system
Given the following post, respond as if you're commenting on Reddit:
human
Could someone step me through the math/deduction/reasoning that Einstein did to come to the conclusion that bodies of mass and energy distort space-time, hence macroscopically observed as gravity?

Large or small, Responses all!
gpt
You need three assumptions:

The laws of physics are the same for all inertial observers.

The speed of light is the same observers.

An observer in freefall is in an inertial frame.

A corollary of the third is that acceleration is indistinguishable from a gravitational field. Imagine being in an elevator in outer space, and suddenly you feel attracted to the floor. You can't tell if the elevator is accelerating one way, or if gravity is pulling you the other way. Now, imagine that there's a hole in the side of the elevator, and someone fires a laser through it. If the elevator is accelerating, the laser will follow a curved path (it's dusty inside, so you can see it). But because acceleration is indistinguishable from gravity, it could also be a gravitational field that is causing the laser beam to curve. But if the speed of light is the same for all observers, it means the path that light takes must be the shortest path, or else you could follow a Euclidean line and get there faster than light. This means that in order to be consistent with these assumptions, the presence of a gravitational field made it so that the shortest path through spacetime, the one that light takes, is curved.