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What meaning/use is associated with the notion of proper length in general relativity? Do you know an example of any quantity that depends on it? I have so far found statements like "the length corresponding to a proper time between two measurement events", or "the length as measured by some observer". However, neither of these ideas seem to work.

For example, if I want to measure the distance from myself to an object, I could do so by bouncing a photon from it and counting my proper time between firing the photon and receiving it. We can do this in the radial direction of the Schwarzschild metric, for example, using Eddington–Finkelstein coordinates (outside of the event horizon, of course). The resulting length (EDIT: derivation at the bottom of this question) depends on the difference in tortoise coordinates, i.e. $$L_{measured} = \frac{c \, \tau_{Round Trip}}{2} = \sqrt{f(r_{observer})} \,\, [r^{*}(r_{object}) - r^{*}(r_{observer})] $$ where

$r^{*}(r) = r + r_{s} \ln (\frac{r}{r_{s}} - 1)$ is the tortoise coordinate,

$f(r) = 1-\frac{r_{s}}{r}$ relates proper time (squared) to coordinate time (squared),

and $r_{s}$ is the Schwarzschild radius. This isn't the proper length between observer and object, which is given by integrating $\mathrm{d}S^{2}$ from $r_{observer}$ to $r_{object}$, with $\mathrm{d}t=\mathrm{d}\theta=\mathrm{d}\phi=0$: $$ L_{prop} = \Big[ r\sqrt{f(r)} + \frac{r_s}{2} \ln \big( r\sqrt{f(r)} + r - \frac{r_{s}}{2} \big) \Big]^{r=r_{object}}_{r=r_{observer}} $$

Unless I'm mistaken, these two expressions aren't the same, despite some superficial similarity.

So the proper length doesn't give the distance measured, and it isn't a distance corresponding to the propagation time of light/information from one point to another. So what exactly is it?

I realise that by fixing the observer and object radii, I'm assuming each has some nonzero proper acceleration... is that the source of the discrepancy?

Questions very similar to this one have been asked before, but the answers have either discussed flat space (this one), or have resolved different but related conceptual issues (such as this one, and this extremely interesting answer).


EDIT: For clarity I'll include my derivation of the measured length.

Eddington-Finklestein coordinates $(u,v)$ are defined by: $$u=ct-r^{*} \,\,\, \text{and} \,\,\, v=ct+r^{*}$$ Radial null geodesics in the Schwarzschild metric correspond to $u=$constant (outgoing rays) and $v=$constant (uncoming rays). Now, letting $r_{observer}<r_{object}$ and taking the photon to be emitted at $t=0$, we have, for the outward trip: $$ u = 0 - r^{*}(r_{observer}) = c t_{reflected} - r^{*}(r_{object})$$ i.e. $ct_{reflected} = r^{*}(r_{object})-r^{*}(r_{observer})$

Now for the return trip, we have $$ v= ct_{reflected} + r^{*}(r_{object}) = ct_{RoundTrip} + r^{*}(r_{observer}) $$ i.e. $ct_{RoundTrip} = ct_{reflected} + r^{*}(r_{object}) - r^{*}(r_{observer}) = 2[r^{*}(r_{object}) - r^{*}(r_{observer})] $

putting this in terms of the proper time of the observer, we have $$c \tau_{RoundTrip} = \sqrt{f(r_{observer})} c t_{RoundTrip} = 2 \sqrt{f(r_{observer})} [r^{*}(r_{object}) - r^{*}(r_{observer})] $$ which we divide by two to get the distance, because we know the photon went there and back.

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The procedure of an observer measuring distance by bounding a photon off a nearby object and timing the return is also called the "radar metric / distance". It is relative to the observer's motion, which is to be expected from introductory special relativity ("length-contraction"). The radar metric is meaningful when defined locally (see Landau & Lifshitz), whence it equals the measurement of idealised rulers moving with the observer, if interpreted carefully. These are equivalent to the proper distance interval $ds$ measured orthogonal to the observer's worldline. However in general, radar distance is meaningless for finite separations.

I expect some readers will find these claims surprising; indeed my understanding changed after studying the topic. I am working on a pedagogical paper about spatial distance generally.

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In both SR and GR time is generally treated as a length by multiplying it by $c$, though we often set $c = 1$ so this isn't immediately obvious. So for example the Minkowski metric is:

$$ c^2d\tau^2 = c^2dt^2 - dx^2 - dy^2 - dz^2 $$

and we multiply $dt$ by $c$ to get a quantity $cdt$ that has units of length and can therefore be sensibly added to $dx$ etc. The quantity $\tau$ is the proper time, which is the time shown by a freely falling clock i.e. a clock following a geodesic.

The proper length is simply the proper time converted to a length by multiplying by $c$. Its physical meaning is that it is the arc length measured along a geodesic. In your examples you are calculating the coordinate length, not the proper length along the geodesic followed by the moving object (the proper length is of course always zero for photons).

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  • $\begingroup$ I've included my derivation of the measured length at the bottom. The logic is: Calculate the Schwarzschild coordinate time of the round trip. Convert that to the proper time of an observer at constant r, where the photon was emitted. This should therefore be the result of their "length" measurement. Can you point out where the error in this logic is? $\endgroup$ – Max Lock Mar 4 '15 at 13:07
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    $\begingroup$ @MaxLock: I can't follow your argument. If you want to calculate the proper length you have to work out the geodesic then integrate $ds$ along it. For the EF metric a null radial geodesic has $d\phi = d\theta = 0$ and because $u$ and $v$ are constant $du$ and $dv$ are zero. So your metric simplifies to $ds = 0$ and the proper length is therefore zero. But then null geodesics are defined as those geodesics that have $ds = 0$, so that's hardly surprising. $\endgroup$ – John Rennie Mar 4 '15 at 16:17
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For example [...] bouncing a photon from [an object] and counting my proper time between firing the photon and receiving it. [...] $L_{[...]} = \frac{c~\tau_{Roundtrip}}{2}$

This would rather be called the "(momentary) chronometric separation" of the "object" under consideration from yourself;
or, if you had found equal values of the ping duration $\tau_{Roundtrip}$ in successive trials, then plainly the "chronometric separation"; cmp. J. L. Synge, "Relativity. The general theory".

Obviously, the "object" under consideration could have carried out this procedure in turn, determining its ping durations wrt. yourself, trial by trial.

Note that even while you had found equal (constant) ping duration wrt. a particular "object" in all trials, this "object" in turn found

  • not necessarily equal (constant) ping durations wrt. yourself, and

  • even if it did find equal (constant) ping duration wrt. yourself (thus obtaining a value of your chronometric separation from itself), then it is not necessarily equal to the constant ping duration which you had found wrt. this "object".

Therefore "chronometric separation" between participants is generally a quasi-distance.

This isn't the proper length between observer and object, which is given by [...]

Apparently the name "proper distance" is reserved for the quantity "$c\int_{\mathcal P}\sqrt{-g_{\mu\nu}dx^{\mu}dx^{\nu}}$", for any (arbitrary) spacelike path $\mathcal P$;
and the name "proper length" is then reserved referring to "proper distance along" spacelike paths which would satisfy more specific requirements (e.g. "extremal conditions").

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  • $\begingroup$ Interesting... I had never heard of "chronometric separation" before. I was aware of the definition of proper distance, and in the expression above, I'm taking the spacelike path $\mathcal{P}$ to be the line going from $r_{observer}$ to $r_{object}$ while keeping the $t$, $\theta$ and $\phi$ coordinates fixed. Do you know what the physical interpretation of this proper distance is? And is there any alternative way of measuring a distance in GR other than chronometry? $\endgroup$ – Max Lock Mar 4 '15 at 14:14
  • $\begingroup$ @Max Lock: "[...] the phys. interpretation [...]" -- AFAIU in order to be attributed a "proper length" value, an everywhere spacelike path would also have to be "straight", i.e. any three of its elements, $a,b,c$, should satisfy $$\sigma[~a,b~]^2+\sigma[~a,c~]^2+\sigma[~a,c~]^2=2~\sigma[~a,b~]~\sigma[~a,c~]+2~\sigma[~a,b~]~\sigma[~b,c~]+2~\sigma[~a,c~]~\sigma[~b,c~],$$ where "$\sigma$" denotes Synge's world function. For two suitable participants, it could then be asked for the "straight path" of max. "proper length" between them. $\endgroup$ – user12262 Mar 4 '15 at 15:48
  • $\begingroup$ @Max Lock: "And is there any alternative way of measuring a distance in GR other than chronometry?" -- Well, the question is rather whether and how "chronometry" is feasable in the first place; i.e. how to determine whether any given clock had been "good" (or how to quantify it having deviated from "being good"), at least in principle. $\endgroup$ – user12262 Mar 4 '15 at 15:51

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