Discret Time

Michele Diego

Although this so called granular or atomistic view of time has been proposed in the past by some thinkers and, according to some, even by Democritus and Plato, it has yet to enter a major physical theory such as Einstein’s theory of relativity or quantum mechanics

What is the smallest fraction of time we have measured in our lives? A second, looking for the exact point at which we interrupted a YouTube video? Maybe a hundredth of a second, to decide the winner of a sports competition or to calibrate the parameters of a camera before taking a shot? A femtosecond (0.00000000001 of a second) working in a laser lab? Whatever our answer may have been, we can be sure that time can be divided into two, resulting in two smaller fractions of time. In principle, this division can go on indefinitely, breaking ever shorter intervals of time in half. If this were really carried on to infinity, we would obtain an infinitesimal fraction of time, which we commonly call an instant.
The scientific article in the previous issue of La Livella [1] was dedicated to the question of what an instant really is and how art and science approach this concept. On that occasion, however, we deliberately stopped at Isaac Newton’s conception of time. The Newtonian structure of time is, on the other hand, also our conception of time, which we use in our everyday life. In this new article, the aim is to pick up where we left off, to show that things are not quite so simple.

Newton’s idea of how time works is based on the assumption that there is a time that is independent from the universe, external to it, absolute, unalterable and that proceeds autonomously, marking out the beats of time like an eternal metronome, second by second, instant by instant. We too, after all, imagine things in the same way: if we watch a film lasting two hours, we take it for granted that two hours will have passed by the end of the film for the rest of the world as well. If we go on a week-long trip, we take it for granted that a week will have passed for those who stayed at home on our return. We somehow imagine that there is a ‘now’ that applies everywhere in the universe and that if we could stop time, we would all be frozen in the exact same moment. In other words, in Newton’s universe, instants are points on a straight line: provided two instants A and B, the result is always A<B (A precedes B) or B<A (B precedes A) or A=B; other options are not obtained: two events happen simultaneously or one of them happens before the other.
Apparently nothing more normal. Yet, Albert Einstein shows us that this assumption is incorrect. The truth is that time is something more complex, and in order to understand the phenomena of the universe, we must rid ourselves of what appears natural and intuitive. Without going into explanations that are beyond the purpose of an informative article, it can be said that Einstein’s great insight was the understanding that the speed of light is constant in any reference system you measure it in. What does this mean? To give a practical example, imagine sitting inside a train moving at 100 km/h, with another passenger walking at 10 km/h in the direction of travel. Since we are inside the train, we see the passenger moving at 10 km/h. However, a man observing the same situation from outside the train would see the man moving at 110 km/h (the sum of the train’s speed plus that of the passenger). The passenger’s speed changes depending on the reference system in which it is measured. An astronaut would have to add the speed of the Earth to that of the passenger and the train, and so on.
In the case of very fast objects, such as light, we realise that this is actually not true: a beam of light moves at the same speed when measured from inside the train, from the outside, or even from space.
From this starting point, reasoning through examples that are not unlike the one just described, Einstein demonstrated that the idea that two events A and B occur either simultaneously or one before the other is false. We shall try to see why. We are again on the train moving at 100 km/h and we stand exactly in the middle of the carriage. We light two torches, one directed towards the head of the train (torch 1), the other directed towards the tail (torch 2). A light beam (beam 1) comes out of torch 1 and reaches the front wall of the wagon (wall 1). Beam 2 comes out of torch 2 and reaches the rear wall (wall 2).
Having positioned the two torches in the centre of the wagon, the time taken for beam 1 to reach wall 1 is the same as the time taken for beam 2 to reach wall 2. The two events are therefore simultaneous. But if we look at the same experiment from outside the train, things radically change. Since we mentioned that the speed of light is the same if measured from inside or outside the train, ray 1 takes longer than ray 2 to reach the respective wall. This is due to the fact that seen from outside, the train is moving and thus wall 1 move away from ray 1, while wall 2 goes towards ray 2. Ray 2, therefore, has to travel less distance and reaches the wall before the other.
The arrival of the light rays on the walls, therefore, from outside the train, is no longer simultaneous. An observer in a second train, moving fast enough in the opposite direction to that in which the torches are positioned, sees the events temporally reversed, i.e. ray 1 reaches wall 1 in a shorter time than it takes ray 2 to reach wall 2.
There is no such thing as one observer who is ‘more right’ than the other about the chronological order of events, we simply have to surrender to the fact that the idea of time as a straight line on which times are ordered one after the other is a preconception that has proven to be untrue. Different reference systems give different chronologies, and no one is ‘truer’ than the other.
With this, moreover, we have by no means resolved the issue. Einstein’s oddities about time go far beyond what has just been described: time flows slower the faster you move (for light, time does not flow at all), time slows down near objects with high gravitational attraction (much of the plot of Interstellar is based on this effect), etc.
Nevertheless, Einstein’s and Newton’s theories have something fundamental in common: they both consider time as a continuous entity, infinitely divisible into infinite instants, between two different instants there is always another instant that could come between them. This, in other words, means that there is no minimum indivisible fraction of time, like a minimum time brick of small but not infinite duration with which time is built.
Although this so called granular or atomistic view of time has been proposed in the past by some thinkers and, according to some, even by Democritus and Plato [2,3], it has yet to enter a major physical theory such as Einstein’s theory of relativity or quantum mechanics. Nevertheless, there are more recent theories that attempt to reconcile relativity with quantum mechanics (which are currently at odds with each other), precisely through this process of atomisation or – as physicists say – discretisation of time.
That said, there is a minimum time below which there is no valid physical theory capable of making predictions about nature. This is called Planck time and corresponds to 5.39×10^-44 seconds. At this scale, space-time in itself loses its physical meaning and the laws of physics are no longer applicable. Therefore, despite the fact that physics has not yet foreseen a real discretisation of time, it is somehow implicitly embedded in the limits of current theories. Under Planck time we are unable to say what happens to reality. Thus, in a certain way, our entire understanding of the world takes place not in a continuous manner, but rather as frames of a film that follow each other in the projection of a movie from the Big Bang until today.

[3] Carlo Rovelli, L’ordine del Tempo, Adelphi (2017).

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