Coincidence

Coincidence, in the strict and idealized sense employed in physics for instance by relativistic thought experiments, is the occurence of several distinct observations jointly in one and the same observational state of a suitable observer; instead of occuring in separate, subsequent states (one after the other), or being made by different observers. Hence, coinciding are certain observations of suitable signal events; and these observations are in turn contained together in one coincidence event.

In applying the idealized notion of coincidence several practical limitations of the given observer will have to be considered, such as the resolution of separate, subsequent (or partly overlapping) states, and sensitivity in recognizing and distinguishing various signals.

In a sharply different, looser sense several distinct events may also be called coincident to each other, if they are considered (by some measure) to be in particular close or unexpected proximity to each other. Finally, coincidence may also denote sheer similarty, especially if it is unexplained and thus at least apparently only by chance.   

On the other hand, the strict sense of coincidence is emphasized by using the term "point-coincidence".

The notion of coincidence in relativistic geometry

The notion of coincidence is introduced as self-evidently reproducible at the outset of Albert Einstein's presentation of special relativity, albeit there only to motivate a definition of simultaneity as a separate, less trivial notion of geometry; s. Paragraph 1 of A. Einstein, ''On the Electrodynamics of Moving Bodies''. Translation from the German article ''Zur Elektrodynamik bewegter Körper'', Annalen der Physik '''17''': 891-921 (June 30, 1905).

While featuring only implicitly, the importance of coincidence can be recognized even in these earliest developments by considering that reference systems are to be set up (Paragraph 2 of [1]) involving several observers (i.e., not just A and B of Paragraph 1), where each participant is therefore naturally expected to judge for each own signal state whether corresponding echoes from various other observers were either received back in coincidence, or else recognize the order of succession in which they were received back.

Only with some delay, however, Einstein applied coincidence explicitly for the definition of geometric relations, in sketching a revised definition of simultaneity: s. Section 8 of A. Einstein, ''Relativity. The Special and the General Theory'', Sect. 8 ''On the Idea of Time in Physics'', Henry Holt (1920). Translation from the German ''Über die spezielle and die allgemeine Relativitätstheorie. (Gemeinverständlich)'', Vieweg & Sohn (1917; submitted Dec. 1916).

Being considered self-evidently reproducible, the notion of coincidence had of course been recognized and employed already independently of Einstein, for instance by Poincaré, Robb, and Kretzschmann.

On general grounds, Einstein recognized coincidence in the strict sense as foundational principle of relativistic geometry; codified as the "point-coincidence principle"; cmp. A. Einstein, ''The Foundation of the General Theory of Relativity'', (Die Grundlage der allgemeinen Relativitätstheorie), Annalen der Physik '''49''': 769-822 (1916).

Accordingly, a central aim of (foundational) relativistic physics is to devise though-experimental descriptions, employing the notion of coincidence, to give definition to various otherwise irreproducible notions (of geometry in classical physics), such as "mutual rest", "free motion" or "inertial motion" or "geodesic motion", "duration", "distance", "speed", "angle", "curvature", "refractive index" and so on.

A decisive reduction of this task was achieved with John Synge's chronometric geometry, in defining geometric notions entirely in terms of "duration" ratios alone, or in some special cases even purely in terms of coincidence determinations. (J. Synge, ''Relativity. The General Theory'', North Holland, 1960).


       (Finally: some highly recognizable sort of humor -- irreverent, yet not too jokey -- that is about something! ... )

Terminology: The notion of observer in relativistic geometry

The choice of the technical term observer for the purpose of this article (and appearing already in Einsteins foundational writings on relativity) appears more appropriate than for instance the terms detector or responder or sensor because

(1) the terms detector or sensor, especially, connote too great specificy of signals.
The described notion of coincidence, for instance, is readily applicable to the observer being a system of a "smoke detector" and a "carbon-monoxide sensor", but not either one alone;

(2) the term observer connotes persistence and individuality (not least by association to persistence and individuality of any human being) which is required to distinguish signals observed in coincidence from signals observed in succession. (This observer-property, by itself, may also be attributed to certain particular worldlines or wordtubes.);

(3) the ability of an observer to function as responder (i.e., its observational states being in turn observable to others) is not required in the above description of coincidence. (Of course, this capability is essential as soon as it comes to geometric relations between several observers, by relativistic definitions. However, property (2) is no less important in these cases.)

In summary, the term observer is suitable for naming any protagonist of signal exchange, and thus of relativistic geometry, as far as it entails individual signal statement, recognition, response, and memory (incl. ordering, or determination of coincidence).