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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".)


23h
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?
To provide a token of my own attempts to work on a solution to the OP question, and an estimate of its difficulty: I've learned (a while ago) about Karlhede's invariant (see arxiv.org/abs/1404.1845 and references there) whose value apparently allows to identify at least some spheres, or some pairs of concentric spheres, in a region of Schwarzschild geometry. (However, I have not yet fully understood either how Karlhede's invariant might be defined, and consequently how its values ought to be determined, strictly in terms of coincidence determinations, as advertised by Einstein.)
1d
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.
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?
[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.
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: "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?,
1d
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
2d
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.
2d
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.
2d
asked Does motion with constant proper acceleration, in a flat region, necessarily mean straight hyperbolic motion?
2d
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.
Sep
2
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.
Sep
1
revised What's the physical meaning of a “sphere”, and a pair (or even a series) of “concentric spheres”, in a region with Schwarzschild geometry?
(v3.141592: Footnote to specify the meaning of "physical meaning" in the context of [tag:general-relativity].)
Sep
1
asked What's the physical meaning of a “sphere”, and a pair (or even a series) of “concentric spheres”, in a region with Schwarzschild geometry?
Sep
1
answered How does work function transform in Einstein's special theory of relativity?
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
revised Angle sum of triangle in Schwarzschild solution
corrected spelling the name of K. Schwarzschild (also in the title);cmp. https://en.wikipedia.org/w/index.php?title=Schwarzschild_Solution
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
suggested approved edit on Angle sum of triangle in Schwarzschild solution