# Entropy and gravity

Entropy, at an intuitive level, is often described as a general level of disorder within a system. For example, I have a gas in a container divided in two areas by a divider, the gas all on one side. I remove the divider and the gas will expand to the whole container, increasing its level of disorder.

Isn't gravity, then, reducing entropy?

If the universe were to slow down expansion (I know that's not true, just making a hypothesis) to eventually contract in a "big crush", the moment of switch from expansion to contraction, would we be starting to see a reduction in the entropy rather than an increase?

Let's assume, for the sake of an argument, that the mass of the universe is large enough that at some point in the future it will start contracting, until it will collapse into a big crunch. We know that entropy at the big bang was at its lowest, what happens to entropy when it starts contracting, will it the start going down, until it will again go to the same level it had at the big bang when it hits the big crush?

• Gravity is not reducing entropy because unlike a wall it can not bind a large number of particles in a limited volume (think about angular momentum conservation and what it means to gravitating systems) and it looks like it can not bind the universe, either. Mar 21, 2016 at 7:21
• Mar 21, 2016 at 8:06
• The effect is very very very negligible , consider 1 mol of Hydrogen gas (for simplicity) now it has a weight of 2g there number of atoms $6*10^{23}$ so the weight of 1 atoms is about $10^{-23}$g therefore you can find force acting on it which is very small.. And already atoms are in very fast motion (elastic as well) so neither the energy is lost nor gravity can affect the motion Mar 21, 2016 at 13:10
• @CuriousOne could you please clarify? Why can't gravity bind particle in a volume? Isn't that what gravity does? Isn't a planet, or a star, just a large amount of gas molecules that have bound together in an orderly manner rather than being spread all over the space in a disorderly fashion?
– user
Mar 21, 2016 at 13:40
• @lemon thanks for the link: math.ucr.edu/home/baez/entropy.html I am still puzzled. If the entropy in the universe keeps increasing, it implies that it had its minimum at the moment of the big bang (see also: physics.stackexchange.com/q/36113). But, at that moment, the gas was so hot that, if I understand the link correctly, entropy would have been instead much bigger than now. What am I missing?
– user
Mar 21, 2016 at 13:48

There are two issues to talk about: gravitation acting within the universe (star formation, solar systems, galaxies and things like that) and the cosmology of the whole universe.

The first is easier. When a cloud collapses by its own gravitation, the net entropy increases. Here is a quote from p.377 of "Thermodynamics, a complete undergraduate course" (Steane, OUP 2018)

"For low enough initial temperature, the self-gravitating cloud cannot be stable because when any given part of the whole cloud loses energy, that part gets hotter and shrinks, while another part gains energy, gets colder and expands. The temperature difference is now enhanced so the process continues. The net result is that the fluctuations in density or temperature in the cloud grow, and the whole process is called condensation. The parts that shrink lose entropy, while the parts that expand gain entropy, and there is a net entropy increase because, as usual, the direction of heat flow at any time is such as to guarantee this. There is no violation of the Second Law. On the contrary: the Second Law is fully obeyed, as it is in all of physics."

If you are puzzled by the statement "gains energy, gets colder" then you need to recall that there is potential energy involved as well.

Now let's turn to the larger cosmological issue which you asked about. Over the past fifty years from time to time there has arisen, in the theoretical physics community, claims along the lines that time would reverse if the universe were to re-contract, and so entropy would decrease. Such claims are based on attempts to think through what general relativity has to say about particle motion. However it is safe to say that those claims were never really established and most of the physics community has found them unconvincing. Certainly quantum field theory would not change if the cosmological scale factor got smaller rather than larger. All of physics would just carry on as before. So there is no good reason to think that entropy would decrease.

The conclusion of the above is that a big crunch would have high entropy not low entropy.

The early universe, by contrast, had low entropy. We can say this because all the processes we have figured out are entropy-increasing processes. Therefore the entropy at early times must have been smaller than it is now. And this is, furthermore, a strong statement: the entropy was very much smaller. To try to get an idea of what it means to say this, one can try to imagine the state-space of the cosmos. Then the statement is that the physical situation occupied, at early times, a tiny volume within this state-space. However I admit that I personally am not altogether sure of how one can be confident of ones reasoning here. The main message is that it is not true to say that the early state was merely amorphous; such statements ignore the great amount of structure residing in the configuration of the quantum fields at very early times, and later in the hot plasma. The very fact that the plasma was highly uniform (without being completely uniform) is the very thing that shows that its entropy was low. This seems surprising if you think of it as like an ideal gas, but owing to the self-gravitation that picture is very misleading.

Thinking of entropy as disorder doesn't always work. Disorder is a vague concept that helps when thinking about the much more technical definition of entropy: The logarithm of the number of possible micro states consistent with the macroscopic observables.

When gravity causes an object to collapse on itself, it reduces the number of possible positions that the particles could be in, exactly analogous to how removing the divider in the cylinder increases the number of possible positions for the gas particles. However, the collapse also heats up the object, and hotter objects have particles with greater momentum, so the number of possible momentum states that each particle can be in increases even more, causing the overall entropy to increase.

On a cosmological scale, we know much less about what would happen to the entropy during a big crunch scenario. There is a chapter in Stephen Hawking's book A Brief History of Time that discusses this question, but there is no mainstream answer. We don't even know why the universe had such a low entropy at the big bang.

• Thank you Travis, this may be the better answer so far. Can you clarify what you mean with: "We don't even know why the universe had such a low entropy at the big bang".
– user
Mar 4, 2017 at 0:18
• The reason why entropy increases towards the future is because the entropy was very low at the Big Bang. Why this is so is an unsolved problem in cosmology. Mar 4, 2017 at 2:53
• My question is: how do we calculate entropy at the Big Bang?
– user
Mar 4, 2017 at 13:31

The truth is noone really knows how to treat the entropy of the gravitational field. Its an open question and requires a good quantum gravity theory to answer. Most agree that gravitational clumping is not an entropy lowering process, so while its true that uniformity in say a gas, is a high entropy state, once you take gravity into account, uniformity of a gravitationally significant amount of an ideal gas actually represents an extremely low entropy situation as gravitational degrees of freedom are not activated. Hence the early universe, while appearing high entropy from a point of view that doesnt include gravity, is actually an extremely low entropy state when including gravity,ie unlikely to the order of one in 10^10^120 according to penrose!

Further to that, a big crunch would not lower entropy as we wouldnt expect it to look like the big bang in reverse, instead we would expect a high amount of activation of gravitational degrees of freedom, ie clumpy messy space time as it approaches the collapse

The second law of Thermodynamics states, Entropy is a measure of disorder or multiplicity of a system, or the amount of energy unavailable to do work. For an isolated system, the natural course of events takes it to a more disordered and higher entropic state.

Gravity, on the other hand, knows attraction only and thus tends to keep the things in orderly state by keeping them close to one another and thus reducing the volume occupied as well as the possibilities of the possible states.

Considering the statistical view of entropy, the number of possibilities of micro-states of information depends on the size of the storage i.e. space considered.

Entropy = (Boltzmann’s constant k) x logarithm of number of possible states

E= K Log(N)

Since the logarithm of a number always increases as the number increases, we see that the more possible states that the system can be in, the greater the entropy. In our case, the larger the space, the larger is the number of possible states and hence the greater is the entropy. Therefore, if an isolated system expands, its entropy increases.

Gravity tries to keep things together through attraction and thus tends to lower statistical entropy. The universal law of increasing entropy (2nd law of thermodynamics) states that the entropy of an isolated system which is not in equilibrium will tend to increase with time.

However, the effect of gravity in a closed volume with an ideal gas, gravity causes the density below to be greater than above, therefore the disorder is less and so is the loss of information. This tells us that gravity subtracts an amount of entropy proportional to the magnitude of said gravity. In formulas it would be S= K ln(N) – K f(g) where g is gravity and f() a function proportional to g and also f(0)=0.

well this is my humble contribution that is surely wrong!

There is no problem about a decreasing entropy. Matter falling into a black hole will lose all information it had before. The entropy is just gone.

The second law of thermodynamics applies to ...: thermodynamics! There is nothing magical about it; its origin is just, that given a lot of microscopical states of a system, which are grouped into macroscopical states (i.e. for a group of internat states you cannot tell them apart), it is much much more probable, that a system will be in a macroscopical state which corresponds to a larger number of microscopical.

And the larger the system, the more "much" you have to add before the "more probable". Thermodynamics deals with large systems per definition, making the system larger is called "taking the thermodynamical limit". There is no concept of entropy in classical mechanics, is there :)

So it's all a question of different concept for different parts of reality. There is only one Nature, of course, but our concepts of it are ( - still, or by design, we don't know) fractional. You have to use the laws that apply to the system you are dealing with. You cannot use classical mechanics for particles, and you cannot use quantum mechanics for your daily life.

To clarify this: even if the world would be completely classical, no quantum uncertainty, we still had to study thermodynamics! Motion would be deterministic in principle, but we have too little information to compute it. You can think of a completely disordered gas with the velocities of particles such, that they all will be in the same place in one second. But if we cannot, it's much more probable, that they will not.