Carroll knows that the weak nuclear force is the only force of nature that is space-time asymmetric. However, in Lecture 7, Carroll unfortunately dismisses the weak nuclear force. "The weak nuclear force has almost no effect on our everyday lives. In fact, it is really weak."
But the fact that weak nuclear force is weak does not mean that it's not the cause of the arrow of time. There are lots of particles interacting in the universe. The arrow of time can be smaller (i.e. weaker) than the gravitational or E&M forces without causing any concern. In fact, if the gravitational or E&M forces were the cause of the arrow of time, we would have a lot of explaining to do because these fields exist everywhere.
The other reason why Carroll doesn't think that the weak nuclear force is the actual cause of the arrow of time that we perceive is that weak nuclear force (according to him) "is just an arrow in which the rates are higher in one direction than another?" (According to Carroll, the weak nuclear force is not the cause of the arrow of time because asymmetric part of the weak nuclear force is only a small component of the force and very small in general.)
While it is true that the time-asymmetric part of weak nuclear force (i.e. the component of the CKM matrix that is time-asymmetric) is a small component of the weak nuclear force (i.e. of all of the components in the matrix), this doesn't mean that the asymmetric component isn't the cause of the arrow of time. Note that this 1 component in the CKM matrix is the only part of a force of nature that is time asymmetric. There are no components in gravity or E&M that are time-asymmetric. There are components in the strong nuclear force that hypothetically could be time-asymmetric, but experimentally have been determined to be near-zero. And all of the time-asymmetric components of the weak nuclear force boil down to the same time-asymmetric component of the CKM matrix. I do not think that it is a coincidence that there is one arrow of time, that the universe is expanding, and that there is only one component of the forces of the nature that is time-asymmetric.
So, let me rephrase this because it's important to highlight. The time-asymmetric part of the weak nuclear force is small compared with the time-reversible parts of the weak nuclear force. Think of this as a small drift velocity sitting on top of a symmetric distribution of velocities in a gas. The drift velocity is small compared with the root-mean-square velocity, but this can cause a large macroscopic effect over time as all of particles eventually move in one direction. So, to be precise, it is one of the (relatively small) components of the weak nuclear force that is the cause of the arrow of time. Unfortunately, Prof Carroll focuses on the whole CKM matrix rather than focusing on the 1 component of the CKM matrix that is time asymmetric.
So, if you are Sean Carroll and if you don't think that the weak nuclear force is the cause of the arrow of time, you have to invent a reason for why there is an arrow of time. So, Sean Carroll brings up the idea of the "Past Hypothesis" (i.e. that the Big Bang was a location of small entropy.) And while the Big Bang definitely was a location of small entropy. This doesn't explain why entropy increases. There are at least three problems with the "Past Hypothesis" as the "cause of the arrow of time." First, it doesn't explain why the entropy was low. Second, it doesn't explain why we are located near the first diffusion away from the low point (rather than living somehow in the fluctuations about equilibrium.) Third, there is no reason to believe that, even if the entropy were lower in the past, that it should increase in the future when the laws of physics are time-reversible and not-probabilistic.
For example, if the Big Bang started off as a bunch of high mass quarks, then these quarks could bounce off of each other; however, the entropy would never increase if there were no time-asymmetric component to the weak nuclear force. The quarks could interact via the strong nuclear force, via the E&M force and via the gravitational force, but nothing would happen in such a system. The heavy mass quarks would remain heavy mass quarks. Entropy would never increase. The information required to describe the microstate of the universe would remain constant. The average curvature of space-time would remain constant, and the size of the universe would also remain constant. The universe would be a nearly dimple-free 3D surface to a 4D sphere. (The radius of the universe might only be a few micrometers.) The high mass quarks would interact, but the size of the universe would remain the same. If we converted the surface area into a 1-D length computer binary code, then the values of 0's and 1's would change, but the overall length of the code would remain constant. The information required to describe the universe would be constant.
What we need to do is to explain (a) why the universe expands, (b) why the amount of information in the universe increases, and (c) why is there is an arrow of time. We can't invent untestable hypothesis, such as the "Past Hypothesis" or multiverses (i.e. multiple universes.) We need to make testable predictions. Everything that I'm arguing for can be tested experimentally. What I'm arguing is that the expansion of the universe, the increase in information, and the arrow of time are the same thing, and they are all linked to the 1 asymmetric components of the weak nuclear force. This is a testable prediction. We can run simulations of the universe using the rules I've stated before (i.e. that the universe expands only because of asymmetric component of the CKM matrix) and we can check these simulations with data from Planck/WMAP/etc... There are testable predictions to the statement that "the weak nuclear force is the cause of the arrow of time and the cause of the expansion of the universe." We would see this asymmetric component of the CKM causing high energy fermions to slowly decay into lower energy particles while also generating neutrinos. There would be a general trend towards a world with more neutrinos and less heavy mass quarks. This shows up as a larger universe because this interaction also expands space-time as it makes neutrinos. The total energy of the universe remains the same, but the radius of the universe increases. (To understand this a little bit better, check out this previous post in which I give an example of a computer code that can increase in length when there are certain rules for how the length of the computer code can increase. In this analogy, the forces of gravity, E&M, and strong nuclear force keep the length of the computer code constant, whereas the 'time-asymmetric' component of the CKM matrix in the weak nuclear force can increase the length of the computer code. So, in this analogy, the length of the computer code is similar to the 3D surface area of our 4D spherical universe. The code increases in size whenever a certain rule is met and requires the length of the code to increase. In our universe, there likely a rule that requires the 3D surface area of space-time to increase whenever there is a certain type of irreversible reaction according to the weak nuclear force.
So, let's step back see another problem with Sean Carroll's logic. He argues that entropy increases naturally with reversible laws of physics. The problem is that we can't increase the amount of information in a system when we only have reversible laws of physics. The amount of information to describe the correct microstate of the universe has been increasing significantly since the Big Bang. There's no way to get this increase in required information from reversible laws of physics. We need a law of physics that can introduce more information, and this is most likely that same component of the CKM matrix that is time-asymmetric. With a time-asymmetric component, the amount of information required to describe the full microstate of a system can be a function of time.
Another mistake to point out in the lecture is that irreversibility has nothing to do with large numbers of particles. The arrow of time is not statistically in nature. For example, you can have a large number of photons travelling from a distant star, but that doesn't make the process irreversible. Or you could have a large number of superfluid helium atoms moving indefinitely in a fountain loop, but the large number of atoms doesn't mean that there is any irreversibility. In Lecture 6: Reversibility and the Laws of Physics, Carroll argues, "If you have a small number of parts, you get reversibility. If you have a large number of parts, you get irreversibility." This is clearly an incorrect thing to say. For example, the amount of information needed to describe N particles is 6N (3N positions and 3N velocities.) You always need 6 values for each particle. If you have a large number of particles, they can move about, but you don't need any more or less information to describe them as they move about if you only have time-reversible laws of physics. What you need is laws of physics that can generate information.
So, I encourage you to watch the DVDs by Sean Carroll. He is a good lecturer, and much more engaging than myself. Just be careful because not everything he says is correct. (Just as I'm sure that not everything I say is correct. Only time and a lot of people thinking about these problems will get us to the point where we can speak clearly on the issues of the arrow of time.) If the arrow of time interests you, but you don't have the time to watch Carroll's full lecture series, I encourage you also to read this post on his blog. Just be careful with statements like the following, "CP violation cannot explain the Second Law of Thermodynamics." What I've been attempting to argue for in this post is that 2nd Law of Thermodynamic is ultimately due to CP (i.e. T) violations in the weak nuclear force. It is these CP / T violations that explain why we aren't already in thermal, chemical and mechanical equilibrium.
As an side, I wanted to point out there are a couple of errors in Lectures 17&19 regarding time travel and space ships. It's called the Twin Paradox, and he unfortunately states the wrong answer to the Twin Paradox. He states that two space ships will measure different times if they take different routes through space-time. However, this is not correct. Carroll's is forgetting about the effect of acceleration on space-time. When you include the velocity and the acceleration on the ships during the travel, the clocks will measure the same exact time when they meet back up at the same location with the same velocity. In other words, you can't go on rocket ship and age less than somebody else. You can't go on a rocket ship at the age of 36 and wait for somebody younger than you (24 years old perhaps) to become 36 years old. When that person is 36 years old, you will be 48 years old, and there's nothing you can do about this.
There's also a major leap of faith in Lecture 22: "It take nothing to create a universe." Or as Alan Guth calls the inflation of the Big Bang the "Ultimate free lunch." Carroll believes that we live in a world with zero net energy. This is faith, not scientific fact.
We don't live in a world with zero total energy/mass. Likely, mass/energy can only be positive, and likely, there is no such thing as negative mass or negative energy. Some physicists and astrophysicists like making up terms like dark energy or negative energy, but there is no experimental evidence for energy that is gravitationally repulsive. Right now, the evidence for dark energy comes from the expansion of the universe. But as has been pointed out earlier in this post, the expansion of the universe need not be due to energy with negative gravitiational energy. It might just be due to a component of the laws of physics that explains how to expand space-time. What I'm saying is that there is not independent evidence for dark energy in the same way that there is independent evidence for neutrinos or quarks.