> heavy-duty use of trig

Trig functions are useful in differential equations, expect them everywhere in differential geometry and geometric analysis.

> How was it derived

If you mean eqn (15) of "the pdf" <https://arxiv.org/abs/2203.12625> (which is what the author at the link at the very top of the discussion was trying to summarize), then it's largely a generalization of Weyl's 1917 "Zur Gravitationstheorie" approach (English translation at sci-hub: <https://sci-hub.se/https://link.springer.com/article/10.1007...>, in particular part B §4-§5 starting at page number 793, pdf page 15 of 32) into higher dimensions.

The particular generalization is described in "the pdf"'s reference 15, "Generalized Weyl Solutions" <https://arxiv.org/abs/hep-th/0110258v2>. Choice quote, "Depending on which feature of Weyl's class one focuses on, there are several directions in which one can try to extend it into higher dimensions." The authors choose an approach which produces a notion of the passage of time for a hypervolume of two fewer dimensions than the whole; contrast this with a notion of the passage of time for a three-dimensional hypersurface of a Lorentzian (3+1, or four dimensions) manifold like our own universe appears to be.

The authors of "the pdf" per the top of II. THE GEOMETRIES focus on five and six dimensions, the former producing following the logic of their ref 15 a set of 3d spatial volumes that are related by a notion of time, which invites comparisons with a Lorentzian manifold like our own, and which they do in III and IV. Their result is that they cannot make their structures observably indistinguishable from Schwarzschild black holes. They do not go further into merely very slowly spinning their topological solitons, and it's not clear what the metric for those would look like, or observationally how they would compare to very slowly spinning Kerr solutions (which become indistinguishable from Schwarzschild).

One of the recurring themes of M-theory/stringy gravity starts with that we see a lot of spinning objects in hydrostatic equilibrium, including relativistic objects like neutron stars (notably millisecond pulsars). We can pick out the orbits of even very small companion objects in close orbits around these (< 2% of mass of Earth, < 1/2 the apoapsis of Mercury <https://exoplanets.nasa.gov/exoplanet-catalog/7134/psr-b1257...>). Those orbits are essentially Kerr orbits (non-gravitational perturbations from e.g. the NS's radiation or magnetic field are small). M-theory/stringy gravity appears largely stuck on recovering spherical symmetry (for black holes and galaxies (e.g. Verlinde's vintage 2011 emergent gravity)) rather than obviously rotating distributions of mass with lumpy and even heavy satellites (from Draugr to the Large Magellanic Cloud and its dozens of friends, to clusters of thousands of galaxies). General Relativity's universality of free fall (UFF or "strong equivalence principle") is extremely well supported at all these scales. That UFF needs to be searched for (often requiring strong suppression of extra gravitational degrees of freedom, except exactly where they want to be different from naive quantization of General Relativity) in these supergravity theoretical ideas mean these theories are maybe-interesting but of no practical astrophysical value at this time.