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The equivalence principle states that there is no difference between an accelerated frame of reference and the acceleration from a gravitational field. Since special relativity is able to describe accelerated frames of reference, why do we need general relativity in order to explain gravity?

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  • $\begingroup$ The equivalence principle talks about uniform gravitational fields. $\endgroup$ – Javier Sep 22 '20 at 18:31
  • $\begingroup$ @Javier If we have a particle in a non-uniform gravitational field, can we not say that the particle moves as if affected by a local and instantaneous uniform gravitational field? That the equivalence principle is valid locally at every instant? $\endgroup$ – E. E. Hammervold Sep 22 '20 at 19:32
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Well, you can use special relativity for gravity! To follow Misner, Thorne, and Wheeler: If a physicist in a box is accelerated, by the equivalence principle, the physicist can not convince himself by local experiment that he is not in a gravitational field. However, by the same principle, a physicist in a box in a gravitational field can not convince himself he is not merely accelerating; under such a delusion, he can work out gravitational problems using the mechanisms of special relativity, so long as he works in small enough areas where spacetime is essentially flat (as the equivalence principle holds only in infinitesimally small areas). Thus, by splitting up spacetime into locally flat places, working with special relativity there, and then putting the pieces back together, one can solve gravitational problems (and this is a viable and used strategy). But in doing this, one finds that the dynamics of curved spacetime emerges naturally! Thus, general relativity both emerges naturally from and is built on special relativity, and one can do general relativity with special relativity, if one is sufficiently careful.

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  • $\begingroup$ That's really interesting. Can GR be fully described by the two postulates of SR plus an equivalence principle postulate? $\endgroup$ – Ryder Rude Sep 23 '20 at 5:23
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    $\begingroup$ @RyderRude one needs one last thing: the local nature of physics. One must deal with small local areas for SR to hold. $\endgroup$ – John Dumancic Sep 23 '20 at 15:59
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The equivalence principle has a very limited scope. SR and GR are equivalent if you are in a box and have no input other than the measured acceleration, which can be either due to gravity or increase of speed. Adding additional information will 'break' the equivalence and cause GR and SR to part ways. The classic example is tidal forces, which would be present when the acceleration is due to gravity but not for change of velocity. A more interesting example would be if you had a clock in you box reporting the time of another non-accelerating reference. In a gravitational field the difference between that clock and your clock would be fixed. The non-accelerating reference would (probably) be faster, but the difference at which is was faster would be fixed. In a truly accelerating reference frame, the clock from the non-accelerating reference would get faster and faster over time. SR would not explain that, while GR would. (the example ignores doppler).

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For example: the equivalence principle says that someone at rest is equivalent to the ground be accelerated upwards at an acceleration $g$.

But the Moon, Sun, and other planets should be left behind after some time, to be consistent with a constant acceleration.

So, the local requirement for the validity of the principle of equivalence also means: short time interval.

I order to be free of that (and also from the spatial) constraint, we need curvilinear coordinates and covariant derivatives. That is the realm of GR.

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You can treat gravity in special relativity as a spin-2 field on a background of flat spacetime: S.Deser, Gen Rel Grav 1 9(1970), http://arxiv.org/abs/gr-qc/0411023 . The resulting theory locally makes the same predictions as GR, but globally it's restricted to spacetimes with the same topology as Minkowski space, which means that it can't describe spacetimes like black holes or cosmological models.

Claudio Saspinski says in an answer:

I order to be free of that (and also from the spatial) constraint, we need curvilinear coordinates and covariant derivatives. That is the realm of GR.

No, GR is distinguished from SR by the flatness of spacetime.

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