Week 30 progress update

On Monday I added another tool to my game engine using Dear ImGui. The tool provides a new window that allows stopping the user time and slowing down or speeding up the evolver time. I used this tool to do the final edits to the TikTok video that I mentioned in previous posts. I published the video on Tuesday. You can find it here .The video had 6K views, more than all of my previous videos but that’s not sufficient to give the project the visibility it needs to survive. I guess it’s fair to say it is a step in the right direction.

I spent the rest of the week investigating the solution to the analytical positioning problem: “connect two points with two pieces of hyperbolic trajectory matching given initial and final positions and velocities”. Solving this problem will allow me to engineer arbitrary relativistic trajectories, and will be very useful in the months to come, when working on the game proper. It would basically give me a curve that is for Special Relativity what a Bézier curve is for computer graphics.

I cracked the 1D relativistic version of the problem back in April this year (1D refers to the number of spatial dimensions.) The breakthrough that allowed me to find the 1D solution was realising that the maths of the problem simplifies considerably if, rather than imposing that the two hyperbolic trajectories have the same proper-duration, I impose that the two trajectories produce the same velocity increment. I define the “velocity increment” as the invariant scalar quantity a 𝜏, where a is the norm of the four-acceleration and 𝜏 is the proper-duration of the trajectory. Note that I use term “velocity increment” because “a 𝜏” has the same units as a velocity, but this quantity has not much to do with a velocity as normally intended.

Despite having found the 1D solution, I struggled quite a bit with the full 3D problem. It is much more complicated than the 1D counterpart. I wasn’t actually sure the 3D problem was analytically solvable or even that there was a simple numerical solution. This is why I am super happy that on Friday 26th of May 2023 I could finally fully crack the problem. It was a culmination of hard work and it is great that a beautiful and easy-to-compute solution was possible.

I still have to publish the solution, although I already shared some of the advances I made in parametrizing hyperbolic motion here (the sections “Coordinate-time parametrization” and “Final velocity as a boundary-condition” are new; the latter section and the section “Four-vector equations” were decisive to make progress with the 3D solution.) I am considering whether to publish a proper article on arXiv, as the work I have accumulated is now quite substantial.

For now, I share with this post a screenshot of one key elements of the solution. The equation shows the relation between Um, the four velocity at the point of intersection between the two hyperbolic trajectories, with the initial and final four-velocities Ui and Uf and the quantity U*, that can be easily computed from the initial four-velocities and four-positions. δ is an unknown. It can be easily determined by imposing that Um really is a four-velocity, i.e. Um^2 = c^2. The resulting equation is quadratic in δ. Once determined, everything else can also be easily found.