Proper length in GR

Question

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. $$L_\text{measured} = \frac{c \, \tau_\text{Round Trip}}{2} = \sqrt{f(r_\text{observer})} \,\, [r^{*}(r_\text{object}) - r^{*}(r_\text{observer})] $$ 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 $\mathrm{d}t=\mathrm{d}\theta=\mathrm{d}\phi=0$: $$ L_\text{prop} = \Big[ r\sqrt{f(r)} + \frac{r_s}{2} \ln \big( r\sqrt{f(r)} + r - \frac{r_{s}}{2} \big) \Big]^{r=r_\text{object}}_{r=r_\text{observer}} $$

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: $$ u = 0 - r^{*}(r_\text{observer}) = c t_\text{reflected} - r^{*}(r_\text{object})$$ i.e. $ct_\text{reflected} = r^{*}(r_\text{object})-r^{*}(r_\text{observer})$

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

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

Answer 1

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.

Answer 2

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).

Answer 3

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").

Answer 4

I would like to add my own answer, even if I believe that @Colin MacLaurin's answer is quite definite.

To try to define a physically meaningful notion of spatial distance in GR is necessary to focus on the general notion of (extended) reference frame.

To this end, let us consider a 4-dim spacetime $(M,g)$, which we also assume to be temporally oriented.

DEFINITION. A reference frame in $(M,g)$ is a pair $R:= (\Gamma,\Sigma)$. Above,

• $\Gamma$ is a smooth family of future-directed timelike curves $\gamma$ such that they do not intersect each other and they completely fill $M$;
• $\Sigma$ is a smooth foliation of $M$ given by spacelike 3-dim surfaces $S$.

As a consequence, an event $e\in M$ stays on a unique $S_e\in \Sigma$. This defines the time location of $e$ in $R$. On the other hand, $e$ meets a unique $\gamma_e\in \Gamma$ and this defines the spatial location of $e$ in $R$.

Along every $\gamma\in \Gamma$, the metric $g$ induces a preferred temporal parameter: the well known proper time. It is nothing but time measured by natural ideal clocks (atomic clocks) evolving along these worldlines.

REMARK In general, the temporal interval between two $R$ rest spaces $S,S'\in \Sigma$ is different if measured along two different curves $\gamma,\gamma'\in \Gamma$. Though this discrepancy does not show up in special relativity when dealing with Minkowski reference frames, it already takes place, for instance, in the Rindler wedge when adopting standard accelerated coordinates associated to the boost.

Usually, it is convenient (if possible) to define $\Gamma$ as the integral lines of a timelike Killing vector field, using the Killing parameter as extended temporal coordinate of the reference frame. Furthermore, in that situation, the rest spaces $S$ are often chosen as constant Killing time surfaces. The arising discrepancy with the proper time is here, in particular, the cause of the celebrated gravitational red shift.

Coming back to the general case of a reference frame $(\Gamma,\Sigma)$, the most intriguing issue is what metric on every general rest space $S\in \Sigma$ has physical significance.

A mathematically minded person (like me!) would probably argue that the relevant metric on the spacelike 3-dim surface $S$ is the one induced by the spacetime metric $g$, exactly as it happens for the proper time! In other words, one sees tangent vectors $V$ at $S$ (i.e., physical rulers on $S$) as vectors in $M$ and he/she defines their length as $\sqrt{|g(V,V)|}$.

Though this procedure induces a well defined Riemannian metric on $S$, the arising notion of spatial length does not comply with Einstein's postulate about the universal constant value of the light speed. (Value measured with a closed path procedure, without a choice of a synchronisation notion, using the proper time coordinate along a curve $\gamma\in \Gamma$)

The physically correct metric $h$ on $S$, is instead the radar metric. That is the unique that produces the right value of the light speed. It has a complicated formula: if $X,Y$ are spacelike vectors tangent to $S$, then $$h(X,Y) = g(X,Y) - \frac{g(\dot{\gamma},X)g(\dot{\gamma},Y)}{g(\dot{\gamma},\dot{\gamma})}.$$ (I assumed $c=1$ and I used the signature $-+++$.)

In any case, this metric produces a notion of distance that is valid for finite separation events on a given rest space $S$. It is however disputable if this notion of finite length has physical significance if the metric changes in time (it does not change if $\Gamma$ is constructed out a Killing vector field and the $S$ are parametrized by the Killing parameter).

A natural issue concerns the possibility to have the radar metric on $S$ that coincides with the metric on $S$ induced by $g$ by the standard pull back procedure of differential geometry.

It is not difficult to prove that is mathematically equivalent to require that the curves $\gamma$ are orthogonal the the surfaces $S$.

In turn, physically speaking, this is the same as requiring that the ideal clocks on "infinitesimally close" curves $\gamma,\gamma'\in \Gamma$ are synchronised by using Einstein's procedure.

A second natural issue is whether or not this requirement can be fulfilled on the full rest space $S$ and not only locally, around a given $\gamma$. That is an integrability issue with negative answer in general. The typical case is the "rotating observer". There, Einstein's synchronisation is not allowable. Disregarding this obstruction may give rise to erroneous interpretations of phtsical phenomena like the Sagnac effect.