Does centrifugal force exist? Of course not, it is just inertia (but try telling that to the observer in the rotating frame). Similarly, at the scale of classical physics, gravity is just inertia in curved space-time. Put another way, when I am in freefall (eg, in orbit) there are no forces acting on me and it can therefore be argued that gravity doesn't exist.
It depends on the frame of reference. Say you were in a spacecraft...If the spacecraft accelerates upwards at 1 G, then you would feel as if you are on Earth. But if the spacecraft is applying no thrust, relative to the spacecraft, it is as if no forces are acting on you.
This is true in Newtonian space-time, but not in general relativity. Consider why the inertial mass is identical to gravitational mass to within as much accuracy as we can measure. Occam's razor would suggest a solution where they are the same because gravity is actually just inertia manifested as a pseudo-force.
Well, if you are in orbit, you would be subject to gravity gradient forces, which are theoretically measurable along the length of your body.
Likewise, the Earth is subject to Sol's gravity gradient, but that would be a very difficult thing to measure in a practical sense. The ancients certainly couldn't do it, but they managed to figure it out by sky observations instead.
Gravity gradients are no separate force to gravity, it is just an artefact that is measurable for (semi)rigid bodies since each particle in the body has its own inertia and they react against each other. And it does not imply that gravity is not just inertia. The same gradients can be observed in centrifugal force by a rotating observer.
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