thedifferentialequationsboy

Hi! The universe is too amazing for us not to get to understand it. See here for random physics ramblings (and ofc lots of differential equations included) but sometimes other things too.

Yes, there are no ideal spherical cows in a vaccum in the Universe. But approximations work to give us results. And that's why theories can allow us to make the calculations to describe the Universe quantitatively.

For instance, yes, the Sun is not a perfect uniform sphere. But with the approximation that it is, we can calculate the trajectories of the planets to meticulous detail. Yes, the interior of a star is technically not a steady-state spherically-symmetric fluid. But with the approximation it is, we can make useful predictions about stellar behavior. Yes, the hydrogen atom is not as simple as a one-electron system under a Coulomb potential. But with the approximation it is, we were able to make predictions extremely close to observed values. And finally, yes, perfect smooth functions do not exist in nature. But using the approximation they are, we can use the power of analytical techniques, calculus, geometry, functional analysis, and differential equations to perform highly-accurate calculations and predictions of incredibly many phenomena. Approximations have worked so well that in many cases we don't even know when they don't work until we uncovered new things that we can't explain using them.

When the OPERA anomaly occured, the scientific team behind it announced their results and nothing more than their results, then did constant check after check, both with their theoretical models and their experiment. By all accounts, what they did was great science - not making exaggerated claims, cross-checking their results, and announcing when their own claims were invalid. It ending up being the result of faulty equipment, while seeming ridiculous, is not in too much sense a difference as forgetting a factor of two in theoretical physics; scientists are human after all. It is also understandable that the result created a media flurry that is essential to get scientific funding. The science was great. Everyone involved was a great scientist. Scientists always want to express skepticism whenever an established theory is overturned or contradicted by results. We need a preponderance of evidence before we make any claims that go against scientific consensus that comes from an abundance of experimental support. That was exactly what was done - the scientists involved made sure they checked, and checked everything they could.

The name is such a misnomer! In fact black holes in our universe tend to be some of the brightest things there are!

Why? because the immense gravity of a black hole tends to pull matter towards it, in tight orbits, and once close enough, into a whirlpool of sorts that enters into the black hole. Matter including things like stars! so a black hole might not itself produce light but it is surrounded by a bunch of things there are. Saying a black hole is a "light ring with a black inside" would actually be more physically accurate.

In addition, that matter, as it accretes around the black hole, it gets super hot and actually glows. This forms quasars, which are (I'm pretty sure) the most energy-efficient way to turn matter into energy in the universe, and hopefully one day we can control this process artificially with mirrors to extract this energy for human use. Here is what quasars look like:

By ESO/M. Kornmesser - http://www.eso.org/public/images/eso1122a/, CC BY 4.0, https://commons.wikimedia.org/w/index.php?curid=15700804

So black holes aren't black. Yes, anything within the event horizon (Schwarzschild radius) is black but black holes in general look more like Gargantua in Interstellar than pure black regions of space.

The Dirac equation is a beautiful thing:

But the equation is math, the interpretation is physics, and here's where it gets confusing:

In regular (relativistic) quantum mechanics, it represents the quantum wavefunction of a spin-1/2 particle with mass (i.e. electrons and their cousins and quarks)

In quantum field theory, it represents a classical complex spinor field that is then quantized to obtain a quantum field

Actuality: ah, but what about QED? hmm this QFT thing looks cool! Also why don't I make pretty notes on them? Oh, definitely have to find the best notation for the born rule! ...

physicist: considering my circumstances, I may model the sensation of cold using a combination of a temperature field and a wind velocity field. I will make the assumption that I may be treated as a particle that does not change the fields themselves. Now I will set up a system of two partial differential equations, a continuity equation for the temperature field representing the flow of heat through air, and a PDE governing wind velocity: