Action at a distance
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Action at a distance is a phrase that, throughout the history of science, has denoted a number of things. Some of these things are not related at all—it was simply a case of re-using an interesting-sounding phrase. If one is totally ignorant, or totally committed to scientific demagoguery, one can conflate the 17th-century meaning with the 20th-century meaning.
There are three phenomena that one needs to consider. Each of them has been described as "action at a distance":
- The possibility that some object can apply a force to another object, even though they are not in physical contact.
- Once one accepts the above, the possibility that this influence across space might travel faster than the speed of light.
- Once one understands the implications of the second question, the possibility that quantum mechanics might appear to violate relativistic causality.
The first of these questions, while trivial by today's standards, was mystifying to 17th-century scientists such as Isaac Newton. Electric and magnetic fields were unknown (though lodestones and compasses were known; they just didn't think sufficiently carefully about the implications of this.) Prior to Newton, people didn't think carefully about the implications of gravity. No one had ever seen an object exert a force on another object except when the objects were not in contact. The pin doesn't go down until the bowling ball hits it. This 17th-century meaning of action at a distance is what Newton was writing about in his 1693 letter cited here. Newton's 17th-century sensibilities had difficulty embracing the notion, even though his theory of universal gravitation had proposed exactly what he was having a hard time accepting. Such is the way of science.
Once one has swallowed that meaning of action at a distance, other phenomena arose in the 19th century—the electric and magnetic fields investigated by Faraday, Maxwell, and others. By this time, the idea that forces could be mediated by a vacuum was well accepted, and the idea of "action at a distance" wasn't considered noteworthy. The term lost its mystique and was not commonly used.
With the introduction of relativity, the question arose: Do these forces transmit their influence at a speed faster than the speed of light? Various thought experiments and actual experiments showed unequivocally that these influences propagate at the speed of light. Prior to relativity, physicists were not aware of any question about this. Newton, and others, assumed that the speed of propagation was instantaneous.
With the introduction of quantum mechanics, the issue was re-opened. Quantum mechanics has presented a number of phenomena that violate common sense or other tenets of physics. For examples, in Quantum mechanics a pair of particles can be "entangled". One of the "interpretations" of entangled particles is that an interaction with one particle causes an immediate state collapse in the other particle, irrespective of distance, aka faster than the speed of light. (There are other plausible "interpretations". Do note that entanglement is testable, but the "interpretations" are not.)
However, this still does not afford the ability to "communicate" or "act" over a distance in any useful way whatsoever - there is no way for one observer to know if they are the one to cause the state collapse, or the one to observe the result of the other observer collapsing the state of the other particle. So far, these entanglement effects can only be seen on systems at the quantum level and extrapolating quantum ideas to the macroscopic world just leads to silliness.
Example of ignorance and/or insanity
Conservapedia's insane attempts to use this concept to discredit Einstein includes statements (hereimg) like "Newton's acceptance of this concept -- which became fundamental to electrostatics and quantum mechanics and has a basis in Christianity  -- was central to this development of his theory of gravity."
- (The first reference says "See, e.g., Jesus's cure of the centurion's slave", (We're not kidding! Conservapedia cites the bible in science articles!), while the second refers to this article, which is addressed above.)
Conservapedia also has a (hopelessly confused, as usual) article on the term here.