Covariant Electrodynamics the Easy Way
michaelallenwarner.github.io
michaelallenwarner.github.io
Something I've been interested in doing is writing a "renderer" that fully simulates Special Relativity including a full treatment of EM such that diffraction, interference, etc... can all be simulated.
This would be an interesting sandpit for testing the limits of our knowledge, a bit of an "acid test" if you like. Such an engine could even be extended to include, say, electroweak theory and then tested to see if it is still consistent.
In principle this ought not to be that hard, but I've found that theoretical physicists don't like to sink down to the levels of numerical simulations. This can leave gaps and issues that are not just difficult to fix, but nearly impossible. One such issue is numerical stability: equations on a page tend to use "mathematical reals", and these often require infinite precision to simulate. However, the real physical world doesn't allow infinite information to be stuffed into a finite space. Similarly, the infinities of the electron self-energy can be handwaved away on paper, but a simulation will just ignore your hand waving and do absurd things.
What do you mean by practical? Do you mean something you can use to calculate the electric and the magnetic fields as if we didn't understand they're the same thing? That exists, they're described by their own fields of science which used to be called "Electricity" and "Magnetism" respectively. Of course, what's the point when you have Maxwell's equations?
Some of these issues are not obvious when manipulating vector algebra purely symbolically. They manifest when it comes time to create a numerical simulation. You suddenly realise that there are practical issues such as high numerical error, conditional statements all over the place, and corner-cases like gimbal lock.
For comparison, something like simulating the full electromagnetic interaction in a 4D spacetime ought to be straightforward with a geometric algebra!
It's just that I haven't seen anyone try. Every EM textbook is the same. They go through the same equations. The same algebra. The same symbolic solutions. With the same corner cases. The same numerical stability issues. The same result: "Oh we can talk about it, but rendering it is too hard!"
I want an "acid test" of EM, a... wavepool to play with. Not a toy model. The full thing, capable of simulating even obscure but experimentally-verified results such such as the Aharonov–Bohm effect. See: https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect
That is, I want something where instead of plugging numbers into a simplified equation to derive such effects, I want to be able to set up a bunch of charges, press "simulate", and watch the test particle get deflected in the area with zero field.
> but we won’t use any index gymnastics or explicit rank-2 tensors; just four-vectors.
For me the whole point of tensor notation is the power of indice gymnastics. I'd love to know the motivation behind this before spending time understanding.
My own notes on covariant notation on this subject: https://0x5.be/physics-notes/four-vectors.html
In general, I’d be wondering how much of what we call physics can actually be factored out as just math, and what we’d be left with.
While it’s technically true that it’s “just using math & computing consequences”, it’s an endeavor of enormous (essential) complexity, warranting focus and effort. Likewise that engineering is “just applied physics”, product design is “just applied engineering”, etc.
Eh hem, That was a computational joke. I know, I know, but anyway, build a Newton’s method for your physics, and the physics is all in the residual (right hand side) with the left hand side purely there to drive it to zero.