## Wednesday, December 21, 2011

### Entropy, information and gravity

Recall that the overall entropy of the universe is always increasing. You could have an "isolated system", like the earth for instance, in which the entropy could decrease, but when you enlarge that system, earth + sun, the total entropy is increasing. So looking at the earth by itself is really not a good idea in defining what is considered to be an isolated system -- a common mistake people make.

Now, if we take a system in which gravity plays no role, then generally speaking, an increase in disorder corresponds to an increase in entropy. For instance, an ice cube left on the table will melt to water and then evaporate as water vapour.

Ice (high order or low disorder, low temperature) → water vapor ( high disorder, high temperature).

We see from that equation, that going from low temperature to high temperature, entropy increases.

However, when gravity is factored in, the reverse is true. For instance, at the Big Bang, matter was at a very high temperature. You would think that like the water vapour, the universe had high disorder. But it didn't. We will show below that such a system has low entropy (read low disorder) and as it expands, its temperature decreases and its disorder increases.

Universe at Big Bang (high order or low disorder, high temperature) → Universe at present times (high disorder, low temperature).

Here, we go from high temperature to low temperature, and entropy increases.

Black holes -- as well as any star, our sun included -- are like the universe. They are in a state of low disorder. With Hawking radiation -- our sun radiates mainly electromagnetic radiation -- they are moving from low disorder to high disorder, that is, entropy is increasing.

Modern theory looks at entropy as the lack of information. So you can substitute in the above equation, the following:

low disorder (high order) → high information

high disorder (low order) → low information.

As the universe is expanding, our knowledge of the universe is decreasing -- entropy (or lack of information) is increasing, that is, we are becoming more and more ignorant. At first sight that might sound paradoxical, but think of it this way, as the universe grows bigger and bigger, it is more difficult for us to keep track of every particle in the universe. In fact many galaxies are so far way, that the light from them will never reach us. And this situation gets only worse as time goes by.

Gravity

When an object is orbiting around a larger one (say a satellite around the earth, for instance) and changes its orbit, it will do that by emitting away some energy. The result is that it will move to a smaller orbit, but its speed will increase. Hence its kinetic energy has increased, even though it radiated some energy away. The reason being the case is that the gravitational potential energy is negative.

E = K.E. + P.E.

E = ½ mv2 – GMm/r

So while the kinetic energy increases, the radius being smaller, the potential has become more negative. In absolute value, the change in potential energy is twice the change in kinetic energy. So whatever gain in kinetic energy, there is twice the loss of potential energy.

ΔE = Δ K.E. + ΔP.E.

ΔE = Δ K.E. – 2 Δ K.E.

= – Δ K.E.

But K.E. = NT, where T is the temperature, and N is any positive constant.

So we must have

ΔT = -ΔE /N

But from the definition of specific heat (c),

ΔE = cmΔT, where m is the mass.

This means that gravitational bound system would have a negative specific heat!!!

To rescue this situation, we must conclude that entropy increases when the system decreases in temperature. The less energy the system has, the higher the temperature. This feature of gravitationally bound systems makes them tricky. Only systems with positive specific heat can be in thermal equilibrium with their environment. So gravitationally bound systems can never be in thermal equilibrium with their environment! They always want to keep shrinking, thus losing energy and, by the 2nd of law of thermodynamics, they increase the entropy. In the satellite-earth system, the satellite will emit energy as its orbit degrades into a smaller one.

NOTE: One might think, is the earth orbit shrinking and will it crash into the sun? The earth's orbit shrinks by 3.5×10−13 m per year. However, the effect on the size of the Earth's orbit is negligible over the age of the universe. Unfortunately, in the case for the many satellites that humans have launched and are orbiting the earth, as time goes by, most of them will have their orbit decaying in a much shorter time, and will meet their fatal fate by crashing into the earth.