I am not sure how the tidal acceleration has anything to do with the equivalence principle.

The point is that it doesn’t.

Why would we need the tidal acceleration to understand the space curvature?

Spacetime curvature, not merely space curvature. GR is a theory of space

*time* curvature.

Spacetime curvature and tidal accelerations is the same thing expressed in different languages. The former in terms of GR and differential geometry, the later in Newtonian mechanics terms.

I’m going to assume that you know what tidal acceleration/gradients are so I won’t define them.

Let’s talk about curvature for a moment. Picture the surface of the earth (i.e. the surface of a sphere). The surface of the sphere is an example of curved surface. Please don’t confuse the fact that the surface of a sphere also has what is called

*explicit curvature* meaning that the curving occurs because the surface curves into a higher dimension. The surface of a cylinder has non-zero explicit curvature and zero intrinsic curvature.

Now go to the equator with a friend. Stand a distance apart along the equator while you each face North. Now both of you start walking north and remain walking in the “straightest possible line”. Eventually you’ll both reach the North Pole where your paths will meet. Thus two geodesics started off parallel but intersected.

From a physical standpoint the deviation of geodesics in spacetime in a gravitational field is what happens when two particles paths converge even though they started out parallel. This is caused by tidal accelerations. If you understand why dropping two apples at the surface of the earth will cause them to accelerate together then you’ll probably understand this argument.

Just out of curiosity: How would you explain the concept of space curvature in a more pedagogical/intuitive fashion?

I’d write a book and illustrate this in diagrams or create a web page with diagrams. It’s easier to see all of this if you visualize it.

What tidal acceleration? In the first case the observer is assumed to be in empty space sufficiently far from any source of gravitational field and moving with an acceleration a=9.8m/s^2.

While that’s true that is not what you described. You repeatedly spoke of this happening on the earth and the gravitational field of the earth is not uniform.

As far as why a certain aspect is correct that's why I provided a link to the book. It's too difficult to explain entirely and correctly in a thread.