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

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?

• An accelerated linear motion is well-covered by special relativity. Commented Aug 26, 2023 at 16:41
• This may help - Why can't I do this to get infinite energy? Commented Aug 26, 2023 at 17:03
• In special relativity, you can easily study an accelerated particle when viewed from an inertial reference frame. If you study anything in an accelerated (non-inertial) reference frame then things get more complicated and you get a formalism that looks indistinguishable from general relativity, in fact, this is precisely how the formalism of general relativity is determined, and the differences between gravity vs inertial frames come down to small but important technicalities such as the behavior of gravitational fields vs reference frames as you go off to infinity. Commented Aug 26, 2023 at 17:21
• Commented Aug 26, 2023 at 17:36
• It doesn’t give rise to general relativity. This is a misunderstanding which unfortunately propagates a lot. General Relativity is a theory of gravity not merely acceleration Commented Aug 27, 2023 at 9:10

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.

• A very nice answer. Commented Aug 26, 2023 at 17:07
• So it´s actually gravity causing accelerations to have different effects from those in flat spacetime? Commented Aug 27, 2023 at 5:03
• @IlGuercio curved spacetime produces tidal gravity. Gravity that changes from place to place.
– Dale
Commented Aug 27, 2023 at 10:01
• So the tidal effects are essential to gravity? Is it the tidal effect that is present only in real gravity, so not in accelerated rockets? If so, doesn't that invalidate the equivalence principle? Or is that a first-order effect? I had an argument with someone. Maybe you can help me out. I said that the only instant where the equivalence principle doesn't hold is in the case of an infinite massive plane. A photon that starts out horizontally stays horizontal, while a massive particle drops down. What do you think? Commented Aug 27, 2023 at 10:12
• @IlGuercio yes, real gravity is characterized by tidal effects which is modeled by spacetime curvature. The equivalence principle is defined as a local principle. It only applies over regions of spacetime small enough that tidal effects are negligible. I haven’t studied an infinite massive plane in that level of detail
– Dale
Commented Aug 27, 2023 at 13:21

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.

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.