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From EP we have that gravity is not a force but a pseudo force, i.e an inertial force due to a gravitational field an accelerating a body independantly of its mass, in other words, the trajectory of any body only depends on its initial position and velocity, but not on its composition/mass.

  • Pseudo forces are well described by newtonian mechanics in the accelerated (non-inertial) frames formalism, so why not gravity? Pseudo forces are taken into account by a change of coordinates, as for example fictitious forces on earth are just the acceleration terms coming from the curvature of spherical coordinates. Therefore these pseudo forces could also be described by a metric and a connection and everything in a manifold...

But then why do we need to describe spacetime as a curved manifold and geometrizing (curvature,metric...) gravity? Can't we stick with the (old newtonian gravity+SR) approach, what makes it false?

  • Is it that this old approach does not fulfill SR? But then SR describes dynamics, so can't we describe gravity as a force in SR (it doesn't work?)? Is it because a newtonian grav. field acts instantaneously (faster than speed of light)? In the end, my question is:

How much of GR formalism is derived/constrained from the EP alone? What's different from (newtonian gravity(contains EP)+SR), why is it better? Or are curvature+metric+connection vs Newton gravity (fictitious force)+SR two ways to describe the same physics?

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  • $\begingroup$ I don't understand why you think the equivalence principle directly gives that gravity is a pseudo force. The use of pseudo force for general relativity like this tends to be sloppy anyway, but I cannot follow either your usage of it here or your train of thought connecting EP and what you're calling pseudo force. $\endgroup$
    – Brick
    Commented Oct 16, 2019 at 21:05
  • $\begingroup$ Explained here: en.wikipedia.org/wiki/Equivalence_principle $\endgroup$
    – Voidt
    Commented Oct 16, 2019 at 22:25
  • $\begingroup$ That says something different than what's in your question, and itself is an example of somewhat sloppy language. $\endgroup$
    – Brick
    Commented Oct 17, 2019 at 12:39
  • $\begingroup$ " Therefore these pseudo forces could also be described by a metric and a connection and everything in a manifold" What do you mean by this? Differential geometry main advantage is that it abstracts away the coordinates from the underlying geometrical concepts. Sure you can use the machinery for curvilinear coordinates on flat manifold, but this can never be made equivalent to curved manifold. $\endgroup$
    – Umaxo
    Commented Oct 17, 2019 at 18:52
  • $\begingroup$ @Brick : I quote : " the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (such as the Earth) is the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference". Now, if you disagree with this, I invite you to say why or give me your understanding of how one gets from the first thoughts of Einstein/postulates to the geometrical descriptions of gravity provided by General relativity. $\endgroup$
    – Voidt
    Commented Oct 18, 2019 at 14:26

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GR does not have to be described in a formalism of curved spacetime. You can have, e.g., Deser's spin-2 field on flat spacetime. This theory is inconsistent until you add corrections, which end up making it equivalent to GR.

The equivalence principle is what prevents us from deciding whether a theory like Deser's is what "really" happens, as opposed to the standard formalism of GR using differential geometry. Ultimately the only way we have of telling whether a ruler is straight is to send a test particle along it. There is no way to tell whether the trajectories of test particles are "really" straight or "really" curved.

Misner, Thorne, and Wheeler has a nice discussion of this sort of thing in ch. 17. They discuss distinguishing features of theories of gravity that include the EP, prior geometry, and the existence of auxiliary fields.

Is it because a newtonian grav. field acts instantaneously (faster than speed of light)?

Yes, this certainly forces us to make a field theory rather than a theory of instantaneous action at a distance.

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