No, this one is new to me. Thanks for bringing it up. Having skimmed through the link, my first impression is that this formalism is not nearly as elegant and intuitive as the standard one (and full equivalence with GR is yet to be shown). I kind of fail to see the advantage, though the point about substructure is interesting.
See studiot’s comments on intrinsic vs extrinsic to begin with. Furthermore, there is not really any force involved in gravity - when you have initially parallel test particles in free fall, and attach an accelerometer to them, it will always read exactly zero, so no forces; nonetheless in the presence of gravity their geodesics will begin to deviate.
Good question! This point is a bit subtle, and really the answer should be “both of the above, depending on context”.
The physical manifestation of curvature is geodesic deviation - meaning that initially parallel world lines will begin to deviate as they extend into the future. It is thus necessary for world lines to have at least some extension in spacetime before “parallel” and “deviate” even make sense - you can’t speak of parallelism at a single event. Thus curvature has measurable meaning only across some distance. I’m highlighting the word ‘measurable’ because counterintuitively the mathematical object describing curvature (Riemann tensor) nonetheless is a local object, like all tensors. For clarification on this point, refer back to the example about calculus in my previous post.
However, there are also scenarios where the effects of gravity are in some sense ‘relative’. Consider a hollow shell of matter, like a planet that has somehow been hollowed out (not very physical of course, but I’m just demonstrating a principle here). Birkhoffs Theorem tells us that spacetime everywhere in the interior cavity is perfectly flat, ie locally Minkowski. There’s no geodesic deviation inside the cavity. Now let’s place a clock into the cavity, and another reference clock very far way on the outside, so both clocks are locally in flat Minkowski spacetime. What happens? Even though both clocks are locally in flat spacetime (no gravity), the one inside the cavity is still gravitationally dilated with respect to the far way one! This is because while both local patches are flat, spacetime in between them is curved - if you were to draw an embedding diagram, you’d get a gravitational well with a ‘Mesa mountain’ at the bottom; and the flat top of that mountain sits at a lower level than the far away clock, thus the time dilation. So in this particular case one could reasonably say that gravitation effects are ‘relative’ between local patches. Or you can put it like this: both regions are Minkowski, but one is more Minkowski than the other
The isn’t very intuitive, but mathematically perfectly consistent - if you look at the world lines of the clocks, you’ll find that while they appear parallel in space (they’re simply at rest wrt to one another), they deviate in spacetime. In GR it is crucially important that one fully understands local vs global, or else there’ll be no end to misunderstandings and problems. This point is where most, if not all, apparent ‘paradoxes’ in GR arise.
In general, no, it’s not a scalar - it’s a rank-4 tensor field, the Riemann tensor. However, you can choose to look at only certain aspects of curvature, such as how volumes change (rank-2 Ricci tensor), or how areas differ from Euclidean counterparts (rank-0 Ricci scalar), or the average Gaussian curvature of a small region of space (rank-2 Einstein tensor). But to capture all aspects, you need the full rank-4 tensor with 20 independent components.
Tensors are not invariant, but covariant - meaning their individual components do vary in just the right ways so that the relationships between the components remain, hence the overall object is the same for all observers. Remember a tensor is all about the relationships between its components.