# GR: "Proper Distance" Meaning and Usage

• pervect
In summary, proper distance is a spatial geodesic measured along a hypersurface of constant cosmic time. It can differ from a defintion that uses Fermi normal coordinate for a "small tube" around a single worldline.
pervect
Staff Emeritus
"Proper distance" in GR

I am aware of two meanings of the term "proper distance" in GR. The first is when you have points in flat space-time, or space-time that's locally "flat enough", in which case it is defined as it is in SR, as the Lorentz interval between the two points. This usage of the term implies that one is considering short distances, or is working in a flat space-time.

The second is a term used by some cosmologists, for instance, Lineweaver, who uses it as a synonym for "comoving distance". See for instance http://msowww.anu.edu.au/~charley/papers/DavisLineweaver04.pdf" . (I often wonder why they don't stick with the term comoving distance, but that's besides the point.)

Are there any other common usages for "proper distance" in GR?

On a related note, what would be the correct terminology to refer to a 'distance' that's measured along a space-like geodesic (specifically a geodesic of the 4-d space-time)?

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for instance, Lineweaver, who uses it as a synonym for "comoving distance"
I think you misread something. They state explicitly "Proper distance is the spatial geodesic measured along a hypersurface of constant cosmic time", which corresponds to your "what would be the correct terminology to refer to a 'distance' that's measured along a space-like geodesic (specifically a geodesic of the 4-d space-time)? "

EDIT: They are actually talking about a geodesic of 3D space. So you're right, "cosmological proper distance" is not measured along a 4D geodesic, which seems to be a cause of confusion especially in the paper you cited. But it's still not "comoving distance".

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I'm assuming you're internet searching for something. If so, you might substitute displacement for distance.

Ich said:
EDIT: They are actually talking about a geodesic of 3D space. So you're right, "cosmological proper distance" is not measured along a 4D geodesic, which seems to be a cause of confusion especially in the paper you cited. But it's still not "comoving distance".
Yes, "comoving distance" between two galaxies moving with the Hubble flow (i.e. at rest relative to the fluid imagined to be filling the universe in the FLRW model) is constant over time, while "proper distance" between the same galaxies grows over time (and is the distance used in the http://en.wikipedia.org/wiki/Hubble's_law]Hubble's[/PLAIN] law), the wikipedia article on 'comoving distance' discusses both. I think proper distance means you just integrate the metric line element along a non-geodesic spacelike curve between the two galaxies, where every point along the curve lies on a single surface of simultaneity in the cosmological coordinate system (chosen so that matter has a uniform density on each surface of simultaneity). This is equivalent to the wikipedia article's notion of imagining a chain of observers between the two galaxies, and each observer makes a local measurement of the distance to the next observer in the chain at a single moment of cosmological time, with the "proper distance" being the sum of all these local measurements.

Not sure about the terminology, but I would guess that if you integrate the metric line element ds along any arbitrary spacelike curve that could be called the "proper distance" along the curve, just like you can integrate ds along any timelike curve and this will be proportional to the "proper time" along the curve ('proportional to' because you have to divide by i*c if ds2 is written with gtt including a factor of -c2)

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Agreed, "proper distance along xxx" seems to be the most general notion.

Okay, I'm going think out loud, and try to add to what other folks have already written.

Suppose a congruence of timelike worldlines of "fundamental" observers is picked out by phyics, symmetry, etc. Consider spacelike curve that intersects each worldlne in the timelike congruence orthogonally, and that has unit length tangent vector. Proper distance for the congruence is given by the curve parameter along such a spacelike curve. Sometimes these spacelike curve are geodesics, and sometimes they are not.

This seems to work for the following congruences of observers;

1) a congruence of observers in special relativity that, in a particular inertial frame consists of the form (t, x, y, z) = (t, X, Y ,Z), where were X, Y, and Z are constants (different values of the constants give different worldlines in the congruence);

2) in special relativity, the congruence of timelike worldlines associated with a Rindler frame;

3) a congruence of "hovering" observers in Schwarzschild spacetime;

4) a congruence of fundamental observers in Friedmann-Robertson-Walker universes.

Note that this definition of proper distance is congruence-dependent, and that in same spacetime, different congruences of observers can have different notions of proper distance; For example, the Milne universe, a subcase of 4) is a subset of Minkowski spacetime, has a different defintion of proper distance than in 1).

This definition can differ from a defintion that uses Fermi normal coordinate for a "small tube" around a single worldline. Fermi normal coordinate always use spacelike geodesics.

Also, the proper distance between observers in the congruence can change with time. Different spacelike curves orthogonally intersect the congruence "at different times".

This seems to work for the following congruences of observers;
Technically, it should work with every congruence. With some funny results if there is vorticity, I suspect.

Ich said:
Technically, it should work with every congruence. With some funny results if there is vorticity, I suspect.

Right. A congruence is hypersurface orthogonal if and only if the vorticity of the congruence vanishes.

George Jones said:
Suppose a congruence of timelike worldlines of "fundamental" observers is picked out by phyics, symmetry, etc. Consider spacelike curve that intersects each worldlne in the timelike congruence orthogonally, and that has unit length tangent vector.

This doesn't work. It should be more like
Suppose a congruence of timelike worldlines of "fundamental" observers is picked out by physics, symmetry, etc. Suppose further that this congruence is orthogonal to a family of spacelike hypersurfaces.

Also,
George Jones said:
Proper distance for the congruence is given by the curve parameter along such a spacelike curve.

Should be more like:

The proper distance between any two points in space at a particular time, i.e., on a particular hypersurface, is the greatest lower bound of the lengths of all the curves in the hypersurface that join the two points.

pervect said:
On a related note, what would be the correct terminology to refer to a 'distance' that's measured along a space-like geodesic (specifically a geodesic of the 4-d space-time)?

Wald gives "length" for any spacelike curve, and "proper time" for any timelike curve (with a minus sign in the appropriate place), and undefined for curves that are mixed spacelike and timelike.

atyy said:
Wald gives "length" for any spacelike curve, and "proper time" for any timelike curve (with a minus sign in the appropriate place), and undefined for curves that are mixed spacelike and timelike.
Other authors do use "proper distance" for the integral of ds along spacelike curves in a non-cosmological context...for example, from p. 824 of Misner/Thorne/Wheeler:
The divergence of $$g_{rr}$$ at r=2M does ''not'' mean that r=2M is infinitely far from all other regions of spacetime. On the contrary, the proper distance from r=2M to a point with arbitrary r is $$\int_{2M}^{r} | g_{rr} |^{1/2} \, dr = [r(r - 2M)]^{1/2} + 2M \, ln |(r/2M - 1)^{1/2} + (r/2M)^{1/2} |$$ when r > 2M ... which is finite for all 0 < r < ∞

atyy said:
Wald gives "length" for any spacelike curve, and "proper time" for any timelike curve (with a minus sign in the appropriate place), and undefined for curves that are mixed spacelike and timelike.

Isn't that a bit of a cop out? The variation around either 'pure' type of curve includes mixed types.

PAllen said:
Isn't that a bit of a cop out? The variation around either 'pure' type of curve includes mixed types.
Since ds^2 must be negative for timelike intervals and positive for spacelike intervals or vice versa (different authors seem to use different conventions), the integral of ds on a purely timelike or purely spacelike curve will be either real or imaginary, so presumably you could define the "length" of a mixed curve as a complex number if you wished.

I wonder, does the mathematical definition of a pseudo Riemannian manifolds only cover manifolds where "length" given by the metric can be both positive and negative, or does the term also cover ones where it can be real, imaginary or complex?

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OK, thanks for all the responses. I believe there is a general consensus, then, that the term "proper length" in GR needs additional specification besides two points: the curve along which the length may be specified, or the hypersurface of "constant time" in which the curve lies might be specified as an alternative, or indirect means might be used to specify the hypersurface (for instance it being orthogonal to a particular preferred family of observers).

When a surface is specified, the curve is specified implicitly as (informally) "the shortest curve connecting the two points" or more formally the distance is specified as the greatest lower bound of all curves connecting the two points.

In SR it can be assumed that given two points one defines "proper distance" by measures the Lorentz interval. This is equivalent to saying that the choice of curve is obvious in SR, one simply chooses the sole straight line connecting the two points in question. In GR, though, this is not in general sufficient.

This is more or less what I thought, but I wanted to make sure I had the details right, especially as I couldn't find any really definitive quotes on the topic from my textbooks.

I believe there is a general consensus, then, that the term "proper length" in GR needs additional specification besides two points
Which leads me to the question if there is a most "natural" definition of distance.
IMHO, the natural curve to connect two events is a geodesic, and if there is only one, this is the natural distance between the events. If there are more, I think it should be the shortest.
Connecting two worldlines needs specifying a time on one of the worldlines, and then it's the geodesic orthogonal to that worldline at that time.

Thoughts? Are there more natural definitions?

Hope this helps. A straight line--the distance between two events in special relativity is extremal--and oddly maximal rather than minimal.

Begin with,

$\Delta s = \sqrt\left( \eta_{\mu\nu} \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{/d\lambda} \right)$

The result, after a variational treatment, is

[tex]\frac{d^2x^{\mu} }{d\tau^2} =0\ \ .[/itex]

A particle moves in a straight line, unaccelerated.

The same square root equation applies in general relativity, by parallel transporting a displacement vector in it's own direction. The path is not unique, as in special relativity, as we would should know from examples of gravitational lensing. The difference between the starting equations is that in special relativity the metric is constant, but in general relativity, it is not. The constant metric eta is replaced with coordinate dependent g.

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Ich said:
Which leads me to the question if there is a most "natural" definition of distance.
IMHO, the natural curve to connect two events is a geodesic, and if there is only one, this is the natural distance between the events. If there are more, I think it should be the shortest.
Connecting two worldlines needs specifying a time on one of the worldlines, and then it's the geodesic orthogonal to that worldline at that time.

Thoughts? Are there more natural definitions?

I thought for a specific spacetime with a specific curvature, there could only be one geodesic that precisely defines the shortest path. And that would also be the most "natural" definition of proper distance.

TrickyDicky said:
I thought for a specific spacetime with a specific curvature, there could only be one geodesic that precisely defines the shortest path. And that would also be the most "natural" definition of proper distance.

In special relativity where spacetime is not curved, there is one path that is longer (not shorter as in Euclidian geometry) than all neighboring paths infinitesimally displaced from it. However, in general relativity, there can be more than one path that is longer than all other neighboring paths. This thing involving neighboring paths is a variational treatment. Generalizing from the idea of a straight line in special relativity, the same variational treatment is applied, and the generalization of a straight line in special relativity is called a geodesic in general relativity.

A simple example is two small objects in circular orbits around a large non-rotating spherical distribution of mass. If the small objects meet on one side of the heavy spherical object and are going in different directions, they will meet again on the opposite side. Their paths through spacetime from one event where they meet to the next are geodesics. They are different geodesics, but because of the symmetry of this scenario, they must have the same proper times. (If one of the paths would have a longer proper time, which one would that be?)

As another example, consider clock 1 in orbit about a spherically symmetric object. At event p, clock 2 is coincident with clock 1, and clock 2 is thrown straight up. Suppose the initial velocity of clock 2 at p is such that clock 2 goes up, falls back down, and is coincident again with clock1 after clock 1 has completed one orbit. Call this second coindence event q. Clocks 1 and 2 both follow geodesics, are both coincident at events p and q, yet the two clocks record different elapsed times between p and g.

As another example, consider clocks A and B in
George Jones said:
In this post, I will summarize the results, and the I will gives an explanation of the results in another post.

Consider a spherical planet of uniform density and five clocks (changing notation slightly):

clock A is thrown straight up from the surface and returns to the surface;
clock B is dropped from rest through a tunnel that goes through the centre of the planet;
Clock C remains on the surface;
clock D remains at the centre of the planet;
clock E orbits the body right at the surface.

Assume that A is thrown at the same time that B is dropped, and that the initial velocity of A is such that A and B arrive simultaneously back at the starting point. The times elapsed on the clocks A, B, and C between when they are all are together at the start and when they are all together at the end satisfy $t_A > t_C > t_B$.

Since A and B are freely falling and C is accelerated, it might be expected that $t_A > t_C$ and $t_B > t_C$, so $t_C > t_B$ seems strange.

Assume that clock E is coincident with clocks A, B, and C when A and B start out. As Fredrik has noted, unless the density of the planet has a specific value, E will not be coincident with with A, B, and C when A and B arrive back, but E will be coincident again with C at some other event. The elapsed times between coincidence events of E and C satisfy $T_C > T_E$. Again, since E is freely falling and C is accelerated, this seems strange.

Again, the two clocks both follow geodesics, yet have different elapsed time between coincidence events that joined by the geodesics.

Thinking about it, it seems that I have no idea what extremal condition specifies a spacelike geodesic in a Lorentzian manifold. Their length function is at a saddle point, right?
This "greatest lower bound" thingy certainly applies only to definite metrics.

Ich said:
Thinking about it, it seems that I have no idea what extremal condition specifies a spacelike geodesic in a Lorentzian manifold. Their length function is at a saddle point, right?
This "greatest lower bound" thingy certainly applies only to definite metrics.

Yes. This is why I applied it to a spacelike hypersurface, on which the spacetime metric is definite.

This is why I applied it to a spacelike hypersurface, on which the spacetime metric is definite.
Yes, I understand that.
What about the first two sentences, am I right stating that we seek neither a minimum nor a maximum, but a saddle point of length?

I've been pondering issues with pseudo-riemannian geodesics on and off for a long time. This seems like a good thread to set my current
thinking down - I propose a possible answer to proper length that I
think is free of any anomalies.

Two approaches to geodesics:

generalize 'straightest possible paths': path that parallel
transports its tangent vector; applies even to affine spaces
constraints on variation, etc. Derive what metric properties
geodesics have in particular circumstances. Call this affine
geodesic.

generalize 'shortest path': variation of invariant interval; issues
with meaning or existence of extrema especially for
pseudo-riemannian metric. Also issues of constrained variation
seem required for pseudo-riemannian metric.

Whichever definition is used, the family of geodesics between two
points can be highly non-unique in certain cases. Consider great circles between poles. For a 2-sphere, this produces a one parameter (direction) family of geodesics (all with the same length). In 4 space, could be 3 continuous parameter family of geodesics between two events in worst case (3 parameters describe direction in
4-space). This case would arise for poles of a 4-sphere, for example.

I think transport definition clearly better for GR. Then define
interval between two events as follows:
Pick either event, compute its forward and backward light
cones. If other event is within the light cones, interval is
proper time, use dt**2 - ; else interval is proper length,
use - dt**2. For proper time, pick LUB of geodesic
times. For proper length, pick GLB of geodesic
lengths. Metric only used to compute interval along affine
geodesics, not for variation.

To use variational definitions, many issues arise:

- Unless you prohibit including null geodesic segments among the
paths varied, you can generate any number of silly variants on a
timelike geodesic with 'away and back' light paths. Each such
excursion makes no change to the path interval. It seems you must
simply ban these.
[EDIT] This is actually a non-issue. These actually decrease the total
path interval, thus would be rejected by seeking the maximum
for a timelike variation.

- You must decide how to treat variations that include negative
contributions to ds**2. J.L Synge proposed that you really vary
integral of sqrt(abs(ds**2)). This leads to the nonsensical result
that there is actually is no upper bound on path interval for timelike
events. You can include as many spacelike excursions as desired,
all adding to the interval. Again, it seems like the best result
is to constrain the variation to exclude spacelike segments for a
timelike variation, and timelike segments for a spacelike
variation.

- Finally, it seems you must preclude backward time excursions, as
these preclude a maximum for timelike case. This is separate form
negative ds**2 contributions, as these have positive dt**2.

Thus, it seems a variational approach requires a somewhat complex set
of constraints to produce meaningful extrema. The affine approach
sidesteps all of this. One might argue that some of these constraints
can be relaxed if they only result in a saddle point in the
variation. But then, one must prove this. I've never seen such a
proof.

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I'd suggest just recognizing from the start that you're looking for the saddle point. Then looking for a "stationary" solution via standard variational techniques will not only work, they can be shown to give you the same solution as the parallel transport method. While "jogging away and back" on a lightlike geodesic doesn't change the path length, clearly it's not a "stationary" solution anymore (i.e. after taking such a detour, nearby curves don't have the same length), so we can rule it out as being a geodesic.

pervect said:
I'd suggest just recognizing from the start that you're looking for the saddle point. Then looking for a "stationary" solution via standard variational techniques will not only work, they can be shown to give you the same solution as the parallel transport method. While "jogging away and back" on a lightlike geodesic doesn't change the path length, clearly it's not a "stationary" solution anymore (i.e. after taking such a detour, nearby curves don't have the same length), so we can rule it out as being a geodesic.

The problem is that the result of the variation has no particular geometric meaning. You have not even a local extremal (in the GR case). On the other hand, using the affine definition you have direct, intuitive meaning: 'straightest possible path', a local condition. Further, you can write the criterion down directly from the properties of covariant derivative, without doing variation. To me, the variational approach is worth it if you actually get an extrema out of it - otherwise it is just a longer route to a curve of unspecified properties.

I notice, for example, that this is approach taken in MTW.

 Let me clarify my preference:

Introduce geodesics using the affine condition, local property of being straight as possible. Then analyze what additional properties they have under certain assumptions. Euler Lagrange only tells you necessary (not sufficient) condition for a local extremal. So, after verifying that, establish that under further assumptions (banning time loops for example), that a timelike geodesic is a local maximum. I believe much confusion is spread by the way books have glossed over these issues. (I only have very old GR books, and none cover this adequately, even MTW).

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The problem is that the result of the variation has no particular geometric meaning. You have not even a local extremal (in the GR case).
Watch http://vega.org.uk/video/subseries/8" . You'll understand why extremal points make sense, and why saddle points have the same importance as real extremae. It's worth the time, I promise!

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I've always found "parallel transport" and "as straight as possible" to be too vague on their own to be satisfying to me. Introducing Schild's ladder made me feel better about using parallel transport.

Wald's approach does take the same path through "prallel transport" but it starts out by introducing a rather abstract notion of derivative operator, and the properties it must have. Not really very intuitive, but the mathematics is thick enough that any confusions introduced by intuition tend to get buried in the sheer difficulty of following what's going on at all. I'm not sure if that's really a recommendation to the approach or not :-)

hmm. what would be the result of parallel transporting a covariant vector?

Phrak said:
hmm. what would be the result of parallel transporting a covariant vector?

Using the abstract approach, you can parallel transport an arbitrary tensor along a curve. If you have some tensor $T^{qwer}{}_{uiop}$ you parallel transport it along a curve with tangent t^a by insisting that along the curve

$t^a \, \nabla_a T^{qwer}{}_{uiop} = 0$

Though as I said, the intuitive meaning, if any, tends to get lost along the way, at least for me.

I would like to propose a conjecture on this. First, I propose that distance between time like events is not meaningful. All observers see the the relation as time like. You must propose some worldline from one on which you pick out an earlier/later event that has spacelike interval to the other. Also, the obvious point that while space like interval is invariant, 'most' observers would call it 'not a distance'. Only an observer using some 'natural' coordinates in which the two events had the same time would be inclined to call it a distance.

Given two events in GR that are space like in their relation (one is outside the light cones of the other), the greatest lower bound of the spacelike intervals of geodesics connecting them is a meaninful definition of their proper distance. Specific conjectures:

- each geodesic distance is a local minimum if backward time
excursions are prohibited (you can always find an arbitrarily small
length path between spacelike points if you allow segments that
are almost backward time null geodesics paired with almost
forward null geodesics ). I also assume time like segements are
prohibited (imaginary contributions).
- where there are 'shorter' and 'longer' geodesics, by analogy to
great circles on a sphere, it is meaningful to pick the shorter as the
distance.
- key conjecture: the proper distance so defined matches matches
the 3-d distance (using the riemannian 3-d sub-metric) on some
3-d slicing of spacetime that includes the events. By construction,
the events are simultaneous in such a slicing (else the prior
statement couldn't be true).

You're just rephrasing what has already been proposed: define some 3-space and measure along a geodesic in that 3-space. You can't define what a "backward time excursion" is unless you define your slicing earlier.

pervect said:
Using the abstract approach, you can parallel transport an arbitrary tensor along a curve. If you have some tensor $T^{qwer}{}_{uiop}$ you parallel transport it along a curve with tangent t^a by insisting that along the curve

$t^a \, \nabla_a T^{qwer}{}_{uiop} = 0$

Though as I said, the intuitive meaning, if any, tends to get lost along the way, at least for me.

It looks like I've mis-phrased another side question. The intended question, shouldn’t be so easy to answer though. I was thinking of parallel transporting a covariant tensor in its own direction. I'm not sure if it makes any easy sense, though--it doesn't live in the same tangent space as vectors.

Ich said:
You're just rephrasing what has already been proposed: define some 3-space and measure along a geodesic in that 3-space. You can't define what a "backward time excursion" is unless you define your slicing earlier.

No, I am proposing that is not necessary to assume any foliation, a priori. That is the whole point of the conjecture. I'm saying you can find the affine geodesics in any frame; compute their proper interval using the (+++-) signature metric; pick the smallest interval; that will be the meaninful proper distance between them.

Then I claim (conjecture) that each of these geodesics is a local minimum under the 'no back time' restriction. This has nothing to do with foliation. Given any frame (whether or not the spacelike events are simultaneous), I propose you need to ban paths in which dt along the curve is ever negative. If you don't, it is easy to construct (physically meaningless) paths with arbitrarily small interval, thus no local minimum property without this assumption.

Nothing so far has assumed any foliation. Now I claim you can always find a foliation in which the endpoint events (assumed spacelike) are simultaneous, and in which the interval so computed matches the 3-space positive definite distance (using 3x3 sub matric of the full metric for coordinates using that foliation). Actually, I now believe that interval computed without any foliation assumption, in any frame, will be such a 3-distance in *any* foliation in which the end events are simultaneous, not just some such foliation.

I have rigorously proved none of this. I am hoping somone might be intrigued enough to at least partially verify this or attempt to find counterexamples.

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Given any frame in which the events are not simultaneous, I propose you need to ban paths in which dt along the curve is ever negative.
I don't know in what way you think a "frame in which the events are not simultaneous" does not define a foliation. Further, the "t" direction is the one orthogonal to the space foliation you chose, so "dt nonnegative" is a concept that depends on the foliation. There is no absolute definition of positive time direction for spacelike intervals.

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