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In Misner, Thorne, Wheeler (henceforth written as "MTW"), "Gravitation", Box 16.4, there's an experimental setup construction (or method) presented by which

"Each geodesic clock is constructed and calibrated as follows: [...]", where "Those that pass through [event] $\mathcal A$ are calibrated [...] against the [specific timelike] standard interval $\mathcal A B$" and "Any interval $\mathcal P Q$ along the world line of a geodesic clock can be measured by the same method as was used in calibration".

Regarding importance and motivation of this method it is stated that

"The Einstein principle that spacetime is described by Riemannian geometry exposes itself to destruction by a "thousand" tests. Thus, from the fiducial interval, $\mathcal A B$, to the interval under measurement, $\mathcal P Q$, there are a "score" of routes of intercomparison, all of which must give the same value for the ratio $\mathcal P Q / A B$.".

Box 16.4 contains two separate sketches of the setup construction; one concerned with interval $\mathcal A B$, the other equivalently with interval $\mathcal P Q$. Explicitly evident in either sketch are two separate participants ("particles", "timelike world lines") who

  • observe pings between each other, and

  • both observe one ping to one particular event (event $\mathcal{B}$ in the first setup sketch; or event $\mathcal{Q}$ in the second setup sketch), where

  • each participant finds some (integer) number of successive pings to the other as same as the one ping to the "external" event ($\mathcal{B}$, or $\mathcal{Q}$). (The "intercomparison" to be achieved then involves counting those successive pings by the two participants.)

There are additional "demands" stated (which the two sketches alone don't make explicit); among them that the participants who took part in event $\mathcal A$ (referring to the first sketch), or in event $\mathcal Q$ (referring to the second sketch) are required to have been "freely falling"; and their world lines correspondingly "paths of freely falling particles" and "geodesics".

Question 1:
Has there at least one such test been carried out already, presumably by "intercomparison" along at least two distinct "routes", by explicitly using the method indicated by MTW ? (i.e. not involving any further "model dependent assumptions" such as in the arguments presented at the end of box 16.4)

Question 2:
Instead of demanding of the setup(s) that certain participants were "freely falling" and obtaining "a "thousand" tests of the Einstein principle that spacetime is described by Riemannian geometry",
could the described method be explicitly adapted in turn, by assuming "the Einstein principle" as outright valid and by suitably combining sufficiently many (if not "a thousand") of the described setups, to obtain a test of whether (or to which accuracy) a given particle had been "freely falling" (or correspondingly, whether, or to which accuracy, a given timelike world line was "geodesic") ?

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Sending some standard clock through different paths and comparing the results is pretty much how relativity gets tested ... that what you mean?

As for the "thousand" vs "sufficiently many" one needs only enough trials in one experiment to satisfy statistics. A single counter-example is usually sufficient to disprove a general theory. The author was using hyperbole to make a point.

The authors proof is showing up a logical inconsistency in the model ... it is like crunching some numbers on an equation and finding it predicts that 0 =-1.

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  • $\begingroup$ Simon Bridge: "Sending some standard clock through different paths and comparing the results is pretty much how relativity gets tested" -- If that's an attempt to address my above questions it should relate more directly to what's evident in MTW's sketches and/or to the detailed terminology of their additional "demands". (Were successive pings observed and counted? Was identifiable equipment "freely falling"? Was "the method of Schild's ladder (Box 10.2)" employed explicitly? etc.). Ultimately: Can this be used to define what you mean by "standard clock" (vs. "any clock whatsoever")? $\endgroup$ – user12262 Jan 16 '14 at 16:16
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    $\begingroup$ sorry - That was an answer to Q1. You did not say you wanted the empirical test in history to have been described by a particular formalism. A quick trawl of the literature shows many experiments involving path comparisons and ways to define standard clocks - I think it is up to you to say how these are or are not satisfactory. I suspect you are over-thinking the statement. $\endgroup$ – Simon Bridge Jan 17 '14 at 4:54
  • $\begingroup$ Simon Bridge: "You did not say you wanted the empirical test in history to have been described by a particular formalism." -- In case you missed it: I've been trying to ask very specificly about the "formalims" and experimenal method(s) laid out in MTW, "Gravitation", box 16.4. (Admittedly, the presentation in that box 16.4 is not perfectly self-contained ...) "A quick trawl of the literature shows [...]" -- Have you been able to look at MTW; especially that box 16.4 ? Do you have access to the reference given there: "Marzke/Wheeler (1964)"; in "Chiu/Hoffman, eds. (1964)"? ... $\endgroup$ – user12262 Jan 17 '14 at 6:32
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    $\begingroup$ Simon Bridge:(cont.): "[...] ways to define standard clocks - I think it is up to you to say how these are or are not satisfactory." -- Certainly wrt. MTW, "Gravitation", Box 16.4 (which titled suitably enough.). Satisfactory there I find especially the explicit use of observing and counting successive pings between various pairs of participants. OTOH, I've been so far unable to find any satisfactory definition of "free fall" and/or "geodesic motion" outside of that box 16.4. Therefore my Q2: does such a definition reside inside that box, if suitably iterated?. $\endgroup$ – user12262 Jan 17 '14 at 7:02
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    $\begingroup$ @magma: "Free fall is defined and explained in paragraph 1.3 pag. 13 [MTW]" -- Defined ?? Even looking beyond the model-dependent drivel of pag. 13 ("free fall of an object" as "its normal track through spacetime"): Pag. 16 gives a kinematic desciption: "it moves {...} in a straight line with uniform velocity." But this still leaves open how to find out whether a given object had so "moved", in a trial under consideration, before and without having an "(ideal) clock" and an "(ideal) rod" in this trial available, which by §16.4 require "free objects" having been identified. $\endgroup$ – user12262 May 10 '15 at 17:53

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