How would you explain to someone the change that Einstein needed in geometry for his new ideas about gravity and spacetime, what did he seek but could not be described by pseudo-Euclidean geometry?
Assuming you're talking to a highschool student who only knows Euclidean geometry, Newtonian physics, Calculus and Cartesian analytic geometry
Edit. So gravity is the reason that geometry of space can't be Euclidean (correction: this is not quite right, as explained by 2 users). I want to ask more specifically, how? Basically I was expecting answer that contrasts the old gravity (wasn't it just a vector field?) and the new gravity (something more complicated)? Einstein found the old geometry somehow insufficient to describe his new idea about gravity, I want to know more about this part.
Edit 2. I think I'm getting close to an understanding from piecing together various answers. So in the Newtonian picture of gravity:
- There's an issue with action at distance (which seems like a huge problem after the postulate about speed of light was made)
- There's also another issue with tidal gravity in a reference frame free falling in gravitational field (in such frame, you can't transform gravity away if space is described by Euclidean geometry because if you could you'd have two objects initially at rest with zero proper acceleration moving towards each other, or, two parallel straight lines intersect)
In Einstein's picture, space is like a trampoline surface, when empty, it's flat, you put a ball on it, the ball curves the surface and the distortion is local and spreads at speed c. Furthermore you can treat gravity like a fictitious force in a free falling frame: two objects initially at rest with zero proper acceleration just follow their geodesics.
But the answer I have been looking for since the beginning is more quantitative than the above.
I'm looking for an explanation that starts with the Euclidean geometry/Cartesian coordinate description of gravitational field, and ends with differential geometry/metric tensor field/(something else ?) description.
I still have a lot of fog about this part but I think I read somewhere that Einstein had some struggles when he needed to describe gravitational field as a mathematical object with 10 components, also what is its connection to coordinate free differential geometry? I want to know more about these points.