General relativity begins from a striking connection between gravity and acceleration.

Inertial and gravitational mass

In Newtonian mechanics, the inertial mass in

$$\mathbf F=m_i\mathbf a$$

and the gravitational mass determining gravitational force are conceptually different quantities. Experimentally, freely falling bodies accelerate alike when nongravitational effects are neglected, indicating their proportionality to very high precision.

Local free fall

Imagine a small laboratory falling freely in a gravitational field. Objects released inside it fall alongside the laboratory, so over a sufficiently small region the occupants can describe themselves as locally weightless.

Conversely, an accelerating laboratory in otherwise gravity-free space can produce effects resembling a uniform gravitational field.

This motivates the equivalence principle: locally, freely falling frames play the role of inertial frames.

Gravity is different from an ordinary force

A freely falling observer does not need a gravitational force to explain nearby freely falling motion. General relativity instead describes free particles as following the natural trajectories of spacetime geometry.

The effects that cannot be transformed away over an extended region are tidal effects: neighboring freely falling trajectories can converge or diverge because the gravitational field varies from place to place.

From special to general relativity

Special relativity describes physics in inertial frames of flat Minkowski spacetime. General relativity allows the spacetime geometry itself to vary with position and time.

At any sufficiently small region around a freely falling observer, the laws of nongravitational physics can be written approximately in their special-relativistic form. Globally, however, no single inertial coordinate system need cover the entire spacetime.

Coordinates are not physics

An apparent gravitational effect can sometimes result from using accelerated or curved coordinates, while genuine spacetime curvature cannot be removed everywhere by a coordinate change.

This distinction is central: general relativity is not merely special relativity written in complicated coordinates. Its physical gravitational content appears in the geometry that remains after coordinate artifacts are separated from invariant curvature.