## Wednesday, December 31, 2008

### The Counterintuitive Time: 3. The Time's Arrows

This is a series of posts about the counterintuitive nature of time in Physics. In this post it is analyzed the difference we perceive between past and future, as it appears in irreversible phenomena.

Seeing that the equations are symmetric at time reversal, we may legitimately wonder why the time has a direction. Boltzmann answered this question long time ago, when he explained the entropy, but since then, many felt that the things are not clear yet.

If at microscopic level the laws are symmetric to time reversal, why are they irreversible at larger scales? At larger scales, two systems which differ at small scale, may look identical. For example, to spheres made of the same material, and of the same radius, having the same density, may be considered identical, although their microscopic structures are far from being identical. Two glass balloons of identical shapes, filled with the same quantity of the same gas, will look identical at macroscopic level, but very different at atomic scale. Boltzmann defined the entropy of a macroscopic state of a system as minus the logarithm of the number of distinct microscopic states that macroscopically look identical to the macroscopic state. This definition fit well the entropy as it was known in Physics, and also has an analog in Shannon’s information theory, which led to an informational interpretation of the entropy. For our discussion, we will deal with its probabilistic meaning. A system tends to evolve to a more probable state, and a state with larger entropy is more probable. This is the key to understanding phenomena which are thermodynamically irreversible, like boiling an egg or breaking a cup.

The entropy will increase only to a maximum value corresponding to the most probable state, after that it will just fluctuate around that value. But then, it seems to follow that the present state is most likely to be one of the most probable, with the maximum of entropy, therefore we should not observe an increase of entropy, and no special arrow of time. The answer is that our present state is one of the most improbable, and therefore the entropy has enough room to increase. Moreover, it appears that the entropy increases since the Big Bang, and at that initial moment the entropy was very low. Very low entropy means very improbable, so the matter distribution at the Big Bang was very improbable. The permanent increase of entropy is explained not by a universal law of Physics, like the fundamental laws, but by a special property of the initial conditions. It is a “historical law”, and not a “universal law”.

The Big Bang itself seems to provide initial conditions improbable enough to activate the Second Law of Thermodynamics, by the simple fact that the matter was all concentrated in a very small region, most probably a singularity. But not all scientists consider this concentration enough. For example, Roger Penrose proposed an explanation of the thermodynamic arrow of time based on the condition that the Weyl tensor canceled. The tensor describing the curvature of the spacetime in General Relativity contains a part corresponding to the energy-momentum tensor, the other part is the Weyl tensor. But the Weyl tensor can be viewed, by the mean of the Bianchi identity, as corresponding to the gravitational field generated by the energy-momentum tensor of the matter. I interpret Penrose’s condition Weyl=0 as simply stating that in gravitation, only the “retarded gravitational potential” should be considered (similar to the retarded potential in Electrodynamics). Therefore, it seems that Penrose’s condition refers to a “radiative arrow of time”. It seems that the Big Bang, the cosmological arrow, is tied with the thermodynamic and radiative arrows.

The psychological arrow of time, corresponding to our minds remembering only the past, is perhaps the most difficult to grasp. It is habitually to be explained by comparing the brain with a computer who, in order to use its memory, needs to heat the environment, increasing the entropy.

I believe that the explanations of the arrows of time are very counterintuitive, and one reason is that they are based on symmetry breaking. The PDE expressing the fundamental physical laws are time-symmetric, but the solutions are not necessarily so. The time asymmetry is related very well with the existence of a special time, of minimum entropy, and that time is, naturally, the origin of time’s arrows. Because of the difficulty in accepting the arrow of time in a world governed by time-symmetric fundamental laws, some physicists try to find fundamental laws which exhibit time-asymmetry. In most cases, the asymmetry is searched in quantum phenomena, especially in the measurement process. But many consider the time arrows explained well enough, not requiring supplemental assumptions.

Yet, if one of the time’s arrows is less understood, I think that this is a psychological one, not necessarily restrained to memory, but to the whole psychological meaning of the words “time flows”. Perhaps the central point of the flow of time is the subject experiencing it, the “I” of each one of us.