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I'm reading the definition of the principle of locality from its Wikipedia page:

The principle of locality states that an object is influenced directly only by its immediate surroundings.

This definition makes sense to me, but I wonder can we interpret this using some common examples in physics? I heard about General Relativity is local. Can I interpret this as any object with mass will cause the curvature of spacetime, which are immediate surroundings that influence the trajectory of the planets? Also, how does the principle of locality is related to action at a distance?

IGY
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  • A (classical) wave equation is completely local. There is no action at a distance. However, there is an emergent wave solution with a finite speed of propagation. Is that what you are asking? How such global solutions emerge from purely local interaction? – FlatterMann Nov 05 '22 at 01:48
  • @FlatterMann Thanks so much for the comment! I'm trying to think of some examples to better understand the principle of the locality from its definition. Could you explain a bit why the classical wave equation is completely local? Also, If we consider the gravitational wave in the universe, is that also local? – IGY Nov 05 '22 at 02:00
  • The classical wave equation is completely local because we defined it that way. General relativity is completely local because Einstein defined it that way. At most you can ask why nature seems to obey local equations rather than non-local ones. – FlatterMann Nov 05 '22 at 02:02
  • @FlatterMann By completely local, do you mean no information can travel faster than the speed of light? – IGY Nov 05 '22 at 02:05
  • No, I don't. The speed of light is a direct consequence of geometry and relativity. It is not a consequence of locality as far as I know. One can define delay equations that are non-local and still don't allow for infinitely fast information transfer. – FlatterMann Nov 05 '22 at 02:07
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    implementationally, locality means whatever algorithm one uses to compute something can only "see" its own properties and its immediate neighbors' properties... It cannot "see" for example what is happening 10 places away, and use this information in any way in its computation. This naturally enforces locality in your physics. – James Nov 05 '22 at 02:10

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The principle of locality is an axiom which asserts that the time it takes for one system to affect another will always be mediated by the distance between them. What general relativity being local means is that no information coming from a system (eg: changes in spacetime curvature/gravity) can propagate faster than the speed of light. Any object with mass (eg: the sun) will curve spacetime, but the time it takes planets to detect any spacetime changes depends on the time it takes for light to travel from point A to point B (which is around 8 minutes between the sun and the Earth).

Unmaxed
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  • Thanks so much for the answer! Then can I say Newton's law is non-local? Because in that theory the earth will immediately move to the tangent direction if the sun disappears. – IGY Nov 05 '22 at 02:23
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    @IGY Newton's law of gravitation is a non-local approximation. One can fix that with a local retarded scalar potential, but that's still not the right theory. – FlatterMann Nov 05 '22 at 02:29
  • @FlatterMann Thanks so much! Is there a reason why we know it's still not the right theory? – IGY Nov 05 '22 at 02:37
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    @IGY https://physics.stackexchange.com/questions/692578/what-s-wrong-with-this-nordstr%C3%B6m-like-scalar-theory-of-gravity – FlatterMann Nov 05 '22 at 02:42
  • Here is a recent question without differential geometry https://physics.stackexchange.com/questions/733441/it-is-possible-to-create-a-relativistic-theory-of-gravity-more-simple-than-gener/733455?noredirect=1#comment1640933_733455 on why a gravitational theory based on a modification Newton's equation directly is not possible. – hyportnex Nov 05 '22 at 11:52
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It's a tautology, obviously an object can only directly be affected by its immediate surroundings.

Picture a table filled with billiard balls and you hit one ball at a time. That struck ball may cause several consequent collisions but only the collisions involving the initial struck ball are directly affected by that ball and only when in the immediate environment of that ball.

fertilizerspike
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    This is wrong; the principle of locality is an empirical statement that is by no means tautological. The example you give only works because billiard balls happen to be described by local laws. – Sandejo Nov 05 '22 at 06:15
  • A tautology would be a statement like "an object can only directly be affected by the things that directly affect it." Saying that the things that directly affect it are only its immediate surroundings is a separate statement. – Sandejo Nov 06 '22 at 05:55