As is known, if body stays on the ground then body is accelerated upwards by the ground.
For example, free-falling observer see that body on the ground move to him with acceleration ##9.81m/s^2##

I have a problem in understanding how it is possible.

If the body is being accelerated by the ground then this is only to prevent body to fall down. But how is it possible that body accelerates (and move) relative to free-falling observer?

Thanks.

You can drop an object and visually see that relative to the object the ground is moving.

Here perhaps a too simple explanation:

Three stones: A and B rest on the ground, C falls.
A: "B is at rest relative to me, we both feel an acceleration (## F/m = g ##), and C falls toward us."
C: "I feel nothing (## F/m = 0 ##), and I see A and B accelerate toward me."

It is possible that a body accelerates relative to a free falling observer simply by that body having a force applied to it. So the free falling observer can light a bottle rocket and off it goes (good trick in vacuum, but I didn't say vacuum).

Differences: Under Newtonian mechanics, gravity is exerting a continuous force on everything near Earth. So a house is stationary because the downward force of gravity is perfectly cancelled by an equal and opposite force applied upward by the ground. The guy jumping off the cliff accelerates due to gravity since nothing counters the force (at first).

Under GR, there is no gravitational force, so the guy jumping off the cliff has no forces applied to him (again, at first), so he's in free fall. No acceleration and no proper acceleration. The house on the other hand has an unbalanced force applied upward to it, so it (and the observer on the ground) all accelerate upwards. They have positive proper acceleration, as can be measured by any accelerometer.

As for reference frames, those two guys each using reference frames where they are stationary and the other guy is moving. Under Newton, the free falling guy is using an accelerating reference frame and the guy in the house is in using an inertial one. Under GR it's the other way around, with the freefalling guy being inertial and the guy on the ground using an accelerating frame.

The bold part answers your question in post 3

You have to specify the reference frame to define motion, that's the key.

Here perhaps a too simple explanation:

Three stones: A and B rest on the ground, C falls.
A: "B is at rest relative to me, we both feel an acceleration (## F/m = g ##), and C falls toward us."
C: "I feel nothing (## F/m = 0 ##), and I see A and B accelerate toward me."

An observer in a locally free falling elevator-cabin can throw a ball an notices, that the ball moves with constant velocity along a straight line, as discribed in Newton's first law. Therefore, the falling frame must be inertial. It has no proper acceletaion.

This video can provide a rough idea:

V = 0 doesn't matter.
Throw C upward: at the top ## v=0 ## for an instant, but C ir's still free-falling all the time (accelerometer = 0).

This video can provide a rough idea:

If C is free-falling why can't we say that Earth moves with acceleration ##9.81m/s^2## relative to stone C?

An accelerometer attached to a body on the ground reads 9.8 m/s^2 upwards. That also has nothing to do with any observer. Every observer will agree that the accelerometer reads 9.8 upwards, so every observer will agree that the body on the ground is non-inertial.

This is proper acceleration, the acceleration measured by an attached accelerometer. This is a relativistic four-vector, and all frames and all observers agree on this vector, even if they use a different basis to describe it.

Let we have water, the motorboat and the boat. Water(blue) and the motorboat (grey) are non-inertial frames. The boat (green) has constant velocity ##V## (inertial frame).
Water flow moves with acceleration ##a## and motorboat moves with the same acceleration. The motorboat is at rest relative to observer on the shore (by analogy with the non-inertial still body on the surface of the Earth). But relative to the man on the boat (inertial frame) the motorboat is still. As is mentioned above, non-inertial frame move with upwards acceleration to free-falling observer. But my example show that non-inertial frame is still relative to inertial frame.

How is it possible that non-inertial body on the surface of the Earth move with upwards acceleration to free-falling observer if non-inertial body is still relative to the surface?

Where did I wrong? Thanks.

The only GR specific part is which frame is considered inertial, but you seem stuck at basic kinematics.

Yes, you obviously have:

Not in the free-falling frame of reference that you were talking about:

Otherwise this thread is just as pointless as your previous threads.

Just as if you are standing on the surface of the Earth, you're not accelerating (in the sense of coordinate acceleration) relative to the Earth. But if you drop a rock, it accelerates (in the sense of coordinate acceleration) downward relative to you.

In both cases, if you use a frame in which the dropped rock, not you, is at rest, you (and the lift/surface of the Earth in your vicinity) will be accelerating upward (in the sense of coordinate acceleration) relative to the rock.

And in both cases, an accelerometer attached to the rock reads zero, and an accelerometer attached to you reads a nonzero upward acceleration.

If you don't understand all of the above, you don't understand Einstein's lift thought experiment, since all of the above points are part of what that thought experiment is intended to illustrate.

Proper acceleration is frame-invariant, all frames agree.

Motion is frame-dependent, different frames disagree.

Dale,
Thanks.

Could you elaborate how does constant upwards acceleration produce motion?

I've found explanation why the body doesn't lose contact with the support and accelerates upwards. Infinitesimally small deformations play key role in the process of the acceleration. These deformations cause atoms to accelerate synchronically and the body accelerates.

I think that since deformations are continuous then we get that body is accelerated upwards. We can consider motion as sum of infinitesimally small displacement that is caused by infinitesimally small deformations.

You can put one accelerometer on rubber and another on steel. Both accelerometers read the same acceleration, but the deformation of the rubber is much more since it is less stiff.