Is my shallow understanding of spacetime curvature effect correct?

Question

in this description of spacetime curvature:

as far as i understand, and in contradiction with what I've seen online, this doesn't change the path of the apple, it rather shows the earth accelerating up until it reaches the apple, and the path of the apple is not affected until the earth reaches it and makes the apple accelerate with it.

and it's curved in such a way, so that this apple on the other side of the planet:

experiences the same thing (earth accelerating towards it).

is my understanding correct?

ScienceClic is an outstanding resource to learn General Relativity, so you are going to the right place. — RC_23, Jan 29 '23 at 17:17
An answer to Why would spacetime curvature cause gravity? has a link to a good animation: General Relativity : Einstein vs. Newton — mmesser314, Jan 29 '23 at 18:48

score 0 · Answer 1 · answered Jan 29 '23 at 14:53

Welcome to the site!

The question regarding what is accelerating towards what needs to be preceded by asking what is acceleration? When you take into consideration General relativity, you need to distinguish Proper acceleration and Geometric acceleration. The curvature of spacetime due to gravity means that we will measure Geometric acceleration of the objects in the gravitational field, but this is different from Proper acceleration in several ways. In particular, the apple can't distinguish between falling in a gravitational field and floating in deep space, at least as long as:

It doesn't hit the ground :) and
The gravitational field isn't so strong as to induce tidal forces across the length of the apple...

Proper accelerations arise from external forces, but Einstein taught us that gravity isn't really a force, hence it doesn't induce acceleration in the classical sense of the word, but it does mean we will measure objects as accelerating even when no external forces are acting upon them, due to the spacetime curvature. This is when we say that such objects follow Geodesics.

score 0 · Answer 2 · answered Jan 29 '23 at 16:46

Yes, your understanding is essentially correct. Accelerometers at rest on the surface of the earth indicate that each part of the surface of the earth has a proper acceleration in the direction away from the center of the earth. Due to curvature this is true even though the surface of the earth is not expanding or the distances between points on the surface is not changing.

This may seem like a strange combination, but it happens even with curved paths in flat spacetime. For example, this situation can be seen on a rotating carousel. There, points in the rim of the carousel have a proper acceleration in towards the center, but because of their curved path the rim of the carousel is not contracting.

Is my shallow understanding of spacetime curvature effect correct?

2 Answers2