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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 http://en.wikipedia.org/wiki/Frame_fields_in_general_relativity and is meant as follow-up to that question; in the attempt to ask perhaps more originally.]

Edit
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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user12262
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4 Answers4

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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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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.

anna v
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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.)

guru
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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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