# Which “space-time coincidences” are described by a “co-ordinate system in which the gravitational field does not appear”?

In Einstein's exposition of the foundations of General Relativity (cmp. http://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity , end of §3) there appears an emphasis on considerations of "coincidences":

All our well-substantiated space-time propositions amount to the determination of space-time coincidences. [...] The results of our measurements are nothing else than well-proved theorems about such coincidences of material points

and their relation to "co-ordinates":

The introduction of a system of co-ordinates serves no other purpose than an easy description of totality of such coincidences.

Subsequently (in §4) one more or less specific "system of co-ordinates" is suggested:

The nature of acceleration of an infinitely small (positional) co-ordinate system is hereby to be so chosen, that the gravitational field does not appear; this is possible for an infinitely small region. $X_{1}, X_{2}, X_{3}$ are the spatial co-ordinates; [...]

Question:
What, explicitly, is contained in the corresponding "totality of such coincidences";
or in some specific subset of this "totality"?,
of which the indicated "co-ordinate system in which the gravitational field does not appear" is a suitably "easy description".

Does this "totality" contain, for instance:

(1)
for any three distinct "material points" (${\textbf A}$, ${\textbf B}$, and ${\textbf P}$, who are assigned the corresponding unequal constant "spatial co-ordinate" triples $\{ \, X_{1}^{\textbf A}, X_{2}^{\textbf A}, X_{3}^{\textbf A} \, \}$, $\{ \, X_{1}^{\textbf B}, X_{2}^{\textbf B}, X_{3}^{\textbf B} \, \}$, and $\{ \, X_{1}^{\textbf P}, X_{2}^{\textbf P}, X_{3}^{\textbf P} \, \}$, respectively),
and for any "coincidence" ${\textbf A}_{\mathscr X}$ in which ${\textbf A}$ had been participating,
the "coincidence" that

${\textbf A}\!$'s indication of having seen that ${\textbf P}$ saw ${\textbf A}\!$'s indication of having seen that ${\textbf B}$ saw "coincidence" ${\textbf A}_{\mathscr X}$
was coincident to
${\textbf A}\!$'s indication of having seen that ${\textbf B}$ saw ${\textbf A}\!$'s indication of having seen that ${\textbf P}$ saw "coincidence" ${\textbf A}_{\mathscr X}$
?

And is "an easy description" of this "totality" afforded by the indicated system for instance in the sense that:

(2)
for any two distinct "material points" (${\textbf A}$ and ${\textbf B}$, who are assigned the corresponding unequal constant "spatial co-ordinate" triples $\{ \, X_{1}^{\textbf A}, X_{2}^{\textbf A}, X_{3}^{\textbf A} \, \}$, and $\{ \, X_{1}^{\textbf B}, X_{2}^{\textbf B}, X_{3}^{\textbf B} \, \}$, respectively) there is another triple, $\{ \, X_{1}^{\textbf M}, X_{2}^{\textbf M}, X_{3}^{\textbf M} \, \}$, reserved as constant "spatial co-ordinates" to denote a "material point" ${\textbf M}$, if it happens to exist, as "middle between ${\textbf A}$ and ${\textbf B}$", described by the "coincidences"

• that for any "coincidence" ${\textbf A}_{\mathscr X}$ in which ${\textbf A}$ had been participating,

${\textbf A\!}$'s indication of having seen that ${\textbf M}$ saw ${\textbf A\!}$'s indication of having seen that ${\textbf M}$ saw "coincidence" ${\textbf A}_{\mathscr X}$
was coincident to
${\textbf A\!}$'s indication of having seen that ${\textbf B}$ saw "coincidence" ${\textbf A}_{\mathscr X}$, and

• that for any "coincidence" ${\textbf B}_{\mathscr Y}$ in which ${\textbf B}$ had been participating,

${\textbf B}$'s indication of having seen that ${\textbf M}$ saw ${\textbf B}$'s indication of having seen that ${\textbf M}$ saw "coincidence" ${\textbf B}_{\mathscr Y}$
was coincident to
${\textbf B}$'s indication of having seen that ${\textbf A}$ saw "coincidence" ${\textbf B}_{\mathscr Y}$, and

• that for any "coincidence" ${\textbf M}_{\mathscr Z}$ in which ${\textbf M}$ had been participating,

${\textbf M}$'s indication of having seen that ${\textbf A}$ saw "coincidence" ${\textbf M}_{\mathscr Z}$
was coincident to
${\textbf M}$'s indication of having seen that ${\textbf B}$ saw "coincidence" ${\textbf M}_{\mathscr Z}$
?

-

What, explicitly, is contained in the corresponding "totality of such coincidences";

Basically these coincidences are collisions of particles. We have world-lines of particles, but he's saying that all physical observations reduce to observations of intersections of these world-lines. The world-lines themselves, and the way they lie in the background of spacetime, are not directly observable. This introduction to the paper is giving a physical motivation for describing spacetime using noneuclidean geometry. In the context of geometry, the word "incidence" or "coincidence" has a specific mathematical definition, which is the intersection of two figures. For example, if two planes intersect in a line, this is called an incidence relation between them. "Coincidence" implies nothing more or less than what you'd expect from the root "coincide."

of which the indicated "co-ordinate system in which the gravitational field does not appear" is a suitably "easy description".

He's saying that locally, we can always choose a frame of reference defined by a free-falling observer. By the equivalence principle, there is no gravitational field detectable by such an observer.

Does this "totality" contain, for instance: [...]

After this point, it seems to me that you've wandered off track. By "coincidences," he means something very specific: the intersection of two world-lines. It's not an observer's "indication of having seen something."

-
"We have world-lines of particles" -- We have Einstein's (and my) insistence on coincidence(s). How do we construct the notion "world-line" from that?: We may need "identifiability" of participants in order to say that (or ask whether) some same "particle" took part in several coincidences; and we may even require some notion of "ordering" those. "It's not an observer's "indication of having seen something."" -- What about the role of "M" here (§8)? Aren't coincidences observable (by other participants)? – user12262 Jul 16 '13 at 22:14
contd.: "a frame of reference defined by a free-falling observer" -- At least I've managed to ask my question without using (and thus presuming to comprehend) the notion of "a free-falling observer"; but rather in the hope to explicate space time coincidences (separately, or in relation to each other) by which to define the notion of "a free-falling observer" in the first place. This seems to be in line with the cited quotations, even if it may be ahistorical. – user12262 Jul 16 '13 at 22:26
p.s.: Frank Wappler wrote (1st comment): "What about the role of "M" here (§8)?" -- Sorry, this and the underlying link were referring to a chapter in a book written by Einstein originally in German. The corresponding translation can be found for instance here, instead. To note (ch. VIII, On the Idea of Time in Physics): "If the observer [M] perceives the two flashes of lightning at the same time, then [...]". Doesn't this describe a relevant form of space-time coincidence, that Einstein meant, and I used in (1), (2)? – user12262 Jul 16 '13 at 22:57
@user12262: Aren't coincidences observable (by other participants)? Yes, but unlike many types of observations they are also observer-independent. We have Einstein's (and my) insistence on coincidence(s). How do we construct the notion "world-line" from that? Because Einstein is assuming a certain level of mathematical literacy, according to which incidence relations are understood to be a geometrical notion. – Ben Crowell Jul 16 '13 at 23:14
"Einstein is assuming a certain level of mathematical literacy" -- Sure. Is it accordingly understood that Einstein is assuming/permitting any participants in any one "coincidence" to be identified and that it may be recognized whether someone (who has been identified as participating in one particular "coincidence") took part in several "coincidences"? (Even though that's admitting a relation between several "coincidences"; in contrast for instance to the "Footnote" remark seen here?) – user12262 Jul 17 '13 at 5:44