# Proper distance in GR

1. ### pervect

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"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 Expanding Confusion.. (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)?

2. ### Ich

1,931
Re: "Proper distance" in GR

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".

Last edited: Oct 14, 2010
3. ### Phrak

Re: "Proper distance" in GR

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

4. ### JesseM

8,489
Re: "Proper distance" in GR

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 Hubble's 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)

Last edited: Oct 14, 2010
5. ### Ich

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Re: "Proper distance" in GR

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

6. ### George Jones

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Re: "Proper distance" in GR

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".

7. ### Ich

1,931
Re: "Proper distance" in GR

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

8. ### George Jones

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Re: "Proper distance" in GR

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

9. ### George Jones

6,474
Staff Emeritus
Re: "Proper distance" in GR

This doesn't work. It should be more like
Also,
Should be more like:

10. ### atyy

11,185
Re: "Proper distance" in GR

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.

11. ### JesseM

8,489
Re: "Proper distance" in GR

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:

12. ### PAllen

5,890
Re: "Proper distance" in GR

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

13. ### JesseM

8,489
Re: "Proper distance" in GR

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?

Last edited: Oct 14, 2010
14. ### pervect

8,155
Staff Emeritus
Re: "Proper distance" in GR

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.

15. ### Ich

1,931
Re: "Proper distance" in GR

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?

16. ### Phrak

Re: "Proper distance" in GR

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.

Last edited: Oct 15, 2010
17. ### TrickyDicky

Re: "Proper distance" in GR

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.

18. ### Phrak

Re: "Proper distance" in GR

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.

19. ### Fredrik

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Staff Emeritus
Re: "Proper distance" in GR

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?)

20. ### George Jones

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Staff Emeritus
Re: "Proper distance" in GR

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