The Dark Arrow of Time

by Massimo Villata

  If we emit a photon from Earth, one second later it will be at distance of around 300,000 km. If at the moment of emission a spaceship passes at any given speed, half of the speed of light, for instance, the photon will also have traveled 300,000 km in one second from the astronaut’s standpoint, and will thus be 300,000 km ahead of the spaceship, if the latter was going in the same direction as the photon. This would mean that the photon is simultaneously located 300,000 km and approximately 450,000 km from Earth, given that in the meantime the spaceship has traveled around 150,000 km. Clearly something doesn’t add up here. In reality, the answer is quite simple, though it might seem strange. What changes is the passage of time, and the measure of the distances. A second measured on board the spaceship is ‘different’ from one measured on Earth. And the distance of 300,000 km measured from the Earth is no longer the same if it is measured from the spaceship. Yardsticks and clocks, space and time, change to adapt to the true universal constant, the speed of light.

  We usually don’t notice, because we deal with small velocities for which the relativistic contraction of distances and dilation of time are too small to be measured.

  And so time is not an invariant. It does not flow smoothly and undisturbed, uniform and identical for everyone. It depends on the state of motion of the observer who measures it. We can make a space voyage at enormous speed, and come home to find that our son is twenty years older than us. And dimensions in space are not invariant either: objects and distances contract in the direction of motion, and their measurements depend on the relative velocity at which we measure them.

  An elementary time interval is designated as dt, while dr is a piece of space, an element of distance. Neither are invariant under Lorentz transformations. Lorentz transformations make it possible to calculate the space-time coordinates of an event in a given inertial frame, starting from the coordinates for the same event in another inertial frame moving at a constant velocity v relative to the first. For small relative velocities, dr and dt are almost invariant, as they appear in our everyday experience. For relative velocities approaching the speed of light, their variation from one frame to the other can become arbitrarily large. A length will contract by a factor γ = (1 − β 2)−1/2 (where β = v/c), and time will slow by the same factor.

  The simplest relativistic invariant, i.e., invariant under Lorentz transformations, is the square of the space-time four-vector, ds 2 = dr 2 − c 2dt 2. In other words, whereas for Galilean physics (which still applies for small velocities), the invariants were dr 2 and dt 2, in relativistic physics, the invariant is a linear combination of the two, where the coefficient c (speed of light) is a universal constant. It should be noted that for a photon we have ds 2 = 0, since dr, or in other words the distance traveled, is equal to the product of speed, c, by the time dt taken to cover the distance. In this case, the space-time interval ds is said to be light-like. Likewise, it is called a time-like interval if ds 2 is less than zero, i.e., when the spatial distance dr is less than that which light can travel in time dt, meaning that the distance can be covered in time dt with a speed less than c, and consequently the two events at the end points of ds can be linked by a cause-effect relationship. By contrast, because no matter and nothing that transports energy or information can have a speed greater than c, the two events at the end points of a space-like interval, i.e., with ds 2 greater than zero, cannot be causally linked, and in other words take place independently of each other.

  We have just said that no matter, i.e., nothing with mass greater than zero, can exceed the speed of light. In reality, it cannot even reach the speed of light. Only an electromagnetic wave, or its quantum correspondent, the photon, which has null mass,4 can, or rather, ‘must’ travel at speed c.

  And so we have this apparently random coincidence: particles with m = 0 also have ds = 0. A close relative of ds is the so-called ‘proper time’, dτ, where dτ 2 = −ds 2/c 2 = dt 2 − dr 2/c 2. It is referred to as proper time because it is the temporal distance between two events that take place in the same point in space, i.e., with dr = 0; in this case, in fact, dτ = dt. In other words, it is the time measured with a clock at rest in the considered reference frame, hence ‘proper time’.

  A photon thus has m = 0 and dτ = 0 (since ds = 0). Normally, we say that a photon does not have a reference frame, as the second postulate of the special theory of relativity (Einstein 1905) states that light travels at speed c in any reference frame. Consequently, there can be no such thing as a reference frame in which the photon is at rest.

  The photon, whatever it is, is certainly something very strange and at the same time fundamental, that our description of physical phenomena can barely hint at. It is a sort of singularity of the physical world, a boundary that cannot be crossed, or at least apparently. It has null mass, as if it did not belong to our world, and null proper time, as if it were unaffected by the flow of time. For the photon, our space does not exist, because the closer speeds come to c, the more space contracts and tends to zero. So in a way it is as if it occupied the entire space in a single timeless event. No sooner has it left than it has already arrived, even if from our point of view it has traveled for billions of years and crossed half of the universe.

  Consequently, having no mass is the essential, necessary and sufficient condition for traveling at the speed of light and not being affected by the passage of time.

  And so how could Helias Kadler and the other travelers, with the whole spaceship and its far from negligible mass, “ride the light”?

  Footnotes

  11 parsec ≈ 3.26 light-years.

  2 c ≈ 299,800 km/s is the speed of light.

  3The distances contract by a factor γ = (1 − β 2)−1/2, where β = v/c. In the case considered here, β = 0.995, so we will have γ ≈ 10.01.

  4Or any other particles with null mass.

  © Springer International Publishing AG 2017

  Massimo VillataThe Dark Arrow of TimeScience and Fictionhttps://doi.org/10.1007/978-3-319-67486-5_2

  2. They Were Flying Over Gentle Crimson Hills

  Massimo Villata1

  (1)Osservatorio Astrofisico di Torino, INAF, Pino Torinese (TO), Italy

  Massimo Villata

  Email: villata@oato.inaf.it

  They were flying over gentle crimson hills dotted with orange shrubs casting long violet shadows in the brooding light that announced the coming sunset. The air was perfectly clear and a brisk wind from the southwest shook the treetops in the woods at the foot of the hills, the tiny leaves shimmering in a broad swath of vibrating color, now yellow, now purple, now maroon, as the wind turned their different surfaces toward the watching eye. The undersides of the leaves were at their brightest yellow when the wind brandished the branches against the sun.

  They were traveling north, almost along the planet’s shadow line. At dusk, they had to increase altitude to clear a mountain range. And then the sun was back, tingeing with pink the immaculate snowfields and the plumes and pennants of clouds floating from the highest peaks. And where the sun failed to reach, the snow was blue and violet in the shadows, much like on Earth.

  For a long time, Helias stood rapt before these breath-taking landscapes, forgetting all the problems and questions crowding his mind.

  Only when darkness finally covered the land below and the first, brightest stars began to appear in the purple and green sky, did he turn from the porthole. Removing his glasses, he looked around the spaceship, his eyes growing accustomed to the dim white light. It was an eight-seater. Two rows of two seats per side, with an aisle down the middle. At the back, the baggage hold, at the front the cockpit with the two pilots. The door to the cockpit was ajar, and from his seat in the first row Helias could see the profile of the serious man, Mattheus, busy in the captain’s seat. He was talking into the tiny mic of his ‘cell’, a miniscule earphone clinging to his earlobe and an invisible rod that supported the mic just to the right of his mouth. He was almost whispering, in a strange dialect. A f
ew more words, interspersed with pauses as he listened, and then what seemed to be a goodbye of some kind and the conversation came to an end.

  On the other side of the cabin, in the second row, a blond girl was drowsing. Hadn’t he seen her before, somewhere?

  Helias fell back into his own thoughts.

  He had been ‘called’, after more than a year on the waiting list. He’d had himself put on the list right after he got his PhD. Shortly before, his girlfriend had left him. He had told her about his plans to leave Earth, and that he wanted to take her with him. She seemed reluctant to go: too attached to her family and her own habits. For her sake, he might even have decided not to leave. But as it turned out, she took things into her own hands, and made the decision for him, telling him it was over—just before his final exam. She didn’t feel she should have to wait until he finally made up his mind, and in any case, she certainly didn’t want to have to blame herself if he gave up his dreams of the future for her.

  For his part, he had no family left. Or almost none. There was a sister, somewhere in the Austrian Alps. They had never had much to say to each other, and from the time she married and moved to Austria he had never found the time—or the inclination—to go visit. They heard from each other from time to time, birthdays or holidays and things like that, but nothing more.

  For the few times he permitted himself a vacation, he preferred the Swiss or Italian Alps. The high peaks and the eternal snows, with his girlfriend. Or Corsica, which he’d always loved, ever since he went, as a boy, with his parents.

  His parents were gone, officially declared missing. They had been two prominent scientists, who worked at the transmitting station orbiting Mars.

  He hadn’t seen much of them, since he started college and was living on his own. They would come and go, staying for a while, busy with their studies, and then leave again. Once they had taken him with them. It was during the school break, and there was an extra seat available on the spaceship. He was sixteen, and he remembered it as the best period of his life.

  Since then, he had had fewer and fewer chances to see them, since they spent increasingly long periods on Mars. Until two years before he finished his PhD. They didn’t come back from their last trip. He never heard from them again.

  Helias’s eyes filled with tears, and his throat tightened.

  But this was no time to get sentimental. And he was used to pulling the plug on his feelings, and pushing everything back down, deep underneath.

  He allowed himself one small liberating sob, and slowly took control of his thoughts again.

  He turned for a moment, glancing back at the seats behind to see if the girl had heard anything and was watching him. Nothing, she seemed completely absorbed, her eyes half closed.

  For good measure, he pretended to cough, just to belie any suspicions.

  He turned again and gazed at girl’s face, as her lips seemed to move almost imperceptibly. It was an oblong face, though not too much so. The eyes seemed narrow, maybe because they were half closed, almost like an Oriental. Despite the fact that they were blue, and despite the fine blond hair that fell to her shoulders. She had a goodish figure, on the tall and slender side, though the loose coveralls made it hard to tell. Not his type, he told himself, though he couldn’t deny a certain attraction.

  The girl turned toward him, and he looked away immediately.

  And now he found himself in this strange, unexpected situation.

  He had been directed to the Kusmiri Center, where, among other things, they did research into alien molecular biogenetics. And where he would have been able to catch up with the field and start something new.

  The last concrete news he had about the Kusmiri Center and, in general, the planet Alkenia, obviously dated back some forty years, though he had heard it slightly before his departure. He hoped nothing fundamental had changed, even if he had to expect that there had been a great deal of progress.

  The information exchanged between the two planets was always twenty years out of date, and was more of historical interest than of any value as information.

  Above all, he knew he would have to assimilate forty years’ worth of new biogenetic research, and he had no idea how much progress the discipline had made.

  Back on Earth, there were simulations that described all aspects of social life and the organization of research on Alkenia in a certain amount of detail. They were based on successive transfers of personnel, information and plans that Earth sent at fairly close intervals, every few weeks or months. Naturally, the simulations were only relatively useful, since they were little more than forecasts starting from the planet’s actual situation as it was twenty years earlier. And they could be confirmed or rejected only twenty years later.

  Up to the time of Helias’s departure there hadn’t been any major surprises.

  But in the meantime, forty years had gone by. And more than anything else, it was practically impossible to make predictions about the advances in research that used ‘raw material’ that didn’t exist on Earth, except in the form of samples taken decades earlier.

  According to the program he’d been given on Earth—every detail of which he had committed to memory—on leaving the station he was to take the shuttle to the terminal in the nearby city of Symiria, where he was supposed to take the first flight for the planet’s capital, which wasn’t far away. Once in the capital, he would have spent the night at the Hotel Starcross, taking the shuttle for the Kusmiri Center the next morning.

  None of this happened. That serious-looking man had immediately blown a hole right through the entire program. He had barely introduced himself, without even saying who he was or what he wanted from Helias. To all appearances, he just seemed to be a nice man who wanted to give him a lift. But his caginess about offering any kind of explanation, and the strange circumstances surrounding their meeting had something mysterious about it that piqued his curiosity. And there was no denying that the man exerted a certain fascination over him, a kind of charisma.

  Upon leaving the emporium, Mattheus had walked off toward the parking lot, without another word. And this attitude had irked him again. The man couldn’t treat him like this, like a child that didn’t deserve an explanation. He had been about to leave without even saying goodbye, but then he ended up trotting obediently along after him, because he ‘felt’ he had to get to the bottom of this.

  The blond girl and the copilot were waiting on the spaceship. Embarking, his impulse was to sit next to the girl, if only in the hope that she might be more talkative than Mattheus. But she hadn’t even turned to see who was coming in and—no surprise—she also seemed caught up in her own thoughts. So he had given up and, a bit huffily, went and sat as far away from her as he could.

  While he was waiting for the flight to arrive, Helias reached into the inner pocket of his jacket and pulled out his little portable, which contained everything that could be digitized, from his childhood memories to all of his studies and research. He glanced at the index of his scientific publications, thinking that by now they were all old, obsolete analyses. With a bit of nostalgia, he looked at pictures from when he was a boy, the favorite photo of his parents, his ex-girlfriend. Pensively, he thought about how everything fades away and disappears, now more than ever before. He had no chance of going back to Earth, but even if he could and had wanted to, what would he have found down there? A seventy year old sister, aunts, uncles and all his relatives long dead, his former girlfriend with white hair, maybe surrounded by half a dozen grandkids and the children from her second marriage. With her first husband’s portrait hanging on the wall, the first husband who, irony of ironies, was lost in space. And almost no memories remaining of their jaunts in the Alps, still so vivid in his mind.

  He was beginning a new life, in every way. With no ties to the past, nothing more than a jumble of memories, old ones by now, and a few pieces of research that had since become meaningless.

  In the midst of these thoughts, a light from outside
caught his attention. A star, brighter than the others, was shining out among the low clouds on the horizon. It was Nasymil, the nearest star, now reflected imperiously on the clouds below, as the spaceship climbed to hurdle the last mountains. It was still mirrored in an immense glacier, brighter than the full moon. Then the lights went out in the spaceship as it began its descent toward the Center.

  This new light shining on his thoughts of a new and unknown life struck him as a good omen, and he cheerfully prepared for the landing.

  The Center appeared suddenly below him, no longer hidden by the looming mountain. And he had plenty of time to admire the architecture as the ship looped around it before landing. In the cold light of Nasymil, which was reflected now on the roofs of the towers, contrasting with the orange and yellow of the artificial lighting outside the building. It was enormous, a stupendous castle overlooking the choppy waters of a lake.

  Despite his good mood, even better after seeing that fairytale landscape, and despite his eagerness to disembark and throw himself into his new life, he decided to keep his feelings to himself, given the lack of interest his traveling companions had shown in him. He was all ready to leave, but stayed glued to the porthole—not that there was much to see anymore—waiting for the girl to go out first. He got down his luggage and—sulkily, to all outward appearances—followed his two fellow travelers, while the copilot remained on board. Nothing, not even a word. What kind of a way to behave was that? Mattheus had barely glanced over his shoulder to check whether he was following. He went first through an ordinary sliding door and walked toward the reception area. Oh! A miracle! Once through the door, the girl seemed to slow down to let him catch up, and she was even turning toward him. With a smile!