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In this (surely notable and earnest) answer to a question about how to measure a certain geometric quantity, which apparently can be defined within the general relativity theory the following step in the (arguably "only thought-experimental") measurement procedure has been suggested:

Draw a circle with yourself at the origin.

I'd like to understand in more detail how this procedural step is to be carried out (at least in thought-experimental principle) in the context of the general relativity theory:
How, specificly, does it amount to determinations of coincidences {...} such as between identifiable material points, as prescribed by Einstein in the Foundation of General Relativity (cmp. the translation) ?

Specificly: Considering four identifiable distinct participants $A$, $B$, $J$, and $K$ (along with additional identifiable distinct particpants, as may be necessary) how ought to be determined, at least in thought-experimental principle, amounting to requisit coincidence determinations, whether these four had been jointly "on the same circle", or not?

(Note: the referenced question stipulates that the requested measurement procedure be carried out in a region with geometric properties corresponding to "FLRW metric". If it is considered necessary or useful for addressing my question, then this may likewise be stipulated here; presuming, of course, that the (thought-)experimental assertion, whether the geometry of a region under consideration corresponds to "FLRW metric", or not, amounts to requisite determinations of coincidences, too.)

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  • $\begingroup$ @claude chuber: Thanks for your altruistic way of drawing attention to my question. Now, Ben Niehoff's recent (attempt to) answer makes clarifications necessary, and I'd like to check and respect your thoughts on that: Do you agree that the sought "circle" is to be constituted as a certain set of identifiable participants, such as "yourself (or myself)"; "making a circle (with definite circumference)", not a sphere? Would you allow ping duration as a quantitative measure of separation between participants (in addition to coincidence determinations)? $\endgroup$ – user12262 Mar 28 '17 at 6:29
  • $\begingroup$ I don't see a big difference between questionning about a (2D) circle or a 3D sphere. I think that Ben's answer part including "then you must first make a choice of spacelike slice" responds to your second question: the distinct participants A, B, J... are bounded by such a constraint, else they might disagree on the circle. The rest of his answer looks clear too. But I agree with you that he does not tell how to do this circle concretely (that is, involving time to construct it, using ping or coincidence methods,...). $\endgroup$ – user130529 Mar 28 '17 at 14:17
  • $\begingroup$ @claude chuber: "I don't see a big difference between questionning about a (2D) circle or a 3D sphere. [...]" -- Well, rather than debating this in another comment I've just submitted an answer to illustrate my thoughts. Perhaps, then, it is best (for me) to leave the question as it is (i.e. as it was when you issued the bounty). And hopefully Ben Niehoff can nevertheless appreciate why I wasn't too enthusiastic about his answer. $\endgroup$ – user12262 Mar 28 '17 at 22:40
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There is an unambiguous notion of what a "circle" is in Riemannian geometry, although one might clarify by calling it a "geodesic sphere", which is the boundary of a "geodesic ball". To construct one geometrically around some point $P$, do the following:

Consider all of the geodesics emanating from $P$. Along each of these geodesics, mark a point at a distance $\ell$ from $P$. A geodesic sphere is the locus of all points thus marked. (Note that $\ell$ must be sufficiently small such that the geodesics emanating from $P$ do not cross each other before reaching a length $\ell$.)

In more everyday language, a "sphere" is exactly what you would expect it to be. If you could anchor a rope of length $\ell$ at $P$, then a sphere is precisely the locus of points that can be reached by pulling the rope taut.

This works well for spheres, but if by "circle" you really mean a circle that lies in a plane, then one must first choose the plane (or more generally, surface) on which this "circle" should reside. In fact, if you want to come up with an operational definition of a "circle" on a pseudo-Riemannian manifold (i.e., with a timelike direction), then you must first make a choice of spacelike slice. But once you do, the definition given above goes through (but you will have to do the process at "one instant of time", so this is only a mathematical definition, not a physical measurement).

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  • $\begingroup$ Ben Niehoff: "[...] a "circle" on a pseudo-Riemannian manifold" -- By the quote and links in the OP we seek participants ("timelike curves", such as "yourself"), who "make a circle", i.e. constituting "the outer shape/shell of a suitable world tube". (I'll try to update the OP accordingly later.) "the geodesics emanating from [...]", "distance", "length" -- The OP, as it stands, was specific about allowing only "coincidence determinations" as basic notons by which to carry out the construction. But this may be too difficult, and not in the spirit of the links. (I'll reconsider ...) $\endgroup$ – user12262 Mar 28 '17 at 5:18
  • $\begingroup$ Ben Niehoff: Regarding my plans to update the OP question I've come to the conclusion that I rather leave the question exactly as it was when claude chuber issued the bounty, and when you submitted (the present version of) your answer. See also the few comments following the OP above. The best illustration of how I intended my question, and of how I interpret it as it stands, is surely the answer I submitted myself here recently. $\endgroup$ – user12262 Mar 29 '17 at 5:37
  • $\begingroup$ So what you're asking for is what type of measurements an observer might do to tell where a "circle" is in space. But I don't think your question as stated makes any sense, because observers are travelling along timelike worldlines, not just sitting at a given event in spacetime. But there is no canonical choice of spacelike slicing, and therefore there is no canonical choice of surface on which your "circle" might be defined. So I don't think there is any meaning to this. $\endgroup$ – Ben Niehoff Mar 31 '17 at 8:18
  • $\begingroup$ Ben Niehoff: "observers are travelling along timelike worldlines" -- Right. The same applies to identifiable crumbs of color pigment which are jointly attached to a "piece of paper", for instance. "But there is no canonical choice of spacelike slicing" -- The question involves giving a general construction, not having to choose. In $(3+1)$ space surely there can be many different $(2+1)$ surfaces constructed/identified which all include the same one given $(1+1)$ worldline as prospective "center of a circle". Besides: We might complain there. $\endgroup$ – user12262 Apr 1 '17 at 6:40
  • $\begingroup$ Not sure I follow you. But a "piece of paper" is a choice of surface! That is what I meant earlier when I said you had to choose a surface in which your "circle" lies. There is no canonical choice of surface! With your piece of paper, you can imagine orienting it in different ways, bending it, etc. In each case what you mean by a "circle" can be drastically different. $\endgroup$ – Ben Niehoff Apr 1 '17 at 9:41
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Preliminary considerations: Identifying a "sphere around a center"

Considering participant $C$ and some other, separate participant $S$ who had been observing each other over the course of an experiment encompassing numerous consecutive mutual signal exchanges ("pings"), such that there is a set of $C$'s signal indications $\{ C_{\sigma} \}$ for which $C$ also indicated its observations of $S$'s corresponding ping echos, $\{ C_{\text{saw that } S \text{ saw that } C \sigma} \}$, then $C$ may observe corresponding ping echos from other participants as well, and determine for which participants these had been indicated in coincidence with its observations of $S$'s corresponding ping echos; i.e. for each signal indication $C_{\sigma}$ determine the set of participants

$$ \mathscr S_{\text{momentary}}[ \, C_{\sigma}, S \, ] := \{ X \in \mathcal P : C_{\text{saw that } X \text{ saw that } C \sigma} \equiv C_{\text{saw that } S \text{ saw that } C \sigma} \}, $$

as "momentary sphere", or "radar bubble", of participant $C$, referring to one specific indication $C_{\sigma}$, "extending to participant $S$";
where $\mathcal P$ denotes the set of all participants under consideration.

Further selecting only those participants which had belonged to all these "momentary spheres", for all of $C$'s signal indications throughout the experiment under consideration, a "permanent sphere" can be determined as

$$ \mathscr S_{\text{permanent}}[ \, C, S \, ] := \bigcap \left[ \left\{ \mathscr S_{\text{momentary}}[ \, C_{\sigma}, S \, ] \right\} \right].$$

By construction, participant $S$ is a member of this "permanent sphere around $C$", and $C$ is not.

Referring again to coincidence determinations there may be additional requirements expressed and evaluated:

  • $\forall X, Y, Z \in \mathscr S_{\text{permanent}}[ \, C, S \, ] :$
    $ \qquad \exists \, C_o \in \{ C_{\sigma} \} \, | \, X_{\text{saw that } Y \text{ saw that } X \text{ saw that } C o} \equiv X_{\text{saw that } Z \text{ saw that } X \text{ saw that } C o} \, \implies $ $ \qquad \forall C_{\sigma} \in \{ C_{\sigma} \} \, | \, X_{\text{saw that } Y \text{ saw that } X \text{ saw that } C \sigma} \equiv X_{\text{saw that } Z \text{ saw that } X \text{ saw that } C \sigma}$,

and

  • $\forall U, V \in \mathscr S_{\text{permanent}}[ \, C, S \, ] :$
    $ \qquad \exists \, C_o \in \{ C_{\sigma} \} \, | \, U_{\text{saw that } V \text{ saw that } U \text{ saw that } C o} \equiv U_{\text{saw that } C \text{ saw that } U \text{ saw that } C o} \, \implies $
    $ \qquad \forall C_{\sigma} \in \{ C_{\sigma} \} \, | \, U_{\text{saw that } V \text{ saw that } U \text{ saw that } C \sigma} \equiv U_{\text{saw that } C \text{ saw that } U \text{ saw that } C \sigma}$,

and moreover

  • $\forall X, Y, W \in \mathscr S_{\text{permanent}}[ \, C, S \, ] :$

    $ \! \! \! \exists \, C_o \in \{ C_{\sigma} \} \, | \, X_{\text{saw that } Y \text{ saw that } X \text{ saw that } C o} \equiv X_{\text{saw that } W \text{ saw that } X \text{ saw that } W \text{ saw that } X \text{ saw that } C o} $
    $ \implies $
    $\! \! \! \forall C_{\sigma} \in \{ C_{\sigma} \} \, | \, X_{\text{saw that } Y \text{ saw that } X \text{ saw that } C \sigma} \equiv X_{\text{saw that } W \text{ saw that } X \text{ saw that } W \text{ saw that } X \text{ saw that } C \sigma}$,

and so on.

These additional conditions may generally not be satisfied for any two separate participants $C$ and $S$, but they are related to (or perhaps even equivalent to, or perhaps even definitive of) the constituents of the (or any) accordingly identified "permanent sphere" being rigid wrt. $C$ and wrt. each other, and characteristic of its intrinsic geometry.

Finally, note an ambiguity of the notation:

If participant $R \in \mathscr S_{\text{permanent}}[ \, C, S \, ]$, where $R$ is distinct from participant $S$, then nevertheless
$\mathscr S_{\text{permanent}}[ \, C, S \, ] = \mathscr S_{\text{permanent}}[ \, C, R \, ]$.

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Identifying a "circle" as intersection of two "spheres":

Given two distinct "permanent spheres" $\mathscr S_{\text{permanent}}[ \, C, S \, ]$ and $\mathscr S_{\text{permanent}}[ \, Q, R \, ]$, constructed as described above, in two "experiments" with significant overlap (encompassing numerous consecutive "pings" between participants of both spheres, as well as between participants $C$ and $Q$), if these two "permanent spheres" have more than one constitent in common (and if they don't have all their constituents in common, which is meant by them being required to be distinct) then their common members constitute a circle; formally:

$$ \mathscr C := \mathscr S_{\text{permanent}}[ \, C, S \, ] \, \cap \, \mathscr S_{\text{permanent}}[ \, Q, R \, ].$$

The additional conditions described above (for each of the two spheres separately), as far as they can be satisfied, along with similar conditions stipulated for all members of both spheres, jointly, in relation to each other, may ensure that the circle obtained as intersection of the two spheres is likewise rigid.

Also note that, if participant $P \in \mathscr C$ then $\mathscr S_{\text{permanent}}[ \, C, S \, ] = \mathscr S_{\text{permanent}}[ \, C, P \, ]$, and $\mathscr S_{\text{permanent}}[ \, Q, R \, ] = \mathscr S_{\text{permanent}}[ \, Q, P \, ]$.

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Identifying the "mid-point" of a given "circle":

Given three distinct permanent spheres (again, with significant common overlap of the experiments in which they were determined), $\mathscr S_{\text{permanent}}[ \, C, S \, ]$, $\mathscr S_{\text{permanent}}[ \, Q, R \, ]$ and $\mathscr S_{\text{permanent}}[ \, M, P \, ]$, such that

  • they have equal (non-empty) pairwise intersection, namely the circle $\mathscr C$,

$M$ finds "ping"-symmetry wrt. the two sphere centers, $C$ and $Q$:

  • $\forall M_{\sigma} \in \{ M_{\sigma} \} : M_{\text{saw that } C \text{ saw that } M \sigma} \equiv M_{\text{saw that } Q \text{ saw that } M \sigma}$,

and surely

  • $\forall P \in \mathscr C, \forall M_{\sigma} \in \{ M_{\sigma} \} : P_{\text{saw that } M \text{ saw that } P \text{ saw that } M \sigma} \text{ before } P_{\text{saw that } C \text{ saw that } P \text{ saw that } M \sigma}$,

as well as

  • $\forall P \in \mathscr C, \forall M_{\sigma} \in \{ M_{\sigma} \} : P_{\text{saw that } M \text{ saw that } P \text{ saw that } M \sigma} \text{ before } P_{\text{saw that } Q \text{ saw that } P \text{ saw that } M \sigma}$,

and perhaps even ("ping"-symmetry of all circle constituents wrt. the two sphere centers, $C$ and $Q$):

  • $\forall P \in \mathscr C, \forall M_{\sigma} \in \{ M_{\sigma} \} : P_{\text{saw that } C \text{ saw that } P \text{ saw that } M \sigma} \equiv P_{\text{saw that } Q \text{ saw that } P \text{ saw that } M \sigma}$,

then $M$ is thereby identified as "mid-point" of circle $\mathscr C$, or in other words: the circle $\mathscr C$ having been "drawn around" participant $M$.

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