121–130 of 839 posts
Entropy. Sometimes you read that it's a measure of randomness; sometimes, information. Aren't randomness and information opposites?
Mach's principle. Why is there a "preferred" rotational frame of reference in the universe? Or as stated in this Wikipedia article, "You are standing in a field looking at the stars. Your arms are resting freely at your side, and you see that the distant stars are not moving. Now start spinning. The stars are whirling around you and your arms are pulled away from your body. Why should your arms be pulled away when th…
It's the same reason the universe has an average speed (unlike what you might expect from special relativity), although it is unclear if this is true for the entire universe or just the portion we can see. We can measure how fast we're moving w.r.t the cosmic microwave background radiation though (it is red-/blue-shifted in a particular direction).
When two particles get closer, their mutual gravitational attraction increases. As the distance approaches zero, the force approaches infinity. In the limit of d -> 0, the energy released -> infinity. Obviously at some scale the notion of a point mass breaks down, but even quantum theory would be problematic if we think of a wave function as describing a probability distribution, wouldn't it? What's the "official" st…
Basically, we just declare that we have no idea what is going on at such short distances, and put in some regulator by hand to get rid of the infinity. One very crude regulator (which nobody uses, but which is suitable for demonstration) would just be to say that particles are simply not allowed to get any closer than some fixed tiny distance.
But what about the effects that occur when particles actually do get that close? Well, in most theories, whatever is happening can be parametrized in terms of a few numbers (e.g. it could shift the observed mass of the particles, or their charge, etc.). Our ignorance of what is actually happening prevents us from computing these numbers from first principles. But we can still make scientific progress, because we can treat them as free parameters and measure them -- and after that measurement, we can use the values to crank out perfectly well-defined predictions.
Repeating this process through several layers was crucial to building the Standard Model, which currently has about 20 free parameters.
Fourier Transforms. I'd wish I had a intuitive understanding of how they work. Until then I'm stuck with just believing that the magic works out.
A function is like a vector, but instead of having two or three dimensions you have a continuous number of them. Adding functions component-wise works just like adding vectors.
Just like regular vectors, you can choose to represent functions in a different basis. So you choose a family of other functions (call it a basis) that's big enough to represent any other you want. For a lot of reasons [1, 2], a very good choice is the set of complex exponentials g_w(x) = exp(2πiwx), for every real w. It's an infinite family, but that's what you need to deal with the diversity of functions that exist.
So you try to find the linear combination of exponentials that sum to your original function. You need a coefficient for each w, so call it c(w) for simplicity. After fixing the basis, the coefficients really have all the information to describe your function. They're an important object, and we call c(w) the Fourier transform.
How do you find the coefficients? Just project your original function onto a particular exp(2πiwx), that is, take the inner product. Usually the inner product is the sum of the products of coefficients. Since functions are continuously-valued, you use an integral instead of a sum. This is your formula for the Fourier transform.
I known there are technical conditions I am glossing over, but this is the intuition of it for me.
[1] There is an intuition for these exponentials. Complex exponentials are periodic functions, so you are decomposing a function in its constituent frequencies. You could also separate the exponential into a sin and cos, and will obtain other common formulas for the Fourier transform.
[2] Exponentials are like "eigenvectors" to the derivative operation (taking the derivative is just multiplying by a constant), so they're really useful in differential equations as well.
Entropy. Sometimes you read that it's a measure of randomness; sometimes, information. Aren't randomness and information opposites?
When I hear explanations like “space is expanding like the surface of a balloon” it’s always confusing. Because a surface is an object, separate from anything on it, but space is the thing we’re all embedded in, so we’re like drawings on the balloon. If space is expanding why aren’t the radii of fundamental particles and their orbits and molecules also expanding? And if that were the case we couldn’t notice space exp…
> If space is expanding why aren’t the radii of fundamental particles and their orbits and molecules also expanding? And if that we’re the case we couldn’t notice space expanding. > Does space only expand somewhere else? Only between me and the Andromeda galaxy, and not _within_ me and the Andromeda galaxy? How would it know to do that? If you start with expanding space in general relativity, and then carefully take…
Bell's theorem. It somehow proves that quantum physics is incompatible with local hidden variables, but I could never see an understandable explanation (for me at least) of just how it works.
Yudkowsky's explanation[1] is the first one that worked for me. I later found Quantum mysteries for anyone[2] helpful. The latter has less soap-boxing. 1: https://www.lesswrong.com/posts/AnHJX42C6r6deohTG/bell-s-the... 2: https://kantin.sabanciuniv.edu/sites/kantin.sabanciuniv.edu/...
Plus, I'd rather expect at least one professional (QED) physicist exists able to explain it and he isn't one. Mermin is, but the explanation is decidedly less clear.
BTW I came here to say Bell's inequality as well. For me it's as baffling as science could ever be.
Entropy. Sometimes you read that it's a measure of randomness; sometimes, information. Aren't randomness and information opposites?
Non-interactive zero knowledge proofs. ZK proofs have a number of good explainers, mostly using graph colorings. Non-interactive versions, however, require quite a bit more than that explanation allows - and despite asking experts, I still haven't found a good, basic explanation.