# Does entropy increase or decrease as our Universe is expanding?

Scientists say that entropy of our universe is increasing as it is expanding and our universe is cooling down gradually from the time of its birth. If something is getting cooler and cooler, then how can it become more random (entropy increase) with the passage of time? According to laws of thermodynamics, at absolute zero temperature, the entropy is zero. That means as we go down and down to the temperature scale entropy must decrease, but why is it not happening in the case of our universe?

The definition of entropy is $$S = -k \log(\Omega),$$ where Omega is roughly the number of microstates (ways of ordering your particles) compatible with the macrostate (what you observe macroscopically).

Intuitively, you can say that, if you have particles inside a box, and you increase the size of the box, you can arrange them in more ways; therefore, the entropy increases. The third postulate tells you that you will never reach 0K, so you can have more and more entropy without a paradox. The entropy density, on the other hand, could be decreasing.

But, beware! Thermodynamics are built assuming there are not long distance interactions, but the evolution of the universe is controlled by gravity, that has infinite range. Therefore, you cannot naively apply any thermodynamical theorem to the universe as a whole. You can physically argue the correctness of many of the postulates, but you are on shaky ground there.

• Gravity does not have infinite range any more than the electromagnetic force does. Both thavel at $c$ and are limited by the age of the Universe. – Thriveth Dec 5 '17 at 18:04
• Up-voted for the definition. But isn't gravity a force and the outcome of distribution of "Energy"? Since "Mass" and "Energy" are related isn't the "Mass" that creates the "Energy" field? So why can't we see the "Entropy" of the Universe as a whole? – Peter Darmis Dec 19 '18 at 20:44

Here is how to see this answer: In the standard model of cosmology, which is given by the FLRW (Friedmann-Lemaitre-Robertson-Walker) solutions of Einstein's field equations, symmetries of isotropy and spatial homogeneity require that such universes be perfect fluid universes. As you know from thermodynamics, perfect fluids have their entropy conserved! So, where does the observed increase in entropy come from in the universe? We know (since we are here!) that the universe also contains matter. So, there are three ways to incorporate this matter content into the FLRW models.

1) The more popular way is to considered perturbed FLRW models, which contain tiny matter perturbations away from isotropy and homogeneity, and one can see directly from these calculations that the entropy of the universe increases since these models allow for matter production as the universe expands from the initial big bang state.

2) Another possible set of universe models consider that because the universe indeed has matter in it, it is really inhomogeneous, so the FLRW models don't apply, they only apply in a dynamical/asymptotic sense. Such models are Swiss-Cheese models and Lemaitre-Tolman-Bondi (LTB), in addition to the G2 class of cosmological models. Since the universe in these models is inhomogeneous, the key point is that The Weyl Tensor is non-zero and increases over time. In fact, this is the essence of Penrose's Weyl curvature hypothesis. Namely, that the observed entropy increase in the universe is because at the initial big bang, the Weyl curvature tensor was zero, and it increases as the universe expands.

3) Yet another alternative, which is something that I have done some work on, is to consider a universe that was anisotropic in its early stages. If there is anisotropy, then, in Einstein's equations, one can add a anisotropic stress term:

$T_{ab} = (\mu + p)u_{a}u_{b} + p g_{ab} + \pi_{ab}$,

and I'll skip the derivations, but... the Einstein field equations give

$\dot{\mu} + 3 \dot{\alpha} \left(\mu + p\right) = \dot{E}$

The left-side of this equation is in fact a change in entropy term:

$dS = dU + p dV$,

and if we let $U = V \mu$ and $V = e^{3\alpha}$ which represents the comoving volume, we see that the change in entropy is:

$dS = V \left[\dot{\mu} + 3 \dot{\alpha} \left(\mu + p\right)\right]dt = \dot{E} V dt$

As the universe expands, $V$ gets larger, and based on physical constraints of not having reversible thermodynamical processes (a bit of a strong assumption, but demonstrates the point), we require $\dot{E} \geq 0$, which implies that $dS \geq 0$.

When people speak about the entropy of the universe increasing, they typically mean in the context of these 3 cases, but it is an active scene of debate in the cosmology community, which of these scenarios is correct! (I personally am partial towards scenario #2)

Imagine slowly (adiabatically) expanding an ideal gas in a cylinder. As its volume increases its temperature decreases, yet its entropy stays the same. If there are other things happening in the gas (chemical reactions etc.) then its entropy will increase as it expands.

The entropy per unit volume approaches zero, but the volume is increasing, which cancels this out. The third "law" is really just a general rule of thumb combined with a definition - it has so many exceptions that the word "law" really isn't justified at all - but in this case it seems to hold.

Doubtless there are important differences between the expansion of the universe and the expansion of a gas in a chamber, but at least this example shows that there's no paradox involved in something cooling as it expands over time.

The entropy of a black hole is proportional to its surface area. If the Universe follows the same rule, then as it expands entropy increases, but entropy per volume might be constant, or even decrease.

For example, if life continues to increase in its ability to efficiently use Gibbs free energy from Sun photons, fossil fuels, and nuclear sources, it might decrease entropy locally in our solar system while releasing enough excess entropy to "allow" the Universe to continue to expand. The net energy balance of Earth's incoming minus radiated photons is approximately zero, but the radiated photons are lower energy and higher in number which means they are more entropy. This leaves the thermodynamic possibility that life can be increasing the ordered arrangement of mass on Earth.

As an aside, the extraction of concentrated geologic resources (metals, minerals, salt, not to mention oil and coal etc) increases Earth's entropy, so it is not clear that the redundancy of many copies of humans and their machines and roads, etc, is a net decrease in Earth's entropy. A more advanced form of life might make a Dyson sphere around the Sun to capture all its energy and even store a lot of it as mass such as spinning flywheels or just spinning itself, if not by creating new atoms, instead of releasing entropy to the Universe. Any thinking or dynamic processes in the system would need to be reversible operations to minimize entropy release. This would make the sun and Dyson sphere dark matter but not a black hole. If such an intelligence stopped the release of entropy, would it in some minuscule sense reduce the expansion rate of the Universe? In an increasing gravitational field does the Hubble constant appear (from that frame of reference) to get smaller and even reverse if the gravitational increase is fast enough (for example if you could be inside a black hole as it collapsed)?

• The first two sentences seem to address the issue at hand, the last two paragraphs seem to be completely irrelevant and largely hypothetical (and likely incorrect). – Kyle Kanos Jul 18 '15 at 14:18
• If life is decreasing local entropy and therefore decreasing entropy per volume, it is relevant to the question. Entropy per volume is more important than total entropy. Since expansion of the Universe is continually reducing the observable universe, entropy per volume is more relevant. If you want to metaphysically include the non-observable Universe and if it is infinite, then again "per volume" seems to be the only way to go (if not "per observable universe). – zawy Jul 18 '15 at 14:39
• Dr. Kohli's 3 cases do not cover the case I am saying that is important to entities in the Universe, i.e. in a fixed volume. I do not see an adjustment for the comoving volume as it becomes red shifted for a stationary observer (i.e. an energy decrease in the observed comoving volume and therefore a reduction in the observable entropy compared to the given equations). If the entropy increase follows the black hole rule of being proportional to (comoving) surface area,the entropy per volume decreases with the radius increase.An accelerating expansion would be a decreasing observable radius. – zawy Jul 19 '15 at 11:17

The amount of 'randomness' is not the only definition of entropy. Entropy is also known as the amount of unusable energy that's present in a system. So if you have a lot of heat energy in a system, since all of that heat energy can be used up, its entropy would be low. If you have less heat in a system, only a small amount of heat in that system can be used up, so it has a high entropy. Entropy is therefore always increasing as the universe cools and expands. Why there was such a low entropy at the beginning (Big Bang) is something we are still trying to figure out...Hope this helped!