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Consider infinitely many distinguishable observers, no two of whom ever meet; and who generally "keep sight of each other", but not necessarily "each keeping sight of all others".
How should they determine whether or not they can be described as being "defined on a Lorentzian manifold"?

[This question refers to terminology of and is meant as follow-up to that question; in the attempt to ask perhaps more originally.]

The phrase "defined on a Lorentzian manifold" appears a very general condition.

To be more specific consider instead the question:
"How should the given observers determine whether or not some subset of the entirety of events in which they (separately) participated can be described as "open set of a 3+1 dimensional Lorentzian manifold"?

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A manifold is basically defined as a space that locally has the same topology as n-dimensional Euclidean space. Examples of things that are not manifolds would include spaces with boundaries and spaces that have a different number of dimensions in different regions. It's hard to say what we would actually observe if spacetime did this kind of stuff, because we don't have any (useful, tested, realistic) physical theory that describes such phenomena.

The fact that spacetime is Lorentzian corresponds to the observational fact that we see one timelike dimension, with the rest being spacelike. Again, it's hard to say what experimental results we'd see if this were not the case. To do an experiment, you have to have a conscious observer who has a memory of his observations. This requires the existence of a timelike dimension.

General relativity, in the standard formulation, works with any signature, but can't describe a change of signature; typically when you get a metric that changes signature, it's a sign that you've chosen an unfortunate set of coordinates, and in some other set of coordinates it doesn't change signature.

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Is a Lorentzian manifold locally Euclidean? – Larry Harson Jul 3 '13 at 1:46
@LarryHarson: Its topology is locally the same as n-dimensional Euclidean space. Its (metric) geometry is different. – Ben Crowell Jul 3 '13 at 2:25
@Ben Crowell "you have to have a conscious observer who has a memory of his observations." -- That's meant to be satisfied by the Q statement of "[all being able to] keep sight of [others]"; corresponding to observers A, B, M etc. of Einstein's thought-experiments. "Again, it's hard to say [...]" -- I'd also accept an answer along the lines of "that's satisfied by definition/statement of the Q". (Is that what you're getting at?) "spaces with boundaries [or having] a different number of dimensions in different regions." -- Please note the recent Edit. (Meta: Rather a new Q?) – user12262 Jul 3 '13 at 4:50

An experimentalist's answer would be :

Repeat the experiments that established that the pseudo-euclidean geometry of the Lorenz transformations fit the measurements.

The constancy of the velocity of light, the E=m*c^2 are established by numerous experiments; one should repeat enough experiments to establish the statistical validity within errors of these crucial elements.

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These experiments wouldn't provide any method, even in principle, for detecting if we weren't in a Lorentzian manifold. For example, if we lived in a manifold with boundary, an experiment disproving the constancy of $c$ wouldn't tell us that. If we lived in a spacetime with signature ++++, we wouldn't be able to carry out the experiment at all, because time wouldn't exist. You can test for Lorentz violation, but suppose you find it. Theories that include Lorentz violation are still expressed in a Lorentzian manifold, but, e.g., may have a vector field defining a preferred frame. – Ben Crowell Jul 3 '13 at 13:37
@BenCrowell The fact that we are talking about Lorenzian manifolds is because experimental data made it necessary. Otherwise we are playing mathematical games detached from the physical world. – anna v Jul 3 '13 at 13:43
"An experimentalist's answer would be : [...]" -- I appreciate the approach in general, but: How, specificly, should a set of observers given in the question go about evaluating and comparing "velocity"? (Not even to get into dynamics/variational calculus.) Also: any particular given Lorentzian manifold may (surely) preserve "its underlying geometric/physical/experimental structure" under coordinate transformations that are considerably more general than "Lorenz transformations" or even "Poincaré transformations". – user12262 Jul 3 '13 at 17:30
@user12262 just saw this comment . It falls within the realm of being able to distinguish between sufficient and necessary conditions. The constant velocity of light is a necessary condition because experiments say so. One can enter theoretically complicated constructs at will as long as the necessary condition is fulfilled. It is up to the theorist to propose extra experiments that will validate the new proposal of more general transformations. If there are no such proposals then one is playing theoretical games. – anna v Oct 2 '13 at 4:38
@anna v: anna v: "experimental data made it necessary [...] experiments say so." -- Referring to experiments is great, but there must be full understanding, agreement and appreciation of what experiments provide: All our well-substantiated space-time propositions amount to the determination of space-time coincidences {such as} encounters between two or more {...} material points I like to know how to express "Lorentz manifold", "velocity" etc. from coincidences (as foundational notion). – user12262 Oct 3 '13 at 8:34

well firstly, i am assuming that the observers you mention are at 'rest' with respect to each other in the sense that their spacial distance does not change. the observers can easily measure the distance between them by sending out light pulses. since they can communicate through electromagnetic signals, any observer can send the time on his watch to another observer. this observer, after accounting for the time taken for the light pulse to travel can set his watch to the appropriate time, in this way they can synchronize their watches. if their watches stay synchronized( this can be determined by repeating the same experiment again), then they are in a Minkowski space(I don't know if this is what you are calling a Lorentzian manifold, I'm not much of a math guy.)

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A Lorentzian manifold is not Minkowski space. It's a curved spacetime that is locally Minkowski. – Ben Crowell Jul 3 '13 at 13:30
@guru Thanks for your answer; sorry that you've been losing points as a consequence. "firstly, i am assuming that the observers you mention are at 'rest' with respect to each other in the sense that their spacial distance does not change." -- Of course this requires a definition on how to measure "distance" in the first place. "any observer can send the time on his watch to another observer" -- Why only " his / one watch"?, and if so: Which one (of "his many")?? ... Anyways: (to be contd.) – user12262 Jul 4 '13 at 17:06
Continued: Anyways, I'm (also) very much interested in how participants who (generally) "keep sight of each other", who can observe and recognize each other, and who each can judge order or coincidence of own observations could on this basis determine and agree on any further relations between each other; such as in the question above, or concerning "(mutual) rest", or determining and comparing "duration" between their various times/indications. I'll try to continue asking questions on this topic or submitting answers accordingly. – user12262 Jul 4 '13 at 17:07

The following describes at least some necessary condition(s); more conditions may have to be considered for arriving at a sufficient/full answer.

Any observer, such as A, should have taken part in at least two distinct events, such as $\!{\mathscr J}$ and ${\mathscr Q}$. (The corresponding indications of observer A I'll denote as "A$\!_{\mathscr J}$" and "A$\,\!_{\mathscr Q}$".)

For any two indications A$\!_{\mathscr J}$ and A$\,\!_{\mathscr Q}$ of observer A there should be (many distinct) observers, such as B who (all) observed event $\!{\mathscr J}$ and whom A observed having observed event $\!{\mathscr J}$ (and in particular recognized having observed A's indication A$\!_{\mathscr J}$) while participating in event ${\mathscr Q}$.

(Also, vice versa, while participating in event $\!{\mathscr J}$ observer A should have seen and recognized no other observers having observed A's indication A$\,\!_{\mathscr Q}$. Observer A, such as any other, would thus be able to order the own indications in "before and after".)

Further, there should be (many distinct) observers, such as N, of whom A "keeps sight between" indications A$\!_{\mathscr J}$ and A$\,\!_{\mathscr Q}$; while any observer N in turn does not "keep sight" of any observer B.

Together, these conditions are meant to correspond to the requirement (necessary of Lorentzian manifolds) that for any two "timelike" related events such as $\!{\mathscr J}$ and ${\mathscr Q}$ there exists a corresponding "causal diamond", with observers B "touching its boundary" and observers N (along with observer A) "tracing its interior".

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There's a difference between preview and post in rendering **A**$\!_{\mathscr J}$, for instance. Any help is appreciated ... – user12262 Jul 3 '13 at 18:42
<b>A</b>$\!_{\mathscr J}$ is rendered equally in preview and post. – user12262 Jul 3 '13 at 18:59

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