If an electric charge suddenly appears will an electric field be present everywhere instantaneously? This question has been asked many times in conjunction with a suddenly disappearing sun. The common reaction to this is that it takes a while for the spacetime curvature to be changed around it. In accordance with special relativity, so it is said. An instantaneous change from curved to flat by means of a disappearing mass doesn't mean though that information is traveling faster than light.
Why not? Simply because you can't remove a mass instantaneously. This renders the question meaningless maybe, but you can do it in the imagination. If the mass suddenly is not there anymore, there will not emerge a gravitational wave. Only when you move it this will happen. Which is needed to transmit information. If I move the mass fast to and fro, a wave will emerge, so you can convey information with it. But not if the mass suddenly ceases to exist.
In connection with charge and the electromagnetic field surrounding this charge, I've read about experiments to uncover if the change in the field surrounding the charge is instantaneous. This means that there is a doubt for that case. The same arguments I gave above can be applied to the electromagnetic field. Will an electromagnetic field instantaneously appear if charges appear in the creation of a positron-electron pair (or disappear when they annihilate)?
See also this article (written in 2006 by W. Engelhardt, retired from Max-Planck-Institut für Plasmaphysik, D-85741 Garching, Germany), in which it is written:

the instantaneous transmission of both energy and information over
macroscopic distances is feasible by using the quasi-static fields which are predicted by Maxwell’s equations.

I don't think that the conclusion drawn there (or by me) is correct though (information can't travel at the speed of light). Even if you could create single electrons (contrary to an electron and a positron, appearing at a point), and the field of that electron would be present at the same time everywhere, you could use them to transmit information: when I create one electron per second it's an A, when I create two per second, it's a B, three per second a C, etc. but is Engelhardt mistaken too? Is it because he's retired..., maybe?
 A: No.
This is a deep principle encoded in different ways in various kinds of theories, but the conclusion is that causality is always respected.

*

*In Maxwell's equations (treated classically), conservation of charge means that a charge cannot suddenly appear. If a charge moves, then the field farway only "learns" that the charge has moved after light has had time to travel from the charge to the observer.

*Gravity is very similar -- in Einstein's equations, conservation of the stress-energy tensory prevents the sudden appearance of mass/energy.

*In quantum field theory, observables at space-like separated positions commute, which prevents observable information from traveling faster than the speed of light.

*In quantum gravity, it is less clear how this should work since the causal structure of spacetime is itself quantum and uncertain. AdS/CFT provides a concrete framework for seeing how this might work -- since the boundary theory is a normal quantum field theory, there is a causal structure on the boundary that is obeyed.

There are solutions in GR that violate causality (have closed timelike loops), but these can be excluded by a "garbage in garbage out" postulate (since they make unphysical assumptions), and there is a chronology protection conjecture that states that quantum backreaction will prevent the formation of closed timelike loops.
A: 
Will an electromagnetic field instantaneously appear if charges appear in the creation of a positron-electron pair (or disappear when they annihilate)?

Pair creation of an electron positron, the first order Feynman diagram

In the quantum field theory of elementary particle interactions, which is very successful in describing and predicting data, the space time algebra is the Lorettz one, and nothing can happen faster than the velocity of light allows.
As mainstream physics posits that all physics partial theories are emergent from the underlying quantum mechanical framework, this should hold for classical  fields too.
In any case , there is no way to make charge appear or disappear  instantaneously, there always must be interactions that are dependent on the speed of light.
A: You are asking about the static EM field (near field) of a charge. This static EM field is always already present around the charge.

The near field and far field are regions of the electromagnetic field (EM) around an object, such as a transmitting antenna, or the result of radiation scattering off an object. Non-radiative near-field behaviors dominate close to the antenna or scattering object, while electromagnetic radiation far-field behaviors dominate at greater distances.

https://en.wikipedia.org/wiki/Near_and_far_field
Please note that although this static EM field (near field) already always exist around the charge, changes in this near field do propagate at the speed of light.
Now you are asking what would happen if a charge appeared because of pair production. Now when the charge does not exist yet, its near field neither does. When the charge appears, you have to apply the law of how fast changes in the near field propagate (spherically outwards from the source). When the charge appears, its near field extends at the speed of light. This is the answer to your question.
Now please note that the appearance of the charge due to pair production is not instantaneous, just like the appearance of a photon while emission happens from an atom. The wavefunction (describing the photon-atom system) smoothly changes from the state of the atom (including the photon's energy) to a state where the atom loses this energy and a separate photon exists.

Over time, c1(t) smoothly decreases from one to zero, while c2(t) smoothly increases from zero to one. So everything happens continuously, and there are no jumps. (Meanwhile, the expected number of photons in the electromagnetic field also smoothly increases from zero to one, via continuous superpositions of zero-photon and one-photon states.)

Do electrons really perform instantaneous quantum leaps?
The same happens with pair production. The wavefunction describing the system of photons smoothly changes from a state of having only photon energy to transforming into a state of an electron positron pair.
This example is for a quantum leap, but it can be analogously applied to your situation, where the expected number of photons smoothly decreases from 1 to 0 and the expected number of electron positron pairs smoothly increases from 0 to 1 via continuous superpositions.
