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I think the missing piece in your puzzle is adding the "time" to space-time. A particle moving through time is "falling" in the time dimension. The curvature of the 4d space makes this movement through time also move through space due to the warping effect of a gravity well.

There is also the asymmetry of time dilation. Someone moving in a rocket ship will be younger than the person staying behind. Isn't their relative acceleration the same though? Why is the rocket man privileged in staying young? This kinda works if you imagine a grid of sand particles being acted upon by either gravity or traversed by the rocket. The rocket traverses many more grids than the person standing on Earth. The person floating in naked space will traverse the fewest grids.

The kicker is that there is no place in space with zero grid traversal. Accounting for all grid traversals becomes a nightmare.

what do you mean by "just sits there"?

Sits where?

In the same position relative to the sun? Relative to the center of the Galaxy? Relative to the local galaxy supercluster?

A particle does not just sit still!

Site there relative to that other mass that wants to accelerate it toward each other.

You’re moving in the time direction — and when spacetime is curved, you end up moving in the space direction as well, to follow the geodesic.

The river model of general relativity.

https://www.youtube.com/watch?v=hFlzQvAyH7g&list=PL__fY7tXwo...

Not arguing for or against this explanation, but it's hard enough (impossible for me) to visualize a curved 3d space, once you try to think of what curved 4d space might be like any intuitive sense of what should happen goes out the window.

Classic diagrams of curved spacetime portray space as a 2d sheet with divots in it. I wonder what a similar visualization of curved 1d time + 1d space would look like. Long valleys for gravitational wells and everything moving along the sheet in the direction of positive time?

> and when spacetime is curved, you end up moving

You did the same as everyone else: You hard waved right past "end up moving". Why do I end up moving instead of just sitting there?

And if I "end up moving" that means I exchange momentum, if it's just bent space that magically makes me move, what did I exchange momentum with, if not via a force toward another particle?

Gravity is a force that acts on energy. If you disagree please give me a counter example.

You do just sit there. In the GR-realism view, the only thing particles _ever_ do is self-propagate forwards in time.

The problem is, this means the particle’s definition of “time”. Spacetime is 4D, and this can be any timelike direction. That’s what causes it to appear to move, from the perspective of someone whose time vector is pointed differently.

Then time can be curved, which reorients said vector as the particle propagates through it. That’s gravity.

> Really? Why exactly will they free fall? Why can't they just stay exactly where they are?

Absent any forces, they can only stay on a geodesic. It’s like the arrow that springs from the bow. No hesitation, no doubts. The path is clear. There’s only ever one geodesic, one path through spacetime that takes no energy to follow.

This is true even if you ignore general relativity and just focus on classical mechanics, as formulated by Lagrange. In Lagrangian mechanics, you must compute a number called the “action” for every path through space that your system could possibly take. Then your system will take the path that minimizes the action. If your system is a particle in space, then it will stay still if that has the minimum action, or it will follow some path through space if that has the minimum action. It can’t ever do both. Only one of them will minimize the action.

> one path through spacetime that takes no energy to follow.

Again, this if fine for an orbit for example, it does not explain how a particle will accelerate, it also does not explain momentum, because when that particle accelerate it exchanges momentum with another particle - it does not give momentum to the "geodesic".

And explain please what information is not captured by saying "Gravity is a force that acts on energy". What nuance is lost by saying that?

Particles in orbit do accelerate. They accelerate continuously, from our perspective, as they go around the orbit. But seen from the perspective of GR, that orbit is really just a straight line through a curved spacetime. It might be easier to visualize though if you think of it as the bottom of a valley. The walls of the valley rise away from the floor as you get closer to the planet you are orbiting, or further away. You can’t climb those walls without energy, so if you have no energy will you stay in the valley forever.

> it also does not explain momentum, because when that particle accelerate it exchanges momentum with another particle - it does not give momentum to the "geodesic".

No one has said that it should. The geodesic is merely the straight path that the particle must follow when it _isn’t_ exchanging momentum with other particles.

> And explain please what information is not captured by saying "Gravity is a force that acts on energy". What nuance is lost by saying that?

You can’t explain time dilation that way, or length contraction.

A charged particle on a straight line does not emit synchrotron radiation. What happens if it is on a geodesic? Do electrons that circle around a black hole emit synchrotron radiation?

That’s beyond my level of expertise, but yes.

Quoting from the abstract of The Motion of Point Particles in Curved Spacetime <https://web.archive.org/web/20151203142700/http://relativity...>

> This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise to a self-force that prevents the particle from moving on a geodesic of the background spacetime. The self-force contains both conservative and dissipative terms, and the latter are responsible for the radiation reaction. The work done by the self-force matches the energy radiated away by the particle.

Note especially that the charged particle won’t actually follow the geodesic. Because the synchrotron radiation carries away some energy, the particle will be continually pushed off of the geodesic; it will spiral inwards instead of orbiting forever.

Does this mean that when the geodesic is a straight line then the particle stays on the geodesic, but when the geodesic is curved the particle interacts with its own e.m. field and deviates from the geodesic because it feels a deflecting force?

Then there would be a fundamental difference between curved and straight geodesics which would be a contradiction to the Einstein Equivalence Principle. How do physicists explain that contradiction?

The equivalence principle only applies to uniform fields and uniform motion. A charged particle in orbit is not in a uniform field and is not experiencing uniform motion.

Hmm, if you look at another comment here: https://news.ycombinator.com/item?id=41971973

It seems like the point is that the particle is actually not accelerating when it follows the one path through spacetime. It only accelerates when it's stopped by something (e.g. the surface of the earth).

Another angle that might help: Newton was correct when he said a particle with no forces on it would keep going in a straight line. He was just missing a dimension in his coordinate system. The geodesic is the straight line in spacetime. And mass distorts the spacetime metric (mostly time) in a nonlinear way, so when you project that straight line back into 3D space, it’s not linear, it’s a curve. An “acceleration” curve.