Suppose you are in a windowless box, your feet pressed against the floor, and anything you let go of falls to the floor. There are two possible situations.
Either the box is sitting on the Earth, or it is accelerating upward at m/s² somewhere with no gravity. Inside the boxes on the left and right of the figure, things fall in exactly the same way.
No experiment inside the box can tell these two apart. That gravity and acceleration cannot be distinguished, at least over a small region, is called the equivalence principle.
Now let the whole box fall freely. You and whatever you were holding fall with the same acceleration, so something you let go of floats beside you.
Inside the box gravity has no visible effect, and it cannot be told apart from floating in space with no gravity. The inside of the International Space Station is weightless because the station keeps falling around the Earth.
Mass appears in Newtonian mechanics in two places: the inertial mass that sets how hard a body is to accelerate, and the gravitational mass that sets the strength of gravity.
The two enter through separate laws, yet they always measure equal. Results from the MICROSCOPE satellite, published in , show agreement to about .
Because they are equal, heavy and light bodies fall with the same acceleration. In Newtonian mechanics this agreement has no reason; the equivalence principle takes it as the starting point.
If the inside of a falling box is the same as a world with no gravity, then falling freely can be taken as the state of feeling no force at all.
Gravity then stops being a force that pulls. A body under no force goes straight through spacetime, and that straight path appears bent near a heavy body.
The theory that writes gravity as the curvature of spacetime is general relativity. The upper row of the figure draws a fall with gravity as a force; the lower row draws the same fall as a bend in the path.
Make the falling box larger and things change. Two bodies falling toward the Earth both head for its centre, so they slowly draw together.
At the Earth's surface, two bodies a distance apart approach each other with acceleration ( is the Earth's radius). For m this is m/s², and the effect is called the tidal force.
The tides arise from the same tidal force, the Moon pulling different parts of the Earth with different strengths. Being removable over a small region but not over a large one is the same property as a curved surface looking flat over a small patch.