Or instead of bodies, let's say have two massive point particles. Does their being local to each other mean that they're infinitesimally close in space and infinitesimally in time? OR does it mean that the spacetime separation between them is infinitesimally small?
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1$\begingroup$ Did you read this somewhere? Can you link for context? $\endgroup$– NJPCommented Jun 5, 2020 at 20:26
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$\begingroup$ I think local to each other can only mean same time and same inertial system , so same "room" , but not necessary same point. $\endgroup$– trulaCommented Jun 5, 2020 at 20:37
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Local means that any deviation from Newton's laws is smaller than experimental accuracy. This can be taken as a definition of infinitesimal in physics, although it is not mathematically infinitesimal.
It does not mean that they necessarily have the same proper acceleration, although if they do have different accelerations then they will typically not remain local to each other.
Local means local in time as well as space. If tidal forces are observed, then the objects are not local. Bear in mind that this generally restricts local to quite small amounts of time. One light second is a large distance by normal measures.
answered Jun 5, 2020 at 20:44
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$\begingroup$ So by your second paragraph, you mean that even if two objects are local to each other, it could also be because they're momentarily at rest w.r.t. each other and happen to be infinitesimally close to each other (both in space and time). Is that correct? In that case for sure they don't necessarily have the same proper acceleration. And by your last paragraph, I suppose then "local" would mean local in space AND local in time, which I guess is a stronger condition than having infinitesimally small spacetime separation. $\endgroup$ Commented Jun 5, 2020 at 21:01
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$\begingroup$ Why Newtonian? I think events can be considered as local if the region of spacetime containing them can be adequately approximated as flat Minkowski spacetime. $\endgroup$– PM 2RingCommented Jun 5, 2020 at 21:27
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1$\begingroup$ @PM2Ring: Hm I guess it's because of the nature of the Minkowski metric. I mean if it were the standard Euclidean metric, then an infinitesimal spacetime separation would imply and be implied by local in space AND local in time. Thanks a ton for clarifying! $\endgroup$ Commented Jun 5, 2020 at 21:42
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1$\begingroup$ It is not necessary for objects to be momentarily at rest for them to be local. They only have to be within a region of spacetime. $\endgroup$ Commented Jun 5, 2020 at 22:04
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1$\begingroup$ @PM2Ring, yes the region is adequately approximated by Minkowski spacetime, but this is determined by the meaning of an inertial frame, which (in its simplest form) is defined empirically by the application of Newton's first law. $\endgroup$ Commented Jun 5, 2020 at 22:06