What is time ...?

Why does time pass by?

We can walk to the corner and come back or go upstairs to the upper floor and return. Why can’t we then travel to the past or to the future and come back?

Let's suppose we make some coffee: we pour the coffee into the cup, add some milk and sugar and stir. Finally we'll obtain a homogeneous liquid: the milk and the coffee will mix up and the sugar will be dissolved. However, if we stir in the opposite direction, we won't be able to split the coffee from the milk again, nor extract the sugar. If we watch a video on coffee making, we'll immediately perceive if we are watching it backwards or in the correct direction: in nature, milk and coffee cannot be separated by stirring, let alone make each of these liquids jump back from the cup into their respective pots when we put them close, or sink a spoon in the liquid and get the dissolved sugar to place it back in the sugar pot. It is clear that time elapses in one direction and not in the other.

Despite the facts recently discussed, for the laws of mechanics, the direction in which time goes by is indifferent: if we watch the movements of the Moon and the Earth on video, either forward or backwards, we will not perceive any difference, except that the bodies will move in opposite directions, which according to physics, is perfectly feasible. Contemplating it in more detail, yes, there is a difference: due to the tidal forces, the Earth slowly diminishes the rotation over its axe (the days get longer) and the average distance between the Earth and the Moon grows. Let's consider for instance the movement of a pendulum: the reach of its oscillations will be slowly reduced because of the friction forces. There always seems to be a subtle consequence that expresses the direction of passing time, although it is not evident in the laws of mechanics

What's so special about tidal forces or friction forces that can make them capable of detecting the direction in which time passes by?

If we consider a particle crashing against another, the first one will transfer part of its impulse to the second one and will change its velocity and direction. In this case, the reverse phenomenon is also possible: we can watch the video backwards and we will not notice anything unusual. However, the situation is very different when we have a great amount of particles.

Let's consider the following case: in the figure, I have tried to represent two containers communicated through a hole. Let's suppose that in one of them, we put an ideal ball, perfectly elastic. Of course, such balls don't exist in nature, but they are an adequate model to represent, for example the molecules of a gas.

If this ideal ball has some initial velocity, it will rebound against the walls of the container forever, going into the other side in either direction every time it goes through the hole. As this is an ideal ball and it doesn't lose energy when it rebounds, we can fast forward or rewind the film without perceiving which direction is correct.

If we add a second ball, we will sometimes have both balls on the right side, sometimes one on each side and sometimes both on the left side. With a third ball, and even with a fourth one, it can still happen that we sometimes have all the balls on the same side. But this probability becomes too small if we have many balls: using our intuition, we can foresee that if we put a hundred balls on the same side, after some time, we will have roughly the same number of balls on each side. And if we started with fifty on each side, it would be extremely rare that after some time they were all grouped on the same side. Therefore, this case is like the milk and coffee one: we can foresee that the balls will go to both containers (and that if we put balls of two different colors, they will ``mix up''), but we cannot expect them to group themselves all on one side.

In absence of dissipative forces like friction, the laws of mechanics can be verified regardless of the direction in which time passes by. This means that if we pay attention to only one of the one hundred balls already mentioned, and we follow it as it rebounds against the container walls and against the other balls, and as it goes to both sides through the hole, we will not be able to tell if the video is being played forward or backwards. There is nothing that can reveal the direction in which time goes by.

Where does the apparent contradiction of the previous paragraph come from? How come the evolution of the aforementioned one hundred balls, considered all together, can point us to the time direction if the movement of any of them, does not?

There is nothing in particular in the hole to make the balls travel into any particular direction: there is exactly the same probability that a ball travels from left to right or from right to left. However, if we start with all the balls on the left, we can bet for sure that we won't see any ball traveling from right to left. There are no balls on the right side that can go through the hole into the left side. Once the first ball has passed into to the right side, the probability of this ball going though the hole again back into the left side is the same of that of any of the balls that remained on the left going into the right. But we have ninety-nine balls on the left and only one on the right, so the chance that the next change takes place from left to right is ninety-nine times higher that the other way round.

Only if we have fifty balls on each side we will we have exactly the same chances of watching a ball passing into either side; otherwise we will always expect more balls to travel from the fuller side to the emptier side. Neither the balls, nor the hole have any preferences as to into which side should be easier to travel, it is just that in one of the containers there more balls rebounding. In other words, more candidates to go through the hole.

The aforementioned examples define what is called the thermodynamic arrow of time. It is clear that although in the world of individual particles, the laws of physics don't distinguish if time passes by forward or backwards, when we consider the macroscopic world, the progress of time becomes evident.

But what do a bunch of balls that rebound against one another have to do with our own perception of time? We remember the things that already happened, but not the ones which are about to happen. The fact that we can remember things depends on the capacity of our brain to store information. But in order not to consider the physical processes that take place in our brain when we memorize something, lets suppose we are absent-minded and therefore write down a telephone number on a piece of paper: the ink will come out of the pen, impregnate the paper and then dry out. But if we rewrite on the numbers in the opposite direction, we will not be able to make the ink absorb the air humidity and return to the pen cartridge, leaving the paper brand new. From the physical point of view, the process of recording a memory requires a certain amount of work (energy) that cannot be recovered by erasing the memory. Some talk about a pyscological arrow of time, which is nothing but the thermodynamic arrow. If we watch a video that begins with a inebriated man and an empty pisco bottle and finishes with a full bottle and the sober drinker, there is no doubt that we have played it backwards. Likewise, in the white coffee example, the pisco is drunk by the drunkard, then goes to his bloodstream to be finally metabolized, but there's no way the ethanol can naturally separate from the canned man, reconstitute the pisco and returns to the bottle like it was before. And if somehow the drunken man gives the pisco away, we should not expect the shopkeeper to give him his money back either.

Finally, the sense in which the universe expands is called cosmological arrow of time, but this has nothing to do with the probability that some balls go through a hole or not: that the universe expands, compresses or twists, does not affect our perception of the progress of time at all.

Translation by Gerardo A. Ostrov - gerarost@yahoo.com

Acknowledgments: To Dr. Sergio Cellone, for his revision and his useful comments and corrections on the original.