# Gravity vs Entropy: Reconciling Two Opposing Trends

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• Frank Peters

#### Frank Peters

Entropy is always increasing, say the thermodynamicists, and the increase will ultimately caue a "heat death" of the universe.

But gravity seems to contradict this. Gravity, by clumping matter together, always engenders a decrease in entropy. Indeed, some cosmologists propose an eventual gravitational collapse of the universe to a state of zero entropy.

How are these two opposing trends reconciled, if they are reconciled at all?

But gravity seems to contradict this. Gravity, by clumping matter together, always engenders a decrease in entropy.
No it doesn't. You are not looking at a closed system when you are looking at the clumped matter. In order to clump, the system needs to get rid of energy in some form, typically radiation. You cannot ignore the entropy carried by the radiation emitted when the gravitational system collapses.

• lomidrevo
Gravity, by clumping matter together, always engenders a decrease in entropy.

What makes you say that? Can you show a calculation? And you do know, of course, that gravitationally bound systems have a negative heat capacity, right?

Here's what I wrote 8 years ago. It's still true:

You have to be very careful about your intuition in this case. The heat capacity of a gravitationally bound system is negative. The first consequence is that every derived equation that assumes it's positive is wrong and you need to go back to first principles. The second consequence is that it cannot collapse unless it can lose energy to an external system. Therefore, it is not a closed system, full stop. The second law does not apply.

I actually know everything in the answers so far, but I still have the same question. If we ignore expansion, because that likely only complicates the issue, and imagine a nearly homogeneous isothermal universe, with only the variations we see in the CMB, and just wait long enough, what happens? If there was no gravity, then entropy concerns suggest the variations would get even smaller with time, because that would increase entropy. But when gravity is turned on, it seems that now we have gravitational instabilities that must find a different path to increasing entropy by increasing density and temperature variations. The ability for gravitational potential energy to drop into a well and create photons, which in turn generate entropy, must somehow compensate for the loss of entropy associated with the density concentrations. But I've never seen this explained clearly, as to how we know that gravity releases enough energy to be able to cause entropy to increase. Were it not so, surely there would not be gravitational instability, and there would not be stars, but I've never seen it concisely explained as to how this is able to work.

Which raises an even more unclear question for me, which is, how does gravitational instability work for dark matter, which does not release gravitational energy in the form of light, but instead can only add kinetic energy to the gas itself? How can we see easily that the gravitational energy released when a dark matter structure shrinks allows it to have access to more states by virtue of its kinetic energy than it loses by virtue of its volume loss?

Indeed, the Sackur-Tetrode (https://en.wikipedia.org/wiki/Sackur–Tetrode_equation) entropy of an ideal gas is a constant per particle plus the natural log of V U3/2, where V is the volume occupied and U is the internal energy. So if gravity shrinks a system and dumps all the energy into kinetic energy, then V U3/2 would decrease with V. That wouldn't obey the virial theorem, so we might expect only half the gravitational energy to show up in U, while the other half is transported somewhere else, but simply transporting kinetic energy somewhere else does not necessarily seem like a good way to increase entropy. It's also not obvious that turning it into light will guarantee that entropy increases. So I'm quite unclear on how gravitational instabilities, for either normal or dark matter, raise entropy, making this a very good question indeed.

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the entropy carried by the radiation emitted when the gravitational system collapses
But, what about within the event horizon of a black hole? No radiation escapes from there...

negative heat capacity
So, if you add heat, the temperature goes down? If heat is lost, it goes up?
How does that work in, say, a planet + moon system?

So, if you add heat, the temperature goes down? If heat is lost, it goes up?
How does that work in, say, a planet + moon system?

Assume the moon is in a circular orbit round the earth. It's orbital velocity is inversely proportional to the square root of its distance from the earth. It you increase its orbital energy it will move away from the Earth increasing it's distance. This means the orbital velocity will decrease ! I.e. adding energy reduced its kinetic energy. Equivalent to heating reducing the temperature i.e. a negative specific heat.

Energy is locally balanced by an increase in it's potential energy.

Regards Andrew

how does gravitational instability work for dark matter

First, the wrong argument: I have a local bound overdensity. Imagine a nearby unbound particle. It can't join the overdensity because when it zooms in, it will zoom right back out again with the same energy.

OK, so what's wrong with that. Two things - one is that you have more than one particle. You can have, for example, 10 particles zoop in, interact gravitationally, and have only 7 zooming out again. Three particles ended up captured, having given some of the energy to the seven particles that escaped. Numerical modeling of this shows that local overdensities grow, and you get a core/halo structure of the dark matter.

The other wrong thing is that this assumes that the potential that the particle sees entering the potential is not the same potential it sees exiting, because of universal expansion. This is the Sachs-Wolfe effect.

My understanding is that the former effect is larger than the latter.

So, if you add heat, the temperature goes down?

That is correct, although it's usually the other way. As a gas cloud loses energy it heats up.

... The ability for gravitational potential energy to drop into a well and create photons...

huh?

or are you talking about gravity exciting the vacuum quantum field of photons such that a creation operation happens?

I meant that mass is falling into the potential well, it's just the normal way stars get bright originally.

OK, so what's wrong with that. Two things - one is that you have more than one particle. You can have, for example, 10 particles zoop in, interact gravitationally, and have only 7 zooming out again. Three particles ended up captured, having given some of the energy to the seven particles that escaped. Numerical modeling of this shows that local overdensities grow, and you get a core/halo structure of the dark matter.
And this is what leads to the question. To keep it simple, against a static background, how can overdensities grow in this way without lowering the entropy? The whole point of a Boltzmann distribution is to maximize entropy given a fixed amount of kinetic energy, and that distribution involves constant density. So it must be that the overdensities release enough gravitational energy to compensate for the non-uniform density, but that crucial aspect of the answer is missing from the initial posts that mentioned either radiation (which is not necessary, as you point out) or the non-closed aspects (but we are interested in the complete system, which is closed).
The other wrong thing is that this assumes that the potential that the particle sees entering the potential is not the same potential it sees exiting, because of universal expansion. This is the Sachs-Wolfe effect.
But you cannot be arguing that galaxies couldn't form in a universe that is not expanding, so the expansion cannot be a crucial aspect of the answer, as you also agree. So it seems that the crux of the answer that the OP is seeking is that gravitational clumping releases enough kinetic energy (whether there is radiation or not) to increase the entropy, overcompensating the drop in entropy associated with the clumping density.

You asked a question, I gave you the answer. I have no intention of arguing.

• weirdoguy
The word "argue" simply meant you cannot be "making the point" that expansion is required for gravitational instability, because that would not seem to be a correct requirement (even if it did play some role in our universe). Ergo, the answer to the OP need not invoke expansion. I'm content the OP has their answer now, if this is indeed correct: entropy increase is provided by the increase in kinetic energy associated with gravitational contraction, despite the entropy drop due to increases in density. The gravitational energy release allows the temperature to rise (the negative heat capacity issue), which means that heat lost from the system always means transport from hot to cool, raising entropy.

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Gravity causes the system's Entropy to go down, but it makes extra energy to heat up the surrounding. Suppose you are cleaning your room. The things were all spread out, but after you clean up it becomes neat and well-ordered. This seems to make the room more "ordered", which means decrease of entropy. However, while you clean up the room, extra heat is emitted as a result of cleaning. What I want to say is that gravity can make particles cluster together and causes a decrease of entropy, but the heat emitted throughout the process more highly increases surrounding's entropy which causes increase of the universe's entropy. If you're curious about entropy, check out <The Fabric of the Cosmos, Brian Greene>. It's well explained.

Potential wells are negative potential energy spaces. So by accumulating particles are falling to a lower energy state. This needs to be weighed against the added entropy of being ordered. I have no idea how to evaluate the two for quantitative comparison but that seems to be the fundamental point OP misses in thinking the grouping increases entropy.