Why does acceleration in special relativity give rise to general relativity (and thus gravity)?

Question

If we include accelerated motion in special relativity, the result is general relativity. But why should that give rise to gravity? Is that only because Einstein introduced the equivalence between acceleration and gravity?

Answer 1

If we include accelerated motion in special relativity, the result is general relativity.

This is simply not true. Special relativity can easily handle accelerated motion on its own. The result of including accelerated motion in special relativity is just special relativity.

But why should that give rise to gravity?

It doesn’t.

Is that only because Einstein introduced the equivalence between acceleration and gravity?

The equivalence principle implies that gravity can be geometrized. Both GR and Newtonian gravity respect the equivalence principle and can formulated in terms of curved spacetime. So the equivalence principle was instrumental in Einstein’s thought process, but does not in itself imply GR.

Answer 2

Acceleration is included in special relativity. Otherwise SR would be very limited.

An uniformly accelerated rocket generates a fictitious force inside, that is exactly like what happens locally at the surface of a planet. But it can be treated with the tools of SR.

A bunch of observers in free fall in a gravitational field measure, at first order, the velocity of the neighbours as constant, no matter the direction of the relative movement. But if they are not so close, small relative accelerations start to be clear. That deviations from constant velocities between free bodies are the object of GR.

Answer 3

Inertial frames in SR are described by rectangular coordinate systems. For the accelerated observers, curvilinear coordinate systems have to be used. Some people call this general relativity, but the spacetime is still flat. The accelerated observer feels the apparent gravity, but the true gravity is due to the curvature of spacetime. The mathematics required to describe curvilinear coordinate systems is similar to the mathematics required for curved spacetime, thus, the ambiguity in the phrase "General Relativity". However, most scientists nowadays reserve the phrase "General Relativity" for situations involving curved spacetimes only.