Geometry? more like hell-ometry, hehehehe

I'm going to take this time to introduce you to some concepts. If you read my earlier posts, you should already be familiar with spacetimes, and how distances are measured in spacetime(go back and repeat it!). Now I'm going to explain why special relativity is used in general relativity, the concept of the three observers and Schwarzschild geometry

The special relativity is simple enough; If you zoom in far enough onto it can appear flat, even the face of a teenager filled with zits.

Since special relativity can only be used in inertial frames of reference in "flat" timespace (as I have described earlier), we can use this zoom-like effect to say the curved timespace, where we have to use general relativity, is made up of an infinite amount of flat pieces, where we can actually use special relativity.

So, Schwarzschild geometry. At the moment, I have only used to geometry forms; Eucladian and Lorentzian. If you have read my blogs, you know what both of these are, even if you might not recognize their names. Eucladian is good old fashioned geometry that you're used to, with Pythagorean theorems and all that jazz. Cats eating and hypotheses and shit. Lorentzian geometry is the geometry used to describe flat spacetime, like \(\Delta S^2 = \Delta t^2 - \Delta x^2\), which you've seen before.

And then we have Schwarzschild geometry, which is used to describe curved spacetime. Pacifically, the ocean of situations that are "in orbit around a gravitationally large object". A line element, or a distance, in Schwarzschild geometry looks kindasorta like this:

\(\Delta s^2 = (1 - \frac{2M}{r})\Delta t^2 - \frac{\Delta r^2}{1-\frac{2M}{r}} -r^2\Delta\phi^2\), in polar coordinates.

This is one of the very few analytic solutions to the Einstein equation that exist (at least that we know of). I'm gonna not show you how this works out because frankly my dear, I don't give a damn.

Now I'll introduce you to the three observers; The free-, the far- and the (whereeeeever you are) shell-observer. The far observer is one who observers so far away that they do not feel any effect of gravity from the massive object and is therefore in an inertial frame of reference. The free observer is in freefall towards the massive thangamajang, spans an infinitesimal space and is therefore in an inertial frame of reference. The shell observer is stuck on a shell around the object, and is not in an inertial frame of reference.
This means that the freefalling observer can use Lorentzian geometry, and the shell observer can use Schwarszchild geometry, but not that the far-away observer can use Lorentzian. This is because while that observer is in an inertial frame of reference, they are still observing curved spacetime, and therefore have to use Schwarszchild geometry. None of them uses Eucladian geometry cuz that shit's for nerds.
A good rule of thumb for checking if you're in an inertial frame of reference is whether something falls when you drop it. This would happen for the shell observer, but not for the far-away one, as there is a negligible amount of gravity there, nor for the free-faling one, as they would fall together and that would be the same as not moving.

Note: Because this object may be a black hole (and indeed often is in these mind-experiments) you can't really measure the radius physically, as any stick that you might use to measure it (as an example) would just disappear into the event horizon, never to be seen again. A physicist can work around this by saying that \(r = \frac{\text{circumferece}}{2\pi}\), where the circumference is physically measurable, at least.