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<h2 class="hd hd-2 unit-title">Time Travel without Paradox</h2>
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<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">What might a system of laws look like if it is to allow for time travel, and do so in a way that steers clear of paradox? <br /><br />It is easy to imagine a system of laws that avoids paradox by banning time travel altogether. It is also easy to imagine a system of laws that avoids paradox in an unprincipled way. It could, for example, postulate an "anti-paradox force" to deflect Bruno's bullet before it hits Grandfather. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">Or it could postulate a restriction on the world's initial conditions, disallowing worlds with problematic time travel. That's not what we're after here. What we'd like to understand is whether there could be a system of laws that allows for interesting forms of time travel but manages to avoid paradox in a <em>principled</em> way. <br /><br />We will explore these issues by considering a "toy model": a system of laws that allows for interesting forms of time travel but is much simpler than the actual laws of physics. This will allow us to bring some of the key issues into focus while avoiding the complexities of contemporary physics. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">As it turns out, there are solutions to Einstein's equations that allow for time travel, in the sense that there are paths through spacetime that form a closed loop and do not require traveling faster than the speed of light. </span><span style="font-family: 'book antiqua', palatino;">(For an amazing example of how this might happen, see the <a href="/courses/course-v1:MITx+24.118x+2T2020/jump_to_id/253b2d92fe91432d9605bfe0893d99d1" target="[object Object]">bonus video</a> at the end of this lecture.)</span></p>
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<h2 class="hd hd-2 unit-title">A toy model</h2>
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<h3 class="hd hd-2">A Toy Model</h3>
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<p><span style="font-family: 'book antiqua', palatino;">The system of laws I would like to discuss is due to philosophers Frank Arntzenius and Tim Maudlin. </span><span style="font-family: book antiqua,palatino;">It governs a world with two dimensions: a temporal dimension and a single spatial dimension. </span><br /><br /><span style="font-family: book antiqua,palatino;">We will represent our world's spacetime using two-dimensional diagrams like this one:</span> </p>
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<figure><span style="font-family: 'book antiqua', palatino;"><img src="/assets/courseware/v1/d016eea7ab76508f3f9ca439f4faf149/asset-v1:MITx+24.118x+2T2020+type@asset+block/TimeTravelFig2.jpg" type="saveimage" target="[object Object]" hight="30%" width="30%" /></span></figure>
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<p><span style="font-family: 'book antiqua', palatino;">The <span class="math inline">\(x\)</span>-axis of the diagram represents spatial change. The <span class="math inline">\(y\)</span>-axis represents temporal change, with earlier times closer to the origin. Events that take place at time <span class="math inline">\(t\)</span> correspond to points on the dotted line. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">Particle <span class="math inline">\(A\)</span>’s location at <span class="math inline">\(t\)</span> is labeled "<span class="math inline">\(a\)"</span>, and particle <span class="math inline">\(B\)</span>’s position at time <span class="math inline">\(t\)</span> is labeled "<span class="math inline">\(b\)"</span>. <span class="math inline">\(A\)</span> is at rest, so its spatiotemporal trajectory is represented as a perfectly vertical line; <span class="math inline">\(B\)</span> is moving rightward at constant speed, so its spatiotemporal trajectory is represented by a diagonal line. (Slower speeds are represented by steeper diagonals.)</span></p>
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<p><span style="font-family: 'book antiqua', palatino;">We shall assume that our particles obey two exceedingly simple dynamical laws:</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Law 1</strong></span><br /><span style="font-family: 'book antiqua', palatino;">In the absence of collisions, particles that are at rest remain at rest, and particles moving at constant speed will continue to move at constant speed.<br />(This is our version of Newton’s First Law of motion. We represent it in our diagrams by ensuring that the spatiotemporal trajectory of a freely moving particle is always a straight line, as in the figure above.)</span></p>
<p style="padding-left: 30px;"><span style="font-family: 'book antiqua', palatino;"><strong>Law 2</strong></span><br /><span style="font-family: 'book antiqua', palatino;">When particles collide, they exchange velocities.<br />(This is our version of Newton’s Third Law of motion. We represent it in our diagrams by ensuring that the spatiotemporal trajectories of colliding particles always form an “X”, as in the figure below.)</span></p>
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<figure><span style="font-family: 'book antiqua', palatino;"><img src="/assets/courseware/v1/332069e8b7208995cd7fa5af5efc2e1b/asset-v1:MITx+24.118x+2T2020+type@asset+block/TimeTravelFig3.jpg" type="saveimage" target="[object Object]" hight="30%" width="30%" /></span></figure>
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<h4><span style="font-family: 'book antiqua', palatino;">Wormholes</span></h4>
<p><span style="font-family: 'book antiqua', palatino;">Now imagine that our world contains a <strong>wormhole</strong>, which causes different regions of our diagram to represent the same region of spacetime, as in the following diagram:<br /></span></p>
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<figure><span style="font-family: 'book antiqua', palatino;"><img src="/assets/courseware/v1/be7f56ed48e8cf7c97652014019a00ef/asset-v1:MITx+24.118x+2T2020+type@asset+block/TimeTravelFig4.jpg" type="saveimage" target="[object Object]" width="30%" /></span></figure>
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<p><span style="font-family: book antiqua,palatino;">The crucial feature of the diagram is that the spacetime points on line <em>W-</em> is identified with the spacetime points on line <em>W+</em>. This identification has two consequences:</span></p>
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<p><span style="font-family: 'book antiqua', palatino;">When a particle approaches line <em>W-</em> from below, it continues its upward trajectory from line <em>W+</em>, with no change in velocity. (See object <span class="math inline">\(A\)</span> in diagram above.)</span></p>
<p><span style="font-family: 'book antiqua', palatino;">In other words: when an object enters <em>W-</em> from below, it leaps forward in time by exiting the wormhole at the corresponding point on <em>W+</em>.</span></p>
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<p><span style="font-family: 'book antiqua', palatino;">When a particle approaches line <em>W+</em> from below, it continues its upward trajectory from line <em>W-</em>, with no change in velocity. (See object <span class="math inline">\(B\)</span> in the diagram above.)</span></p>
<p><span style="font-family: 'book antiqua', palatino;">In other words: when an object enters <em>W+</em> from below, it leaps backwards in time by exiting the wormhole at the corresponding point on <em>W-</em>.</span></p>
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<p><span style="font-family: 'book antiqua', palatino;">Our toy physical model is now in place. Let us use it to illustrate some surprising consequences of time travel, as it occurs in this setting.</span></p>
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<h2 class="hd hd-2 unit-title">Indeterminacy</h2>
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<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"> In the absence of wormholes, our world is fully deterministic. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">In other words: one can use a specification of the positions and velocities of the world’s particles at a given time to determine the positions and velocities of the world’s particles at any other time. To see this, consider the following diagram:<br /></span></p>
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<figure><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"><img src="/assets/courseware/v1/332069e8b7208995cd7fa5af5efc2e1b/asset-v1:MITx+24.118x+2T2020+type@asset+block/TimeTravelFig3.jpg" type="saveimage" target="[object Object]" hight="30%" width="30%" /></span></figure>
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<p><span style="font-family: 'book antiqua', palatino;">At time <span class="math inline">\(t\)</span>, <span class="math inline">\(A\)</span> is at rest and occupies spacetime point <span class="math inline">\(a\)</span>, and <span class="math inline">\(B\)</span> is traveling leftward with speed 1 and occupies spacetime position <span class="math inline">\(b\)</span>. To determine the position and velocity of <span class="math inline">\(A\)</span> and <span class="math inline">\(B\)</span> at other times, we draw a straight line through each of <span class="math inline">\(a\)</span> and <span class="math inline">\(b\)</span>, at an angle corresponding to the velocity of the object at <span class="math inline">\(t\)</span>. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">As long as no collisions take place, the spacetime trajectory of a particle is given by the straight line that intersects its position at <span class="math inline">\(t\)</span>; when a collision takes place, the spacetime trajectories of the two particles swap, with each continuing along the straight line corresponding to the trajectory of the other prior to the collision. The spacetime trajectory of a particle can then be used to determine its position and velocity at any given time. </span></p>
<p><span style="font-family: 'book antiqua', palatino;">The particle’s position is given by the point at which the particle’s spacetime trajectory intersects the horizontal line corresponding to the relevant time; the velocity is given by the angle of the particle’s spacetime trajectory at that point.</span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">Now for the interesting part: when a wormhole is introduced, determinism is lost. </span></p>
<p><span style="font-size: 12pt; font-family: 'book antiqua', palatino;">Consider,<span style="font-size: 12pt;"><span style="font-size: 10px;"> </span></span>for example, the wormhole in the following diagram:<br /></span></p>
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<figure><span style="font-size: 12pt; font-family: 'book antiqua', palatino;"><img src="/assets/courseware/v1/4a0b85cc31bf1a05a0498582a48bb6ba/asset-v1:MITx+24.118x+2T2020+type@asset+block/TimeTravelFig5.jpg" type="saveimage" target="[object Object]" hight="30%" width="30%" /></span></figure>
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<p><span style="font-family: 'book antiqua', palatino;">At time <span class="math inline">\(t\)</span>, no particles exist. If we had determinism, this would be all the information we would need to determine how things stand at every other time. In particular, we should be able to figure out how many particles exist in the ``wormhole region'' (i.e.~the spacetime region between <em>W-</em> and <em>W+</em>). </span></p>
<p><span style="font-family: 'book antiqua', palatino;">But our laws do not determine an answer to this question. As far as the laws go, there could be no particles in the wormhole region, or one, or two, or any number whatsoever, as long as there aren't too many of them to fit into spacetime. (The diagram above depicts the case in which there are exactly two particles between <em>W-</em> and <em>W+</em>.)</span></p>
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<h2 class="hd hd-2 unit-title">Paradox</h2>
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<p><span style="font-family: book antiqua,palatino;">Now suppose that our world contains a "mirror" : a stationary object that reflects particles by inverting their velocity.</span><br /><br /><span style="font-family: book antiqua,palatino;">We will use the mirror to construct a version of the Grandfather Paradox in our toy world. The relevant scenario is depicted in Figure 1 below. Particle \(A\) is on a "paradoxical path''. It travels rightward, passes through spacetime point \(a\) and enters the wormhole at spacetime point \(b\), jumping to the past. It exits the wormhole and continues its rightward trajectory until it reaches the mirror at spacetime point \(c\).<br /></span></p>
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<figure><span style="font-family: 'book antiqua', palatino;"><img src="/assets/courseware/v1/9fcb41a0d0bee1e6ecf6e17b15015350/asset-v1:MITx+24.118x+2T2020+type@asset+block/TimeTravelFig6.jpg" type="saveimage" target="[object Object]" hight="30%" width="30%" /> </span><figcaption><span style="font-family: 'book antiqua', palatino; font-size: 10pt;">Figure 1. After passing through point <span class="math inline">\(a\)</span>, and crossing through the wormhole at point <span class="math inline">\(b\)</span>, particle <span class="math inline">\(A\)</span> is reflected by a mirror at point <span class="math inline">\(c\)</span></span></figcaption></figure>
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<p></p>
<p><span style="font-family: book antiqua,palatino;">But what happens next?</span></p>
<p><span style="font-family: book antiqua,palatino;">It is not clear that we could draw out \(A\)'s trajectory in full without violating the dynamical laws of our toy model. Notice, in particular, that if we attempted to complete \(A\)'s trajectory as depicted in Figure 2, particle \(A\) would be blocked from entering the wormhole region at point \(a\) by a future version of itself. (Not just that: it would be blocked from crossing point \(b\) on its way to the mirror by a later version of itself, and it would get blocked from crossing point \(b\).</span></p>
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<figure><span style="font-family: 'book antiqua', palatino;"><img src="/assets/courseware/v1/76cb7f1066e3248cbe064fb8cc197403/asset-v1:MITx+24.118x+2T2020+type@asset+block/TimeTravelFig7.jpg" type="saveimage" target="[object Object]" width="30%" /> </span><figcaption><span style="font-family: 'book antiqua', palatino; font-size: 10pt;">Figure 2. Particle <span class="math inline">\(A\)</span> is prevented from entering the wormhole region by a future version of itself.<span class="math inline"></span></span></figcaption></figure>
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<p></p>
<p><span style="font-family: book antiqua,palatino;">So does particle \(A\) of Figure 1 succeed in entering the wormhole region or not? If it does enter, it shouldn't have, since its entry should have been blocked by a future version of itself. And if it doesn't, it should have, since there is nothing to block it. In other words: the scenario depicted by Figure 1 is paradoxical, given the laws of our toy model.</span></p>
<p><span style="font-family: book antiqua,palatino;">Notice, however, that the laws do allow for a situation in which particle <em>A</em> starts out traveling on the "paradoxical path'' of Figure 1 but is prevented from entering the wormhole region by a <em>different</em> particle. One way in which this can happen is as follows: </span></p>
<center>
<figure><span style="font-family: 'book antiqua', palatino;"><img src="/assets/courseware/v1/80175e3a441df6956c0a80fdf41c3c6d/asset-v1:MITx+24.118x+2T2020+type@asset+block/TimeTravelFig8.jpg" type="saveimage" target="[object Object]" width="30%" /> </span><figcaption><span style="font-family: 'book antiqua', palatino; font-size: 10pt;">Figure 3. Particle <span class="math inline">\(A\)</span> fails to enter the wormhole region, after colliding with particle <span class="math inline">\(B\)</span> at spacetime point <span class="math inline">\(a\)</span>. Particles <span class="math inline">\(B\)</span> and <span class="math inline">\(C\)</span> are each caught in a loop within the wormhole.</span></figcaption></figure>
</center>
<p><span style="font-family: 'book antiqua', palatino;"><br />Note that Figure 3 is very much like Figure 2 above, but it represents a situation in which there are two additional particles living within the wormhole region and one of them blocks particle \(A\) from entering. (Think of this as analogous to a version of Bruno's story in which consistency is achieved because something blocks Bruno from carrying out his assassination attempt.)<br /><br />Here is a more detailed description of the behavior of each of the particles in Figure 3:<br /></span></p>
<ul>
<li><span style="font-family: 'book antiqua', palatino;">Particle \(A\) moves rightward until it reaches spacetime point \(a\), where it collides with \(B\) and bounces off, moving leftward.</span></li>
<li><span style="font-family: 'book antiqua', palatino;">Particle \(B\) is trapped in a loop. It departs spacetime point \(b\) at the time of <em>W-</em> moving leftward, until it collides with particle \(A\) at spacetime point \(a\). It then bounces rightward until it reaches spacetime point \(b\) at the time of <em>W+</em>. Two things happen next. First, particle \(B\) collides with particle \(C\), and bounces off leftward. Second, \(B\) enters the wormhole, and jumps back in time.</span></li>
<li><span style="font-family: 'book antiqua', palatino;">Particle \(C\) is also trapped in a loop. It departs spacetime point \(b\) at the time of <em>W-</em> moving rightward, until it collides with the mirror at spacetime point \(c\). It then bounces leftward until it reaches spacetime point \(b\) at the time of <em>W+</em>. Two things happen next. First, particle \(C\) collides with particle \(B\), and bounces off rightward. Second, \(C\) enters the wormhole, and jumps back in time.</span></li>
</ul>
<p></p>
<h4><span style="font-family: 'book antiqua', palatino;">Escaping Paradox</span></h4>
<p></p>
<p><span style="font-family: book antiqua,palatino;">Figure 3 demonstrates that our toy model allows for consistent scenarios in which particle \(A\) starts out on the "paradoxical path'' of Figure 1. But one might worry that there is something artificial about this way of escaping inconsistency. One might think it amounts to the postulation of a "pre-established harmony" whereby there happens to be an additional particle in just the right place to avert paradox. </span></p>
<p><span style="font-family: book antiqua,palatino;">Such harmony could certainly be imposed by restricting the range of admissible "initial conditions" of our toy model so as to allow for situations in which particle \(A\) starts out on a paradoxical path only when another particle is ready to block its entry to the wormhole region. But the resulting theory would be horribly unprincipled. </span></p>
<p><span style="font-family: book antiqua,palatino;">(It would be a bit like saying that we should allow for "initial conditions'' whereby Bruno to go back in time in an attempt to kill Grandfather but only if the circumstances are such as to ensure that the assassination attempt would be derailed.) </span><br /><br /><span style="font-family: book antiqua,palatino;">When one thinks of the toy model in the right kind of way, however, there is no need for a postulation of pre-established harmony. </span></p>
<p><span style="font-family: book antiqua,palatino;">The trick is to be careful about how one characterizes one's worlds. One does not characterize a world by <em>first</em> deciding how many particles the world is to contain (and assigning them each a position and velocity at a time), and <em>then</em> using the dynamical laws to calculate the spacetime trajectories of these particles. Instead, one characterizes a world by <em>first</em> drawing a family of spacetime trajectories that conform to the dynamical laws and <em>then</em> using the laws to determine how many particles the resulting world must contain. </span></p>
<p><span style="font-family: book antiqua,palatino;">On this way of thinking, it is a mistake to think that one can characterize a world by stipulating that it is to contain a single particle traveling as in Figure 1 and then ask what happens when the dynamical laws are used to calculate the particle's spacetime trajectory. Instead, one draws a family of spacetime trajectories such as the one depicted in Figures 2 and 3, and one uses the laws to reach the conclusion that a world with those spacetime trajectories must contain three different objects (as in Figure 3).</span><br /><br /><span style="font-family: book antiqua,palatino;">This delivers a very satisfying result. We get a system of laws that allows for interesting forms of time travel and yet has a principled way of avoiding paradox. </span></p>
<p><span style="font-family: book antiqua,palatino;">Notice, moreover, that one is able to explain why there is no world in which particle \(A\) completes a "paradoxical path'', and that the explanation is analogous to our earlier explanation of why one couldn't draw a figure that is both a circle and a square. Just like there is no distribution of ink-blots on a page that yields a figure that is both a circle and a square, so there is no lawful distribution of spacetime trajectories in our toy model that is both such that a particle is on a paradoxical path and such that there isn't another particle there to block it.</span><br /><br /><span style="font-family: book antiqua,palatino;">There is a final point I would like to emphasize before bringing our discussion of the toy model to a close. We have seen that our laws allow for the situation depicted in Figure 3, in which particle \(A\) starts out on a paradoxical path but is prevented from entering the wormhole region by another particle. It is essential to keep in mind that the presence of the additional particles in the wormhole is not <em>caused</em> by particle \(A\)'s paradoxical path. The additional particles are not an "anti-paradox'' mechanism that is activated by particles on paradoxical paths. What is going on is simply that there is no lawful distribution of spacetime trajectories that is both such that a particle is on a paradoxical path and such that there isn't another particle there to block it.</span></p>
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<span style="font-family: 'book antiqua', palatino;">I noted earlier that the number of particles living in the wormhole region between <em>W-</em> and <em>W+</em> is not settled by facts prior to the appearance of the wormhole.</span>
</p>
<p>
<span style="font-family: 'book antiqua', palatino;">For this reason, the situation depicted by Figure 3 is not the only way of restoring consistency to a situation in which particle <span class="math inline">\(A\)</span> is the only particle that exists prior to the appearance of the wormhole, and is traveling at constant speed on a path that would (in the absence of collisions) lead to <span class="math inline">\(b\)</span> via <span class="math inline">\(a\)</span>.</span>
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<span style="font-family: 'book antiqua', palatino;">Describe an additional resolution.</span>
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<h2 class="hd hd-2 unit-title">Conclusion</h2>
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<p><span style="font-family: book antiqua,palatino;">We have been trying to get clear on why the Grandfather Paradox is supposed to be problematic. In this section we considered the question of whether the problem is that the Grandfather Paradox makes it hard to see how a system of laws could rule out paradoxical time travel in a principled way, without banning it altogether. We tackled the question in a simplified case, by identifying a toy model that appears to allow for interesting forms of time travel while avoiding paradox in a principled ways.</span></p>
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