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The following questions (in no particular order) which I had submitted have been "Deleted by Community":

2. Is average speed an invariant?

Is the value of average speed an invariant?,
specificly the value of the average speed, with respect to suitable(1) specific participants, say $P$ and $Q$, of some specific participant, say $A$, as $A$ moved from $P$ and $Q$?

Expressing the value of the average speed of $A$ wrt. $P$ and $Q$ as

$$v_{PQ}[~A~] := c~\beta_{PQ}[~A~],$$

where $c$ denotes the signal front speed, and $\beta_{PQ}[~A~]$ is a specific real number,
and where the average refers to the trial from $P$ and $A$ having departed from each other until $P$ and $A$ having reached each other,
does the value of $\beta_{PQ}[~A~]$ depend on the assignment of coordinate values to the relevant unique events $\varepsilon_{AP}$ and $\varepsilon_{AQ}$ (and/or to other events)?

Does the real-number value $\beta_{PQ}[~A~]$ change if coordinate values which are assigned to event $\varepsilon_{AP}$ are being changed, or if coordinate values which are assigned to event $\varepsilon_{AQ}$ are being changed?

Note also, that the real-number value $\beta_{PQ}[~A~]$ can be expressed in terms of intervals "between" certain pairs of the relevant events, e.g.

$$\beta_{PQ}[~A~] = \frac{s^2[~\varepsilon_{AP}, \varepsilon_{AQ}~] - s^2[~\varepsilon_{FQ}, \varepsilon_{AQ}~]}{s^2[~\varepsilon_{AP}, \varepsilon_{AQ}~] + s^2[~\varepsilon_{FQ}, \varepsilon_{AQ}~]},$$

where event $\varepsilon_{FQ}$ denotes the (unique) event of the future ("forward") light cone of event $\varepsilon_{AP}$ in which $Q$ took part (in coincidence with some suitable participant $F$); and that (presumably) the values of intervals are invariant.

(1: Specifily, $P$ and $Q$ remaining separate and at rest with respect to each other; i.e. constituting members of an inertial system in the sense of Rindler: "simply an infinite set of point particles sitting still in space relative to each other".)


7h
comment Does motion with constant proper acceleration, in a flat region, necessarily mean straight hyperbolic motion?
Jerrold Franklin: "For motion along a circular trajectory the acceleration changes its direction" -- That's certainly true of the acceleration vector evaluated in reference to any inertal system (since the members of an inertial system, in a flat region, are not rotating). But if we're considering proper acceleration, i.e. the acceleration of the participant under consideration, at some instant, wrt. the corresponding instantaneously comoving inertial system then ... the notion of "(constancy, or change of) direction" may be more difficult; perhaps just a matter of convention.
11h
comment What's the physical meaning of a “sphere”, and a pair (or even a series) of “concentric spheres”, in a region with Schwarzschild geometry?
[contd.] ... in a region of Schwarzschild geometry. (Also not yet addressed is: which particular value $r \gt 0$ then ought to be assigned to which particular "sphere"; but this part "how we define the $r$ coordinate" is not my interest/question at all.) "You seem to know enough about GR that surely you know this." -- I'm certainly not aware that I'd know how "we" determine which events/worldlines belong to the same sphere, and which don't. If you do indeed know how to answer this (do you?) but you think it wouldn't benefit me personally, you should still provide it to benefit others.
11h
comment What's the physical meaning of a “sphere”, and a pair (or even a series) of “concentric spheres”, in a region with Schwarzschild geometry?
John Rennie: "Are you asking how we define the $r$ coordinate?" -- Well, since you seem to insist on involving coordinates, and specifically "the $r$ coordinate": In your answer, as it stands, you have already given some relevant part of "how we define the $r$ coordinate", namely, AFAIU: to all elements of the same one "sphere" shall be assigned the same value $r>0$. But what's not yet addressed (and that's my actual question, besides any coordinate assignments): How do we define which events (or which wordlines) contitute or belong to the same one "sphere" in the first place?,
21h
comment Continuum analogue of $ \langle \psi | \psi \rangle = \sum _i a_i^* a_i$
Anuar: "I think I understood your comment." -- I think so, too; thanks. As the (my) concluding insight I'd summarize that the question "How do you express the continuum analogue of a sum over discrete summands?" is plainly but exhaustively answered by "Replace summation by integration, and replace the value of the sum by the value of the integral. (The expression of the integrand is then only a formality.)" Btw. the requirement (2) and its replacement are surely relevant, and they're implied: summation is over disjointly indexed summands; integration is over an ordered set of variable values
1d
comment Squaring a Vector?
sTr8_Struggin: Regarding notation: note that the norm of a vector $\vec x$ is often written as $$\| \vec x \|,$$ expressing the conceptual distinction to the absolute value of (real, or complex) numbers.
1d
comment Continuum analogue of $ \langle \psi | \psi \rangle = \sum _i a_i^* a_i$
Anuar: "Can you suggest me a book [...]" -- I can readily suggest the Wikipedia page on the "Fundamental theorem of calculus" which makes plausible that, in a suitable sense, the integrand term of an integral can be thought of, and be expressed as, the derivative of the (definite) integral value. My point was mainly to criticise the problem statement for using the integrand term $\psi^*(x')~\psi(x')$ without definition. Indeed, the introduction of $a(x)$ in your solution provides such a definition in the first place, too.
1d
comment What's the physical meaning of a “sphere”, and a pair (or even a series) of “concentric spheres”, in a region with Schwarzschild geometry?
John Rennie: "One of the lessons of relativity is that we are free to choose any coordinates to describe the geometry of spacetime" -- The lesson is much stronger: we can describe the geometry of spacetime without referring to coordinates at all, but (merely) to determinations of physical coincidence. "some coordinates make more physical sense than others" -- The more thorough technical statement is that coordinates may or may not be affine to a physically given suitably generalized metric space. So you should address how coordinate values $r$ are to be assigned to any suitable worldlines.
1d
comment Continuum analogue of $ \langle \psi | \psi \rangle = \sum _i a_i^* a_i$
Anuar: The solution is of course just the repacement $$\langle \psi | k \rangle ~ \langle k | \psi \rangle \mapsto \frac{d}{dx}[~\langle \psi | \psi \rangle~]~\mid_{x'},$$ along with replacing the discrete sum (over the spectrum of eigenstates of any suitable operator, which I labelled with index $k$) by a corresponding integral over a continuum of eigenstates $x'$. The abbreviations $$\langle \psi | k \rangle ~\langle k | \psi \rangle \equiv a_k^{*}~a_k $$ and $$\frac{d}{dx}[~\langle \psi | \psi \rangle~]~\mid_{x'} \equiv \psi^{*}(x')~\psi(x')$$ tend to obscure the simplicity of the problem.
Aug
13
comment Problem conserving 4-momentum at CoM frame in an inelastic collision
udr: "[...] correct, provided [...]" -- Actually, as far as I understand, the two formulas of my above comment are correct in general; and therefore also in the special case $\mathcal S\mapsto Lab$, $\mathcal K\mapsto \text{ COM frame of the participants in the collision}$, and $\Omega\mapsto\text{ the participants in the collision}$; whereby $\vec p_{\text{COM}}[~\text{participants}~]\mapsto\vec 0$. "you may want to simplify your notation a bit using \beta-s, \gamma-s, & so on" -- I rather avoid introducing notions and abbreviations (trivial as they may be) which the OP didn't use already
Aug
12
comment Problem conserving 4-momentum at CoM frame in an inelastic collision
where $\vec p_{\mathcal S}[~\Omega~]$ denotes the momentum of object $\Omega$ with respect to reference system (inertial frame) $\mathcal S$, $\vec p_{\mathcal K}[~\Omega~]$ denotes the momentum of object $\Omega$ with respect to reference system (inertial frame) $\mathcal K$, $E_{\mathcal S}[~\Omega~]$ denotes the energy of $\Omega$ with respect to $\mathcal S$, $E_{\mathcal K}[~\Omega~]$ denotes the energy of $\Omega$ with respect to $\mathcal K$, and $\vec v_{\mathcal S}[~\mathcal K~]$ denotes the velocity of (each member of) $\mathcal K$ with respect to $\mathcal S$.
Aug
12
comment Problem conserving 4-momentum at CoM frame in an inelastic collision
$$\vec p_{\mathcal K}[~\Omega~]= $$ $$\vec p_{\mathcal S}[~\Omega~]+\vec v_{\mathcal S}[~\mathcal K~]~\left(\frac{(\vec p_{\mathcal S}[~\Omega~]\cdot\vec v_{\mathcal S}[~\mathcal K~])}{|\vec v_{\mathcal S}[~\mathcal K~]|^2} \left( \frac{1}{\sqrt{1-|\vec v_{\mathcal S}[~\mathcal K~]|^2/c^2}}-1\right)-\frac{E_{\mathcal S}[~\Omega~]}{c^2~\sqrt{1-|\vec v_{\mathcal S}[~\mathcal K~]|^2/c^2}}\right),$$and$$E_{\mathcal K}[~\Omega~]=\frac{E_{\mathcal S}[~\Omega~]-(\vec p_{\mathcal S}[~\Omega~]\cdot\vec v_{\mathcal S}[~\mathcal K~])}{\sqrt{1-|\vec v_{\mathcal S}[~\mathcal K~]|^2/c^2}},$$
Aug
12
comment Problem conserving 4-momentum at CoM frame in an inelastic collision
udr: "$\beta = p_{Lab}c/E_{Lab}$" -- That's essentially the solution, +1; the velocities $\vec v_{Lab}[~frame~1~] := \frac{M \vec w + m \vec u}{M + m}$ and $\vec v_{Lab}[~frame~2~] := \frac{M^* \vec v + m \vec u'}{M^* + m} := \overline s^*$ from the OP are only different "classical" approximations; neither "frame 1" nor "frame 2" are therefore the COM frame, in general. p.s. "The new energies and momenta read: [...]" -- You seem to be considering only a 1-dimensional problem; $(\vec w \cdot \vec u)^2 = (\vec w \cdot \vec w) (\vec u \cdot \vec u)$, etc. Consider instead:
Jul
30
comment Angle sum of triangle in Schwarzschild solution
@John Rennie: "How are you going to define the straight lines that make up the sides of your triangle?" -- This is a worthwile question by itself; and it is closely related to the OP question as well as to the (arguably more basic) question "What is the notion of a spatial angle in general relativity?" (PSE/q/108359). (The latter question has been answered already, possibly providing the means of addressing the OP question directly).
Jul
30
comment Is momentum an invariant?
@Danu: "This is a trivial question for first year courses [...]" -- Your assessment suggests that you (too) know a plain, unambiguous answer to my question. So which is it: "Yes.", or "No."? If "No." then please consider expanding and submitting this as an answer. (The answer "Yes." has been issued already.) "[...] The "usual notation" is clearer and you would do well to learn to use it." -- What exactly do you consider the "usual notation" for denoting the momentum of one specific particle (such as a $\Lambda^0$) with respect to one specific reference system (such as "the lab"), please?
Jul
29
comment In a CMCS 2-body system, why does the speed of the particles after collision stay the same?
Greg: "Changing to a moving coordinate system, the Center-of-mass Coordinate System (CMCS), we now have [...]" -- The equations and the speed values $v$ and $v_c$ given in your question don't seem to involve any coordinates at all. So instead of referring to coordinate systems, perhaps you mean a comparison of two reference systems; i.e. first considering the inertial reference system of which particle $m_2$ was a member, and then changing to considering the Center-of-mass Reference System (CMRS).
Jul
29
comment Will two clocks moving in opposite directions measure the same time as one at rest?
Well, what could you know or demand of the tick rates of various clocks at all if you didn't understand and use the relativistic methods for comparing them in the first place?
Jul
29
comment Why is a silicon ball with an exact number of atoms a bad measure of mass?
ton.yeung: As @Shaurya Bhave's answer indicates, the more pertinent question might be to ask, whether and why it is a bad idea to attribute some particular value of "mass" so some (non-zero) number of artefacts (concretely: to attribute a "mass of 12 grams" to the Avogadro number of C12 atoms. (And I'd say that's bad because it precludes asking and measuring whether two exemplars of "C12 atoms" had equal "mass", or whether the "mass" of one particular exemplar of "C12 atom" had remained constant, or not.)). Anyways: +1 to your question.
Jul
29
comment Can you recover the values of spacetime intervals $s^2$ from given causal relations between events?
dmckee: Re your recent edit, with several "counter examples": Yes, these seem indeed examples of events whose causal relations do not imply specific interval ratios. (I'd foremost think of "arbitrarily many events which are pairwise timelike to each other".) But may I remind you: I had been asking in the OP about a suitable set of events $\mathcal S$, i.e. if such a set may be thought of at all. Taking (again) a hint from Synge's more or less well-known "five point curvature detector" (GR, p. 408), I'd guess that such a set should have a whole lot more than 4 elements, suitably related.
Jul
28
comment Can you recover the values of spacetime intervals $s^2$ from given causal relations between events?
@Slereah: "As the causal relationships will be identical for two conformally related metrics" ... perhaps related to "$\Omega^2(x) \gt 0$" in the other answer mentioned above ... "the interval will also be conformal, since it is ~ g". -- Can you prove that (in the sense of my question, the corresponding interval values due to either metric tensor are "scaled isometric", with a non-zero proportionality constant) even allowing that neither $g$ nor $\Omega$ are necessarily constant? If so, please consider submitting that as an answer; and please don't forget the "how to go about".
Jul
28
comment Can you recover the values of spacetime intervals $s^2$ from given causal relations between events?
dmckee: "[...] but the intervals in the second set are 4 times as large." -- Please note the phrase "up to some (non-zero) constant" in the OP question statement. The two example cases described in your answer are in so far to be considered equivalent (with the applicable constant of value 4, or 1/4); and your answer is therefore (trivially) incorrect.