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Apr
17
comment Binding energy per nucleon in radioisotopes of hydrogen
@Saprativ Ray: "[...] elaborate on "the binding energy per nucleon is not necessarily the full measure of whether a nucleus is stable or not."" -- For example, tritium decays by a weak process. "what the values of binding energy per nucleon actually imply?" -- Comparison to all individual nucleons being separated from each other. (Btw.: yet another/more relevant notion of "stability" is: Which nucleus is the most resilient against gamma-induced fission?.)
Apr
17
comment Why positron emission is unlikely to occur for nuclei with an excess of neutrons?
@dmckee: "Follow the logic. [...]" -- Logic allows at least to fill in the blanks you left: "Positron emission can only occur when a _$p$_ is converted into a _$n$_ inside the nucleus" -- (That's by plain charge conservation; a gimmee.) "but in a neutron rich nucleus adding a _$n$_ takes more energy than you get from removing a _$p$_ so [...]" -- So the presumed logic holds. But why does converting $p$ to $n$ in an already $n$-rich nucleous take more energy than the reverse?? (Why "valley of stability" rather than "ridge of instability"??) And: Does the OP ask this question?
Apr
16
comment Why positron emission is unlikely to occur for nuclei with an excess of neutrons?
@Farcher: "Possible duplicate? Why beta+- decay occurs?" -- Not really: at least the OP question title implies an inquiry to justify, roughly, "why there is a line/valley of stability in the isotope chart (rather than, say, a line/ridge of instability)". However, admittedly, this request is not (yet) spelled out in the OP question text.
Apr
14
comment Acceleration of particle “held in place” at $x = 1$
jld: "Did you invent this notation yourself or did you get it from some other source? If the latter please forward it" -- Gladly; see my recent comments there. "the question as posed assumes that lengths are dimensionless. [...] So you could say the question was expressed badly." -- That's what I was trying to point out. Of course, my complaint is therefore not only to OP writer @user265817 but also to those responding without being critical; and, to be fair, surely to certain people who inspired such a badly expressed question being asked.
Apr
14
comment How to express in terms of spacetime intervals whether two participants in a flat region were at rest to each other
Finally (I suppose): I don't know any precedence, other than my own writing, for the notation of events (i.e. the arguments of the spacetime interval funct) I used above (which explicitly lists the participants having been coincident). However: it aims to adhere explicitly to Einstein's prescription that "All our space-time verifications invariably amount to the determination of space-time coincidences {such as} meetings between two or more material points".
Apr
14
comment How to express in terms of spacetime intervals whether two participants in a flat region were at rest to each other
The use of square brackets for enclosing (a comma-separated list of) arguments to a function corresponds to Mathematica (TM) style (StandardForm); see e.g. mathematica.stackexchange.com/questions/20166/… (Btw., this notation style allows to reserve parentheses for the purpose of grouping alone.) [... continued]
Apr
14
comment How to express in terms of spacetime intervals whether two participants in a flat region were at rest to each other
@jld: "Did you invent this notation yourself or did you get it from some other source?" -- The symbol $s^2$ for denoting spacetims intervals is standard (cmp. link in the OP question), as is the notation expressing it being a function from pairs (of elements of a set) into the set of real numbers; cmp. en.wikipedia.org/wiki/Distance#General_metric [... continued]
Apr
13
comment Acceleration of particle “held in place” at $x = 1$
jld: "I'm not familiar with the notation you're using [...]" -- Does this present a difficulty in matching my notation to notation which you might be using for the same purpose? Or perhaps instead your being not familiar with the purpose itself? This question and these preliminaries may be helpful. "If you're working in more common units [then ...]" -- Surely a particular value of a dimensionful quantity is independent of particular choices of units; and is not just any coordinate value either.
Apr
12
comment Acceleration of particle “held in place” at $x = 1$
jld: "When the metric is Minkowskian the operator $v \cdot \nabla$ reduces to $d/d\tau$" -- Well, that's not quite the same as applying $d/d\tau$ outright in any case. "The quantities appear dimensionless because in GR we tend to work in geometric units where $c=G=1$." -- But, notably, re-introducing "$c$" as a (non-zero) dimensional symbol in the above dimensionless formulas doesn't seem to restore the appropriate dimension of $\mathbf a$. (And surely similarly for $G$, too.) So what's missing?
Apr
12
comment Acceleration of particle “held in place” at $x = 1$
jld: "In general: $\mathbf{a = (v \cdot \nabla) v}$" -- Above we had (coordinate four-velocity) $u^{\mu}$ with coordinate dependent components; while I had introduced $\mathbf v$ in contrast. Let me spell out its coordinate independent norm for the flat case: $$\| \mathbf v_{PQ}[~A~] \| := \text{lim}_{\{s^2[~\varepsilon_{AP},\varepsilon_{AQ}~]\rightarrow 0\}}[ \sqrt{(\text{Max}_{\{J,K\}}[~s^2[~\varepsilon_{PJ},\varepsilon_{QK}~]~])~/~s^2[~‌​\varepsilon_{AP},\varepsilon_{AQ}~]}],$$ for participants $P,Q$ at rest wrt. each other (i.e. in particular with constant and equal mutual ping durations).
Apr
8
comment Distance in General relativity
... not yet be satisfied with my answer as it stands (apart from a typo which I'll correct shortly): values of distance, or of Lorentzian distance, are not plainly given (as I had put in my answer), but they need to be measured; so it still needs to be answered how they are supposed to be obtained (at least up to a non-zero constant factor), by definition, in GR, explicitly from "determination of space-time coincidences {such as} encounters between two or more material points", as Einstein prescribed.
Apr
8
comment Distance in General relativity
@The Homeworker: I'm glad that you found my answer acceptable, thanks; in turn I should regard your question as sufficiently useful and clear (+1) even though you didn't change the last few words of your question statement, as I had suggested in my above comment from yesterday. However: Surely we cannot yet be satisfied with my answer as it stands [... continued]
Apr
7
comment Acceleration of particle “held in place” at $x = 1$
jld: "[...] $a_{\mu}a^{\mu}$ is also invariant" -- Ok, so my "coordinate acceleration" was unjustified. (My apologies, and thanks for having helped me to this insight). It (the "four-acceleration magnitude squared") is indeed an unambiguous and (regarding the OP stipulations) specific quantity. Or (recalling the earlier version of your answer): is it perhaps just a plain ("dimension"-less) real number?? I still wonder how/whether it relates to the (dimensional) quantity $$\|~\frac{d}{d\tau}[~\mathbf v~]\mid_{\mathbf v = \vec 0}~\|$$ (but this may not necessarily have been the OP question).
Apr
7
comment Acceleration of particle “held in place” at $x = 1$
jld: "I derived [...] the physical (proper) acceleration (magnitude of 4-accel)." -- I'm not convinced that's equivalent. You seem to start out with a manifestly coordinate-dependent quantity: $$u^{\mu} = \frac{dx^{\mu}}{d\tau}.$$I don't see what this "coordinate four-velocity" might have to do e.g. with $$\gamma_{\mathbf v}(c, \mathbf v);$$and consequently I don't see what the quantity $|a|$ you calculate from $u^{\mu}$ might have to do e.g. with $$\| \mathbf a \| := \|~\frac{d}{d\tau}[~\mathbf v~]\mid_{\mathbf v = \vec 0}~\|.$$ p.s. +1 for your way of addressing a homework-like question.
Apr
7
comment Acceleration of particle “held in place” at $x = 1$
@garyp: "Please do not post complete solutions to homework-like questions. [...]" -- Please note that jld's answer (in the present version above) is quite deliberately left incomplete by including "If you work out the Christoffel symbols (I'll leave that to you) ...". Even more importantly, jld's answer seems to be (only) deriving a value of coordinate acceleration, and therefore appears not to be addressing the OP by user265817 at all (which is asking instead to determine acceleration, and force).
Apr
6
comment Distance in General relativity
The Homeworker: Your question seems to try to distinguish between one particlar quantity related to "time" and "time-like curves", and another particular corresponding quantity (about which you mean to ask specificly). May I suggest that the latter quantity is related to "space" and "space-like curves", and that you might replace the last words of your question accordingly? Specificly, you might ask about the "(intrinsic) path length of a segment of a (everywhere) space-like curve"; to be contrasted with the "(intrinsic) duration of a segment of a (everywhere) time-like curve".
Apr
6
comment Distance in General relativity
miha priimek: "Points in spacetime are labelled by coordinates, which can be arbitrarily chosen" -- Yes, coord.s for labelling events may be arbitrarily chosen (at least: one-to-one); or events may also be labelled directly by who took part and what they observed. "proper time along a time-like curve" ... rather: its duration (or generally: the path length) ... "is not the same as coordinate time, although they converge in the Newtonian limit." -- This applies not to any arbitrarily chosen labelling of events by coordinate tuples, but only to so-called "good" or "affine" coordinates.
Jan
4
comment Comparing durations for two simply described motions in Schwarzschild geometry
John Rennie: "[...] watching from afar (which makes you a Schwarzschild observer)" ... surely there's an appropriate more specific (and coordinate-free) description of what you mean by "Schwarzschild observer (wrt. object $M$)" ... "if you start your stopwatch when [you saw that] $A$ passe[d] $B$ then the next time [you saw that] $A$ passe[d] $B$ your stopwatch will show a time $t$. The equations for the time measured by $A$'s clock and $B$'s clock are given [as ...]" -- No, no, no! Instead: If these "t" values of these three clocks were related as stated then they ran equally.
Jan
4
comment Comparing durations for two simply described motions in Schwarzschild geometry
John Rennie: "An object is following a geodesic if it is falling freely. [...]" Alright, thanks. Sorry for responding so late; partly due to the holiday break, partly because my question is quite shallow and rather an expression of my surprise about ... the remarkable inequality of durations arising in such a seemingly simple setting. I'll rather reward (and/or question) your more detailed derivation elsewhere. Here's just a general objection: [continued]
Nov
27
comment Angular velocity from orientational displacement
@Markus Fjellheim: "I found a solution where" -- Thanks for letting me know. I plan to look and possibly comment or vote on your answer later today. Without having done so, for now just some thoughts: "the difference vectors can be [...] zero" -- Interesting idea. If only one difference vector is zero, but not the other, then I suppose/believe that the constant vector gives the rotation axis. If both difference vectors are zero then angular speed seems ambiguous by some multiple of $2~\pi / \tau$, and the direction of rotation is completely undetermined.