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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
13
asked How to express in terms of spacetime intervals the average speed of one participant having moved between two suitable others
Apr
13
answered How to express in terms of spacetime intervals whether two participants in a flat region were at rest to each other
Apr
13
asked How to express in terms of spacetime intervals whether two participants in a flat region were at rest to each other
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
revised Distance in General relativity
(v3.141592653: copy-n-paste-typo corrected.)
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
8
revised Distance in General relativity
(v3.14159265: major revision (coordinates are even less useful than surmized).)
Apr
7
revised Distance in General relativity
(v3.141592: struggling with coordinates (as usual) ...)
Apr
7
answered Distance in General relativity
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]