Why does the Earth accelerate upward, according to Einstein? I recently watched a video on the YouTube channel PBS Space Time which was called "Is Gravity an Illusion?". In this video, the host explains that Einstein claimed that it is not the apple that accelerates towards Earth but the other way around. In Newton's theory, the apple accelerates down because the Earth's gravity is pulling it down, but in GR, the Earth accelerates up because of––what?
I am familiar with geodesics and spacetime warping in GR, but am new to this concept of "Earth accelerating upward" and do not understand it very well.
 A: In general relativity, gravity is not a force. Any object will follow the path of a geodesic if there are no forces acting on it. Such is the case of the apple, which follows the path of a geodesic when it moves towards the Earth. In the video, what they refer to as acceleration is actually the four-acceleration. The apple has nonzero coordinate acceleration (which is what you observe), but it has zero four-acceleration (this is the definition of a geodesic).
So why do they say that the Earth is accelerating upwards? First, I ask, if you examine a portion of the ground laying on Earth, what are the forces acting on it? In a Newtonian sense, you would say there is a normal force upward which balances the downwards gravitational force. In a relativistic sense, however, gravity is not a force! So the only force acting on that portion of the ground is the normal force upwards, which gives rise to a nonzero four-acceleration, with zero coordinate acceleration.
A: At each point on the surface of the Earth, the local ground is being pushed outwards by compression forces in the rocks, whose ultimate cause is electromagnetic repulsion and quantum effects such as the Pauli Exclusion Principle.
A: It depends upon the frame of reference. 
If one adopts the frame of reference of the apple then the Earth accelerates towards it. 
If one adopts the Earth's frame of reference then it is the apple that accelerates towards it. 
In GR, there is an equivalence to frame of references and so either scenario is correct.
An object moves up or down relative to another object and so the Earth "moving up" is the result of the apple's frame of reference being adopted.
A: Well, the thing is, if we want to analyse the motion an object, we need an inertial frame of reference. By equivalence principle, free-falling objects are inertial frame of reference.
So we have to analyse it from that point of view. But one of them should move, right? To get the visual. So, it's supposed that the earth moves upward with acceleration $g$.
But, the theory of relativity which predicts that gravity is just a pseudoforce, also tells us that there is no absolute space.
It's just Newtonian Classical Mechanics breaks down and much better GR comes up.
Still, you can ask, then why don't we move up, if there is only normal force to push us up and no gravity to equate it. But your spatial coordinates may not change even if you are accelerating. This is also the result of GR.
You can watch a Veritasium video on the same topic for more understanding.
