Questions on Spacetime Curve/Gravity

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Questions on Spacetime Stretching/Gravity

Is it true that the apple didn't fall to Newton's head, but that Newton, the base of the tree, the ground and everything else rose up to meet the apple? I know it is true to some extent, depending on whose perspective, but I am looking to see if it would be true that space stretching or swelling (expanding) and gravity are basically one and the same thing. If space is expanding, stretching, then when we let go of something and let it "drop", is it really that we're observing space expanding upwards from the Earth all around it's sphere, making what we dropped "appear" as though it were falling toward the earth?

 

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Is it true that the apple didn't fall to Newton's head, but that Newton, the base of the tree, the ground and everything else rose up to meet the apple? I know it is true to some extent, depending on whose perspective, but I am looking to see if it would be true that space stretching or swelling (expanding) and gravity are basically one and the same thing. If space is expanding, stretching, then when we let go of something and let it "drop", is it really that we're observing space expanding upwards from the Earth all around it's sphere, making what we dropped "appear" as though it were falling toward the earth?

As you remark, it's a matter of perspective as to who is moving and who is staying still. In "apple" coordinates, it's the Earth that's moving.

There is not, as far as I know, however, any widely used coordinate system in which all apples are at rest - we can find a local coordinate system in which any individual apple is at rest fairly easily, but it is not so easy to imagine a global coordinate system in which all apples are at rest (or at least moving with constant velocity).

Even the local coordinate system of the apple tends to have unusual properties, such as a choice of not covering all of space-time, or the choice of being non-orthogonal.

"Apple coordinates" are usually used only near the apple, they are not intended to describe the universe, and in many cases they can't be made to describe the universe as some points in the universe are not included in "apple coordinates".

The "expanding space" idea would only make numerical predictions in a universal generalization of "apple coordinates" in which all apples were stationary, f I'm interpreting your remark correctly. It's not clear that such a coordinate system exists. If it does exist, it probably can't be constructed with all the nice properties of coordinate systems that you are used to (i.e. the nice proprerties of orthonormal coordinates).

So at this point it's unclear to me if the "expanding space" idea can be made to work or not, though I lean towards the "not" position (if one demands that the "expanding space" coordinates be orthonormal).

It is fairly clear, though, that "expanding space' is not a widely used idea for actual calculations - it's not how GR is currently done. GR takes the approach that coorrdinate systems are irrelevant, and allows one to use whatever coordinate systems are convenient.

As a practical matter, it tends to be convenient to keep the orthogonality property, but sacrifice the normality condition - i.e. distances and times have a "scale factor" that depends on position, so that coordinate time is not equal to the time reading given by local clocks.