one end of a CTC must emerge at the other end with identical properties. Therefore, a particle emitted by the machine with a probability of one half would enter the CTC and come out the other end to flip the switch with a probability of one half, imbuing itself at birth with a probability of one half of going back to flip the switch. If the particle were a person, she would be born with a one-half probability of killing her grandfather, giving her grandfather a one-half probability of escaping death at her hands—good enough in probabilistic terms to close the causative loop and escape the paradox. Strange though it may be, this solution is in keeping with the known laws of quantum mechanics.

In their new simulation Ralph, Ringbauer and their colleagues studied Deutsch's model using interactions between pairs of polarized photons within a quantum system that they argue is mathematically equivalent to a single photon traversing a CTC. "We encode their polarization so that the second one acts as kind of a past incarnation of the first,” Ringbauer says. So instead of sending a person through a time loop, they created a stunt double of the person and ran him through a time-loop simulator to see if the doppelganger emerging from a CTC exactly resembled the original person as he was in that moment in the past.

By measuring the polarization states of the second photon after its interaction with the first, across multiple trials the team successfully demonstrated Deutsch's self-consistency in action. "The state we got at our output, the second photon at the simulated exit of the CTC, was the same as that of our input, the first encoded photon at the CTC entrance," Ralph says. "Of course, we're not really sending anything back in time but [the simulation] allows us to study weird evolutions normally not allowed in quantum mechanics."

Those "weird evolutions" enabled by a CTC, Ringbauer notes, would have remarkable practical applications, such as breaking quantum-based cryptography through the cloning of the quantum states of fundamental particles. "If you can clone quantum states,” he says, “you can violate the Heisenberg uncertainty principle,” which comes in handy in quantum cryptography because the principle forbids simultaneously accurate measurements of certain kinds of paired variables, such as position and momentum. "But if you clone that system, you can measure one quantity in the first and the other quantity in the second, allowing you to decrypt an encoded message."

"In the presence of CTCs, quantum mechanics allows one to perform very powerful information-processing tasks, much more than we believe classical or even normal quantum computers could do," says Todd Brun, a physicist at the University of Southern California who was not involved with the team's experiment. "If the Deutsch model is correct, then this experiment faithfully simulates what could be done with an actual CTC. But this experiment cannot test the Deutsch model itself; that could only be done with access to an actual CTC."

Alternative reasoning Deutsch's model isn’t the only one around, however. In 2009 Seth Lloyd, a theorist at Massachusetts Institute of Technology, proposed an alternative , less radical model of CTCs that resolves the grandfather paradox using quantum teleportation and a technique called post-selection, rather than Deutsch's quantum self-consistency. With Canadian collaborators, Lloyd went on to perform successful laboratory simulations of his model in 2011. "Deutsch's theory has a weird effect of destroying correlations," Lloyd says. "That is, a time traveler who emerges from a Deutschian CTC enters a universe that has nothing to do with the one she exited in the future. By contrast, post-selected CTCs preserve correlations, so that the time traveler returns to the same universe that she remembers in the past."

This property of Lloyd's model would make CTCs much less powerful for information processing, although still far superior to what computers could achieve in typical regions of spacetime. "The classes of problems our CTCs could help solve are roughly equivalent to finding needles in haystacks," Lloyd says. "But a computer in a Deutschian CTC could solve why haystacks exist in the first place.”

Lloyd, though, readily admits the speculative nature of CTCs. “I have no idea which model is really right. Probably both of them are wrong,” he says. Of course, he adds, the other possibility is that Hawking is correct, “that CTCs simply don't and cannot exist." Time-travel party planners should save the champagne for themselves—their hoped-for future guests seem unlikely to arrive.

The Quantum Physics of Time Travel   (All-Access Subscribers Only) By David Deutsch and Michael Lockwood

Can Quantum Bayesianism Fix the Paradoxes of Quantum Mechanics?

Astrophysicist J. Richard Gott on Time Travel

Quantum Time Travel: Understanding the Physics Behind Time Jumps

Quantum mechanics, the cornerstone of modern physics, has revolutionized our understanding of the fundamental nature of reality, challenging our intuitive notions of time, space, and causality. Among its many intriguing implications, the concept of quantum time travel stands as one of the most captivating and enigmatic. This theoretical possibility, entwined with the fabric of quantum theory and general relativity, explores the potential for time jumps, time dilation, and the complex interplay of quantum phenomena that could allow for journeys through time. In this article, we will delve into the fascinating world of quantum time travel, unraveling its theoretical foundations, paradoxes, and the profound insights it offers into the nature of time, reality, and the universe.

Introduction to Quantum Time Travel

Quantum time travel delves into the quantum mechanical phenomena, spacetime geometry, and the theoretical frameworks that explore the potential for time travel, time loops, and time dilation within the context of quantum mechanics, general relativity, and the mysteries of the cosmos.

• Time Dilation and Relativistic Effects: Time dilation, a consequence of Einstein’s theory of special relativity, describes the slowing down of time for an observer moving relative to another inertial frame or experiencing gravitational effects, revealing the interconnectedness of time, space, and motion within the fabric of spacetime.
• Quantum Mechanics and Time Evolution: Quantum mechanics, governed by the Schrödinger equation and the principles of superposition, entanglement, and quantum states, offers insights into the non-deterministic, probabilistic nature of quantum systems, time evolution, and the quantum phenomena that challenge our classical understanding of time, reality, and the quantum landscape.

Wormholes, Black Holes, and Spacetime Geometry

Theoretical constructs, such as wormholes and black holes, serve as gateways to exploring the potential for time travel, time loops, and the intricate spacetime geometry that shapes the cosmic landscape and the possibilities for traversing through time and space.

• Wormholes and Einstein-Rosen Bridges: Wormholes, hypothetical passages connecting distant regions of spacetime, offer potential pathways for time travel, enabling journeys through time and space, linking distant cosmic regions, and revealing the interconnectedness of the universe through the fabric of spacetime and the curvature of reality.
• Black Holes, Event Horizons, and Cosmic Singularities: Black holes, regions of spacetime exhibiting strong gravitational effects that nothing can escape from, including light, represent cosmic laboratories for exploring extreme gravitational fields, time dilation, and the potential for time loops, revealing the mysterious nature of black holes, event horizons, and the cosmic singularities that define the cosmic landscape and the boundaries of our understanding of reality.

Quantum Paradoxes and Temporal Dilemmas

Theoretical paradoxes, conundrums, and thought experiments, such as the grandfather paradox, the twin paradox, and the Möbius strip of time, challenge our intuitive notions of causality, time directionality, and the consequences of time travel within the quantum realm.

• Grandfather Paradox and Causality Loops: The grandfather paradox, a classic time travel dilemma, explores the potential consequences, contradictions, and causal loops arising from changing the past, altering timelines, and the implications for causality, free will, and the interconnectedness of events within the fabric of time.
• Twin Paradox and Relativistic Effects: The twin paradox, a consequence of special relativity, illustrates the differential aging, time dilation, and the relativistic effects experienced by observers traveling at different velocities or experiencing different gravitational potentials, revealing the dynamic nature of time, spacetime geometry, and the interconnectedness of time, motion, and reality within the cosmic landscape.

Quantum Mechanics, Entanglement, and Time Symmetry

Quantum mechanics introduces the concepts of entanglement, superposition, and time symmetry, offering new perspectives on the nature of time, reality, and the quantum phenomena that challenge our classical understanding of time’s arrow, quantum states, and the mysterious quantum correlations linking distant events and particles within the quantum realm.

• Quantum Entanglement and Nonlocal Connections: Quantum entanglement, a phenomenon where particles become interconnected and the state of one particle instantly influences another, illustrates the nonlocality, quantum correlations, and the mysterious quantum connections that transcend classical boundaries, challenging our understanding of spacetime, locality, and the nature of reality within the quantum landscape.
• Time Symmetry, Quantum States, and Timeless Quantum Mechanics: Time symmetry, quantum superposition, and the principles of quantum mechanics challenge our classical perceptions of time’s arrow, temporal directionality, and the timeless nature of quantum states, revealing the interconnectedness of time, reality, and the quantum phenomena that shape the quantum landscape, cosmic dynamics, and the mysteries of the cosmos within the fabric of quantum time travel, quantum entanglement, and the quantum realm’s profound insights into the nature of time, reality, and the universe.

Quantum time travel, with its theoretical richness, paradoxes, and profound implications, stands as a testament to the intricate interplay of quantum mechanics, general relativity, and the mysteries of the cosmos, unveiling the potential pathways, cosmic connections, and the dynamic interplay of time, spacetime geometry, and the quantum phenomena shaping the cosmic landscape.

As we explore, investigate, and unravel the mysteries of quantum time travel through scientific inquiry, theoretical physics, and the pursuit of knowledge, we embark on a journey of discovery, exploration, and enlightenment that transcends boundaries, deepens our understanding of the universe’s complexity, beauty, and the mysterious interplay of quantum forces, spacetime curvature, and the cosmic dynamics shaping our cosmic journey, destiny, and the eternal quest for truth, meaning, and the timeless wonders that inspire wonder, curiosity, and a renewed appreciation for the grandeur, diversity, and interconnectedness of the cosmos, quantum phenomena, and the boundless realms of the universe and beyond.

Read More: The Goldilocks Zone: Finding Planets Just Right for Life

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Quantum Physics

Title: quantum theory looks at time travel.

Abstract: We introduce a quantum mechanical model of time travel which includes two figurative beam splitters in order to induce feedback to earlier times. This leads to a unique solution to the paradox where one could kill one's grandfather in that once the future has unfolded, it cannot change the past, and so the past becomes deterministic. On the other hand, looking forwards towards the future is completely probabilistic. This resolves the classical paradox in a philosophically satisfying manner.

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Physicists Say Time Travel Can Be Simulated Using Quantum Entanglement

“whether closed timelike curves exist in reality, we don’t know.".

The quantum world operates by different rules than the classical one we buzz around in, allowing the fantastical to the bizarrely normal. Physicists have described using quantum entanglement to simulate a closed timelike curve—in layman’s terms, time travel.

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Before we proceed, I’ll stress that no quantum particles went back in time. The recent research was a Gedankenexperiment , a term popularized by Einstein to describe conceptual studies conducted in lieu of real tests—a useful thing when one is testing physics at its limits, like particles moving at the speed of light. But a proposed simulation involves “effective time travel,” according to the team’s recent paper in Physical Review Letters, thanks to a famously strange way that quantum particles can interact.

That interaction is called quantum entanglement , and it describes when the characteristics of two or more quantum particles are defined by each other. This means that knowing the properties of one entangled particle gives you information about the other, regardless of the distance between the two particles; their entanglement is on a quantum level, so a little thing like their physical distance has no bearing on the relationship. Space is big and time is relative, so a change to a quantum particle on Earth that’s entangled with a particle near a black hole 10 billion light-years away would mean changing the behavior of something in the distant past.

The recent research explores the possibility of closed-timelike curves, or CTCs—a hypothetical pathway back in time. The curve is a worldline—the arc of a particle in spacetime over the course of its existence—that runs backwards. Steven Hawking posited in his 1992 “Chronology protection conjecture” paper that the laws of physics don’t allow for closed timelike curves to exist—thus, that time travel is impossible. “Nevertheless,” the recent study authors wrote, “they can be simulated probabilistically by quantum-teleportation circuits.”

The team’s Gedankenexperiment goes like this: Physicists put photonic probes through a quantum interaction, yielding a certain measurable result. Based on that result, they can determine what input would have yielded an optimal result—hindsight is 20/20, just like when you can look over a graded exam. But because the result was yielded from a quantum operation, instead of being stuck with a less-than-optimal result, the researchers can tweak the values of the quantum probe via entanglement, producing a better result even though the operation already happened. Capiche?

The team demonstrated that one could “probabilistically improve one’s past choice,” explained study co-author Nicole Yunger Halpern, a physicist at the National Institute of Standards and Technology and the University of Maryland at College Park, in an email to Gizmodo, though she noted that the proposed time travel simulation has not yet taken place. 

In their study, the apparent time travel effect would occur one time in four—a failure rate of 75%. To address the high failure rate, the team suggests sending a large number of entangled photons, using a filter to ensure the photons with the corrected information got through while sifting out the outdated particles.

“The experiment that we describe seems impossible to solve with standard (not quantum) physics, which obeys the normal arrow of time,” said David Arvidsson-Shukur, a quantum physicist at the University of Cambridge and the study’s lead author, in an email to Gizmodo. “Thus, it appears as if quantum entanglement can generate instances which effectively look like time travel.”

The behavior of quantum particles—specifically, the ways in which those behaviors differ from macroscopic phenomena—are a useful means for physicists to probe the nature of our reality. Entanglement is one aspect of how quantum things operate by different laws.

Last year, another group of physicists claimed that they managed to create a quantum wormhole—basically, a portal through which quantum information could instantaneously travel. The year before, a team synchronized drums as wide as human hairs using entanglement. And the 2022 Nobel Prize in Physics went to three physicists for their interrogation of quantum entanglement, which is clearly an important subject to study if we are to understand how things work.

A simulation offers a means of probing time travel without worrying about whether it’s actually permitted by the rules of the universe.

“Whether closed timelike curves exist in reality, we don’t know. The laws of physics that we know of allow for the existence of CTCs, but those laws are incomplete; most glaringly, we don’t have a theory of quantum gravity,” said Yunger Halpern. “Regardless of whether true CTCs exist, though, one can use entanglement to simulate CTCs, as others showed before we wrote our paper.”

In 1992, just a couple weeks before Hawking’s paper was published, the physicist Kip Thorne presented a paper at the 13th International Conference on General Relativity and Gravitation. Thorne concluded that, “It may turn out that on macroscopic lengthscales chronology is not always protected, and even if chronology is protected macroscopically, quantum gravity may well give finite probability amplitudes for microscopic spacetime histories with CTCs.” In other words, whether time travel is possible or not is a quandary beyond the remit of classical physics. And since quantum gravity remains an elusive thing , the jury’s out on time travel.

But in a way, whether closed-timelike curves exist in reality or not isn’t that important, at least in the context of the new research. What’s important is that the researchers think their Gedankenexperiment provides a new way of interrogating quantum mechanics. It allows them to take advantage of the quantum realm’s apparent disregard for time’s continuity in order to achieve some fascinating results.

The headline and text of this article have been updated to clarify that the team describe a way that time travel can be simulated; they did not simulate time travel in this experiment.

More: Scientists Tried to Quantum Entangle a Tardigrade

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Only a Matter of ‘Time’: On Einstein, Negative Mass, Time Travel and Aliens

In 1957, the astrophysicist Herman Bondi  wrote   a paper in which he considered the possible existence of a negative mas s in Albert Einstein’s theory of gravity. A negative mass would repel a positive mass away from it. Given that, a pair of positive and negative masses could accelerate together up to the speed of light. The negative mass would push away the positive mass which in turn would pull the negative mass for the ride. The runaway pair would accelerate indefinitely, without any need for fuel or a propulsion system. Energy conservation would not be violated because the sum of the two masses is zero.

However, in a recent discussion I had with my brilliant coll aborator , Mark Hertzberg, we agreed that the existence of a negative mass in nature would allow for a time machine in which one can visit the past. The reason is simple. A positive mass delays the arrival of light that passes near it, as recognized in a seminal paper by my distinguished colleague, Irwin Shapiro.

The “Shapiro time delay” near a positive mass reverse s sign and turn s into a “Shapiro time advance” near a negative mass. If the sender is moving sufficiently close to the speed of light, the Shapiro time advance would mean that a light signal can be reflected of f a mirror and arrive back to the sender when the sender’s clock shows a time earlier than the original transmission time of the same signal .

Effectively, the Shapiro time advance is equivalent to communicating with a faster-than-light particle which violates causality. Within Einstein’s gravity, the technical term for a time machine of this type is a “closed timelike curve.”

Within the standard framework of Einstein’s gravity, a negative mass would allow children to prevent their parents from giving birth to them, a logical inconsistency. Time machines can also be created by traversable wormholes which , as Stephen Hawking recognized in a seminal paper about “chronology protection” from 1992, require negative energy as well.

“ It has been suggested that an advanced civilization might have the technology to warp spacetime so that closed timelike curves would appear, allowing travel into the past, ” the abstract of Hawking’s paper begins, a nd thereafter concludes by saying, “ These results strongly support the chronology protection conjecture: The laws of physics do not allow the appearance of closed timelike curves. ” Hawking’s conclusion was supported in a subsequent paper by Bernard Kay, Marek Radzikowski and Bob Wald , but the final verdict on its validity is awaiting a better understanding of quantum gravity.

If time machines are absent, what are the most extreme distortions of spacetime that our Universe is known to display?

In Einstein’s gravity , the Earth orbits the Sun because the Sun curves space. A n analogy is offered by the surface of a trampoline which is curved by a bowling ball, causing a marble with the proper speed to move in a circle around the heavier ball . Removing the bowling ball w ould release the marble on a straight trajectory on the flat trampoline . Similarly, discarding the Sun w ould free Earth on a straight path toward  interstellar space.

The spacetime curvature created by the Sun at our location is characterized by a length scale that is equal to the inverse of the square root of ( GM/r^3c^2 ), where G is Newton’s constant, M is the mass of the Sun, r is the Earth-Sun separation, and c is the speed of light. Plugging in the relevant numbers yields a curvature length of 10,000 times the Earth-Sun separation. Another way to derive this number is by recognizing that the Earth’s speed around the Sun is 10,000 smaller than the speed of light, c .

Other astrophysical objects display a stronger spacetime curvature over a shorter length scale . The largest distortion s of spacetime are expect ed near a black hole. L imiting our attention to the accessible region outside the event horizon, th is curvature length scales in proportion to the black hole mass. The most massive black holes in the Universe have a mass of ten billion solar masses and a curvature scale of 1,000 times the Earth-Sun separation. T he curvature length shrinks by billion s of times for black holes or neutron stars formed from the collapse of massive stars .

Diary of an Interstellar Voyage: Parts 28-30 (June 25-26 2023)

What is the la rgest spacetime curvature that we observed so far in the Universe? It corresponds to a length scale of about 24 kilometers, the size of a large city on Earth , and is realized on the surface of neutron stars which carry about twice the mass of the Sun at nuclear density.

The same spacetime curvature characterizes an atomic nucleus because it has the same matter density as a neutron star. But a curvature length of 24 kilometers is 19 orders of magnitude larger than the size of an atomic nucleus and therefore has no significance when nuclei are smashed into each other in CERN’s Large Hadron Collider .

In conclusion , based on all we know from observations or theory within the standard model of physics, negative masses are forbidden for the same reason that time machines are not allowed. If aliens were able to create negative masses and use them for travel , then they would have also been able to travel back i n time.

If time travel had  been feasible, then future generations of quantum-gravity physicists could have been able to visit Earth and rewrite our history as they wish. For example, they could have killed Adolf Hitler and allowed six million Jews to thrive in Europe between 1939 and 1945. The fact that this never happen ed suggests that either time machines will never be constructed in our future or that their creators have no interest in saving human lives .

A vi L oe b is the head of the Galileo Project, founding director of Harvard University’s – Black Hole Initiative, director of the Institute for Theory and Computation at the Harvard-Smithsonian Center for Astrophysics, and the former chair of the astronomy department at Harvard University (2011-2020). He is a former member of the President ’ s Council of Advisors on Science and Technology and a former chair of the Board on Physics and Astronomy of the National Academies. He is the bestselling author of “ Extraterrestrial: The First Sign of Intelligent Life Beyond Earth ” and a co-author of the textbook “ Life in the Cosmos ”, both published in 2021. His new book, titled “ Interstellar ”, was published in August 2023.

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Superradiant atoms could push the boundaries of how precisely time can be measured

by University of Copenhagen

Superradiant atoms can help us measure time more precisely than ever. In a recent study, researchers from the University of Copenhagen present a new method for measuring the time interval, the second, mitigating some of the limitations that today's most advanced atomic clocks encounter. The result could have broad implications in areas such as space travel, volcanic eruptions and GPS systems.

The second is the most precisely defined unit of measurement, compared to other base units such as the kilogram, meter, and degree Kelvin. Time is currently measured by atomic clocks in different places around the world, which together, tell us what time it is. Using radio waves , atomic clocks continuously send signals that synchronize our computers, phones and wristwatches.

Oscillations are the key to keeping time. In a grandfather clock, these oscillations are from a pendulum's swinging from side to side every second, while in an atomic clock, it is a laser beam which corresponds to an energy transition in strontium and oscillates about a million billion times per second.

But according to Ph.D. fellow Eliot Bohr from the Niels Bohr Institute—great-grandson of Niels Bohr—even atomic clocks could become more precise. This is because the detection laser, used by most modern atomic clocks to read the oscillation of atoms, heats up the atoms so much that they escape—which degrades precision.

"Because the atoms constantly need to be replaced with fresh new atoms, while new atoms are being prepared, the clock loses time ever so slightly. Therefore, we are attempting to overcome some of the current challenges and limitations of the world's best atomic clocks by, among other things, reusing the atoms so that they don't need to be replaced as often," explains Bohr who was employed at the Niels Bohr Institute when he did the research, but who is now Ph.D. fellow at the University of Colorado.

He is the lead author of a new study published in the journal Nature Communications , which uses an innovative and perhaps more efficient way of measuring time.

Superradiance and cooling to absolute zero

The current methodology consists of a hot oven that spits roughly 300 million strontium atoms into an extraordinarily chilly ball of cold atoms known as a magneto-optical trap, or MOT. The temperature of these atoms is approximately -273°C—very near absolute zero—and there are two mirrors with a light field in between them to enhance the atomic interactions. Together with his research colleagues, Bohr has developed a new method to read out the atoms.

"When the atoms land in the vacuum chamber , they lie completely still because it is so cold, which makes it possible to register their oscillations with the two mirrors at opposing ends of the chamber," explains Bohr.

The reason why the researchers don't need to heat the atoms with a laser and destroy them is thanks to a quantum physical phenomenon known as "superradiance." The phenomenon occurs when the group of strontium atoms is entangled and at the same time emits light in the field between the two mirrors.

"The mirrors cause the atoms to behave as a single unit. Collectively, they emit a powerful light signal that we can use to read out the atomic state, a crucial step for measuring time. This method heats up the atoms minimally, so It all happens without replacing the atoms, and this has the potential to make it a more precise measurement method," explains Bohr.

GPS, space missions and volcanic eruptions

According to Bohr, the new research result may be beneficial for developing a more accurate GPS system. Indeed, the roughly 30 satellites that constantly circle Earth and tell us where we are need atomic clocks to measure time.

"Whenever satellites determine the position of your phone or GPS, you are using an atomic clock in a satellite. The precision of the atomic clocks is so important that If that atomic clock is off by a microsecond, it means an inaccuracy of about 100 meters on the Earth's surface," explains Bohr.

Future space missions are another area where the researcher foresees more precise atomic clocks making a significant impact.

"When people and crafts are sent out into space, they venture even further away from our satellites. Consequently, the requirements for precise time measurements to navigate in space are much greater," he says.

The result could also be helpful in the development of a new generation of smaller, portable atomic clocks that could be used for more than "just" measuring time.

"Atomic clocks are sensitive to gravitational changes and can therefore be used to detect changes in Earth's mass and gravity, and this could help us predict when volcanic eruptions and earthquakes will occur," says Bohr.

Bohr emphasizes that while the new method using superradiant atoms is very promising, it is still a "proof of concept" which needs further refinement.

The research was conducted by the team of Jörg Helge Müller and Jan Thomsen at the Niels Bohr Institute, in collaboration with Ph.D. students Sofus Laguna Kristensen and Julian Robinson-Tait, and postdoc Stefan Alaric Schäffer. The project also included contributions from theorists Helmut Ritsch and Christoph Hotter from the University of Innsbruck, as well as Tanya Zelevinsky from Columbia University. This work underscores the importance of international collaboration in the sciences.

Journal information: Nature Communications

Provided by University of Copenhagen

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Highly precise atomic clocks could soon get even better. Here's how

• Oops! Something went wrong. Please try again later. More content below

The use of a special type of atom could make even the most advanced atomic clocks more precise, scientists believe.

If confirmed, this breakthrough that could lead to more accurate GPS systems and better atomic clocks for use in space travel — it could even lead to devices that can detect earthquakes and volcanic eruptions with a higher level of accuracy. And fascinatingly, one of the researchers behind the development has a familiar name, based on a fitting family legacy rooted in the cutting edge of atomic science: Eliot Bohr. He's Neils Bohr's great-grandson.

Related: Atomic clocks on Earth could reveal secrets about dark matter across the universe

Of all the units humanity uses for measurement, the most precisely defined is the second , a fundamental unit of time. Crucial to this and all types of time measurements throughout history are different kinds of oscillations. Just as grandfather clocks use oscillations of a pendulum to measure time, atomic clocks define a second as 9,192,631,770 microwave oscillations of a cesium atom as it absorbs microwave radiation of a specific frequency.

Many modern atomic clocks use oscillations of strontium atoms rather than cesium to measure time; the most precise of these is accurate to within 1/15,000,000,000 of a second. This means that, even if it had been running since the dawn of time around 13.8 billion years ago, the clock still wouldn't have lost or gained a full second. Yet, for the majority of atomic clocks, which are used to keep Universal Coordinated Time (UTC) from positions around the globe and make sure our cell phones, computers and GPS tech is synchronized, there is still some room for improvement.

That's because the laser used to read the oscillations of atoms in atomic clocks heats up those atoms while doing so, causing them to escape the system. This can create some discrepancy, albeit incredibly slight. Still, researchers from the Niels Bohr Institute thinks they have found a way to eliminate the laser altogether, thus avoiding atomic heating and potential degradation of precision. It is an institute named for Eliot Bohr's great-grandfather, and one Bohr himself is affiliated with.

"We found that it is possible to read out the collective state of an atomic ensemble, as is required in atomic clocks and sensors, at an enhanced rate and with minimal heating using superradiance," lead researcher Eliot Bohr, who was a Ph.D. fellow at the institute, told Space.com. "There is a threshold for superradiance to occur for our chosen experimental geometry, and we can leverage this threshold in a clock sequence."

Atomic clocks could be cooler

In current atomic clocks, 300 million or so hot strontium atoms are spat into a magneto-optical trap located within a vacuum chamber. This trap is a ball of atoms cooled to temperatures near absolute zero , the theoretical temperature at which all atomic movement would cease. Because of these temperatures, the introduced atoms lie almost still. This makes it possible for two mirrors with light between them to register their oscillations.

"In traditional atomic clocks, the detection heats up the atoms, requiring atoms to be freshly loaded," Bohr said. "This loading takes a while and causes downtime in the atomic clock cycle, limiting precision." The team's sort of "paused" atoms that have been cooled so immensely, however,  can be reused. This means they wouldn't need to be replaced as often, therefore leading to more precise atomic clocks.

Bohr explained that superradiant atoms are atoms that exist in a collective quantum state and are excited by the addition of energy in the form of photons , or particles of light. When the atoms release the photon-induced energy, or "decay," they all emit light in the same direction and at an enhanced rate.

"One cannot fundamentally distinguish which atom emitted which photon. They emitted them together, collectively," he added. "This enhanced emission rate allows for photons to be emitted much quicker from the type of atomic transitions that are used in atomic clocks."

This powerful light signal can be used to read out the atomic state of the collective strontium atoms, which means a laser isn't actually needed in the first place. And, again, because this process happens without the superradiant atoms being heated more than a very minimal amount, they won't need to be replaced.

Not only would doing away with the laser make more precise atomic clocks, but it could result in devices that are simpler and more portable.

"State-of-the-art atomic clocks are now so precise they are sensitive to gravity," Bohr said. "There are proposals that if we have atomic clocks that are portable and precise enough, we can place them strategically and better predict earthquakes and volcano eruptions by measuring certain variations in gravity."

Revolutionary atomic science is the family trade

Coming from a line of scientists who have been influential in our understanding of the subatomic world, Bohr may well have this sort of research in his blood. Most famous in this lineage is his great-grandfather, Niels Bohr, one of the fathers of quantum physics and a scientist who made a huge contribution to the understanding of the atomic structure, without which research like this couldn't happen.

In 1913, Niels Bohr, along with Ernest Rutherford, presented a model of the atom, suggesting it to be a dense nucleus surrounded by orbiting electrons. Though this "Bohr model" of the atom is now considered relatively simplistic compared to the detailed diagrams we have now, 111 years after its inception, it is still used to introduce students to the concept of the atom in classrooms across the globe.

Eliot Bohr's family's connection to the atomic structure goes deeper than this, too.

His grandfather is Aage Niels Bohr, who in 1975 was awarded the Nobel Prize in Physics along with Ben Roy Mottelson and James Rainwater for their discovery of the connection between collective motion and particle motion in atomic nuclei. This led to the development of an improved theory of the structure of the atomic nucleus. "Both my great-grandfather and my grandfather inspired me greatly," Bohr said. "They both worked on theoretical work, understanding the atom and nucleus. My great-grandfather's theory that atoms can absorb a photon of a particular wavelength and go to an excited state, or emit a photon and decay to a lower state, is precisely what we do in our lab each and every day using lasers."

Bohr added that it is the open-mindedness demonstrated by his great-grandfather and colleagues that he finds particularly inspiring.

"The concepts are completely non-intuitive, but through rigorous data and debates, they accepted these new 'quantum' rules," Bohr said. "We now accept them and use them in many of our modern-day technologies. I hope to contribute to developing the next quantum technologies which will benefit society."

Related stories:

— Sending atomic clocks close to the sun could unlock the secrets of dark matter

— High-energy 'sun goddess' particle opens possibilities for new physics, exciting scientists

— Earth's atmosphere could be turned into a giant dark-matter detector

As for his superradient atomic clock research, Bohr said there are lots of possibilities for future advancements. The group he was part of in Copenhagen is now continuing to understand various properties of superradiant light to see how it can be harnessed for other situations.

Meanwhile, Bohr has started a postdoctoral research position at JILA, a joint institute between the National Institute of Standards and Technology (NIST) and the University of Colorado, Boulder. This is a lab that also studies superradiance and other collective atomic effects for next-generation quantum sensors.

"I plan to continue researching collective quantum effects which can be used in clocks and sensors," he concluded. "We have some ideas for further refining the method, such as finding the optimal parameters and understanding and reducing the noise level in the superradiant signal.

"There are a lot of possibilities to use superradiance to advance clocks and sensor technology."

The team's research was published in February in the journal Nature Communications.

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Apple's second-generation AirPods Pro are back down to their lowest price ever

Apple's AirPods Pro are back down to the lowest price we've seen.

The best face wash of 2024 to cleanse away dirt, makeup and past mistakes

Cetaphil, Cerave, Youth to the People, Tata Harper, Tatcha and The Outset top our list and tackle every problem and skin type.

Bucks All-Stars Damian Lillard, Giannis Antetokounmpo reportedly in doubt for elimination Game 5 vs. Pacers

The Bucks could be without Giannis Antetokounmpo and Damian Lillard for an elimination game.

2025 Jeep Grand Cherokee rumored to get 2.0-liter four as base engine

New rumor says the 2025 Jeep Grand Cherokee gets a 2.0-liter four as the base engine. The 2-row keeps its 4xe option, 3-row only gets 2.0-liter.

U.S. Soccer, Mexico pull 2027 Women's World Cup bid, will instead aim for 2031

The U.S.-Mexico bid for the 2027 Women's World Cup was half-hearted. With a FIFA vote approaching, it has now been formally withdrawn, leaving two contenders.

Made in the shade: This $150 gazebo from Wayfair — it's over 50% off right now — will transform your summer

Got a graduation, Father's Day or 4th of July bash coming up? This early Way Day deal is your secret weapon.

UConn adds former Saint Mary's guard Aidan Mahaney from the transfer portal

UConn is adding Saint Mary's transfer Aidan Mahaney. The junior guard played the past two seasons with the Gaels, leading them in scoring as a sophomore.

McDonald's faces weaker foot traffic, value proposition as it sets to report Q1 earnings

Wall Street is waiting to hear more about how McDonald's consumers are faring.

Tesla stock surges on 'watershed' 'Full Self-Driving' approval in China

Tesla (TSLA) stock soared on Monday following reports that CEO Elon Musk won Chinese approval to deploy the automaker’s Full Self-Driving (FSD) autonomous software on the mainland.

NASA releases ‘unprecedented’ Horsehead Nebula image captured by James Webb Space Telescope. What to know about the stellar nursery.

The image of the Horsehead Nebula captures the iconic nebula in a whole new light.

Recent College Football Playoff complaints, Dylan Raiola shines and Damien Martinez picks Miami

Dan Wetzel, Ross Delleneger, and SI's Pat Forde recap the 2024 NFL Draft, reacted to recent College Football Playoff complaints, reacted to news of Damien Martinez picking Miami, and discussed Dylan Raiola's impressive spring game.

Memorial Day sales 2024: Everything we know, including early deals you can shop now

Mark your calendar: The holiday weekend runs Friday, May 24 through Monday, May 27 — here are some early savings you can score right now.

Mike Tyson’s fight against Jake Paul will be a sanctioned professional bout after all

Mike Tyson will be 31 years older than Jake Paul when they step into the ring this summer.

NYT learns how GM car buyers wound up in a program that raised insurance rates

The NYT discovers how dealer salespeople were opting buyers of GM cars into a the Smart Driver+ program that could raised a driver's insurance rates.

NFL Draft grades for all 32 teams | Zero Blitz

Jason Fitz and Frank Schwab join forces to recap the draft in the best way they know how: letter grades! Fitz and Frank discuss all 32 teams division by division as they give a snapshot of how fans should be feeling heading into the 2024 season. The duo have key debates on the Dallas Cowboys, New York Giants, New Orleans Saints, Los Angeles Rams, New England Patriots, Las Vegas Raiders and more.

What does PITI mean, and how does it affect your mortgage?

PITI is an acronym for the four factors that make up your monthly mortgage payment: principal, interest, taxes, and insurance. Learn how to calculate your PITI.

Wolves chase Suns out of NBA Playoffs, Knicks keep rolling & Harden carrying the Clippers

Vincent Goodwill and J.A. Adande talk their way through the most interesting NBA Playoffs action from over the weekend.

'Comfortable right out of the box': These supportive sandals are on sale for just $20 — save up to 30%

'I wore these brand-new and walked over 12,000 steps a day for four days with no issues,' raved a happy shopper.