A PHILOSOPHY OF THE FOUR-DIMENSIONAL SPACE-TIME
The Worldview of Relative Simultaneity (MURAYAMA Akira)
CHAPTER III The Problems of Time Representation
4. Time Travel
Now, let me examine the subject of time travel, which is quite a common topic in science fiction. In this context I will logically examine time travel from the perspective of time theories.
Dividing time travel into two categories is necessary: time travel to the future and time travel to the past. Time travel to the future does not involve any problem in principle. The theory of relativity shows that in almost flat space-time, a person who travels far away and returns to where he or she started experiences a shorter passage of time (proper time) than another person who does not move in such a way. This is the so-called twin paradox. When a twin older brother returns to the earth after having traveled in space, he ages more slowly than the younger brother staying at the earth and feels that he has traveled to the future. Movement is relative. Based on this assumption, if you consider that the twins are distant from each other for a while and meet again, one possible interpretation is that the younger brother on the earth has made a round trip to the older brother aboard the spaceship, which is symmetrical condition for both brothers. The reason for the asymmetrical phenomenon (paradox) in which the older brother feels himself aging more slowly by traveling to the nearby star is that he returns to the earth after having experienced an accelerated movement at a distant spatial point in almost flat space-time. It is conceivable that during this period of decelerated and accelerated movements, the older brother’s line of simultaneity rapidly changed its slope and the younger brother grew older in a flash. There are many other explanations, but it has been theoretically demonstrated that this kind of phenomenon can occur. Even if someone travels to the future at his or her own risk, there will be no problem concerning causal relationships.
Now, I will go into a more detailed explanation of the twin paradox. This paradox is based on the assumption that the brothers on both sides of earth and spaceship appear to go far away and return. The two situations are seemingly equal, but it is paradoxical that more time has passed on the earth, not vice versa.
With regard to this point, some people have given ambiguous explanations of this paradox and you should be careful not to be confused. For example, some argue that one brother experiences an accelerated movement but the other does not. Others claim that one brother experiences an inertial force as apparent force but the other does not. Others explain that the entire universe appears to be moving for one brother but it does not move for the other. They conclude that the two brothers are not on an equal footing and no contradictions are involved. These descriptions just mention the inequality between the two brothers but do not provide the right answer to the question of the difference in time passage. In addition, short-distance acceleration has only a small effect and returning to the earth by using gravitational fields may not involve any inertial force. Therefore, the explanations leave much room for misunderstanding. The twin paradox cannot be understood without specifically examining the situation according to the relativity of simultaneity and the measurement structure of space-time. There are many conceivable ways of approaching this situation. The best method in general versatility is to integrate proper time on the basis of the geometrical measurement of space-time. In this context, I will introduce explanations based on the relativity of simultaneity and the Doppler effect.
Let me explain by focusing on two specific individuals. For example, if the two people move far away from each other at a speed of 60 percent of light, both feel that their unit time stretches by 25 percent. That is, this means both consider that when one person’s clock gains ten years, the other person’s clock gains only eight years. Direct observations of this phenomenon show that one receives the light signal indicating passage of ten years from the other 20 years afterward and vice versa. Each person feels that the light signal must reach much earlier if the opponent's clock has not delayed. (See Figure 3-4-1.) The two people are on an equal footing with regard to this point.
Now, suppose that the spaceship suddenly turns around 30 light-years away in the earth system and returns in the reverse direction at the same speed. In the earth system of coordinates, the spaceship turns its direction 50 years afterward and the time passage of the ship is late, with only 40 years passing for one way. In the meantime, from the standpoint of the ship, while it is turning around (that is, slowing down and accelerating in the reverse direction), the line representing the simultaneity of the spaceship system drastically changes its slope. During that period of time, 36 years have elapsed from the 32nd year to the 68th year in the earth (the other person) system of coordinates. During the subsequent 40 years in the ship’s system of coordinates, only 32 years pass in the earth system of coordinates and they meet again. In the ship’s system of coordinates, while it is in uniform motion, the passage of time in the earth system of coordinates also slows, but while the ship is increasing speed, a dramatic increase occurs in the passage of time in the earth system of coordinates. Consequently when the ship returns to the earth, the person is 20 years younger than the other on the earth. However, it is important to note that while the ship is turning around, activities on the earth do not appear to move faster as would a video tape being fast-forwarded. The following explains how the activities appear. (See Figure 3-4-2.)
In the spaceship’s system of coordinates, the frequency of its reception of earth time signals differs between before and after it turns around. At the point in time at which the ship turns around in the 40th year, only 20 years have passed on the earth, but after the ship has turned around, 80 years appear to have passed in the earth system of coordinates during the 40 years on its return trip. That is why when the ship reaches the earth, the person on the spaceship directly knows the fact that 100 years have passed on the earth during its 80-year round trip. Meanwhile, from the earth’s standpoint the situation appears as follows: The earth receives sparsely the time signals which were periodically sent from the spaceship, and at a temporal point at which 80 years have passed, the earth receives the message that 40 years have passed in the ship’s system of coordinates and it will turn around now. Subsequently, the earth receives a sudden increasing frequency of signals and during a short period of 20 years in the earth system of coordinates, it receives the rest of the ship’s 40-year reports. As a result, when the spaceship returns to the earth, 100 years have passed on the earth and the person on the earth directly knows the fact that the 80th year has just begun in the ship’s system of coordinates. When the person on ship and the other on the earth meet again, they have quite the same facts, but the time passage they observe, that is, how they appear in each other’s system of coordinates, clearly differs. In the ship’s system of coordinates, the earth time appears to progress slowly until just before it turns around, and just after that, the earth time appears to suddenly become faster. Conversely, in the earth system of coordinates, the ship’s time continuously appears to progress slowly and quite a while after it receives a message when the ship turns around, the ship’s time appears to suddenly become quicker. This difference in signal frequency between leaving and approaching is the Doppler effect. This physical law states that when a source of waves (in this case the signal radiator of the ship or the earth) is moving toward an observer, the frequency of the waves increases, whereas when the source is moving away from the observer, the frequency decreases. The phenomenon is well known, as for example, the difference in the pitch of a siren before and after it passes by you. The spaceship perceives the Doppler effect before and after it turns around. In the meantime, the earth perceives no change when the entire universe and itself turn around in relation to the spaceship. At a certain point of time at which the earth receives the message of turning around from the ship a while after the ship has turned around to the earth, it perceives the Doppler effect. This critical difference between the two coordinate systems on the basis of direct observational phenomena makes it clear whether it is the earth or the spaceship that actually turns around in relation to space-time with almost the same gravitational field (In this context, my argument is based on the assumption that the ship’s turning around can be quickly accomplished to facilitate more simple calculations. In fact, directional changes usually take some amount of time and one must take a line integral of proper time in the spaceship’s system of coordinates for the sake of performing strict calculations.)
In addition, it is very difficult to achieve this effect with current technology. For example, consider the case in which the spaceship continues to move away for 25 years at a speed of approximately 640 kilometers per second (16 times as fast as the planetary explorer Voyager) and returns to the earth from about 53-milli-light-years away for 25 years. The astronauts grow older more slowly than people who spent 50 years on the earth, but that gap is at most one day. However, if the spaceship travels at 99.9 percent of the speed of light and moves away from and toward the earth for the respective 25 years, the turnaround point will be about 500 light-years away from the earth. When the ship returns, approximately 1,000 years will have passed in the earth system of coordinates. In this scenario, time passes very slowly on the earth while the ship is in uniform motion, but a long period of time, close to 1,000 years on the earth, passes in a great stride at the turning point.
In this way, if you want a noticeable experience of the Doppler effect, you have to travel at an exceedingly high speed to a distant place in the universe. Theoretically, it is definite that you can get to the future of the earth in a relatively short period of time, but you cannot get back to what the earth used to be.
A similar effect may be obtained if someone is put in a totally frozen state of suspended animation for a long period and is reanimated, if possible. During that frozen period, the person will have no sense of time passage, but this case is a matter of consciousness. Even if the body remains young, the person has the history of living on the earth for a long time at least at an atomic level and just lives longer, staying young while sleeping. Meanwhile, in the case of space travel, delays in the passage of time, that is, time itself, occur for all objects, including human consciousness and considering these two different cases cannot be considered in the same way as a trip to the future.
Worth reconsidering is if the space travel simulation can be truly described as a trip to the future. In the scenario, the astronauts make their time progress slower than on the earth. Objects aboard the ship, including their consciousness and body, progress more slowly than the equivalent on the earth, but this is because they quickly scanned earth time. Their proper time passes by just as it goes. The astronauts did not travel to their own future.
If the astronauts return to the earth, they may see others or their descendants who have grown old earlier than they have, but the astronauts will never see themselves of the future. If they see themselves of the future on the earth at all, they face the question, “Who in the world am I?” For their being on the earth, they recognize that their other self traveled in the universe and returned younger. Still, is the other self a complete stranger? This is impossible and utter nonsense to contemplate seeing the self of the future.
Then, if you try to travel to your own future, what kind of situation can be realized? Suppose that you are a high school student fresh from junior high school. If you travel to your post graduation future by taking three years, no one will think that you are in the future. If it can be considered as travel to the future, everybody is traveling to the future. If you can become yourself after graduation in a moment soon after enrollment, people will recognize that you have traveled to the future. Next, suppose that you become yourself after graduation in a moment. If you have no memory of your three-year high school life and just have memory of your life up until your junior high school, is it possible to say that you have become your future self? At the very least, those around you will think that you have amnesia. Then, what if you can instantly put yourself in a condition in which you have all memories of your three-year high school life? It is just your future self. In this scenario, is it possible for you to think that you have instantly made a trip from three years ago? At least, you can wallow in the nostalgic mood that these three years have passed very quickly. (Now, what kind of time is the time referred to as “instant” in this context?)
Now, let me consider travel to the past in this section.
We sometimes wish we could get back to a certain point of time in the past to start all over again. For example, I hear that many people think this after having bought a betting ticket for a horse race and losing the race. If you knew how you could even slightly exceed the speed of light, you could get back to the point in time just before you bought the ticket.
Some people may wonder that even if you could travel at a little faster than the speed of light, you could just go farther more quickly and why you could get back to the past by that. I will explain in the following section.
For Figure 3-4-3, the starting point O represents the spatiotemporal point of the present me after the race (the S system of coordinates). The point P represents me at a certain point of time before I buy the race ticket. Suppose that someone travels to the spatiotemporal point Q faster than the speed of light from the starting point O. (In this simulation, the travelling subject does not always need to be a person. It may be a microchip involving information about a winning horse or a kind of wave motion.) From my standpoint, the travelling subject just moves farther more quickly than the speed of light. For someone in another system of coordinates (S' system) that is moving at a pace close to the speed of light, the line of simultaneity differs and the travel from the spatiotemporal point O to point Q is judged as travel to the past. This gives me no advantage at all. However, this travel to the past is possible from the perspective of someone in the S' system of coordinates and hints about the possibility of travelling to the past in the opposite spatial direction. Then, suppose that it is possible to travel from point Q to point P carrying information about a winning horse. In this coordinate system, the subject moves in the opposite spatial direction at the same speed directed to the past. In addition, there is another coordinate system in which travel from point Q to point P can be judged as an ultra-light-velocity travel directed to the future. The utilization of this coordinate system makes it possible to transfer information from point O to point P without using a vector directed to the past. In this way, the information about a winning horse can reach the spatiotemporal point P before the purchase of a race ticket, which will definitely enable what I have now to switch from a losing ticket to a winning ticket (if I in the past made the right decision.) This is quite a happy story. Even if you do not directly travel to the past, you can open up the path to the past as long as you can exceed the speed of light.
Then, how can you exceed the speed of light? It is at least impossible in principle to exceed light velocity by just accelerating. The velocity-addition formula in the theory of relativity (refer to (4) in Section 3 of Chapter 1) shows that any acceleration possible does not enable you to exceed light velocity. An object ejected at the speed of light from a vehicle moving at the speed of light is travelling at the speed of light, not double the speed of light. In addition, the theory of relativity suggests that kinetic energy represents an increase in the mass of a substance and that the closer it gets to light velocity, the larger it becomes endlessly. Mass means how difficult an object can accelerate and the closer an object gets to the speed of light, the more difficult it becomes for the object to accelerate due to an increase in its weight. The object diverges from an infinite mass at the speed of light and it is virtually impossible for the object to exceed the speed of light by accelerating. In physics, a hypothetical ultra-light-velocity object called “tachyon” is sometimes considered, but it is conversely considered to be unable to move below the speed of light. Therefore, it is unthinkable to elaborate a way to use it as a vehicle to carry information. No practical ultra-light-velocity means of information transference has been discovered thus far.
Some physicists are persistent in seeking the theoretical possibility of travelling to the past. They depend on the general theory of relativity for their scientific pursuit. The theory indicates that space-time is curved where mass and energy are congested. The ultimate example is the black hole. The physicists study the theoretical possibility of subjects to instantly travel to a different universe or a different place in the universe, which could even include a place that subjects cannot reach below the speed of light in their whole lives and the universe in the past, instead of being just crushed and trapped if the black hole is under good conditions, such as its high-speed revolution. This is a purely mathematical study. If the black hole is not sufficiently large, your body will be torn apart and crushed as you get closer to the hole. Unless there exists such an ultra-large black hole in which at least more than tens of thousands of the sun are compressed, you cannot easily approach it. How you can get through the world alive where even atoms maintain their forms is totally unimaginable.
This tunnel that instantly connects separate spatiotemporal points is called the “wormhole.” This wormhole is considered to be exceedingly small and lasts for only a short time even if it exists, and it is impractical unless you can manage to continuously open it for a long time. In response to this situation, a way to open the wormhole by generating negative gravitation from negative energy is being devised. However, this tunnel is just confirmed as having a very short period of lifetime in the form of quantum fluctuations and technological challenges concerning the practical wormhole are far beyond our imagination.
In one of his books, How to Build a Time Machine (*4), Paul Davies explains the following plan about his vision for building a time machine.
To briefly summarize, the first step is to create the bubbles of quark-gluon plasma of ten trillion degrees centigrade by crashing heavy atomic nuclei by the heavy ion accelerator. The second step is to compress them into an ultra-minute sphere with high density. Although its technological critical factor is yet to be specified, you make a minute black hole or wormhole by turning the sphere into an extremely high density that reaches even one trillion-fold of one trillion-fold of one trillion-fold of one trillion-fold of one trillion-fold of one trillion-fold of one trillion-fold of one trillion-fold of ten kilograms per cubic meter. The third step is to stretch the small wormhole by negative energy. The fourth step is to make the time passage of one side of the wormhole’s openings slower by using the accelerator or strong gravitational fields near a neutron star and to create time gaps between the two openings of the wormhole. This is a kind of ultra-giant structure to deal with vast space or space-time itself, rather than a machine like a vehicle.
This is an unbelievable story that you cannot follow even if you give a sigh trillions of times, but some physicists seriously try to address this subject in pursuit of theoretical possibility. (As a matter of fact, their real purpose seems to study the theoretical challenges of modern physics, such as the quantum gravitational theory, through these thought experiments. The time travel theory using a wormhole originated from Kip Stephen Thorne’s announcement in 1988. In addition, there are several types of theoretical examination of past travel based on hypotheses, such as John Richard Gott III’s cosmic string.) A wise way of thinking seems to be giving up changing a losing race ticket into a winning ticket by getting back to the past and bet on the next race. From the perspective of space-time determinism, it has been pre-determined whether to buy a ticket for the next race or not, which ticket to buy and how the race will end, but it is guaranteed that you cannot see how the race will end until it ends and that you can enjoy having expectations and dreams until it ends. Of course, you have the option to stop betting on the race and consider your family life.
I cannot specifically judge whether or not these theories on time travel are valid and proper. But even if something or someone that can return to the past exists in theory, I am not convinced of the idea that such existence can change its mind on its own will even by returning to the past. As many people point out, this simulation involves serious problems concerning causality. One of their examples is a simulation that what will happen if a child returns to a certain point of time in the past and kills his future mother or father. In this way, contradictory problems arising in causality are serious, but fundamentally, I think that getting back to the past means that you get back to the past together with your consciousness and that changing your own will in the past is self-contradiction and nonsense. It is consistent if you travel to the past together with your awareness. That is, you get back to a state of mind that you will travel to the past now for the first time in your life. In this scenario, even Friedrich Nietzsche (1844–1900) would be astounded by perfect Eternal Recurrence, in which everything is repeated in the completely same way, and it is completely consistent.
If you travel back to the past with the memory that you came from the future, strictly speaking, we cannot say that you have returned to the past. This is because the past is not an area in which you yourself with such memory exist. In addition, if you have any different will or act differently from the past, this can never be considered as the past. Some people may think that it will be no problem if you behave carefully so that you will not cause any serious causal contradictions. In fact, science fiction stories depict the spatiotemporal police keeping things under strict control. However, this is quite human-centered thinking. The mere slight movement of a small stone can affect the fate of many microorganisms. Egocentric thinking is to focus on keeping your own destiny safe. The world is a complex system. A small change in the fluttering of a butterfly’s wings can cause subtle ripples and can lead to an enormous hurricane somewhere else and large death tolls.
Essentially, if you acknowledge that the past can be somewhat changeable, doing so means that you can travel to another space-time and another universe. In this case, you would choose the many-worlds interpretation as a world model. This book does not totally deny the theoretical possibility of the many-worlds interpretation but is based on the assumption of not taking that stance. Therefore, I will not argue about this subject any more.
As long as you consider things on the basis of the unitary world theory, a person who has a memory that he has come from the future is judged as suddenly appearing in this world and disappearing as such as the person having such memory. That person must never have experienced a state in which he would travel back to the past now for the first time in his life.
Let me return to the story about the horse race ticket. One day when a man goes out to buy a horse race ticket, another person who says that he is a messenger of the first person from the future appears and hands him a sheet of paper containing information about the winning horse in the race suggesting that he bet boldly on that horse. He does not believe the messenger’s dubious suggestion and buys a different ticket in accordance with his own expectation. In the end, he knows that if he had taken the messenger’s advice, he would have won in a big way. Then, he meets another man. This man can send a message to himself, the himself of the past. This man is the very same person who gave him the advice about the winning horse. He gives that person a message and asks the man to give it again to himself of the past. The messenger travels back to the past and meets himself of the past before buying a ticket. Subsequently, these same processes are repeated over and over again. Indeed, this involves no contradictions.
In addition, let’s think about another story. One day, a man declares that he has invented a medicine that enables you to travel back to the past. This special medicine is primarily potassium cyanide. He takes the medicine and travels back to the past. This means that the man returns to a certain point of time, at which he has no idea that he would travel back to the past. He is involved in many activities and begins to hope to get back to the past from a certain point of time. Then, the day comes when he invents a special medicine that enables him to travel back to the past. There is no proof that human consciousness that scans four-dimensional space-time is cyclic, but there is also no proof that denies that. Therefore, it is difficult to declare that such thing logically cannot be established. You may think that Sisyphus in Greek mythology, who endlessly repeats the work of carrying up a huge rock to the top of the hill only to have it roll down again, is luckier. However, it could be said that the man who invents a medicine enabling him to travel back to the past has a very exciting life. He has no memory of the future and there is no possibility that he is tired of repeating the same pattern. In addition, possibly, you do not need to take such a radical medicine to travel back to the past. When you wake up in the morning, you may find yourself being yourself from the future, although your memory goes only as far back as yesterday.
Furthermore, when you travel back to the past, you do not need to be back in your own consciousness. You can get back to another person’s state of mind until that moment and be that other person. In a strict sense, this is quite the same as a certain condition in the past and involves no theoretical contradictions. When you wake up in the morning, you may find yourself as another person from the future. But that person has lost all memories of himself and has placed himself in your body with your memory until yesterday. “What? Who am I then?”
(*4) Refer to Paul Davies (2002), How to build a time machine, chapter 3 How to build the time machine p.77-103
translated by Hajime Hayashi(2003), Shisosya, p.107-113