The easiest way to start conceptualising gravity as a geometric effect is to ponder the simplest toy model: the convergence of great circles on a sphere. Two nearby meridians are “as close to parallel as they can be” at the equator, and ships/planes that follow them will certainly not be accelerating sideways, but they will nevertheless draw closer together.
That takes the geometry for granted not explaining why (or in precisely what manner) space-time geometry is curved by the presence of mass, but it captures the essence of the kinematics. The worldline through spacetime of a free-falling body is a geodesic, and the *failure* to follow a geodesic, which involves what’s known as a “proper acceleration”, requires a force, and is perceived as weight.
The weight you feel as you stand, motionless, with respect to the Earth is due to the fact that your worldline is curved compared to the spacetime geodesic that would take you towards the centre of the Earth. But in principle, that’s no stranger than the sideways force required to keep a ship/plane at a constant (non-equatorial) latitude, rather than following a great circle.
Surprisingly, if you are on Earth and we let a stone free fall and it hits your head, it is not the case that the stone is accelerating, it is following a geodesic: you are accelerating towards the stone!
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Author of the notes: Antonio J. Pan-Collantes
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