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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?


9h
comment Acceleration is zero, for non-zero net force
To expand on "Mew's simple solution" (in the above comment) "the table is on an angle and you are pushing upwards against gravity": $$\mathbf F_{app} := m \, (\mathbf g\cdot\mathbf v)\mathbf v/\text v^2;$$ $$F_{app} := m \, g \, \text{Sin}[ \phi ].$$ If it seems somehow counter-intuitive that the applied force doesn't depend on the velocity $\mathbf v$ remember that the power does: $$P = (\mathbf F_{app}\cdot\mathbf v) = F_{app} \, \text v.$$ (But of course it remains to be seen whether the OP had this sort of "solution" in mind at all.)
10h
comment Are events in this experiment simultaneous if observed in platform's frame?
bright magus: "[...] rays that get scattered [...]" -- There is a decisive difference between saying (as Einstein did in his thought-experimental description) "Lightning has struck railway tie $A$ and locomotive front buffer $A' \, $" and on the other hand (what seems to preoccupy you) "to see lightning in the general vicinity surrounding embankment and train". Do you recognize and appreciate this difference?
23h
comment Are events in this experiment simultaneous if observed in platform's frame?
bright magus: "[...] chasing me all over the forum?" -- You must be referring to our previous correspondence there. Well, someone better; since the "drive-by community" apparently wouldn't point out to you that even 1905 "the accelerations are measured in the stationary system K"
23h
comment Are events in this experiment simultaneous if observed in platform's frame?
bright magus: "[...]" -- Get a grip on the thought-experimental foundations of (S)RT!: "two events (e.g. the two strokes of lightning A and B) {...} the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the mid-point M {...} an observer sitting in the position M' {...} will see the beam of light emitted from B earlier than he will see that emitted from A.". [to be contd.]
1d
comment Are events in this experiment simultaneous if observed in platform's frame?
bright magus: "Nobody (or no device) can see light that is not sent directly to its eyes (receiver)." -- That's surely correct. However, it is generally assumed in the thought-experiments of RT that any events (and all separate indications of the various participants in any one event) are perfectly visible; giving off sufficient "light" to be seen and recognized by multiple observers (who are not necessarily at rest to each other). Yes, RT may be generalized by weakening this idealization. But whether your argument relates to the OP's question is dubious.
Apr
12
comment About the speed of light
The actual reason for the rocket under consideration not going faster than $c$ (i.e. maximum signal speed) between a given starting gate and a given finish line is of course that the rocket also presents a signal having been sent from starting gate to finish line (and not even necessarily a signal of the highest speed). p.s. Concerning the well-known formula for adding speeds in the same direction it is worth noting that $$\frac{u + v}{1 + u \, v} > 1$$ for instance if $$ v > 1 > u > 0.$$
Apr
12
comment How a accelerated object sees another accelerated body in Special relativity?
bright magus: On the main point (1) you don't seem to concede that "accelerating objects" could be among the "natural phenomena" to be considered in the SRT?? Well ... Concerning you recent EDIT: If you're objecting to the cited formulation "... an accelerating object [...] at any instant, we can say that the object is located in a particular inertial frame" I agree that this formulation is objectionable. Though such objection may be better expressed elsewhere, e.g. at physics.stackexchange.com/q/3193 and requires of course a foundation in the axiomatics of (S)RT (point (2) above).
Apr
11
comment How a accelerated object sees another accelerated body in Special relativity?
bright magus: ... And 2. A good start for trying to understand Einstein (and what he might have meant be "law", "uniformly moving" etc.) is Einstein's maxim that "All our well-substantiated space-time propositions amount to the determination of space-time coincidences {such as} encounters between two or more recognizable material points".
Apr
11
comment How a accelerated object sees another accelerated body in Special relativity?
bright magus: "As I understand Einstein, SR is [...] about "natural phenomena running their course with respect to K' according to exactly the same general laws as with respect to K"." -- 1. Among those "natural phenomena" can be "accelerated objects" (and specificly, in the sense of the OP, "objects in hyperbolic motion"); can't they? And 2. [... to be continued]
Apr
11
comment Zero photon energy in moving frame
@kηives (@kives ?): "There is no photon rest frame" -- Correct; compare with this question about phtotons and mutual rest. "the princip[al] postulate of SR: [\begin paraphrase] the speed of light is 'c' in all frames of reference [\end paraphrase]." -- Emphasis on inertial frames of reference; i.e. any system whose members are mutually at rest. There is a system "comoving with the photon"; it's just not an inertial system.
Apr
10
comment How a accelerated object sees another accelerated body in Special relativity?
bright magus: "I'm [...] talking about the axiom of the SR theory." -- Well, it's surely correct and noteworthy that Einstein's SR postulates don't explicitly mention objects in accelerated motion (but to the contrary, explicitly mentioned are only inertial systems, i.e. sets of participants who were at rest wrt. each other.) Nevertheless, this terminology which is used and presumed (as axiomatic) for expressing these postulates allows to speak just as easily about "participants who were not at rest wrt. anyone else" and to quantify their geometric-kinematic relations to inertial systems
Apr
10
comment How a accelerated object sees another accelerated body in Special relativity?
bright magus: "[...] Therefore [...] SR precludes acceleration." -- In the answers linked above by the OP as well as in this answer it is explicitly shown that and how accelerating objects can be described in terms of the SRT. And that seems more compelling, and teachable, and comprehensible than claims to the contrary; whatever their grounds might be.
Apr
8
comment Breaking the speed of light relative to a moving object
@User58220: I like the emphasis on _measurement_, contrasting with observation (i.e. collecting observational data such as identifying the participants and registering their indications) from which measurements would be derived (e.g. values of pairwise mutual speed, in _pairwise mutual agreement_). It should be emphasized that $A$ and "you" had _measured_ their speed (0.75 c) and that $B$ and "you" had _measured_ their speed (0.75 c) just as $A$ and $B$ _measured_ 0.96 c. "S-A and S-B are approaching each other at 1.5c" -- That's a third party opinion (a.k.a. "improper").
Apr
8
comment Breaking the speed of light relative to a moving object
Daniel Geisler: "I used the term relative velocity, where the speed of light c is set to 1." -- Yes you did. But then you should be consequent and drop the "c" from your expression "$\frac{.75+.75}{1+.{75}^2}$ = .96 c" as well.
Apr
7
comment Travel at the speed of light
@WetSavannaAnimal: (Oops!, btw. I set $c = 1$ above) ... And I'd be happy to agree on saying that "the rest mass (and even the inertia) of the box-with-photon-system is larger than that of the box alone". "In frames other than the box's, there isn't a single frequency [...] superposition of a redshifted "right-to-left" running state (if the box is moving rightwards) and a blueshifted "left-to-right"" -- This certainly doesn't quite fit my "standing wave" intuition. But to learn and resolve that I really should read the two links you indicated carefully again, and possibly comment there.
Apr
7
comment Travel at the speed of light
WetSavannaAnimal: "You asked about wavelengths and frequencies of the light, which are frame dependent." -- True. (So is $E$.) But starting from my first comment I only intended to consider "the box at rest". (Due to being limited to 600 characters.) I recognized later that you're used to considering box and fields in other setups. (Me too, btw.) "[...] The rest mass is then the value of this inertia for changes from the box's rest frame." -- To me, non-zero rest mass is $$E \times \sqrt{1 - \left(\frac{v}{c}\right)^2}.$$ And [...to be contd]
Apr
6
comment Travel at the speed of light
WetSavannaAnimal: "I categorically did not say the acquired inertia was frame dependent." -- Then the start of your first comment "That of course depends on the reference frame." was a poor choice of wording. To repeat: I object to the phrase ""rest mass" is acquired by a particle with a rest mass of nought when [...] confined" of your answer above; and I ask you again which value of wavelength $\lambda$ "in the box picture" you suppose for a confined photon of energy $E = h \, \nu$.
Apr
6
comment Why does a cup with 100 g water float when placed on another cup with 50 g of water?
@DumpsterDoofus: Since you seem to have the capability to make and include drawings in your answer it might be nicer (and more inviting to practical replications) to draw two equal, slightly conical cups; both similar to this one, together with the appropriate amount of water included. And the corresponding discussion might benefit from mentioning (and even using) "hydrostatic pressure" ...
Apr
6
comment Distance Between Two Photons Calculated in Different Inertial Frames
Nuclear_Wizard: "it's fine for an interval to move at c" -- An interval moving?? A pair of events moving?? Anyways, my main objection was against the OP's question itself speaking of "distance between photons" (I blame the text being studied; not the messenger(s).) Distance is between "[pairs of] points [as pairs of] stationary particles" you considered; but not "even when the points are travelling at the speed of light." Now: "[...] constant in all inertial reference frames, there cannot exist a frame where the speed of a photon is zero" -- Why not some "non-inertial frame"?!?
Apr
6
comment Distance Between Two Photons Calculated in Different Inertial Frames
Nuclear_Wizard: "imagine them as stationary particles" -- +1 for matching Einstein's "space-time propositions amount to the determination of space-time coincidences {such as} encounters between two or more recognizable material points". "photons do not have a rest frame." -- Right. But: "light speed is constant in all reference frames" -- No: in inertial frames. Finally: "even when the points are travelling at the speed of light." -- -1.