A Journey Through Time (Column 33)
From Sukkot to an ontic conception of time
The column opens with Hazal’s reading of ״ושמחת בחגך״: not only to rejoice during the festival, but to rejoice in the festival itself, meaning in time and not only in events that happened within it. Precisely Sukkot, which does not straightforwardly sit on a historical event that occurred in Tishrei, highlights that time itself has standing. From here the rabbi moves to a halakhic conception of time as a real entity, and brings halakhic support for it, such as the possibility of attaching a vow to a fast day. Against this he sets Kant, for whom time and space are only human forms of intuition and not things that exist in themselves.
Why the computerized ‘solutions’ solve nothing
The column explains that the grandfather paradox does not disappear because software managed to generate bizarre but supposedly consistent scenarios, such as a case in which the granddaughter becomes her own grandmother or in which other figures also travel through time. Friedman himself, in the original paper, offers at most a simulation method in computer science, and perhaps that has value; but the journalistic report presents it as though a philosophical-physical problem has been solved. Even without software one can generate infinitely many ‘solutions’ of this sort, including resurrection, birth after death, or a physical prevention of any contradiction-producing act.
The grandfather paradox is only a symptom: returning to 1966 is itself unclear
The column argues that even without murdering one’s grandfather, time travel raises a conceptual difficulty. If I return to 1966 and my presence there changes the circumstances, is this still 1966? If I retain my memories from 2016, it seems I am not really in 1966; and if I lose them, then it is no longer me. Even if we move all of 2016 back to 1966, we effectively get 2016 under another name. Questions such as the speed of the trip only sharpen the confusion: if the trip happens, from my perspective, in an instant, it is not clear in what sense one is even speaking about passage between times. So the grandfather paradox is not the problem but only a vivid illustration of it, and the same goes for traveling forward in time.
The world as a function of time, and what Kant does not allow
Physics and common sense tend to describe the circumstances of the world as a function of time: C=F(t). On such a description, different times may share the same circumstances, but one and the same time cannot have different circumstances. Therefore, if returning in time changes what exists in 1966, then apparently we are no longer in that same year. In the Kantian picture the problem is even more severe: time is defined through the circumstances, so if the circumstances have changed there is no meaning at all to the claim that we returned to the same time. By contrast, if one adopts an ontic conception of time, one can at least in principle say that we are at the same t even when the circumstances differ, because time is not identified with what happens in it.
Why ordinary time travel remains unsaved even if time is ontic
Here the column sharpens the claim: even if time is a real entity, motion on it is not like motion in space. Motion in space is described as a change of place along time. But when one tries to describe motion in time, the variable and the parameter are the same thing, and so one gets either a tautology of the form ‘at time t I am at time t’ or a contradiction of the form ‘at time t1 I am at time t2’. Therefore ‘traveling to the past at a later time’ is a meaningless string of words, not merely a process that is hard to perform. In this context the column mentions the symmetry of time and space in relativity and Einstein’s static picture of the world, in which the distinction between past, present, and future is to a large extent illusory.
The only way out: two time axes, external and internal
To define time travel coherently, the column proposes a model with two time axes. The external axis t is the objective time in which the world’s events occur; the internal axis τ is the person’s biography, age, or experienced time, and it always moves forward. In backward time travel, the person reaches an earlier t with a later τ. So someone who returns after his wedding to his bar mitzvah does not return to the same full time-point, but to an earlier t with a more advanced τ: for example {t=1984, τ=24} rather than {t=1984, τ=13}. In this model, the circumstances are a function of both variables, C=F(t,τ), and t itself can be a function of τ that sometimes decreases.
How the grandfather paradox disappears, and what remains open
Within this framework, the grandfather paradox is no longer a logical contradiction. One can describe a case in which I am born, grow up, return to 1943 at a later τ, and kill my grandfather; from that point onward I may disappear from the scene, but there is no single full time-point at which two contradictory truths hold. Every change occurs forward along τ, even if it changes what happens along t. So the model does not prove that time travel is physically possible; it only turns the question from self-contradictory verbiage into an empirical and physical question. The column ends by saying that this matters not only for physics but also for halakhah: in the sugyot of condition and bereirah there is a kind of retroactive legal ‘time travel’, and there too one needs a precise conceptual language in order to analyze the mechanism without confusion.
With God’s help
On the eve of Yom Kippur I received by email a link in which a science website (cosmosmagazine) reports on a computer scientist from the Interdisciplinary Center in Herzliya (Doron Friedman) who solved, using a computer program, the grandfather/father paradox that accompanies the issue of time travel. Upon receiving the link, I was reminded of the beautiful song by Avraham Tal and Shlomo Artzi, Elohai (the lyrics are here), which also speaks about time travel, the bond between a father and his son, and even a desire for a relationship with God. For some reason, today my son showed me a YouTube clip in which a person speaks with himself twenty years older/younger. That too is a kind of time travel.
When I read the report, I saw that it is highly problematic. This has to do with fundamental questions touching on our conception of time and its relation to reality and to the human being, questions that connect directly to the approaching festival of Sukkot. I therefore thought that this would be a good time to touch a bit on these issues. I should preface this with a warning to readers that this column has a somewhat philosophical and mathematical character. I did try to avoid formalism and to explain the ideas in a way that does not require prior knowledge in order to understand them, but I assume some readers will find it difficult or simply will not want to follow. My apologies to them. I will return to the regular format later on.
"And you shall rejoice in your festival"
In the Torah portion Re’eh, the Torah commands rejoicing on the festival of Sukkot (Deuteronomy 16:13-15):
You shall observe the festival of Sukkot for seven days, when you gather in from your threshing floor and from your winepress. And you shall rejoice in your festival—you, your son and your daughter, your male servant and your female servant, and the Levite, the stranger, the orphan, and the widow within your gates. For seven days you shall celebrate a festival to the Lord your God in the place that the Lord shall choose, for the Lord your God will bless you in all your produce and in all the work of your hands, and you shall be altogether joyful.
In the simple sense, "And you shall rejoice in your festival" is a commandment to rejoice, with "in your festival" indicating the timing. The commandment is to rejoice during the festival. But the Talmud (Moed Katan 8b) understands this differently. It derives from this verse a prohibition against marrying during the festival:
For Rabbi Daniel bar Katina said in the name of Rav: From where do we know that one may not marry women on the intermediate festival days? As it is said, “And you shall rejoice in your festival”—in your festival, and not in your wife.
In the Talmud here, the verse is read differently. The word "in your festival" indicates the cause of the joy, not its timing. What gladdens us is the festival, not marriage or some other cause. When we marry during the festival, that joy is problematic, because the cause of the joy is not the festival but something else. Perhaps this can also be linked to the words at the end of the passage above: and you shall be altogether joyful, meaning that one should be joyful only in the festival and not in something else.
What is time?
In what, then, are we to rejoice? In the festival. What there is supposed to be joyous? Ostensibly, the events that occurred in it. Admittedly, Jewish law establishes the commandment of rejoicing on all three pilgrimage festivals, but in the Torah it is written specifically in the passage that deals with Sukkot. It turns out that Sukkot, specifically, is a festival not instituted to commemorate events. Passover was instituted in memory of the Exodus from Egypt, and therefore the joy of it can refer to those events. Shavuot was instituted as a commemoration of the giving of the Torah (though here too that is not entirely precise), and so there too there is something to rejoice over. But Sukkot commemorates the clouds of glory and the booths in which the Holy One, blessed be He, caused us to dwell when we left Egypt, and all of these are events that did not occur in Tishrei but in Nisan. So in what exactly are we meant to rejoice on Sukkot?
Sukkot is unique in that it is the season of our rejoicing, and in light of what we have seen, the meaning is not the events that occurred then but time itself. It thus seems that on Sukkot we are meant to rejoice in time itself and not in any events that happened in it.[1] This conception assumes that time exists in itself and not merely as an axis that serves us for marking various events. In several sources in Jewish law one can see a conception of time as an existing entity. For example, the Talmud in Shevuot 20a (and in Nedarim 12a) states that one can invoke the day on which Gedaliah ben Ahikam died as the basis for a vow. According to most medieval authorities (Rishonim), such invocation of a vow can be made only with regard to some object upon which a prohibition has taken effect (such as a sacrifice or another vowed object). It seems that the Talmud sees the fast day as a kind of object. In mathematical terminology one can say that the dependence of the obligation of rejoicing on Passover (F) is on the events of the Exodus from Egypt (X), which themselves occurred at some time (t). This is a dependence of the form: F(X(t)). See note [2]. There is no explicit and direct dependence here on time, but only through the events. By contrast, the dependence of the commandment of rejoicing on Sukkot (which falls at a time when no historical events took place) is of the form: F(t). In our book there we brought additional examples of explicit and direct dependence on time, usually of the type: F(X(t),t) (that is, dependence on events and on time itself). Whenever we find explicit dependence on time (not only through events that themselves depend on time), that is a sign that time is a real existent, for otherwise the dependence would always have to be on the events. One cannot make something depend on a thing that is not an existent. Such a thing cannot be the cause of anything else (a cause is something that truly exists, since something that does not exist cannot cause anything).
Set against this stands Kant’s conception, according to which space and time are only forms of our apprehension of reality (categories by means of which we organize it in our consciousness) and not existing entities. In the Kantian picture, without human beings there is no time and no space.
For what follows, let us call Kant’s conception a subjective conception of time, and the halakhic conception presented above an ontic (= entity-based) conception of time.
Well then, the time has come to return to our journey through time.
The solutions proposed in the report
As noted, the report describes a solution to the grandfather paradox by means of a computer program. Many people describe the difficulty of time travel through a situation in which a person goes back in time and kills his grandfather. For example, Tamar goes back fifty years and murders her grandmother before she met her grandfather. In such a case Tamar’s mother could not have been born, and therefore Tamar could not have been born either. But if Tamar was never created, then who exactly killed the grandmother? And if there is no one to kill the grandmother, then she does meet the grandfather, and the whole line of descendants does indeed come into being. The conclusion is that time travel is paradoxical and therefore probably impossible. This is a proof by contradiction that backward time travel is impossible.
The report describes software that examines these situations by simulating virtual reality. It turns out that the software finds several solutions that allow travel backward in time while leaving the situation consistent. Two examples are brought there. The first is a situation in which Tamar becomes her own grandmother, and together with her grandfather, in a heroic act of incest, they produce her mother, who then gives birth to her, and everything remains consistent and kosher (though not according to the most scrupulous standard, of course). The second solution is given on the assumption that the grandmother too can travel forward in time. She then produces Tamar’s mother, afterward goes back in time, and is murdered by Tamar, who to her misfortune also returned to that same time. You can already imagine all the other “solutions.”
Because of the absurdity of the matter, I went to read the original article. There I saw that his intention was to define a method for carrying out simulations of time travel. The goal was within computer science, not philosophy or physics, and for that purpose perhaps it has value (I do not know exactly what is of interest from a computer-science standpoint). He does hope that this method will also contribute to solving the philosophical and physical problem, which I very much doubt, but Friedman himself was at least clear-headed about the meaning of the article. The report, by contrast, describes it as though the article offers solutions to the grandfather paradox. In that sense, there is nothing here, of course. None of these things really solves the paradox. Moreover, even without the software one could easily have found these “solutions,” and many others besides. Truth be told, I do not understand why one needs to go so far, instead of offering a “solution” in which the grandmother is indeed murdered but immediately rises from the dead; or alternatively, she gives birth to the mother while dead; or perhaps the mother herself (or Tamar) is created by another couple. Or perhaps we should speak of a situation in which she both exists and does not exist at the same time. Alternatively, if she goes back in time, she is simply prevented from doing something that would create an inconsistent situation (such as murdering her grandmother). In such a case the murder would fail. Either the gun would jam, or the grandmother would merely be wounded and not die, and so on.[3]
The conceptual difficulties
Beyond the question of the article’s meaning and the “solutions” it proposes, there is room to examine the problem itself further. Up to this point I assumed that time travel is conceptually well defined, and I discussed solutions to the grandfather paradox. But the grandfather paradox is only a concrete illustration of the conceptual difficulty raised by time travel. In fact, such travel is paradoxical by definition, regardless of murders and grandmothers. Therefore, solutions such as those we saw above, even if they had substance (and they do not), do not really show that time travel is possible.
A return to an earlier time is defined as follows: entity X is at time t2 and afterward returns to an earlier time t1. The sting in this definition is the word “afterward.” Without that element, time travel is trivial. On Tuesday I was on Tuesday, and two days earlier I was on Sunday. That is not time travel. Time travel is when the return to Sunday (the earlier Sunday) happens after my being on Tuesday. But if I am on Sunday, then by definition that is a time earlier than Tuesday, so in what sense can this happen “after” I am on Tuesday?
Let us assume I am twenty years old in 2016 and that I have gone back fifty years to 1966. As is well known, in that year I did not yet exist (because I was born in 1996). But after I return there, a different reality now prevails in 1966 from the one that prevailed there in the past. I too am now there. Is this still 1966? After all, in 1966 I did not yet exist at all. So I have returned to a different reality from the one that prevailed in the original 1966. Have I really returned to 1966? And from another angle: is the one now in 1966 really me? After all, I did not exist then. But if he is someone else, then what is the problem if he murders my grandfather and I am not born? The loop does not continue, and the situation is entirely consistent. The problem is not the logical loop, but whether a return in time is conceptually defined at all: it is unclear whether this is me, and it is unclear whether the time I have reached really is 1966.
Now think of a situation in which I murdered my grandfather and he died. That too changed the reality in 1966, for now my grandfather does not exist there, although originally in 1966 he was alive and kicking. Incidentally, the same is true if I murdered someone else who is not my grandfather, or if I gave charity to a poor person. All these things change the circumstances that prevailed then, and now that same year (1966) looks different from the way it originally looked. So did I really return to 1966? After all, in the original 1966 the poor man was poor, my grandfather was alive, and I did not exist. The circumstances that now prevail are different from those that prevailed in 1966. So have we really returned to it?
To sharpen the difficulty, let us look at it from another angle. Suppose we return everyone alive in 2016 back to 1966. You know what? Let us return all of reality (and not just people) as it is now to 1966. What does 1966 look like now? We will see there the reality of 2016. Nothing has changed. So in what sense are we in 1966? You might say that if we ask the people, they will tell us that it is now 1966, and that is the difference. But that would only show that they are confused, no? Alternatively, if they really would say that it is now 1966, then we have not actually returned them, since the people in 2016 would tell us that it is now 2016. That is, the people with whom we speak after the return in time do not give the same answers as those who left 2016. This is not the return of the same people to an earlier time, but a change in the people. From yet another angle: suppose we implant by hypnosis into the minds of all the people in 2016 that it is now 1966. The situation would then look completely identical to the situation after a return in time to 1966. We would now have the same reality with the same people, and they too would tell us that the year is 1966, just as in the situation we described after the return in time. So what is the difference between hypnosis and returning in time? Did a return in time occur here at all, or is it merely hypnosis?
And what about my memories? Have I returned to 1966 with memories that also include events from the year 2000? How can that be? Those events have not yet occurred. If I remember them, then apparently I am not in 1966. But if I do not have those memories, then it is not me. Alternatively, the return did not happen “after” 2016—and therefore this is not really a return in time. And so too regarding the second case (returning all of reality backward). What about the history of the people who returned to 1966? Is it the same history they had in 2016? If not, then these are not the same people. If so, then again there is no way to determine that we really have returned to 1966.
Thus, return in time raises conceptual difficulties entirely apart from the murder of the grandfather. The paradox of killing the grandfather is meant only to illustrate the problematic nature of time travel, but it is by no means needed in order to define it. Hence even if we succeed in solving the grandfather paradox, that will contribute nothing to the question of returning in time. It is like the dog that bites the stick that strikes it rather than the hand that holds the stick. With the grandfather or without the grandfather, the return in time remains as self-contradictory as before. The grandfather paradox is a result of the problem, not the problem itself.
Incidentally, all these problems exist in travel forward in time as well, even though the paradox of killing the grandfather does not arise there. There is nothing special about travel backward, and it is easy to see that any deliberate change of time suffers from the same conceptual problems.
What is the speed of such a transition?
Another interesting question is: what is the speed of such a temporal transition? Are there physical limitations on the distance traversed in a transition backward in time (so that the speed not exceed the speed of light)? That is, can one be transported to a very distant place (outside the solar system, at a distance of more than fifty light-years)[4] such that the quotient of the difference in places by the difference in times will be greater than the speed of light?
Can one speak of speed at all, when the time is not the amount of time it took me to make the journey, but the difference in time between its two endpoints? One should remember that from my perspective the transition happens in a single instant. The assumption is that when I arrive in 1966 I will be exactly twenty years old, just as I was in the year from which I departed (and not minus thirty, which is the difference between the time values at the ends of my trajectory).
A mathematical look: the circumstances as a function of time
Physicists are used to describing the relation between time and the circumstances prevailing in it in the form of a function. The circumstances C are a function of time t, thus: F(t)=C. The meaning is that given the time, which is the independent variable, the function gives us the circumstances that prevail in it. Those circumstances include all of reality at that time: a precise description of every person and object, indeed every particle, its position, its velocity, and a complete description of its state in every respect. It seems to me that this is how ordinary people see it as well.
What is special about functions is that there can be two different time points at which the same circumstances prevail, but there cannot be two different circumstances at the same point in time. From this it follows that if the circumstances are different, then necessarily it is not the same time. This is the mathematical description of what we saw in the previous section: returning in time involves changing the circumstances of that time, and therefore it is not really a return to that same time.
Return in time in the subjective conception and in the ontic conception
The conclusion is that if we want to speak about a return in time, we must give up the functional conception. The relation between the circumstances and time is not a function. Physicists will probably object to this, since in physics that conception is a foundational principle, but a mathematician would probably accept it with complete equanimity. Why assume in the first place that the relation between the circumstances and time must be a functional relation? But one who is sensitive to the philosophical aspects will notice that there is here a nontrivial metaphysical claim that we must examine.
We saw above that there are two philosophical conceptions of time. The ontic conception sees it as an existent in reality itself, but Kant proposed a subjective conception that sees it only as one of our organizing categories. According to the subjective conception there is no such thing as time. Time serves us only to organize our impressions and cognitions. But then the time-point is in fact defined solely by the circumstances prevailing in it. There is no meaning to the year 1966 apart from the fact that this is the year-label that marks the existence of this particular set of circumstances (the circumstances that prevailed then).
Suppose that the circumstances that prevailed in the original 1966 were X1, and in 2016 the circumstances were Y. We saw above that after my return to 1966 the circumstances there are different (for example because I now exist there, or because my grandmother was murdered), and let us denote them by X2. This is a situation of two different systems of circumstances (X1 and X2) at the same time-point (1966). As we saw, a functional relation does not allow such a situation.
Now let us look at the return in time from the opposite angle. Suppose we returned all of reality as it is in 2016 (circumstances Y) back to 1966. If so, the circumstances now prevailing there are Y, and this means that we have created the very same system of circumstances at two different times. According to Kant, it follows that we have not really returned there, but are still in 2016, since that is the year that marks the system of circumstances Y, and as noted, on his view time is defined through the circumstances that prevail in it. 1966 marks a different system of circumstances, X1.[5]
Thus, in Kant’s subjective conception of time, return in time is not conceptually defined. Only in the ontic conception of time can one speak about return in time. If we returned to 1966, the circumstances now prevailing there (X2) are different from the circumstances that prevailed there originally (X1). We asked above: in what sense are we in 1966 if the circumstances are different from what existed there originally? The answer now is that being in 1966 is not defined by the circumstances but by the time axis itself. We are in 1966 even if the circumstances are different, simply because the time axis points to 1966. It is admittedly somewhat difficult to imagine how one could know what time one is in without recourse to the circumstances, but in the ontic conception there is at least a theoretical way to determine what time we are in that does not proceed through the circumstances. One simply measures directly the time one is in. The conclusion is that in the ontic conception a return in time can at least be defined. Even if I returned all of reality as it is to 1966, the question why we should not think that we are in 2016 does not arise. Time is defined in itself and not only through the circumstances that prevail in it.
The hardest problem of all
Up to this point we have not seen insoluble conceptual problems raised by a return in time. The question was whether time really exists or not, but as we saw, at least in the ontic conception one can speak consistently about a return in time. But now we shall see that even in the ontic conception there is an insoluble problem. The problematic nature of motion backward in time lies in the question whether there is a difference between time and space. Are these variables that behave similarly or not? If motion backward in time is impossible, what emerges is that in space one can move forward and backward, but in time one cannot. That is, there is an asymmetry between time and space.
The motion of a body describes a change of place along the time axis: X=F(t). At every time t the body is in a different place X. Any dynamics or change are described in the same way. When one empties or fills a pool, the amount of water depends on time in the same way. And so too when one repents (see the previous column) or deteriorates, the spiritual condition depends on time in that fashion. Change or motion always take place along the time axis. The conclusion is that the motion of a body is described as follows:
(a) At time t1 the body was in place X1, and at a later time t2 it is in a different place X2.
A similar sentence will describe a change in our spiritual condition during repentance or decline, or the amount of water in the pool while it is being filled or emptied.
Forward movement (to the right) in space (positive velocity) is described by such a sentence in which X2 > X1. Backward movement in space (to the left) is described by the same sentence when X2 < X1. How are we to describe backward and forward movement in time? In order to do this we must construct a sentence that describes movement in time. In such a sentence the variable X also describes time. But if we make the appropriate substitution in sentence (a), we obtain only trivial sentences, like this:
(b) At time t1 the body was at time t1, and at a later time t2 it is at a different time t2.
If t2 < t1 this is movement backward (to the past), and if t2 > t1 this is movement forward (to the future). But these sentences do not describe motion; they describe identity. It is obvious that at every time ti the body is at time ti. This is not the motion of the body along the time axis, but the motion of the time axis itself.
Our own movement along the time axis is described by a change in some variable along the time axis. When I move from the year 2016 to the year 1966, this is described by the following sentence:
(c) At time t1 I am at time t2, and at time t2 I am at time t1.
If the previous sentence was trivial, this claim is simply self-contradictory nonsense. How can I be at time t2 when the time is t1? The dependent and the independent variable in this sentence are the same, and therefore their values cannot differ. Time is a very simple function of time: the identity function. There is no possibility of a different dependence of a variable on itself. This is really the expression of a very basic conceptual difficulty raised by time travel. If I have returned to 1966, I am in it before 2016 and not after it. But as we saw, time travel is defined only when the earlier time appears after the later time. But if it is earlier on the time axis, how can it be regarded as later?
So perhaps the way out is to try to describe the movement of the time axis itself. It is not I who move along the time axis; rather, it itself flows backward. Now the question arises: over what does it flow? As we have seen, everything that changes flows over the time axis, but it is not clear over what the flow of the time axis itself takes place.
A natural solution for examining the symmetry between time and space is to return to the sentences brought above and exchange the roles of time and space. After the exchange we ask ourselves whether the resulting sentence is reasonable (logically or physically consistent). Let us take sentence (a), substitute into it values that describe backward movement in place, and exchange the roles of time and space. What we obtain is the following sentence:
(d) At place X1 the body was at time t1, and at a later place (to the right) X2 it is at an earlier time t2.
This sentence is an innocent one that actually describes motion to the left in space, only in reverse phrasing. It is a perfectly legitimate sentence, and this means that whatever can be said about space can also be said about time. The symmetry between them is complete. Does this mean that one can travel backward in time? Certainly yes—but only in the trivial sense described here. This is really motion to the left in space. As we have seen, motion backward in time means arriving at an earlier time at a later time, but that is not defined at all. It is simply an empty expression. The conclusion is that traveling backward in time in the accepted sense is impossible, because it is not conceptually defined in the first place. It is a meaningless collection of words. If you have returned to an earlier time, then you are before the later time and not after it; that is, you have not traveled backward in time in the accepted sense.
Let me remind you again of what I already noted. The very same problem exists in travel forward in time as well, even though the paradox of killing the grandfather does not arise there. Even when I travel forward in time, when I arrive in 2030 I will be in 2030 and not in 2016. There is no travel in time here, but simply ordinary flowing along with it. Time travel has no meaning at the conceptual level.
A further note on space-time symmetry
At first glance, it might seem possible to show one asymmetry between time and space. One state can exist with respect to space and not with respect to time. If we look at sentence (a) when the places are identical and the times are different, the meaning is that the body is standing still. By contrast, in that same sentence, when the times are identical and the places are different, this seems altogether impossible (a body cannot be in two different places at the same time).[6]
But this too is a mistake. The version with different places and identical times is indeed possible if we are dealing with an extended body (which occupies both places together at the same time). But for a point-body this is impossible. Yet conversely, for a temporally point-like body (one that exists for only a single instant and is immediately destroyed), even persistence through space is impossible as well (sentence (a) with identical places and different times). The conclusion is that, surprisingly enough, the symmetry is preserved here too. Time and space are completely symmetrical.
This symmetry is the foundation of Einstein’s theory of relativity. In relativity, Einstein presents events over four variables (three spatial and one temporal), and his relation to space and time is identical. They function in the same way and even exchange roles. This symmetry finds expression in Einstein’s attitude toward time. He sees it as static and frozen exactly like space. An amusing expression of this attitude is the letter of condolence that he sent to the Besso family, after the death of his friend from the Swiss Polytechnic, in which he writes:[7]
Now he has departed from this strange world a little before me. That has no significance. People like us, who believe in physics, know that the distinction between past, present, and future is nothing but a stubborn illusion.[8]
The meaning of these words is that Besso’s death on such-and-such a date is a true fact, and it was always true. Nothing changed when Besso died on date X, for the statement that Besso died on date X was true also ten years or a thousand years earlier. This is a way of viewing the time axis as though it all lies before us in a static fashion, with nothing really changing, exactly like space.
Two time axes
The necessary conclusion from all this is that time travel is an oxymoron, and this is true even in the ontic picture of time. The verbal description of time travel is a contradictory collection of words that expresses nothing. But it turns out that the discussion until now was not in vain. There is a way to define time travel consistently, and in light of what we have seen it seems to me both necessary and unique. For this, we must assume that there are two different time axes, one of which flows over the other.[9] This picture is presented in detail in the fourth book of the Talmudic Logic series, where it serves us in analyzing Talmudic topics of condition and breirah (retroactive determination); both of these are time travels in the legal sphere. Here I will touch on it only briefly, in order to complete the discussion of time travel.[10]
First, let us recall that the fundamental problem was this: over what does time itself flow? Or, in other words, how can one define being at some time on the time axis itself? When I say that I am at the later time t2, and ‘afterward’ I arrive at the earlier time t1, I need to define in what sense this happens ‘afterward.’ In what sense does the earlier time t1 come ‘after’ the later time t2? Along what axis is the one later than the other (and this cannot be along the ordinary time axis, since there it is earlier than it)? The earlier time, by definition, is earlier and not later. In order properly to define the process of returning in time, we must understand through what lenses we are measuring the connective expressions, such as ‘afterward’ and ‘beforehand,’ in the previous sentences. Suppose I have passed my Bar Mitzvah, reached the time of my wedding, and now I return again to the earlier time of my Bar Mitzvah. The time axis, as it were, ‘bends’ and folds back upon itself. I am at the same time-point twice: once at the Bar Mitzvah, and the second time after the wedding. How can one be at the Bar Mitzvah that took place at age thirteen when one has already passed the age of twenty-two?
As stated, the conclusion is that in order to speak about returning backward in time, we must introduce some axis that is independent of the ordinary time axis. This axis will define in what sense the Bar Mitzvah (which is earlier on the ordinary time axis) is ‘later’ than the wedding. This axis is what will be responsible for the connective terms in the sentences above. When we say that Reuven returned ‘after’ his wedding to the time when he celebrated his Bar Mitzvah, we mean that the Bar Mitzvah indeed occurred before the wedding on the ordinary time axis. But there is another time axis relative to which the arrival at the Bar Mitzvah occurred after the wedding.
We can identify these two time axes as follows:
-
First, there is the ordinary, external, objective time axis in which the various events in the world and in our lives occur. We have already denoted it by the letter t.
-
In addition, there is also an internal time axis that accompanies the person from within (= his age, the number of years that have passed since his birth), and it always flows forward. Along this axis he never goes backward, and everything that happens in the world is a function of it. Let us denote this axis by the Greek letter τ (tau).[11]
A detailed description of return in time in this model
Let us now see how we describe return in time coherently. When a person is born, his internal time axis is τ = 0. The year in the world is t = 1971. We denote this as follows: {τ = 0, t = 1971}. Thirteen years later he reaches his Bar Mitzvah, and then the times are: {τ = 13, t = 1984}. After another 9 years he marries, and then the times are: {τ = 22, t = 1993}. Now he waits another two years and returns backward in time to the period of his Bar Mitzvah. What now happens according to the description we are proposing is: {τ = 24, t = 1984}. That is, his internal time has advanced by another two years, for it never goes backward (this is his ‘real’ age). All these processes take place along it, and it is the independent variable. But the external time is what we ordinarily think of as the time axis. This is the time I am in after the journey backward in time, and therefore I have in fact gone back eleven years here, to 1984. In terms of the events in the world, we are in 1984, but my internal age is 24.
In this language, we can express any idea that touches on return in time, and it will be well defined. Of course, this does not mean that the thing is possible; it means only that there is a consistent and well-defined logical formulation of the language in which one can conduct discussions on these matters. When a person returns backward in time, the meaning is that he reaches a time t that is earlier while his τ is higher. By contrast, progression forward is an increase in t together with the increase in τ.
Now the dependence of the circumstances on time is a function (that is, a single-valued dependence), for in order to speak about events in the world we must give both times, and not only t. Given the value of both times, the world and the events taking place in it are unique and well defined. The relevant function of the circumstances as dependent on time is: C=F(t,τ).
The relation between the two time axes
Returning backward in time is in fact a folding of the time axis backward. One can describe the relation between these two times in the structure of a function, in which the independent variable that constantly flows forward is our personal biography (our age), τ. When we return backward in time, the meaning is that an increase in τ causes a decrease in t. If so, we have one function: t(τ), which describes how τ determines t. This is the function that determines the return in time. If there is return in time, this means that this function decreases at some point along τ and returns to an earlier t.
In a normal situation (without return in time), the shape of this function is:
The time t and the time τ flow at exactly the same rate, and if one is 13 years, so is the other.
By contrast, in the case in which there is a return in time (as in the example described above, of returning after the wedding to the time of the Bar Mitzvah), we obtain the graph:
At some value of τ (in the above example: τ = 24, two years after the wedding), the time t drops from its value (which was also 24) and returns to a lower value (13, the age of the Bar Mitzvah), and from then on it continues to flow forward at the same rate as τ.
It is important to understand that in both of these types of cases, every value of τ defines uniquely what t is. As we defined above, t is a function of τ. Therefore, when we know τ, we know everything that happens in the world, which as we have seen is a function of these two times: f(t,τ), or more precisely f[t(τ),τ].
At present there is also no obstacle to returning backward to a time before I was born. If the return in time were a folding of history as it actually was, such a folding could never bring us to a situation in which we had not existed in the past. But now, since we are not taking history as it was and ‘folding’ it backward, but rather folding the function t(τ), there is no obstacle to returning even to a time in the distant past. The time τ will move forward with my age, but along it I may visit the eleventh century, or even earlier.
In this description we are speaking of a return to earlier times while our memory contains events from the future. We return to our own Bar Mitzvah, but unlike the original Bar Mitzvah, we now have memories of our wedding. Therefore this is not the very same time-point, except in a partial sense (from the external aspect, not the internal one). We return in t but not in τ. That is exactly what changed by introducing an additional time axis into the language we use in treating these problems. In the example brought above, this is how one can describe arrival in 1966 while we ourselves are age 20 and in the very same state as in 2016. The two types of circumstances do not characterize the same time, but only the same t. But full time is (t,τ), and it does not repeat itself. Dependence on time (t,τ) is a perfectly legitimate function.
And back again to time travel and the grandfather paradox
The definition we have proposed solves all the conceptual problems raised by the notion of ‘return in time.’ The concepts are now well defined, and the grandfather paradox does not arise.
For example, if a person returns backward in time, can an event from the future affect what he will do? Obviously there can be no influence from the time {τ = 24, t = 1984} to the time {τ = 20, t = 1980}. Such an influence cannot exist because it is not even defined. As we defined it, the τ-axis constantly flows forward. Therefore we can now at least formulate the question coherently: can one influence from the time {τ = 24, t = 1984} to the time {τ = 24, t = 1980}, or to the time {τ = 26, t = 1980}? The first is an immediate influence (in τ-time), and the second is a causal influence forward (in τ-time).
Now, once the question is well defined, the answer belongs to the domain of the physicists. In our present state, the language is coherent and the concepts are well defined. The result—whether one can return in time or not—is a factual question, not one that can be answered by tools of logical analysis. It depends, of course, also on the factual-physical question whether there really are two such time axes or not.
Why does the paradox of killing the grandfather not arise at all in this model? Let us describe the chain of events as follows:
-
{τ = 0, t = 1930} My grandfather is born.
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{τ = 13, t = 1943} My grandfather has his Bar Mitzvah.
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{τ = 63, t = 1993} I am born.
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{τ = 76, t = 2006} I have my Bar Mitzvah.
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{τ = 80, t = 1943} I return in time to my grandfather’s Bar Mitzvah and kill him. All those present are struck with astonishment (for their invitations were issued validly, at a time when my grandfather was still alive, and therefore it is obvious that they are present there).
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{τ = 81, t = 1944} My grandfather no longer exists in the world, and everything else proceeds as usual.
The question now is whether there is a point in time (that is, in both t and τ) at which the situation is not single-valued. If there is such a situation, then our definition is problematic; or alternatively, this would constitute a proof by contradiction that indeed one cannot return backward in time (because the opposite claim leads to paradox).
The answer is that there is no such problem. At every point in time there is only one true situation, and therefore there is no obstacle even to killing the grandfather. It is true that at the time t=1990 there is multivalence, for my grandfather is still alive (he is 60 years old), but after my return in time I kill him, and in that very same year, 1990, when we examine matters we discover that he is dead. But these two events occur at different τ, and therefore for us this is not the same point in time, and so it does not impair the single-valuedness of the function.
What is true is that after my return in time, my grandfather is dead, and therefore I too die. That is, beginning from the time τ = 80 I disappear from the scene (for if my grandfather dies before his wedding and before he has children, that means that the entire chain that came out of him also dies). If I were to kill him after he married and fathered children, the resulting situation would be that my grandfather dies but my parents and I remain alive. But then it would not be I but someone else, and then the question would arise: where did he come from? Therefore the reasonable interpretation is that in such a situation I disappear.
Thus, the picture is consistent and free of paradoxes. All influences are only prospective, that is, they are carried out forward in time, except that this is so only from the standpoint of τ-time. With respect to t-time there is no necessity for prospectivity, and there return in time can indeed occur.
Return in time in the legal sphere
You may ask: why is the logical and conceptual definition of return in time important when it is not at all clear whether physics permits such a thing at all?[12] First, as we have seen, even a discussion in physics must begin with well-defined concepts. Without that there is no possibility of carrying out the scientific inquiry and reaching a conclusion as to whether physics in fact permits such processes or not. Second, in the fourth book of the Talmudic Logic series we applied these models to the topics of condition and breirah. In both of these topics, which appear in dozens of different Talmudic passages, precisely this logic is used. There is time travel there, except that it occurs in the legal sphere and not the physical one. For example, a man betroths a woman and says to her: “Behold, you are betrothed to me on condition that you give me two hundred zuz in a week.” If she gives him the sum, it turns out retroactively that she was betrothed to him. If she does not give it to him, then she was never betrothed to him. This is an example from the discussions of condition. In the discussions of breirah one can also see time travel of a somewhat different sort.[13]
It is clear that even if in the legal sphere there is no problem of physical constraints, it is still very important that the concepts be defined so that conclusions can be drawn from any method. In these topics too there are not a few confusions because the concepts are not always well defined. The logical model presented here is vital for analyzing both of these topics, and it has a number of halakhic and analytic implications.
[1] I deal with the conception of time in my essay Middah Tovah on the third and thirteenth roots (currently being edited for publication as a book). There the sources are brought, as well as additional articles that deal with the topic of time in Jewish law. See briefly also here on the site. These matters are also discussed in the first part of the fourth book in the Talmudic Logic series, Logic of Time in the Talmud.
[2] This is a very important distinction in mechanics. A force that depends explicitly on time does not conserve energy. Conservation of energy exists only when the force depends on time only through another variable (for example, the place X), even if that variable itself depends on time. Think of a particle moving along some topographical profile. The force acting on it depends on its location (a steep slope exerts a strong force, and vice versa), and if it moves then its location depends on time. This is a conservative force, and its energy is conserved throughout the motion. But if the profile itself changes with time, that means that the force at the same point changes with time, that is, there is explicit dependence on time. In such a case there is no conservation of energy.
[3] It should be noted that such a possibility is also mentioned in the article itself in the name of Lewis (in his 1976 article).
Incidentally, unlike all the preceding ones, there is a slightly different version of this proposal that can be taken more seriously. It is reasonable to assume that no creature can perform a step that creates a logical contradiction (if I can create a wall that stops all shells, then clearly I cannot at the same time create a shell that penetrates all walls). This proposal is fully parallel to the proposal of R. Shimon Shkop for solving a Talmudic loop, which we expanded (in the eighth chapter of the fifth book in the Talmudic Logic series, Resolving Conflicts and Normative Loops in the Talmud) to many additional loops. R. Shimon proposes a meta-halakhic principle according to which if there is a halakhic situation in which a loop is created, the loop can be stopped when a step is taken that creates inconsistency (that is, destroys itself). See a detailed analysis in the eighth chapter there.
[4] To get a sense of proportion, the sun is at a distance of about 8 light-minutes from us.
[5] This is a subtle point, since a functional relation does permit such a situation. We saw above that given a functional relation, there can be two different points in time with the same system of circumstances. So why did I write that according to Kant’s conception this aspect too is problematic?
To understand this, we must note that Kant’s conception indeed requires a functional relation, but it is a very special functional relation, one in which there is no situation of two time-points at which the same system of circumstances prevails (in mathematics this is called a one-to-one relation). That is, the mere existence of a functional relation does not necessarily imply a Kantian conception. Time travel violates the assumption of a functional relation in one respect (the existence of two different systems of circumstances at the same time) and Kantianism in two respects (the existence of two systems of circumstances at the same time, and the existence of two different time-points with the same system of circumstances). The Kantian conception in fact requires a one-to-one and onto function, for only then is it an invertible function, and then one can truly define time through the circumstances as follows: t = F-1(C).
[6] An interesting question is whether this is a physical inability (because it would involve infinite speed, beyond the speed of light), or a logical problem (if the body is here, it is not there, by the principle of non-contradiction). In my second notebook I argued that infinity is a potential and not a concrete concept, and therefore it seems to me that the problem here is not physical but logical.
[7] See chapter three of the fourth book in the Talmudic Logic series, Logic of Time in the Talmud.
[8] See:
Freeman Dyson, Science and the Search for God: Disturbing the Universe, New York : Lantern Books, 2003, p. 24
[9] In order not to torture the readers, I decided not to enter here into the different possibilities as to which of them is ontic and which is subjective.
[10] See chapter five of the book Logic of Time in the Talmud, where these matters are presented in detail. There the equivalence to a description in terms of many worlds, which was proposed as an interpretation of quantum theory, is also discussed; the literature on return in time uses it as well (it is also mentioned in the report above).
[11] The notation is taken from the theory of relativity. There the letter τ is used to describe the interval. It has dimensions of time, but it is not conceived as time. In the proposal here I rely on a model developed by Prof. Larry Horwitz of Tel Aviv University and Bar-Ilan University, who in a series of articles over quite a few years rebuilt all of physics while assuming the existence of two different time axes as described here.
[12] There is, of course, the important practical ramification for a woman’s betrothal, into which I will not enter here.
[13] Regarding the difference between condition and breirah, see Gittin 26a in Rashi and Nachmanides, and in our aforementioned book.
Discussion
Miki:
Hello, honored Rabbi,
The article is fascinating, as usual for you. On the other hand, it reminds me of the joke about the scientists in the hot-air balloon (the one on which you based the title of the book Two Carts and a Hot-Air Balloon): anyone who has read literature dealing with time travel knows that there are two time axes – the objective-world axis and the subjective-personal axis. What you did in the article is a formalization of our “gut feeling,” which is an important task in itself, but it does not introduce anything substantial regarding the topic of time travel.
In my humble opinion, there is also a substantial difficulty in the move made here. According to what emerges in the article, it seems that two people who perform time travel do not affect one another. What happens if two people of the same age return to the same point in time and meet there? Suppose they meet at the same point – (20,2000), that is, for both of them τ has the same value. On the other hand, the value of τ is personal. Are they really at the same point in time according to your definitions? Do they really meet? Perhaps they undergo different experiences stemming from different time axes? And what are the theoretical implications if these points are indeed different? And what happens if afterward they again decide to meet at the same time (40,2000) (“after twenty years”) with time travel 🙂 ? Do they meet the same people? Do they meet themselves? Who is “themselves”?
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Rabbi:
Hello Miki. There are formalizations that are nothing more than a worthless translation, and there are formalizations that clarify some important aspects. I think and hope that this belongs to the second type. For example, the grandfather paradox does not arise in this formulation.
By the way, the personal axis is not subjective, otherwise we would not really have solved the problem. It is the most objective axis there is (tau in relativity is the interval, something absolute). The obsession with the paradox and the fixation on it prove that people do not really understand this point.
And from here you can also see that your own remarks reflect the same problem. They indeed both return to the same point in time, and I do not see why they also cannot affect one another. Remember that this is not a subjective axis.
As for the question of continuity of identity, it is not really relevant here. You can ask the same question about a certain person now and a moment later: is it the same person? There is a rather silly article by Adi Tzemach (in Marcelo Dascal’s collection, The Just and the Unjust) where he bases moral duty toward the other on the fact that concern for myself a moment from now is also concern for some other person.
Yariv:
Hello,
An interesting topic, and interestingly written. The conceptualization and the mathematical model proposed here are very good and elegant.
I think it is worth noting that despite the reservations discussed in the article, time travel to the future in the model presented is completely possible on the basis of relativity. In particular, travel to the future. The more you accelerate, the more your internal clock slows down while world time continues on. The implication is that fast travel over time keeps your internal clock slow while time in the world proceeds at its own pace. If I remember my physics calculations correctly, a minute at the speed of light means four years in Earth time.
The problem is of course the issue of backward time travel. As I understand it, there is no physical law that prevents it, and yet the only way to do this, at least according to known physics, is through wormholes (Hawking dealt with this quite a bit and gave many interviews on the subject; over time he changed his mind regarding the possibility of sustaining time travel). However, wormholes and the event horizon at their boundary fall under the physics of relativity on the one hand and quantum theory on the other. At present there is nothing definitive that can explain what happens there. Still, wormholes are supposed to connect points in spacetime and hence to allow time travel.
As for the paradoxes, the only explanation that is consistent with existing laws of physics is the theory of parallel universes, in which every entry into a wormhole creates a duplication of the current universe and thereby solves the paradox problem. It creates problems of its own in physical and conceptual contexts that have not yet been exhausted.
I highly recommend Michio Kaku’s book Physics of the Impossible. He surveys there a broad set of technologies and ideas that at present seem impossible. Popular science, but nice…
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Rabbi:
Hello Yariv.
It seems to me that in relativity the meeting between systems moving at different velocities is problematic, and it is not clear exactly how it takes place (how the synchronization of the clocks is created). It seems to me that this is not exactly time travel, but rather different measurements of time.
At the speed of light one arrives at an infinite rate of time flow (not 4 minutes). There is a factor in the denominator that goes to zero.
As for wormholes too, in my opinion the translation into our sense of time is not well defined. It is a loop in spacetime, but it is not clear what exactly its meaning is in terms of what we call time.
It seems to me that my model provides an explanation for time travel that does not require parallel universes. As I showed, this model is needed even apart from paradoxes (as I explained, the paradoxes are not the basic problem in time travel. The problem is conceptual).
I have Kaku’s book. Thank you.
From one interesting matter to another (not) of the same matter
Are the things published here correct
Time as a natural phenomenon
About a hundred years ago, the Jewish scientist Albert Einstein developed the theory of relativity of time, according to which time is “fluid” and flexible. According to Einstein’s explanation, time depends on a person’s frame of motion. The implication is that the faster we move, the more slowly time will pass for us relative to someone who remains stationary. A person moving at such speeds will not notice any change at all in the flow of time for himself, and it will appear to him completely normal.
To bring this closer to our understanding, we must understand that time is one of the natural phenomena in the world. Just as the force of gravity operates only on Earth and is nullified in space, so fast travel, close to the speed of light (no matter where), changes the state of a person’s time relative to his state on Earth, where he was stationary.
The twin paradox
The best-known example illustrating the phenomenon is called the “twin paradox,” in which we consider two twin brothers who are twenty years old. We give each of them accurate watches showing the year, the hour, and today’s date. One brother remains on Earth, while his twin, the astronaut, departs on a flight away from Earth at a very high speed. The flight time is set for only one year. From the point of view of the astronaut pilot, only 12 months pass during the flight, as the watch on his hand will show.
If the spaceship’s speed is 99.5 percent of the speed of light (300 kilometers per second), then after one year according to the watch of the brother in the spaceship, when he returns to Earth and meets his twin brother, he will discover to his surprise that while he has aged by only one year, his twin brother on Earth has aged by ten years. And if the astronaut brother returns from a fast trip after five years according to his own watch, he will find that fifty years have passed on Earth.
Therefore, if we succeed in flying a spaceship at a speed close to the speed of light, and send it on a journey of ten years in space, the astronaut inside it will discover, upon returning to Earth, that a hundred years have passed.
Although it is very difficult to grasp the meaning of this, dozens of experiments conducted from then until our own day have proved it. (True, nowadays there is no spaceship capable of flying at such a speed, because the amount of energy required to create such a speed is roughly the entire amount of electrical energy that exists on the whole Earth. To approach the speed of light, one would need to circle the Earth seven times in one second. The highest speed achieved was 40,000 km/h, aboard Apollo–10. To travel in time, one needs to fly at a speed 2,000 times greater than Apollo 10.)
But experiments connected to time distortion have proved it. For example, in an experiment conducted a few years ago, two highly precise atomic clocks were synchronized to the same time with maximum precision. One clock was left on Earth and another clock circled the Earth twice on a jet plane (motion with acceleration). After landing, the clocks showed different times. The clock that had been on the plane was “behind,” and thus one could see a distortion of time as a result of the rapid motion.
Time dilation is a phenomenon that always occurs, but we do not feel it because the speeds to which we are accustomed in daily life are very small relative to the speed of light. For example, the watch of a pilot in a fighter jet whose average speed is 1,800 km/h (1.5 times the speed of sound) will lag, after two hundred flight hours, by only a millionth of a second.
Even the greatest physicists, who have devoted most of their energy and time to researching the phenomenon of time, admit that they still do not fully understand this dimension. And it is hard to understand that time is a natural phenomenon.
The scientist Stephen Hawking brings proof of this from the global satellite positioning system (GPS), which shows that time passes faster in space. For inside every spacecraft there is a very precise clock. But despite its being so precise, there is a gap of about one-third of a billionth of a second every day. Therefore the system must be corrected in order to avoid changes in the times; otherwise, these tiny differences would disrupt the entire system and cause every GPS device to deviate by 10 kilometers each day.
From a quick skim – yes.
So what is the explanation for a thing like this: a person disappears from us with his newborn baby
and returns after 13 years
with a baby who is only one year old, and he himself too has aged by only one year.
Just because he flew at tremendous speed into space and back?
Is there any explanation that is satisfying?
You already asked questions like this in the past, and I will answer you what I answered then: there are explanations in terms of scientific theories, and if you want them you need to study those theories. If you want an explanation in terms of your simple intuitions, you probably won’t find one, because they are mistaken.
I found an explanation from your former student
Part A:
https://theslightestclue.com/?p=6602
Part B:
https://theslightestclue.com/?p=6678
Here, I copied it
Understanding the Twin Paradox
May 15, 2023Shai Yefet
According to Einstein’s theory of relativity, time travel is definitely possible, at least to the future. The classic scenario involves two twins, one of whom travels in space close to the speed of light and returns to Earth. But in the above scenario there is a paradox (apparently). What is the paradox? And what is its solution? All the answers are in the present article.
Einstein’s special theory of relativity brought about a real revolution in everything having to do with the way we physically perceive space and also time. According to Einstein, space and time are not absolute concepts; that is, they do not exist in reality in an absolute manner. The opposite is true: space and time are relative concepts, and they depend on the speed at which we are moving. In everyday life you move at very low speeds (relative to the speed of light), but if you move at a speed that approaches the speed of light, then you will discover that time and space begin to “contract.”1
One moment… contract? What does that even mean? So let us bring things down to earth with a simple example, admittedly unrealistic, but enough to illustrate the principle:
• Imagine that you were running a marathon at a speed of 60% of the speed of light. At such a high speed, you would discover that the length of the course would shrink by 20%, and it would no longer be 42 km, but 33.6 km. I mean this seriously: space itself would contract; in the direction of your running, all the objects would become thinner, or somewhat squashed. I know this sounds very strange, but what can you do—this is a result that follows directly from the mathematics of relativity.2
• In addition, time itself would contract, and the whole world around you would appear to be moving in slow motion. For example: if while running you looked at a large wall clock mounted on the front of one of the buildings on the street, you would discover that the hands of that clock are ticking more slowly compared to the movement of the hands of the watch on your wrist. Here too, the rate of time in the whole world around you would slow by 20%; that is, in the time that your wristwatch ticked 5 seconds, the clock on the wall of the building would tick only 4 seconds. Again, this sounds very strange, but according to relativity this is reality.3
Run fast enough and all of reality will change. Seriously.
Source: Arthur Sasse, Public domain, via Wikimedia Commons
It is no surprise that Einstein’s theory very quickly gave rise to strange and bizarre scenarios that at first glance seem very hard to digest. One of those strange scenarios—and perhaps the most famous of them—is called:
The twin paradox.
If you have heard before about the paradox but felt that the topic went over your head, and you never managed to understand what exactly it is about, where exactly the paradox lies, and what its solution is, then this post is for you.
Before I describe the paradox and its solution, it is important to remember that Einstein’s theory of relativity has stood—and still stands today—the test of empirical experiment. Thousands of experiments conducted from the beginning of the 20th century to the present repeatedly confirm relativity. Therefore, even if the conclusions of relativity seem strange to us, they should nevertheless be taken seriously.4
Old age suddenly overtook him
Let us begin with a detailed description of the paradox itself:
• Meet Sela and Ofek, two twin brothers who were born on the same day on Earth.
• Sela had always preferred to stay at home and hardly ever went outside to play.
• Ofek, on the other hand, was always looking for adventures and dreamed of becoming an astronaut from first grade onward.
• One day Ofek decided to set out on a journey in a spaceship from Earth toward a distant star.
• Sela, as expected, stayed on Earth.
• Throughout the journey, Ofek’s spaceship travels through space at a very high speed, a speed close to the speed of light.
• The moment Ofek reached the star, he decided not to stay there, and so immediately turned the spaceship around and began returning to Earth at the same speed at which he had traveled on the outbound leg.
• Ofek’s journey ends when he returns to Earth and meets his twin brother Sela.
At that point a very strange phenomenon occurs:
Ofek discovers that his twin brother Sela is much older than he is.
In simple words: a situation arises in which Ofek aged more slowly but Sela aged more quickly, and therefore at the end of the journey Ofek is much younger than his twin brother Sela. How can this be?
The twin paradox: when the twins part, they are the same age. But when the twin who set out on the journey returns to Earth, the twins discover that the one who returned has barely matured, while the twin on Earth has aged significantly.
Okay, before we proceed, we need to pause for a moment and emphasize that this fact in itself is not considered a “paradox.” At most, it is a very strange phenomenon, one we are not accustomed to in everyday life, for clearly in our ordinary experience twins age at the same rate and always remain the same age. In any case—whether we like it or not—Einstein’s theory of relativity predicts that Ofek and Sela will not age at the same rate. What can you do, this is the prediction of Einstein’s theory of relativity, and if the theory is correct, then that is what will happen in reality.
So then, what exactly is the paradox in all this?
Everything in life is relative
Well, the paradox lies in the fact that the physical concept of speed is a relative concept. In simple words: when two bodies move at constant speed relative to each other, there is no way to determine which body is “really” in motion and which body is “really” at rest. Physically, one can determine that body A is at rest and body B is moving relative to it, but one can also determine the exact opposite (body A in motion and body B at rest), and the two determinations are equivalent.
If we translate this idea into the twin paradox scenario, we arrive at the following conclusion:
• Sela—the twin on Earth—claims that he is at rest, and his brother Ofek—who set out on the spaceship journey—is the one in motion.
• But from Ofek’s perspective, the spaceship is what is at rest, and in fact his twin brother Sela is the one in motion. From Ofek’s perspective, Earth is what set out on a distant journey through space and eventually returned to the spaceship!
I know what you are thinking… Ofek’s point of view seems completely absurd, since we are accustomed to the fact that Earth—unlike spaceships—does not set out on interstellar journeys. But remember that Earth is not necessary in this case; it is there only to make the story nice. We can get rid of Earth entirely, and imagine as though each twin is inside his own spaceship. Both of them are floating together in empty space, far from any planet, until one of them decides to set out on a journey and return. No matter which of the brothers sets out on the journey, in any case each of them will think that he himself is at rest and that it is his twin who is in motion.
If it still bothers you that only one of the twins actually started up the spaceship engines, never mind… we can also imagine a scenario in which both brothers set out on journeys in opposite directions, and after some time one of them turns around, returns, and catches up with his brother. None of this changes anything; the bottom line will be the same:
When Sela looks out the window of his spaceship, he sees Ofek receding, disappearing, and after some time approaching again. But equally, Ofek too looks out the window of his spaceship, and he too sees Sela receding, disappearing, and after some time approaching again.
It follows, then, that according to relativity each of the brothers ought to be older than his twin, for of each of them it can be said that he is in motion relative to his twin.
That is the paradox.
Even if we accept the (strange) fact that according to relativity the twins will not be the same age, still at the moment of meeting only one of them can be older. But which one? It cannot be that the principles of relativity lead to contradictory conclusions.
Breaking the symmetry
So what is the solution?
Well, the solution lies in understanding that the twins’ situation is not really symmetrical, and their points of view are not equivalent. The symmetry is broken not because one twin went on a journey and is therefore in motion, for one cannot determine who is “really” moving. The symmetry is broken because only one of the brothers (Ofek in this case) changed the direction of his velocity at the point where he turned around to return to Earth. At first glance this may seem like a small detail, but that is where the whole matter is buried:
A change in velocity creates acceleration.
It follows from this that although each twin sees himself at rest and his brother in motion, only one twin is actually under the influence of acceleration, and the effect of acceleration cannot be missed.
We all know the phenomenon from everyday life: when we travel in a car at constant speed in a straight line, we do not feel anything special, and it makes no difference whether we are driving at 100 km/h, 200 km/h, or not driving at all. By contrast, if the car makes a sharp turn, we immediately feel ourselves being pressed sideways. If the car makes a sudden stop, we find ourselves thrown forward. In all these cases we have a clear indication that the system is under acceleration. In simple words: acceleration breaks the symmetry in the twin paradox.5
At this stage you may feel that you still have not received an explanation, and you are right. The puzzled reader is probably still scratching his head and does not understand exactly how acceleration causes the age difference. What is the precise mechanism?
Well, the annoying answer is that there is no choice: one has to do the calculation. In other words, one must formulate the relevant equations of relativity, solve them, and see that the acceleration of the twin in the spaceship is what causes him to be younger than his brother on Earth.6
It is obvious to all of us that digging through mathematics is not the purpose of the blog. Therefore the natural question is this:
Can one explain how the age difference between the twins arises in a simple and clear way without getting into complicated calculations?
The answer is:
Yes, one can.
So I thought about how one might take the cumbersome mathematical solution and transform it so that we could describe the solution verbally, step by step, from the point of view of each twin separately. That is exactly what I will do below: I will spell out how the rate of time changes between the two brothers לאורך the journey, and more importantly: why, at the moment of meeting, they will both indeed agree that the twin on Earth is the older one.
In any case, bear in mind that it takes a little time to digest all the stages in the description, so you may need to read the sequence of events two or three times before the penny drops. Either way, it is a nice challenge and healthy exercise for the brain. In the end the reward is worth it, because this is one of the best ways to internalize the principles of relativity.
Before I begin, an important technical note: note that I will measure time in years, and distance in light-years. This is a very convenient method for describing bodies moving at high speed; for example:
• A body moving at the speed of light will travel in one year a distance equal to one light-year.
• A body moving at half the speed of light will travel in two years a distance equal to one light-year.
• A body moving at a quarter of the speed of light will travel in one year a distance equal to a quarter of a light-year.
• And so on, and so on…
The sequence of events in the twin paradox
To set the stage for the explanation, let us assume that each of the brothers has a wall clock showing how much time has passed. At the beginning of the journey, both brothers reset their clocks and synchronize them. In addition, each brother has a camera that records his own personal clock and sends the recording continuously to the other brother. Therefore at every given moment, each brother sees not only his own wall clock, but also a live broadcast on a screen of the other brother’s wall clock. It is important to remember: the camera’s transmission is sent by radio waves moving at the speed of light, and as is well known this is a finite speed!
Immediately after the twins reset their clocks, Ofek sets out in a spaceship on a journey at a speed of 60% of the speed of light toward a star located 6 light-years from Earth. Sela remains on Earth.
First question: how long does the journey to the star take from the point of view of each of the twins?
• From Sela’s point of view, Ofek’s spaceship has to cover a distance of 6 light-years at a speed of 60% of the speed of light, so Ofek will reach the star after 10 years. But Sela sees the moment Ofek reaches the star on the screen only after 16 years from the start of the journey! Why? Remember that from Sela’s point of view it took Ofek 10 years to reach the star, but the broadcast of the moment of arrival took 6 years to return to Earth.
• From Ofek’s point of view, the journey to the star took only 8 years. Why? As I explained at the beginning of the article, one who moves at a speed close to the speed of light sees distances contract along his line of motion. Therefore from Ofek’s point of view, the distance to the star is shorter by 20%, and is not 6 light-years but only 4.8 light-years. If Ofek moves at 60% of the speed of light, then 8 years are enough for him to cover a distance of 4.8 light-years.
Second question: what does each twin see when he looks at his own clock and at his brother’s clock?
• From Sela’s point of view, when his wall clock shows 16 years, that is the moment when he sees on the screen Ofek reaching the star and Ofek’s wall clock showing 8 years. Therefore Sela concludes that Ofek is younger.
• From Ofek’s point of view, when his wall clock shows 8 years, that is the moment when he reached the star. But when at that same moment Ofek looks at the screen in the spaceship, he sees that Sela’s clock shows only 4 years! Why? Do not forget that it takes the radio waves of the transmission 6 years to travel the distance from Earth to the star! Therefore even before Ofek reached the star, a transmission had already left Earth showing Sela’s clock 4 years after the start of the journey, and that transmission “chases” Ofek and reaches the star together with Ofek! Therefore Ofek concludes that Sela is younger.
Here is a summary table of each twin’s clock displays after the first half of the journey:
What the private clock shows What the clock on the screen shows
Sela 16 8
Ofek 8 4
Let us pause for a moment at this point. Notice how the fundamental principle of relativity appears here in all its glory: each of the brothers thinks that the other brother is younger by a factor of 2! The reason is simple: the symmetry has not yet been broken… Ofek has not yet turned back toward Earth. If Ofek had not stopped at the star but continued traveling, each of the brothers would think that the other was younger, and this would continue as long as Ofek kept moving away from Earth. As for the question of who is younger, the twins would have to agree to disagree. This is somewhat strange, but according to special relativity this is reality.
Third question: what is the state of the clocks at the end of the journey, when Ofek returns and meets Sela?
The moment Ofek reached the star, he turns around and returns to Earth at exactly the same speed. Let us proceed, then, and think about what happens in the second half of the journey from the point of view of each twin, and what the clocks show at the time of the meeting between the twins.
• From Ofek’s point of view, the return journey to Earth is no different from the outbound journey to the star. The distance contracts by 20% and comes to 4.8 light-years, exactly as before, and it takes Ofek 8 years to make the return journey. Therefore when Ofek arrives at Earth, his wall clock shows 16 years.
• From Sela’s point of view, Ofek has covered a total round-trip distance of 12 light-years at a speed of 60% of the speed of light, so Ofek will return after 20 years from the start of the journey. When Ofek returns to Earth, Sela’s wall clock shows 20 years.
Here is a summary table of each twin’s clock displays at the end of the journey when they meet:
What the private clock shows What the clock on the screen shows
Sela 20 16
Ofek 16 20
And there we have it. At the moment the brothers meet, Sela has aged by 20 years, but Ofek has aged only 16 years. They have nothing to argue about, because that is exactly what their wall clocks show.
Did you get it?
Summary
If you have made it this far, the whole thing may seem like one big mental failure. What connection is there between the ticking of clocks and aging?
Einstein’s answer is that clocks measure time. If they are not broken and are not faking due to an internal malfunction, then all they do is track the flow of time by means of some cyclical process. But one must remember that the clock itself is a system based on physical processes. A clock is not some magical creature that came from another world; a clock is composed of atoms and molecules, just like our bodies.
If the clock ticks more slowly, there is no reason why our “bodily” clock should not tick more slowly as well. In other words: all the physical, chemical, and biological processes in your body will also obey the change in the flow of time, and their rate will slow down (of course, in comparison with the rate of the processes in the body of someone who is not in motion).7
In any case, the twin paradox proves that at least time travel to the future is definitely possible:
Ofek traveled 4 years into Earth’s future!
1. Throughout the present article I will present things only within the framework of special relativity. Therefore from here on, wherever I write “the theory of relativity,” the intention is only the special theory of relativity. [↩]
2. The speed of light is 300 thousand km per second, and therefore 60% of the speed of light is 180 thousand km per second. If you ran a marathon at such a high speed, you would finish the marathon within 140 microseconds. This is a very short time, but it does not really matter, because one can imagine a world in which the value of the speed of light is 15 km/h, and in such a world running at 60% of the speed of light would mean running at 9 km/h, which is a typical running speed for an average person. In such a world, all the strange phenomena of relativity would manifest themselves even during ordinary running. In any case, in everyday life we do not come close at all to the speed of light. Even the speed of the International Space Station, which is 7.66 km per second, is only 0.0025% of the speed of light. [↩]
3. It should be emphasized that someone standing on the sidewalk and not running would see exactly the same phenomenon. In other words: if you hold a watch in your hands while running, then someone standing on the sidewalk will actually see your watch as ticking more slowly compared to his own wristwatch. [↩]
4. All the strange phenomena of the relativity of space and time can be derived directly from Einstein’s famous determination that the speed of light is finite and constant for every observer in every frame of reference. [↩]
5. It should be noted that the above explanation—that acceleration breaks the symmetry—is valid for special relativity (flat spacetime) and only for the scenario presented in the present article. However, it is important to emphasize that using acceleration as an explanation for the paradox is not valid in general, that is, it is not valid for every possible twin-paradox scenario. The twin paradox can also be described in another way; for example, both twins set out on a journey from Earth, both accelerate in opposite directions, and both return to Earth, except that one twin returns long before the other. In this case both twins experience acceleration, so acceleration as such is not the root reason for the solution of the paradox. In addition, the twin paradox can be represented not only within the framework of special relativity (flat spacetime) but also by scenarios within the framework of general relativity (curved spacetime). Therefore in the general case it is not acceleration that determines which of the brothers will be younger, but the length of each twin’s worldline within the spacetime through which he moves. All these concepts are more complex and will be explained in a separate post. [↩]
6. An excellent article on the subject with a detailed calculation can be found at the link here. [↩]
7. Again one must remember: this is a relative phenomenon. When two bodies move relative to one another, each one thinks he is at rest, and so each one sees the other as slowing his rate. [↩]
Understanding the Twin Paradox (Much Better). \\
May 29, 2023Shai Yefet
In the previous post we learned about the twin paradox and its solution. However, the solution presented there is relevant to a very specific and simple scenario of the paradox. The real solution is more complicated, and to understand it we will need to learn additional principles in relativity, principles that revolutionized the world of our physical concepts.
In the previous post we learned about the famous twin paradox, which occurs within the framework of Einstein’s theory of relativity. The classic description—and the simplest one—of the paradox includes two twin brothers, Sela and Ofek, where Sela remains on Earth while his twin brother Ofek sets out on a journey through space at a speed close to the speed of light. At the end of the journey, Ofek returns to Earth and meets his twin brother Sela. To the twins’ surprise, Ofek discovers that he is younger than his brother Sela.
As we saw, the explanation for Sela’s faster aging and Ofek’s slower aging does not lie in the fact that Ofek traveled through space at a very high speed, for each of the brothers can claim that he is the one at rest and that in fact it is his twin brother who is in motion; as we recall, speed is a relative concept, and one cannot determine which of the brothers is “really” in motion. Seemingly, we get an absurd situation: according to relativity each of the twins ought to be younger than his brother at the moment of meeting.
The reason why דווקא Ofek aged more slowly than Sela lies in the fact that only Ofek was under the influence of acceleration; unlike Sela, who remained on Earth, Ofek had at some point to slow down, stop, turn around, and accelerate again in order to return to Earth. It follows from this that acceleration is the factor that breaks the symmetry between the twins, and that is why Ofek is younger than Sela.
The purpose of the present post is essentially to tell you this:
Forget this whole story about acceleration.
Wait, wait… so everything I explained in the previous post can be thrown in the trash? If acceleration is not really the reason why דווקא Ofek is younger, then what is?
Well, the answer is this: it is true that in the classic scenario of the paradox we find that acceleration is the root cause of the solution, but that is only in the specific case in which Sela remains on Earth and Ofek goes on the journey.
But if you think about it deeply, quite quickly you can reach on your own the conclusion that acceleration cannot be the root cause of why דווקא Ofek is younger than Sela, and not the other way around. The reason is simple: what exactly prevents Sela from being under the influence of acceleration just like his twin brother Ofek? In simple words: why is only Ofek given a spaceship and sent on a journey? Why not give a spaceship to Sela as well, eh? Eh?
Consider, for example, the following scenario:
• Each of the twins enters his own spaceship.
• Both Sela and Ofek set out on journeys through space in opposite directions, each twin toward a different star.
• Somewhere in mid-journey Sela regrets it, and decides to stop and return to Earth, before reaching his destination star.
• Ofek, by contrast, completes the journey to his star, and only then turns around and returns to Earth.
In this scenario too, the mathematics of relativity predicts that at the moment the twins meet on Earth, they will not be the same age. Except that now both of them were under the influence of acceleration! If in the classic scenario we pinned the blame on the acceleration that acted only on Ofek and only that caused him to age more slowly, what shall we do now?1
So if you want to understand what really causes the age difference between the twins, and what is the root reason that determines which of the twins will be younger, this post is for you. As you will see below, the answer will lead you to more bizarre and more interesting regions in our conception of reality.
God’s living room
In order to understand what really determines the age difference between the twins—in any scenario you like—we must understand a basic concept in relativity, and a revolutionary one in the history of physics:
Spacetime
Until Einstein’s time, physicists perceived reality in the same way that each and every one of us perceives it. In accordance with our everyday experience, all the bodies in the universe—from a tiny grain of sand to the great sun—move within three-dimensional space, while time flows in the background at a constant rate without interruption. This is the routine conception of all of us, isn’t it? It seems obvious that we may all move within a space that has forward-backward, up-down, right-left. This space is enormous—perhaps even infinite—and contains within it all the planets, stars, galaxies, and black holes in the universe, everything without exception. In addition to this vast space, it seems that there exists some cosmic clock ticking away in the background, tick-tock, tick-tock, and thus time passes within that vast space in which everything exists, in accordance with the hands of that invisible cosmic clock.
This is in fact the Newtonian conception of reality, and it can be presented as follows:
All of space is like an aquarium in God’s living room, and time is measured by a large wall clock hanging above the aquarium.
In addition, it is obvious that everything inside this aquarium—the planets, suns, and galaxies—does not affect the aquarium itself, and certainly not the clock. The aquarium exists on its own, and the clock ticks of itself.
And then Einstein came and turned this whole conception upside down.
Reality as Plasticine
According to Einstein’s relativistic conception, the aquarium and the clock became one entity. All of reality is a kind of four-dimensional block in which time has become another axis alongside the three axes of space. There is no longer space and time; there is: spacetime.
If that were not enough, then according to Einstein spacetime is not a “rigid” block; it is more like plasticine. In other words, spacetime can stretch and warp. Who is responsible for that? Who bends and shapes the fabric of this spacetime? Well, all of us. Me, you, the stars, and the galaxies. In short, everything that fills spacetime affects its shape.
According to relativity, matter in the universe, such as the sun (represented by the white sphere at the center of the system) bends the space and time around it.
Source: Lucas Vieira Barbosa authored the original OGV. Selected frames (10% of the original) extracted and re-timed by Stigmatella aurantiaca, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons
Einstein showed that in this plasticine spacetime, it is entirely possible that events which from your point of view have already happened in the past still lie in my future, and vice versa. There is no longer an absolute past, present, and future that are valid everywhere. Therefore that little hyphen between the two words space and time has very great significance: spacetime is one entity, a four-dimensional block in which the past and the future are “real” exactly like right\left, forward\backward, and up\down. To put it in the words of one of the greatest physicists (and mathematicians) of the 20th century:
In spacetime, things simply are; they do not happen.2
Hermann Weyl (1885-1955)
There is distance, and there is distance
All this is very nice and terribly cool, but what does a four-dimensional spacetime have to do with the twin paradox? How exactly does this change in perspective help us understand the real reason for the age difference between the brothers?
Well, the upheaval I described above in our conception of reality gave rise to another conceptual change connected to the concept of:
Distance
Here too, we all understand intuitively what is meant, right? What is the problem: take a string, stretch it between two points in space, and the length of the string is the distance from here to there—nothing could be simpler. But the everyday intuition all of us have regarding the concept of distance is an intuition relevant to reality according to the Newtonian conception. According to Einstein’s relativistic conception, however, reality is a four-dimensional spacetime, and the concept of distance takes on an entirely different meaning:
• According to the Newtonian conception, distance is measured between two points in space.
• According to the relativistic conception, distance is measured between two events in spacetime.
This sounds a little strange, I know, but it is important to emphasize that distance in Einsteinian spacetime is not the same distance as in Newton’s “ordinary” space! Even mathematically, each distance has a different method of calculation! If you have the nerves and patience, then at the link here I explain more about the mathematics of the whole business, but the bottom line is this: distance in relativistic spacetime is defined and calculated differently than distance in “ordinary” Newtonian space.3 The simplest example one can think of in order to internalize the fact that these are indeed two different kinds of distance is when you are sitting in place and not moving at all:
• According to the Newtonian conception, as long as you sit in place you cover a distance equal to zero. This is simple and clear, since you are not moving.
• According to the relativistic conception, even if you are sitting in place and not moving a millimeter, you are still moving through spacetime, and therefore in spacetime you cover a distance that is not zero; it is simply a different kind of distance.
A page from a presentation by Hermann Minkowski (1864-1909) in which one can see a diagram describing the idea of spacetime as well as a description of the trajectories of bodies in spacetime. Minkowski was a professor of mathematics at the Zurich Polytechnic when Einstein studied there. Minkowski did not publish his discoveries immediately, but waited in order to develop the idea more comprehensively; therefore Einstein preceded him in publication. Minkowski never claimed “priority” but gave Einstein full credit.
Source: Hermann Minkowski, died 1909, Public domain, via Wikimedia Commons
Figure eights in space
Okay, so how exactly is this insight relevant to the twin paradox?
Well, every scenario of the twin paradox you can think of will always be bounded by two events in spacetime:
1. The first event is when the twins synchronize their clocks and part from one another.
2. The second event is when the twins meet again and compare their clocks (of course there is no necessity that they meet in the same place where they parted).
Between these two events, each twin can do whatever he likes: stay in place, or go on a journey in whatever direction he wants, for however long he wants, and at whatever speed he feels like. Each twin can slow down, accelerate, stop, turn around, and even do figure eights in space. In general one can say that each twin can move along whatever winding path he wants; it makes no difference.
But! At the moment when the twins meet again and compare clocks, there is only one thing that will determine the age difference between them: the distance each twin traveled in spacetime. In simple words:
The twin who traveled a longer distance in spacetime will be the older brother. Always.
Did you get it? Distance in spacetime is the real reason for the age difference between the twins, and it is what determines which twin will be younger and which older. Clearly the spacetime distance traveled by each twin is affected by factors such as acceleration, speed, and duration of the journey, but none of the above factors determines on its own which twin will be younger.4
Summary
As we saw above, the simplest scenario of the twin paradox—the one they will present to you everywhere—is the scenario in which one twin remains at rest on Earth and the second twin goes on a journey through space and returns to Earth. In this simple scenario we find that the twin who went on the journey is the younger one, and it seems (apparently) that the reason is because only he was under the influence of acceleration.
Now you know that this is only incidental… acceleration is not the exclusive factor that decides who will be younger; what determines it is solely the distance each twin traveled in spacetime. And again, remember that this is not “ordinary” distance in space as we perceive it in daily life, but an entirely different concept; for example, it can be shown that there are scenarios in which even if one twin does not move a millimeter from his place, sometimes he will be younger and sometimes older.
In order not to overload the post too much, I elaborate on those cases at the link here, from which one can see clearly that acceleration does not determine the matter: in one scenario we discover that the accelerating twin remains younger than his brother, but in another scenario the accelerating twin actually becomes older than his brother.
1. One can take the scenario to an even more extreme case; for example, if we assume that Sela regrets his journey more than once. In other words: on the way to the star Sela can turn back toward Earth, then regret that and turn again toward the star, and afterward turn yet again and return to Earth. It follows that in total Sela was under the influence of acceleration for a longer time than Ofek, and still we will find that at the moment of meeting, Ofek is the younger one. [↩]
2. This is a free, non-literal translation of the original quote: “The objective world is, it does not happen” [↩]
3. This is despite the fact that the two kinds of distance have the same units. In addition, it should be emphasized that the relativistic distance mentioned here is called: Proper distance, and should not be confused with: Proper length, or: Coordinate distance. [↩]
4. Surprisingly, it can be proved that in the classic twin-paradox scenario, it is precisely the twin who remained on Earth who traveled a longer distance in spacetime. [↩]
Rani:
Regarding your revised explanation,
I still don’t understand how it is well-defined.
If the circumstances are determined only by the two variables in question (t, T), then how can I return to the year 1985? The circumstances in 1985 are not defined at all; to define them one needs the two variables, and if we use the two variables for the return, then we are back at the starting point.
In my opinion there are many other interesting points to raise, but this is a point that needs to be clarified first.
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Rabbi:
Indeed, one does not return to the same circumstances. That is the whole idea of how to formalize time travel when the circumstances at that time are different from what originally prevailed there. Without duplicating time axes this is impossible (at least within a framework of functional dependence), and therefore I needed another time axis.
When there are two time axes, the circumstances at the point (1966, -30) are as they originally were, whereas the circumstances at the point (1966, 20) are as they were then except that I am present there (with my memories) and my grandfather is dead. Time travel according to my proposal is return along the t axis, but not to the same circumstances (because tau is different). And that is indeed what people are talking about in these contexts. Clearly one cannot go backward on both time axes, otherwise what have we gained?
By the way, in the original article he really does use two time axes, but it seems like a computational necessity and not the result of a conceptual philosophical analysis.
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Rani:
I still don’t understand how this reflects time travel.
Obviously this is a definition, but it has to reflect time travel.
If we take the pair (1985,22), which comes after (21, 2002), what does 1985 reflect? Why does it reflect a return? After all, 1985 by itself reflects no circumstances at all; it is just a number. If I have no circumstances to return/move to, then where do I arrive?
If these are new circumstances, then this reflects no time travel at all.
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Rabbi:
Rani, you’re getting carried away.
Clearly 1985 reflects something. But not all the circumstances, only part of them (in fact almost all of them, except that I am there and my grandfather is dead). This is time travel only with respect to the t component, and it happens without a full return of the circumstances (only approximately). The claim of the model I proposed is that our experience of time expresses only one component of the full physical time, and only in that way can we speak of time travel.
If you define time travel as necessarily also being a return of the circumstances, then of course that is impossible (except in the trivial sense of being there when you are there). That was the problem I tried to deal with here, and if you refuse to give up anything then of course you will remain without a solution. That is precisely my claim.
Therefore I proposed a close definition that gives a consistent meaning to the concept of time travel, where the concept of “time” reflects what we experience but is partial (one component) compared to time in its full physical sense (which includes both components).
By the way, the claim that there are several time axes and that what we experience is not necessarily physical time already appears in many places (for example, basic physics treats time as directionless, but in our consciousness/experience it has a direction). If you want an accessible and pleasant presentation, see Avshalom Elitzur’s little book, Time and Consciousness, published by The Open University’s broadcast university series.