EDIT: my mind is totally blown by this; it put together some pieces of general relativity in a new way for me today. thanks HN!
Here's a way of thinking about what's going on.
Consider a large mass, like the Earth. That mass curves spacetime in the vicinity of the Earth. A small object, like a satellite, simply moves completely freely in a "straight line" (i.e., geodesic path) according to that curved geometry. The curved geometry is such that those geodesics are just the satellite orbits that we see. In other words, the satellite isn't affected by any "force", it's just moving in a straight line in a geometry that happens to be curved, so we see it doing circles around the Earth.
It's exceedingly neat: in John Wheeler's great phrase, matter tells spacetime how to curve, and spacetime tells matter how to move. No forces required! Of course, after the fact you can tack on a notion of "force", but it's in no way fundamental.
Same thing is going on with projectile motion here on Earth.
Once you internalize this point of view, the statement about holding your arm in place becomes a lot clearer. The "natural" force-free thing your arm wants to do is to move along geodesics of spacetime, which means falling toward the Earth at an acceleration of g. But if we exert a (muscular) force to keep it up, then we can hold it in place. In short, in this point of view, forces are things which cause deviations from geodesic motion.
But, as you say, I haven't seen better.
See "equivalence principle".
I never really understood this part. General relativity gives us one explanation of gravity (mass curves space-time), while quantum theory tells us that gravity arises from the exchange of particles. So... how do we reconcile? I imagine if we ever discover a unified theory, it'll manage to deal with that. But how does say string theory (for lack of other examples) reconcile these two views?
...or is it something as simple as the exchange of particles causes the warping of space >.>
No, it doesn't. That statement is wrong on two levels:
1) We don't have a confirmed quantum theory of gravity
2) In quantum field theories in general it's not really accurate to say that "[forces] arise from the exchange of particles". In some QFTs, in some situations, you can perform a perturbative expansion which maps onto the Feynman diagram view of particle exchange, but that's not universally possible. They really are quantum _field_ theories, not quantum particle theories.
The first definition is tautological: First, you define a curved space, accounting for gravitational forces such that a small object will follow a certain geodesic. Then, you say that since it does indeed follow said geodesic in that geometry without deviation, gravity is not a force. Well of course not!
With the same same hand-wavy argument, you could define the curvature of space based on electromagnetic field strength, for example, and argue that electromagnetism is not a force.
>With the same same hand-wavy argument, you could define the curvature of space based on electromagnetic field strength, for example, and argue that electromagnetism is not a force.
No. You're very wrong here. You (irritatingly) seem to think that because you have only been exposed to popularized hand-wavy arguments, that physicists necessarily rely on them.
The vast majority of possible forces (including Newtonian gravity and electromagnetism) can not be explained the way gravity is explained. Your confusion probably comes from not understanding how much smaller the space of possible geometries is than the space of possible forces.
In fact, the stunningly beautiful thing about gravity is that it can be derived from knowing only that it is a geometric effect and obeys a few symmetries. Even if you relax some of these symmetries (e.g. Brans-Dicke theory) the resulting parameterizable space of possible geometries is very restricted.
Not really so much, since different objects couple to electromagnetism differently based on their charge. All objects follow the same geodesics in general relativity.
As far as we know .. I'm trying to imagine particles that do not [Higgs-teflon engage] I'm assuming they would be extra-dimensional (by definition when considering geodesics).
If you decouple particles from the Higgs field where do they go.
When I walk I push the earth around, and drag the universe along with it a little.
Another way of looking at it is to imagine that you are standing in an elevator in space that is accelerating with the speed of 1g. That scenario will feel exactly the same as if that elevator was standing still on the surface of the Earth, but it's more clear that the elevator is pushing you, and not the other way around.