I asked AI this question about GR:

[Alan Grayson]

Do Free-Falling Objects Accelerate According to General Relativity?

It listed about 30+ links and said it would have to research them and would then give me the full report. About 30 minutes later, I got an 11 page report, which summarized its result. I will post it here if there's any interest. In summary, it wrote:

In general relativity, free-falling objects do not accelerate in the sense of feeling a force; instead, their seemingly accelerated paths in space and time are manifestations of their straight-line (geodesic) motion in a curved spacetime whose geometry is set by matter and energy. Free fall, therefore, is the natural state of motion, and apparent acceleration is a coordinate effect, not a felt force.  

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I didn't find this response useful in that it didn't tell me anything I didn't already know, and it didn't explain "coordinate effect". IMO, for what it's worth, if we don't change the definition of acceleration, test bodies do in fact accelerate in GR if we're willing to use external observers, rather than limiting ourselves to observers subject to the gravitational effects of GR who cannot make external observations. That is, the magnitude and/or direction of motion changes -- which defines acceleration -- even though the internal observers do not feel any force of gravity. This gave me an idea; namely, if test objects are confined to curved surfaces in GR and follow the curvature of these surfaces (aka pseudo Riemannian manifolds) while in free fall, a change of direction is due to curvature, and replaces in part, the directional change part of acceleration. So far, I can't account for changes in magnitude, but it's a work in progress. What I want to know is why geodesic motion is accelerated motion. I might be halfway there.

Finally, I have to say that Brent's diagrams have a fundamental deficiency IMO, since when comparing two spacetime paths which have different lengths, but share the same endpoint events, any two paths have to share something in common to make any conclusion about the comparative elapsed proper times. What they share is the same spacetime velocity, so if their spatial lengths differ, the elapsed times can be compared provided we assume the time differential is converted to spatial length by multiplying it by c, assumed to be unity. Here I am not nitpicking, but I like complete proofs, and this requires statements about key facts.

AG