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I have read this question:

How fast does gravity propagate?

where hawkeye says:

So what does that mean? It means that the "speed of gravity" is the speed of light … technically. Changes in the geometry of spacetime actually propagate at the speed of light, but the apparent effects of gravitation end up being instantaneous in all real-world dynamical systems, because things don't start or stop moving or gain or lose mass instantaneously for no reason. Once you factor in everything you need to in order to model a real system behaving in a realistic manner, you find that all the aberrations you might expect because of a finite speed of light end up canceling out, so gravity acts like it's instantaneous, even though the underlying phenomenon is most definitely not.

When electron moves constantly, it's electric field moves with it instantly?

where Albert says:

Yes, in a sense, the field "instantly" moves together with it's source (if this source moves uniformly). That does not mean that the force propagates infinitely fast. The force on a test particle at any given instant is due to the electromagnetic field in the immediate vicinity of the particle at that instant.

Let's say that the Sun moves away from us, this should cause a change in gravitational force, when will this change be noticed by us?

where Ben Crowell says:

If something starts applying a force to the sun at a certain point in time, then there will be effects from this that will be detectable at the earth 8 minutes later.

So one says that when the source of the force field moves, the static field moves with it instantly. But this is not a violation of SR, since the field exists around the particle (that the field interacts with) already.

The other one says that the if something applies force at the Sun (the source of the gravitational field) then the static field's change will only be felt 8 minutes later on Earth.

This is a contradiction, because the static gravitational field already exist around the Earth too, so it is in the immediate vicinity of Earth, so when it moves instantly together with the source (Sun), we should feel it too on Earth.

Is this because EM static fields are different from gravitational, or are both fields working the same way in this context?

Question:

  1. Does the static gravitational field move with its source instantly?
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To the order in $v/c$ necessary to answer your question, EM fields and gravitational fields behave in exactly the same way. Both of them display a speed-of-light delay and cannot "detect" changes in the source's behavior until light has had enough time to propagate between the source and the point in question.

I don't know what you mean by the "static field generated by a moving source," which seems to me like a contradiction in terms. Either a field is static and no sources are moving, or it isn't. So your question doesn't really make sense. You seem to be imaging some dichotomy between "static" fields that respond to their sources instantly, and "non-static" fields that respond with a speed-of-light delay. But there is actually no such dichotomy. All fields can be written as responding to all sources with a speed-of-light delay.

Hawkeye's statement that

you find that all the aberrations you might expect because of a finite speed of light end up canceling out, so gravity acts like it's instantaneous

is highly misleading; the correct statement is that the aberrations cancel out to first order in $v/c$, but there are nonzero speed-of-light corrections starting at second order in $v/c$.

The key phrase in Albert's statement is "in a sense". His statement isn't "really" true in general. The field's apparent instantaneous motion only holds for objects moving with uniform velocity. The instant they start to accelerate at all, the results of that acceleration only propagate outward with a speed-of-light delay.

You can find more details in my answer here.

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Let's work through the cases:

  • Source isn't moving: Both the electric and gravitational forces point back to the force always. Hopefully that's what you expect.

  • Source is moving in constant straight line, and has been doing a really long time: Both the electric and gravitational forces point back to the force always. Hopefully that's also what you expect. If not, to see it, consider it from a reference frame moving with the source: It's just sitting there, emitting a constant field, which you observe somewhere.

  • After moving constantly for a while, the source is accelerated: Here, it takes some time for the forces to change at the distant observer. Until $d/c$ time has elapsed, the force will still be from the constant motion earlier; after that, the force will be due to the accelerated motion.

These three cases correspond exactly to the three quotes in the Question, they're just (hopefully) worded more clearly to make the physics more visible.

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