# user12262

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The following questions (in no particular order) which I had submitted have been "removed from PSE for reasons of moderation":

1. Which geometric relations obtain between two distinct rest systems?

Consider, as a thought experiment, a set of participants who measure throughout the experiment having been at rest to each other; among them explicitly participants ${\mathbf A}$, ${\mathbf B}$ and ${\mathbf F}$ who determine the ratios of their (chronogeometric) distances between each other as real number values $\frac{{\mathbf A}{\mathbf B}}{{\mathbf A}{\mathbf F}}$, $\frac{{\mathbf B}{\mathbf F}}{{\mathbf A}{\mathbf F}}$, and $\frac{{\mathbf A}{\mathbf B}}{{\mathbf B}{\mathbf F}} = \frac{{\mathbf A}{\mathbf B}}{{\mathbf A}{\mathbf F}} / \frac{{\mathbf B}{\mathbf F}}{{\mathbf A}{\mathbf F}}$.

Further let there be another set of participants (of which neither ${\mathbf A}$, nor ${\mathbf B}$, nor ${\mathbf F}$ are a member) who measure throughout the experiment having been at rest to each other as well; among them ${\mathbf J}$, ${\mathbf K}$ and ${\mathbf Q}$, who determine the ratios of their (chronogeometric) distances between each other as real number values $\frac{{\mathbf J}{\mathbf K}}{{\mathbf J}{\mathbf Q}}$, $\frac{{\mathbf K}{\mathbf Q}}{{\mathbf J}{\mathbf Q}}$, and $\frac{{\mathbf J}{\mathbf K}}{{\mathbf K}{\mathbf Q}} = \frac{{\mathbf J}{\mathbf K}}{{\mathbf J}{\mathbf Q}} / \frac{{\mathbf K}{\mathbf Q}}{{\mathbf J}{\mathbf Q}}$,

such that

• ${\mathbf J}$ passed ${\mathbf A}$, then passed ${\mathbf B}$,

• ${\mathbf A}$ passed ${\mathbf J}$, then passed ${\mathbf K}$,

• ${\mathbf Q}$ passed ${\mathbf F}$, in coincidence with ${\mathbf Q}$ and ${\mathbf F}$ observing ${\mathbf J}$ and ${\mathbf A}$ having passed each other,

• ${\mathbf B}$ and ${\mathbf F}$ determined that ${\mathbf B}$'s indication of the passage of ${\mathbf J}$ was simultaneous to ${\mathbf F}$'s indication of the passage of ${\mathbf Q}$, and

• ${\mathbf K}$ and ${\mathbf Q}$ determined that ${\mathbf K}$'s indication of the passage of ${\mathbf A}$ was simultaneous to ${\mathbf Q}$'s indication of the passage of ${\mathbf F}$.

Question:
Is thereby guaranteed that for these distance ratios obtains

(1)
$\frac{{\mathbf A}{\mathbf B}}{{\mathbf A}{\mathbf F}} = \frac{{\mathbf J}{\mathbf K}}{{\mathbf J}{\mathbf Q}}$ ?,

and (moreover)

(2)
$\left( \left(\frac{{\mathbf B}{\mathbf F}}{{\mathbf A}{\mathbf F}}\right)^2 + 1 - \left(\frac{{\mathbf A}{\mathbf B}}{{\mathbf A}{\mathbf F}}\right)^2 \right) \left( \left(\frac{{\mathbf K}{\mathbf Q}}{{\mathbf J}{\mathbf Q}}\right)^2 + 1 - \left(\frac{{\mathbf J}{\mathbf K}}{{\mathbf J}{\mathbf Q}}\right)^2 \right) = 4 \left( 1 - \left( \frac{{\mathbf A}{\mathbf B}}{{\mathbf A}{\mathbf F}} \right) \left( \frac{{\mathbf J}{\mathbf K}}{{\mathbf J}{\mathbf Q}} \right) \right)$ ?

Or otherwise:
What could be concluded if (1) and/or (2) were not found satisfied?

# 1,219 Actions

 2d comment Does the definition of the SI unit “second” require that possible perturbation of primary frequency standards should be measured? p.s. On first looks this answer of yours, especially the "Addendum" seems very interesting, thoughtful, substantial, and ... well ... very much debatable/contestable. (Indeed closely along the lines of physics.stackexchange.com/q/78684 where I put my two cents already). Indeed too interesting for me to get hung up too much about your preceding comment. (And, therefore, it may take rather until mid next week to express a reply; perhaps best in form of an own separate answer to my question; of which I plan to notify you explicitly, because, thankfully, you seem interested in the Physics.) 2d comment Does the definition of the SI unit “second” require that possible perturbation of primary frequency standards should be measured? Emilio Pisanty: "Yes, that was intentional." -- Are you suggesting that I was lying in my preceding comment, and/or that I didn't in good conscience carry our correspondence right below the OP question since yesterday?? () Well, no. Think again about the working conditions and habits of PSE contributors. I had been curious (and I had asked) when you mentioned "your edit" the first time (I considered whether you had referred again to that); the second time "your edit" came up I discovered this answer here right away. 2d comment Does the definition of the SI unit “second” require that possible perturbation of primary frequency standards should be measured? Emilio Pisanty: Very sorry, only just now I noticed that you had posted this lengthy answer. (Btw., AFAIU its URL is specificly " physics.stackexchange.com/a/184999 "). So, thanks for that; I'll need to take a while to read it ... And, due to weekend etc. I may be able to submit some additional comment (as I surely may want to) only late on Monday. 2d comment Does the definition of the SI unit “second” require that possible perturbation of primary frequency standards should be measured? @Emilio Pisanty: "something constructive like adding a clear description of Marzke-Wheeler clocks to Wikipedia." -- Certainly not to Wikipedia at its present rigor. Much rather to PSE. "[...] You may need to scroll slightly." -- Hmm... If I scroll slightly, I read "The M-W construction makes no appeal whatsoever to rods and clocks of atomic constitution. ...". "I will go back to quietly mourning [...]" -- I mourn, too; each time NIST asserts "accuracy" of their contraptions. Now let me dig out that good old/well-worn "Bevington" of mine ... 2d comment Does the definition of the SI unit “second” require that possible perturbation of primary frequency standards should be measured? @Emilio Pisanty: "[...] I have provided an extensive edit addressing these concerns." -- You did what, where? Can you provide a URL-link, please? "a simple bibliographic request" -- Now that you put it this way (thanks for clarifying!), I hope I've understood your request (better) and I've tried to meet it by the recent EDIT. p.s. "abrasive reply [...] your tone [...]" -- If you believe that you're capable of discerning and judging any nuances of "tone" in my hardly 600-character-comments then you might as well call it an "abrasive tone honed by lacking substatial response". 2d comment Does the definition of the SI unit “second” require that possible perturbation of primary frequency standards should be measured? @Emilio Pisanty: "If you expect every physicist in the world to have immediate access to MTW [...]" -- What's really sadly disappointing: As of just now, there's still no in-depth-analysis, nor at least some brief description, nor at least any mentioning of "Ideal clocks" (or "Geodesic clocks") constructed by "the procedure of Marzke and Wheeler" to be found at Wikipedia. (And even where this procedure is mentioned there in other contexts they spell the last name of Robert Franklin Marzke "Märzke".) 2d revised Does the definition of the SI unit “second” require that possible perturbation of primary frequency standards should be measured? (v3.1: Added excerpts of MTW Boxes 16.4 and 10.2; in response to comments.) May20 comment Does the definition of the SI unit “second” require that possible perturbation of primary frequency standards should be measured? @Emilio Pisanty: "you ask an atomic physics question and then expect the answerers to be fully conversant in GR." -- Anyone dealing with (assertions of) geometric or kinematic relations ("durations of oscillation periods", "orbital radii", "speeds") better be fully conversant in the applicable "geometric-kinematic part of" GR. No? "to address a passage in MTW then you will need to provide it in full." -- I'm looking forward to that. Surely PSE-MathJax will support "pstricks" commands any time now, too, so we can collaboratively work out and improve/adapt all those helpful illustrations ... May20 comment Is it possible to speak about changes in a physical constant which is not dimensionless? Emilio Pisanty: We should distinguish two sorts of "dimensional physical constants": Those which are purely formally symbolic, with no intrinsic property than being distinctive and non-zero, vs. those which are artefact based (model dependent, incidental). An example of the former would clearly be the symbol "$c_0$" which appears in the chrono-geometric definition of "distance" (and which only subsequently is identified as "signal front speed"); examples of the latter are any Standard Model particle parameters (charges, masses). Surely it's absurd to consider "changing a mere symbol by 10 %". May20 comment Does the definition of the SI unit “second” require that possible perturbation of primary frequency standards should be measured? @Emilio Pisanty: "There's no way to compare a clock's output with what it gave out yesterday," -- MTW Box 16.4, "Ideal Rods and Clocks Built from Geodesic Worldlines", seems to suggest otherwise. Though I haven't been able to discern whether and how MTW supposed to distinguish "geodesic worldlines" from "non-geodesic worldlines", or how to distinguish an "affine parametriziation" of a worldline (required in Box 10.2) from any "non-affine parametriziation", before and without having "ideal rods and clocks" available already. "external assumptions" -- Hardly; only coincidence determinations. May20 comment Lorentz Transformations and time of event Emilio Pisanty: "on all serious SR literature, 'event' is essentially synonymous with a $(t,x)$ pair." -- No, that's at most how 'event' is presented in all serious literature of how certain mathematicians construed SR (or Einstein's Theory of Relativity overall); to the expressed bewilderment of Einstein; until the underlying coordinate-free physics was at least partly recognized (e.g. as "point-coincidence argument"). The OP, too, wrote that "[one participant] strikes [the other]". I've awarded the OP +1 just for this insight. May20 comment Lorentz Transformations and time of event Emilio Pisanty: "Your second point is pure semantics. [...]" -- My complaint is that you (and others) refuse to explicitly name the (generally) several distinct parts of any one event (any one "encounter between two or more recognizable material points"); namely the distinct individual indications ("positions of the little hands" etc.) of these distinct individual participating "recognizable material points", at that one coincidence event. If you don't even admit this notion by (distinctive) name then there's no telling whether you understand it; let alone teach Einstein's method. May20 comment Lorentz Transformations and time of event Emilio Pisanty: "It is meaningless to talk about the objective temporal order of spacelike-separated events." -- Right; objective; sure. "It is perfectly possible to talk about the temporal order each observer sees." -- So what's this vague/evasive language ("seeing temporal order") supposed to mean?? Do you understand and agree that the objective temporal order of indications of suitable pairs (i.e. pairs being at rest to each other) of participants in spacelike-separated events is measured by Einstein's method? May20 comment Lorentz Transformations and time of event 2: "The only way for events to be timelike-separated and simultaneous is [...] that they are the same event." -- Your phrase identifies plural ("several distinct events") with singular ("one and the same event"); that's to be avoided. Also, arguably, this uses the notion of simultaneity reflexively ("any one event is simultaneous to itself"); that's at least a dangerously degenerate use of the notion "simultaneous", being implicitly dismissive of the distinction to "being coincident", and of the underlying notion "middle between" certain participants. May20 comment Lorentz Transformations and time of event Emilio Pisanty: 1:"it is meaningless to talk about the temporal order of spacelike-separated events." -- I agree. Do you agree that this includes talk of "simultaneity of (spacelike-separated) events" being meaningless, too? Then why do you phrase your answer from the outset and throughout as if pretending that such talk were meaningful ?? (Viz. "[...] need not agree on the simultaneity of two events, or on their temporal order."; "[...] for whom [events] $A$ and $B$ are simultaneous"; "[...] for whom $A$ happens before $B$", etc.) [contd.] May20 revised Lorentz Transformations and time of event (v3.14159265: more minor copy-editing.) May19 answered Lorentz Transformations and time of event May13 comment Time flow difference for satellites AV23: "[...] not sure of the reasons behind the difference [...]" -- And I didn't mean to be critical of your derivation as much as of the (seeming) lack of care and responsiveness of certain other contributors. "I've updated in a way that I believe clarifies [...]" -- I still recognize the offending choice of phrase, and you didn't use my suggested substitution. But you added: "where the notion of simultaneity is given by the $t$ coordinate" -- Hmm: "simultaneity" in a region which is explicitly not flat?? (I'd like to pursue this further but I'll get a chance only in about a week). May13 comment Time flow difference for satellites p.p.s. Oops! ... Well, in my preceding comment I had mistakenly been thinking of geostationary satellites. So, instead, express "the answer" rather as: $\qquad~\qquad~\qquad~\qquad~\qquad~\qquad~\qquad~\qquad~\qquad~\qquad$ "It takes a GPS satellite $\approx~(1 + 1.35 \times 10^{-8})$ as long from one passage right above a particular constituent of Earth's surface, until the next, as it takes this constituent from one passage right underneath the GPS satellite, until the next." May13 comment Time flow difference for satellites AV23: "the passage of time is therefore faster for [...]" -- What an ugly, possibly coordinate dependent choice of phrase! (Half the blame goes to the OP question.) Better to make a (plain) comparison of (definite) durations; e.g. "It takes a GPS satellite $\approx~(1 + 1.35 \times 10^{-8})$ as long to complete a full revolution around Earth than a constituent of Earth's surface." p.s. Any thoughts on the difference (albeit it as good as negligible for the problem at hand) between the exact expressions of your solution and of this accepted answer?