# Principle of locality

In physics, the principle of locality states that an object is directly influenced only by its immediate surroundings. A theory that includes the principle of locality is said to be a "local theory". This is an alternative to the older concept of instantaneous "action at a distance". Locality evolved out of the field theories of classical physics. The concept is that for an action at one point to have an influence at another point, something in the space between those points such as a field must mediate the action. To exert an influence, something, such as a wave or particle, must travel through the space between the two points, carrying the influence.

The special theory of relativity limits the speed at which all such influences can travel to the speed of light, ${\displaystyle c}$. Therefore, the principle of locality implies that an event at one point cannot cause a simultaneous result at another point. An event at point ${\displaystyle A}$ cannot cause a result at point ${\displaystyle B}$ in a time less than ${\displaystyle T=D/c}$, where ${\displaystyle D}$ is the distance between the points and ${\displaystyle c}$ is the speed of light in a vacuum.

In 1935 Albert Einstein, Boris Podolsky and Nathan Rosen in their EPR paradox theorised that quantum mechanics might not be a local theory, because a measurement made on one of a pair of separated but entangled particles causes a simultaneous effect, the collapse of the wave function, in the remote particle (i.e. an effect exceeding the speed of light). But because of the probabilistic nature of wave function collapse, this violation of locality cannot be used to transmit information faster than light. In 1964 John Stewart Bell formulated the "Bell inequality", which, if violated in actual experiments, implies that quantum mechanics violates either local causality or statistical independence, another principle. The second principle is commonly referred to as free will.

Experimental tests of the Bell inequality, beginning with Alain Aspect's 1982 experiments, show that quantum mechanics seems to violate the inequality, so it must violate either locality or statistical independence. However, critics have noted these experiments contained "loopholes", which prevented a definitive answer to this question. This might be partially resolved: in 2015 Dr Ronald Hanson at Delft University performed what has been called the first loophole-free experiment.[1] On the other hand, some loopholes might persist, and may continue to persist to the point of being difficult to test.[2]

## Pre-quantum mechanics

In the 17th century Newton's law of universal gravitation was formulated in terms of "action at a distance", thereby violating the principle of locality.

It is inconceivable that inanimate Matter should, without the Mediation of something else, which is not material, operate upon, and affect other matter without mutual Contact…That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it. Gravity must be caused by an Agent acting constantly according to certain laws; but whether this Agent be material or immaterial, I have left to the Consideration of my readers.[3]

— Isaac Newton, Letters to Bentley, 1692/3

Coulomb's law of electric forces was initially also formulated as instantaneous action at a distance, but was later superseded by Maxwell's equations of electromagnetism, which obey locality.

In 1905 Albert Einstein's special theory of relativity postulated that no material or energy can travel faster than the speed of light, and Einstein thereby sought to reformulate physical laws in a way that obeyed the principle of locality. He later succeeded in producing an alternative theory of gravitation, general relativity, which obeys the principle of locality.

However, a different challenge to the principle of locality subsequently emerged from the theory of quantum mechanics, which Einstein himself had helped to create.

## Relativity

Locality is a key axiom of Einstein's theory of special relativity, where it is essential to causality that effects do not propagate faster than the speed of light.

Einstein's quantum theory (currently termed the old quantum theory) is said to be relativistic because it does not violate either his general or special theory of relativity: speed of light is a limiting factor.

In Einstein's theory, two observable objects are localised, each within its own distinct spacetime region (frame). When regions are separated from each other in space, effects pass from one object to the other at the speed of light or slower. This is a key property of spacetime flowing from the special theory of relativity.

A solution of Einstein's field equations is local if the underlying equations are invariant (a condition where the laws of physics are invariant—that is, the same—in all frames that are moving with uniform velocity with respect to one another).

Alternatively, a solution of Einstein's field equations is still local if the underlying equations are co-variant: i.e. if all (non-gravitational) laws make the same predictions for identical experiments taking place at the same time in two different inertial (that is, non-accelerating) frames; such that the variations from the resting state are the same (i.e. vary equally) for each frame.

## EPR argument

Locality and free will are the basic assumption of the EPR logical argument.