An object sat on the surface of the earth experiences a 9.8m/s^2
True, but it also experiences a 9.8m/s^2 acceleration downwards due to gravity. The two balance out and so the object doesn't move.
Lets be clear here, the accelerometer, in its own frame of reference, experiences
an acceleration upwards - that's all it knows (apart from the tidal forces, but they
are much much smaller).
In our frame of reference we also experience an acceleration upwards - we are in
the same frame, to a first approximation, so see the accelerometer as stationary.
Nothing is or appears to be accelerating downwards from this frame. The force from
the floor is not cancelled out by anything, its a true upwards acceleration we feel
from our feet.
This is the key insight behind general relativity, that inertial and gravitational mass
are identical and apparent forces due to gravity are inertial forces.
The fact it doesn't appear to move due to this acceleration is because we are accelerating
No. We aren't accelerating and neither is the object. If we were accelerating we'd be moving.
see answer above.
There are two opposing forces on the object. The earth's gravity trying to pull it towards the centre of the earth and the force of the ground pushing it up by exactly the opposite amount. If the two forces aren't balanced, the object will be moving.
There is no force due to gravity, there is only curved space, described by the
The fact that we stay the same distance from the centre of the earth is because
space is curved.
No. The curvature of space around earth is caused by the earth's mass and the result of the curvature is what we perceive as gravity.
Yes the curvature is caused by the mass, and YES that's why we don't appear to move
relative to the earth's centre. We perceive the force from the floor, which is due to
the earth being an elastic body, resisting the deformation that the curvature gives it.
An object in free-fall has zero acceleration
That is a contradiction. An object in free fall is, by definition, accelerating.
An object in free fall is free of all forces, and therefore subject to zero acceleration
in its own frame - this makes it an inertial frame (apart from those pesky tidal forces,
themselves the signature that space is curved).
I recommend a course on general relativity. Its been validated by several elegant
experiments since Einstein proposed it. Without it you cannot explain why atomic clocks run
apparently faster at altitude - with it you realise you have clocks in different accelerating
frames hence the difference