By the divergence theorem, the integral of the Berry curvature over any closed surface = the divergence of the Berry curvature over the volume bounded by that surface.

However, the divergence of the curl of a vector field is necessarily zero (and the Berry curvature is the curl of the Berry connection).

Put another way, by the curl theorem, the integral of the curl of a vector field over any closed surface is zero, since

The integral of the curl of a vector field over a surface = the integral of the vector field along the edge bounding that surface (which for a closed surface is non-existent)

So how is it possible that C_S can have a non-zero value?

]]>I literally had the exact question about the charge and the rod in my mind for 6 years. Thank you for the explanation! I’d love to learn more about this. Any suggestions for further reading?

]]>Much better than https://en.wikipedia.org/wiki/Moving_magnet_and_conductor_problem

Amazing work

]]>Until reading this article, I thought there was a real time dilation between A and B, ie. you could go away from earth with high speeds, and then come back to find your peers older than you are. Now I think that time dilation is only a visual illusion.

The reason of my thought is, speed is relative. Imagine in the case that we’re proposing, A also having a rod and particle for theirselves. Our case has just became symmetrical. Who is actually moving, Team A or Team B?

I think the answer is clear, there is no “actual”, “rightful” mover. They’re only moving relative to each other, both are observing each other in slow motion. And even without the rod and the particle in team A, this fact must stay the same.

Then, which one is the actual “slow” team? Who is the time bender? Again, the answer is the same, there is no slow team, and no time bending. It is just that they experience each other’s movements as slow.

So I think the term time dilation is dangerously named, it is not easy to see that it is only a perceived dilation, not a real one.

Now some questions:

1- Is my interpretation above correct? I am sure that the topics are well settled in science communities, but ordinary people like us still have to reinterpret it.

2- (another topic) How the hell there are some objects that haven’t made their light reach to us yet (outside observable universe)? Isn’t every particle (that emits light) already emitting light towards us since the beginning of the universe? I think it is only the case that there are some “stars” that haven’t made their light reach to us yet, since stars started to form much, much later, and they are still forming. When we look near the edges of the observable universe, we should be perceiving such an early era of the universe that the stars are just starting to form. That should be the reason why it is the edges of what we call observable, because if there are no stars, there is not much of a source of light. What do you think?