The Dark Arrow of Time

by Massimo Villata

  “So, young man. Sit down. Let’s get back to business.”

  Helias sat in one of the armchairs arranged in a half circle around the screen, now off.

  “Professor, could you do me a favor?”

  “Tell me. If I can….”

  “Don’t call me ‘young man’ anymore.”

  The professor stared at him, astonished, with his eyes bulging out of his head in an utterly idiotic expression.

  “What? Why?”

  “Because I find it very irritating.”

  The professor continued to goggle at him.

  “Are you nervous, young man? No, what’s your name again? Kadler? Are you nervous, Dr. Kadler?”

  “Nervous? Me nervous? Of course I’m nervous! Why shouldn’t I be nervous? Is there any reason at all that I shouldn’t be nervous?”

  Helias rose from the chair and began to pace back and forth, nervously.

  “Give me a reason I shouldn’t be nervous. Give me just one damn reason I shouldn’t be nervous.”

  Throughout this rant, he had been stabbing his finger at the professor who, in the meantime, not knowing what to do, had sat down, embarrassed and flustered.

  “A few days ago I said goodbye to everybody and everything. Just like that! Who gives a damn anyway! I get tossed onto this fucking planet. I don’t even know how. I don’t even know if I’ve still got all my cells in the right place. Maybe one fine day I’ll realize some of them are missing, floating around in space. The minute I arrive I get practically kidnapped, oh, all very politely…but not really all that politely. Maybe I needed a little time to get used to things here. Who knows? I certainly don’t, I didn’t even have time to ask myself before I get caught up in who knows what fiendishly complicated plot, and discover I’ve got a mission or something like that, what it is I don’t know, I don’t know a thing. I find myself having to be a secret agent, without knowing for who or for what noble cause. Me, who never even did so much as sneak jam from the cupboard as a kid. I get three or four good frights, from people who materialize from behind the walls, or creep around in the dark, dragging their feet in the fog. And you want to know if I’m nervous? Damn right I’m nervous. Very nervous. Nervous, pissed off and tired. Good lord.”

  Helias sat back down, head in hands. The professor was about to say something, but stopped halfway, his mouth ajar.

  “A beautiful woman, from another planet, says she loves me. She says she also knows how many hairs I’ve got on my head. And to know what I’m thinking. Even my most intimate thoughts. A woman like that can steal your soul, don’t you think? Or frighten you. Or both. I don’t know. All I know is that she’s always in my thoughts. Like an obsession. A wonderful obsession. Or an anchor, a haven. Because she’s all I’ve got, in this whole crazy story. And now she’s gone. I can’t even so much as see her.”

  Helias fell silent for a moment, eyes downcast.

  “Sorry. I didn’t mean to tell you these things. Maybe I just needed to get it off my chest….”

  “You’re tired. Have you eaten?”

  “Not much, I wasn’t hungry.”

  “Would you like something? An ice cream, perhaps?”

  “I don’t know, I might….”

  “Wait for me here a moment. I’ll be right back.”

  The professor went out for a few minutes.

  “Here’s the ice cream. I always keep a stock in the freezer.”

  Helias looked at the ice cream. He took it. And he devoured it.

  “Thanks. Could you tell me where the lavatory is, please?”

  When he returned, the professor was lost in thought.

  “Mattheus.” he said, addressing the active screen.

  “Yes?”

  “Mattheus, could you please check whether Kathia is available? Tell her someone would like to see her.”

  “She’s in her office. I’ll have her call you back shortly.”

  “I’m going to the other room for a moment, young… Dr. Kadler. I have a few things I have to do….”

  And for the first time he looked him in the eyes, and smiled.

  Helias looked at him in turn. Actually, those bags under the eyes weren’t that ugly, really. The wattles, no, those he really couldn’t take.

  “Thanks.” murmured the young man.

  Kathia’s face was drawn. She was trying to smile, but couldn’t hide her fatigue and worry.

  “Hello, Helias.”

  “Kathia, how are you?”

  “Fine.”

  “Are you sure?”

  “Yes, I’m fine.”

  She seemed embarrassed. And she kept glancing off to the side, as if there were someone beyond her screen.

  “Is there someone with you?”

  She nodded assent.

  “A little further over there.”

  “Can you talk?”

  She essayed a pallid smile and dropped her eyes.

  “More or less.”

  “What’s going on?”

  “The professor will explain it to you. We can’t talk much. The line is secure, they can’t listen in on it, but if we use it for long they might be able to discover it and trace our location.”

  “Who?”

  “We don’t know. Or rather, we’re not sure.”

  “Are you in danger?”

  “N…no. Maybe….”

  She had lowered her eyes. She was lying.

  “Is there anything I can do?”

  “You’re already doing a lot….”

  “For you, I mean.”

  “Yes, Helias, you can do a lot for me….”

  “What?”

  “Things, unfortunately, have spun out of control….”

  “What is it that I can do?”

  “You can look at me, Helias. Look me in the eyes. For these few seconds that remain to us.”

  Kathia had brought her hand close to the screen. Helias did so too. Their fingers seemed almost to touch.

  Helias looked her intensely in the eyes.

  “Be seeing you, Helias.”

  “When will I be able to see you again?”

  “Soon, I hope. But I still have to find a way to do it.”

  “What do you mean?”

  “We’ve run out of time. Look at me again.”

  Kathia smiled. But her eyes were sad. Then the image vanished.

  The professor reappeared after a few minutes.

  “How’s it going now?”

  “A bit less nervous. But upset.”

  “Perhaps what you need is a good meal, and some rest.”

  “Is she in danger? How much? In what way?”

  “We don’t know. In any case, she’s safe where she is. At least for the moment. She might have to leave. They may have found her, but we don’t know what their intentions are. We don’t even know exactly who they are. We’re trying to understand. Sooner or later they’ll have to come out in the open. We’re hoping they’ll make a false move. For now, we only have the ‘sensation’ that Kathia has been found out. And so we have to be extremely cautious. We had to ‘isolate’ her, or rather, get her entirely out of the picture, as if she had never existed. There’s no longer any trace of her, not even in the archives. If our sensation is wrong, so much the better. When we’re sure, she could even come back. Conversely, if our suspicion is grounded, they’re looking for her and they’ll have found she’s been ‘erased’. They could ask for a ‘back-up’ of the archives, but that would tip their hand, so they probably won’t do that. We don’t know whether they are informed of our station in the mountain, nor whether they know the people Kathia had dealings with, like you, Mattheus, me and the others. All we can do is reduce the interactions between us to the bare minimum, and not put ourselves in situations where someone could easily read our thoughts.”

  “How would they have been able to find her out?”

  “Are you sure you don’t want to rest a little, young… Dr. Kadler?”

  “Perhaps you’re right. When can we continue?”
r />   “Tomorrow, I’d say. You need a good supper and a good night’s sleep.”

  “All right. I thank you.”

  “I’ll see you as far as the corridor. Remember, don’t let yourself be approached by anyone, if you can avoid it, not even in the dining hall. If you can’t avoid it, think as little as possible, or even better, think of ‘innocent’ things, confusedly.”

  They returned to the office and the professor turned on a monitor.

  “There’s no one in the corridor. Wait, I’ll go out first.”

  The professor, unexpectedly, offered his hand. Helias shook it.

  “So, see you tomorrow, young…. Sorry….”

  “It doesn’t matter. Go ahead and call me whatever comes naturally.”

  “Turn on the device. I’ll ‘follow’ you back to your room. When you go out again, leave it behind, it’s more prudent.”

  The professor, after one last look at the monitor, went out in the corridor.

  “All clear. See you tomorrow.”

  Helias went straight back to his room. No detours, even when passing the lavatories.

  Helias had returned to his room. Shortly thereafter, he had gone out for supper, steering clear of other people as much as possible. He had had a hard time falling asleep, and when he did he slept uneasily at first. Then he had slept heavily until eight thirty, and woke feeling decidedly refreshed, ready for new battles. After breakfast he had contacted the professor and gone to his office, with the usual precautions. Here the professor had explained why they suspected that Kathia had been found out.

  The previous morning Kathia had gone to the optometrist’s office for her appointment. There was a younger doctor there, not the same one she had seen at her earlier eye examination. Perhaps the latter had taken the two days off for the ‘jaunt’ to the capital and asked his assistant to stand in for him. Kathia had explained her problem and the young optometrist, without saying much, disappeared into the adjacent office for a few moments. Returning, he was suddenly talkative, chattering away about everything and nothing, as if to distract attention. He was looking at Kathia’s eye in the screen, talking nonstop, when the girl had felt, in the midst of that smokescreen of words, a thought, something along the lines of ‘Bingo! Finally!’. Kathia had had the distinct sensation that she had been found out. The examination was over and the young doctor was by now in full verbal flood, but they both watched each other guardedly, trying to conceal their suspicion. “Come back in two days, the new lenses will be ready.” the doctor had said as he handed her the card with her appointment. “And the other ones, what was wrong with them?” Kathia had asked, but the doctor had purposely framed his answer in incomprehensible technical jargon.

  “Will you be here in two days?”

  “Not necessarily.”

  In practice, he continued not to answer.

  “Your name, please?”

  The young man was still looking at her guardedly, without responding. Apparently his thoughts were elsewhere. At a certain point Kathia ‘felt’ that he was ogling her, as if he had stripped her bare with his thoughts and then started to touch her. She knew she mustn’t react: it was a trap designed to unmask her once and for all.

  “For reference. In case you’re not here.”

  “My name is on the door, next to my colleague’s.”

  “All right. Goodbye.”

  As she went out, Kathia had taken note of the two names on the door, calling Mattheus without delay. Mattheus had immediately checked the archives: neither of the two doctors matched the description. Then they had broken off the call. Kathia had gone to her quarters, to gather up everything that might be compromising. In the meantime, Mattheus had ‘erased’ her from the archives, and checked that no one had asked for information about her. And then Kathia had vanished inside the mountain. And Mattheus had gone to her room, to remove all trace of her presence.

  The professor had then asked Helias to tell him what he already knew, and Helias had given him a brief summary.

  “So you don’t know what the object you’re safeguarding contains?”

  “No.”

  “Are you interested in knowing?”

  “I’m interested in knowing everything. From A to Z.”

  “Very good.”

  They were in the ‘secret’ room. The professor checked the time.

  “It’s a long story and it’s almost time for lunch. Let’s wait until later. You think better on a full stomach.”

  Looking at the professor’s prominent paunch as he rose, Helias mused that he must do a lot of thinking.

  “So, down to business, young man.”

  The professor had assumed a markedly academic air, most likely induced by ongoing digestion. The term ‘young man’ must have been part of his lecture-room manner, and evidently an inseparable part.

  He had had a very abundant meal brought in, sharing it, rather sparingly, with Helias. Afterwards he appeared to have blissfully put all fleshly desire to rest.

  “Right. You remember your Lorentz transformations, young man?”

  “Vaguely.”

  The professor took a pen and a tablet and wrote something.

  “Screen.”

  The writing on the tablet appeared on the screen.

  “”

  “Yes, I remember. That’s the covariant formulation.”

  “Remember this too? Which derives from the quadratic form’s linearity and invariance requirements.”

  “”

  “More or less.”

  “Now for the important part. If we disregard the improper transformations, or those with negative determinant, we are left with the proper subgroup, where ‘’. The latter includes the orthochronous transformations ‘’ and the antichronous transformations ‘’, which do not preserve the sign of the time component. Like the improper transformations, the antichronous transformations are also usually considered to have no physical meaning, or in other words, as not applying to any phenomenon of the physical world as we know it.”

  “Unless we consider the existence of antimatter.”

  Clearly startled, the professor swiveled toward Helias, a pleased look on his face.

  “Exactly, my boy, exactly. But how…?”

  “I don’t know, maybe I heard people talking about it in the past. Or maybe I just connected it with what Kathia was saying….”

  “Ah. They had told me you were an intuitive type, son.”

  ‘Son’? ‘My boy’? What was happening to the professor? Not just vocabulary, also the way he looked at Helias had changed. Now he was looking at him with a sort of scientific curiosity, pleasantly surprised.

  “Go on….”

  “No, no. You go on, professor, please.”

  “Right. Take the quadratic form ‘’ and its canonical conjugate ‘’. They are invariant quantities—scalars—of the theory. Now let’s take something traveling at speed ‘’, where c is the speed of light in vacuum (equal to one in our units, where space and time, as well as mass, energy and momentum, have the same dimension), invariant in any inertial frame. These particles thus have zero proper time and rest mass: ‘’. In addition, they cannot be considered at rest in any reference frame, as their speed is always c.”

  “Okay. This is something I’ve never understood. How can ‘something’ not have its own reference frame?”

  “Yes. In reality, it’s a sort of dialectical contradiction, but it’s only apparent. The contradiction is resolved by the plusvariant formulism, which you don’t know, where the reference frames are ‘replaced’ by reference ‘states’, which take all the intrinsic variables into account, for the real as well as the imaginary parts. The reference frames are simply trivial approximations of a simplified ‘metric’ for these states. Obviously, I can’t go into the details of a formalism you don’t know. We’ll talk about it again in the future, if you like. Consider, in any case, that a hypothetical photon at rest would have zero proper time, as well as zero mass, as we’ve just seen. It would
be a reference frame in which everything, space, time and energy, is reduced to zero, becomes unmeasurable: and what kind of reference frame would that be? But let’s continue with the formalism you’re familiar with.”

  Helias had coiled himself up in the armchair, rubbing and tugging his nose.

  “For the moment, we’ll ignore the singular case of photons, the case where ‘’. We’ll consider the other particles, the ‘normal’ ones, that can only travel at subluminal speeds, less than c. They thus have ‘’ and ‘’. In particular, we’re used to dealing with positive proper times and masses. But nothing prevents there from being particles with ‘’ and ‘’.”

  “Antimatter, in other words?”

  “Precisely, except that in Lorentz’s and Einstein’s day, antimatter was not known to exist, and so there was no awareness of this kind of particle and the associated antichronous transformations. Later, when antimatter was discovered, it was more convenient—with the mathematical formalism used in those days—to consider the antiparticles as strange particles, opposite to the known ones, but still ‘traveling’, together with us, forward in time. Instead of considering them as particles that are perfectly normal, but travel in the opposite time direction. It was only a question of convenience in describing physical events mathematically. Or even an involuntary, primitive refusal to consider that two arrows of time could exist? A sort of unconscious repugnance? A bit like when astronomers didn’t want to acknowledge that the Sun was at the center of the solar system, which made describing the motion of the planets so simple and direct, but preferred to ‘invent’ curious and complicated epicyclic motions, all so as not to change reference frame, all so as not to admit that the reference frame they found themselves on was not what the preferential one. A sort of intellectual laziness? Or a recondite—but not all that recondite—ancestral anthropocentric wishfulness? Was it the same for there being two arrows of time? Why was it so hard to admit that there was another one, that wasn’t ours? And that everything had to be ‘viewed’ and ‘measured’ from what was in the middle, from the standpoint, or system, or reference state from which everything is most symmetrical? The Sun, in one case. The ‘photon’ in the other, because it’s in the middle, between the positive and negative masses, and doesn’t need space and time in order to exist.”