Action at a distance (physics)
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In physics, action at a distance is the interaction of two objects which are separated in space with no known mediator of the interaction. This term was used most often with early theories of gravity and electromagnetism to describe how an object could "know" the mass (in the case of gravity) or charge (in electromagnetism) of another distant object.
According to Albert Einstein's theory of special relativity, instantaneous action-at-a-distance was seen to violate the relativistic upper limit on speed of propagation of information. If one of the interacting objects were suddenly displaced from its position, the other object would feel its influence instantaneously, meaning information had been transmitted faster than the speed of light.
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Electricity
Coulomb's law in electrostatics appears to be a theory with action-at-a-distance - Coulomb's law deals with charges which have always been static. Efforts to develop a theory of interaction between moving charges, electrodynamics, led to the necessity to introduce the concept of a field with physical properties. In the theory of electrodynamics as formulated in Maxwell's equations, interactions between moving charges are mediated by propagating deformations of an electromagnetic field. These deformations propagate with the speed of light and therefore do not violate special relativity. The deformations of the field can carry momentum independently, thus facilitating conservation of angular momentum.
Gravity
Newton
Newton's theory of gravity offered no prospect of identifying any mediator of gravitational interaction. His theory assumed that gravitation acts instantaneously, regardless of distance. Newton had shown mathematically that if the gravitational interaction is not instantaneous, angular momentum is not conserved, and Kepler's observations gave strong evidence that in planetary motion angular momentum is conserved. (The mathematical proof is only valid in the case of a Euclidean geometry)
A related question, raised by Ernst Mach, was how rotating bodies know how much to bulge at the equator. How do they know their rate of rotation? This, it seems, requires an action-at-a-distance from distant matter, informing the rotating object about the state of the universe. Einstein coined the term Mach's principle for this question.
Einstein
One of the conditions that a relativistic theory of gravitation must meet is to be mediated with a speed that does not exceed lightspeed. It could be seen from the previous success of electrodynamics that the relativistic theory of gravitation would have to use the concept of a field or something similar.
This problem has been resolved by Einstein's theory of general relativity in which gravitational interaction is mediated by deformation of space-time geometry. Matter warps the geometry of space-time and these effects are, as with electric and magnetic fields, propagated at the speed of light. Thus, in the presence of matter, space-time becomes non-Euclidean, resolving the apparent conflict between Newton's proof of the conservation of angular momentum and Einstein's theory of special relativity. Mach's question is resolved because local space-time geometry is informing a rotating body about the rest of the universe. In Newton's theory of motion, space acts on objects, but is not acted upon. In Einstein's theory of motion, matter acts upon space-time geometry, deforming it, and space-time geometry acts upon matter.
Quantum mechanics
Current physical theories incorporate the upper limit on propagation of interaction as one of their basic building blocks, hence ruling out instantaneous action-at-a-distance. At the same time however, instantaneous action at a distance appears to be an essential feature of some very fundamental quantum mechanical effects like entanglement and quantum nonlocality. For these implications of quantum mechanics, Einstein coined the term "spooky action at a distance".
The question of whether this 'spooky action' at a distance constitutes a violation of the relativistic upper limit on the propagation of interaction is not straightforward. According to the laws of quantum mechanics, entanglement cannot be employed for relaying information from one place to another.