Exploring the Definition of "Proper Distance" in Cosmology

[pervect]

http://arxiv.org/PS_cache/astro-ph/pdf/0310/0310808.pdf

uses the term "proper distance", but doesn't define it. Presumably this must be a standard defintion. So far, though, I have not been able to track down a definitive defintion (I'm still looking).

My guess is that this distance is integrated along a spatial geodesic (as contrasted to comoving distance, also used in the above paper, which is integrated along a curve of constant cosmological time).

However, I'd much rather not guess :-).

Also, if it turns out that the distance measured along a geodesic is NOT proper distance, does this sort of distance have a name?

Alternate names and the most common usage for "proper distance" are also of interest. I've seen a sourced claim that Weinberg uses "proper distance" as a synonym for "comoving" distance, for instance - unfortunatley I don't have that textbook to check the source. This seems to conflict with the Lineweaver & Davis usage, as well.

[add]
In SR there isn't any confusion as to what proper distance means. It's a lot less clear what that term means in GR or cosmology. Being literal minded, it seems to me that any distance measured by integrating the Lorentz interval along a curve could be called "proper".

 

[marcus]

http://arxiv.org/PS_cache/astro-ph/pdf/0310/0310808.pdf

uses the term "proper distance", but doesn't define it.

It actually IS defined in that paper at the top of page 17
(and a formula is given at the bottom of page 16 in terms of quantities based on the RW metric).

the context is Gen Rel, so what they mean by the term is not the same as in special relativity.

"Proper distance is the spatial geodesic measured along a hypersurface of constant cosmic time (as defined in the Robertson-Walker metric). It is the distance measured along a line of sight by a series of infinitesimal comoving rulers at a particular time, t. "

 

[pervect]

It actually IS defined in that paper at the top of page 17
(and a formula is given at the bottom of page 16 in terms of quantities based on the RW metric).

the context is Gen Rel, so what they mean by the term is not the same as in special relativity.

"Proper distance is the spatial geodesic measured along a hypersurface of constant cosmic time (as defined in the Robertson-Walker metric). It is the distance measured along a line of sight by a series of infinitesimal comoving rulers at a particular time, t. "

Hmm. Well, then, what's going on with figure 1 on page 3?

If we look at figure 1 on page 3, we see that we have a plot of light cones and the event horizon with the x-axis labelled "Proper distance D". Right below it is a different graph, with the x-axis labelled "Comoving distance $$R_0 \chi$$

The y-axis is scalled identically. Both plots are different. This suggest that proper distance D is something different from comoving distance. This is in fact what convinced me that they were different.

However, the defintion given seems to be the same as the standard defintion for comoving distance (per Ned Wright & other sources).

If we go to the text you point out:

Proper distance is the spatial geodesic measured along a hypersurface of constant cosmic time (as defined in the Robertson-Walker metric). It is the distance measured along a line of sight by a series of infinitesimal comoving rulers at a particular time, t.

Equation 12 says$$D = R \chi$$. Except for the subscript, this is identical to the equation for comoving distance.

Furthermore, the curve at a hypersurface of constant cosmic time is not a spatial geodesic.

Let's look at the flat FRW metric:

ds^2 = -dt^2 + a^2(t) * (dx^2 + dy^2 + dz^2)

The Christoffel symbols can be computed to be
$$\Gamma^t{}_{xx} = \Gamma^t{}_{yy} = \Gamma^t{}_{zz} = a \, \frac{da}{dt}$$

Thus the geodesic equation for the t coordinate is:
$$\frac{d^2 t}{d \lambda^2} + a \frac{da}{dt} \left( \frac{dx}{d\lambda}^2 + \frac{dy}{d\lambda}^2 + \frac{dz}{d\lambda}^2 \right) = 0$$

It is clear that if $$a(t) > 0$$ and $$\frac{da}{dt} > 0$$ then $$\frac{d^2 t}{d \lambda^2} < 0$$ if any of the squared terms are nonzero.

Thus a surface of constant t does not describe a geodesic for a line in (for instance) the x direction.

 

[SpaceTiger]

Equation 12 says$$D = R \chi$$. Except for the subscript, this is identical to the equation for comoving distance.

The subscript is important, though.

$$D=R(t)\chi \neq R_0\chi$$

At the current time, the two quantities are equal because R(t_0)=R_0. Since R(t) is a non-trivial function of time, the two graphs ought to look different.

 

[pervect]

OK, that makes sense. So is this usage in Linewaver & Davis standard usage - proper distance between comvoing particles varies with time while comoving distance remains constant, and is equal to the proper distance "now"?

Also, does anyone define a distance measure measured along a spatial geodesic? So far I haven't run across anyone who does this.

 

[oldman]

I think I grasp the distinction that has been drawn here between comoving and proper distances --- "comoving" being constant coordinate-label differences, "proper" being distance summed by a long line of infinitesimally spaced observers equipped with even more infinitesimal rigid rulers --- with the two kinds of "distance" agreeing right now. But I don't know how one would in practice show that comoving distances are constant or find that proper distances do vary.

Indeed in practice we no longer define distance with rigid platinum rulers or gauge blocks; we do it with radar and use a committee-defined and assumed constant-everywhere value of c. An atomic clock is the only physical standard, apart from the committee.

To me it seems slightly less fanciful to imagine long-lived observers that measure with radar distances to objects partaking of the Hubble flow or distances in curved space sections near a black hole.

Would they measure proper distance correctly?

 

[smallphi]

It seems like 'proper distance' is the integrated spacetime interval along the radial line connecting two points with the same cosmic time in the usual FRW coordinate system.

The quantity is grossly coordinate dependent. First 'the same time' depends on the chosen coordinate system (with the abstract metric tensor remaining the same) i. e. on the chosen foliation of spacetime into space and time. One can always transform the coordinates so points with the same coordinate time previously are not 'at the same time' in the new coordinates. Even in SR, 'simultaneity' is a coordinate dependent concept.

Second, as pervect pointed out, the line connecting the events on the hypersurface of the chosen 'same cosmic time' is not even a geodesic in spacetime and therefore is completely arbitrary chosen line. On the other hand, the line looks like geodecis in the flat metric induced on the time slice. Therefore the only element of arbitrariness seems the choice of time slicing.

 

[pervect]

It's nice to be agreed with.

I've always found the name "proper distance" in cosmology to be rather unfortunate, because it invites students to confuse the "proper distance" in cosmology with "proper distance" in special relativity.

And they're really not the same at all, though some authors give at least the appearance of downplaying the difference.

It's pretty standard to define the distance between two objects as the shortest curve connecting them on a surface of "constant time" - there may be exceptions to this general rule, but I can't think of any offhand. But the definition of the surface of constant time is a coordinate dependent choice.

 

[MeJennifer]

But the definition of the surface of constant time is a coordinate dependent choice.

Not really, a surface of constant proper time is a coordinate independent concept. In FRW type models co-moving observers enjoy the same proper time (e.g. cosmological time), this is a propery of the metric not of any particular chart.

 

[smallphi]

The surface of constant proper time will depend on your choice of the set of observers whose proper time is measured. For example, if you choose non-comoving observers in FRW spacetime, their proper time won't have anything to do with the cosmic time.

One could say, well choose free fall observers to define the constant proper time surface. Indeed, the comoving observers in FRW are also free-fall observers i.e. following geodesics. Unfortunately, they are not the only ones possible observers in free fall. Through each point of spacetime, pass a gazillion of free fall observer geodesics and only one of them gives a comoving observer. The proper times of the others are connected to the comoving one by the usual "time dilation" Lorentz factor of SR (since GR is locally SR for observers in free fall) determined by their relative speed with respect to the comoving one when they pass through the spacetime point. Thus, choosing arbitrary free fall observers in general won't generate the same constant proper time surfaces as the comoving observers.

So the constant proper time surface depends both on the metric and your choice of gazillion observers to define it.

 

[MeJennifer]

Smalphi, yes of course there are different classes of observers.
In that case we can say that the surface of proper time for any particular class of observers is not a coordinate dependent choice.

By the way, as you undoubtedly know, those observers must reside on or be considered test particles since all mass objects in the FLRW metric are co-moving.

 

[smallphi]

My point was that in GR just like all coordinate systems are equal, so are different sets of observers. If you define a concept with respect to coordinate system, you have to exaplain why you picked it. Exactly the same way you have to explain why from all possible sets of observers you chose the comoving.

Picking up those comoving observers is equivalent to picking up 1time+3space foliation of spacetime where the spatial hypersifaces are orthogonal to the comoving observers world lines. That is exactly the usual FRW coordinate system so essentially you are picking coordinate system and correspondingly the quantity defined in that coordinate system, 'the proper distance', is just a quantity defined in a specific coordinate system defined by the comoving observers. I can always pick different coordinates and the 'proper distance' in those coordinates would be totally different.

The point of that thread is that 'proper distance' is not a geometrically invariant quantity defined irrespective of any coordinates or observers, like the spacetime distance between two close spacetime points is. In cosmolgy texts they often imply that you see it is the 'physical' distance between the objects abosolutely independent of any coordinates, as if you are stretching a ruler from point A to point B or something.

 

[MeJennifer]

My point was that in GR just like all coordinate systems are equal, so are different sets of observers.

You are free to use any kind of coordinate chart in GR that might or might not completely cover what you are trying to investigate.

However it is incorrect that all classes of observers in GR are equal. For instance in the FLRW metric all classes of observers that are not co-moving are not inertial but accelerating.

If you define a concept with respect to coordinate system, you have to exaplain why you picked it. Exactly the same way you have to explain why from all possible sets of observers you chose the comoving.

Yes ok. Explain that to cosmologists who talk about "expansion of space" as something physical, or relativists who talk about "stretching of electromagnetic waves".

Picking up those comoving observers is equivalent to picking up 1time+3space foliation of spacetime where the spatial hypersifaces are orthogonal to the comoving observers world lines.

Right, that is typically how cosmologists like to do it.

That is exactly the usual FRW coordinate system so essentially you are picking coordinate system and correspondingly the quantity defined in that coordinate system, 'the proper distance', is just a quantity defined in a specific coordinate system. I can always pick different coordinates and the 'proper distance' in those coordinates would be totally different.

Hold on smallphi, I was commenting on the surfaces of equal proper time not proper distance.

I simply made a comment with regards to some inaccuracy presented in this topic. If you disagree with it then we can debate that.

 

[smallphi]

However it is incorrect that all classes of observers in GR are equal. For instance in the FRLW metric all classes of observers that are not co-moving are not inertial but accelerating.

There are free fall (inertial) observers that are not comoving. They have a nonzero speed with respect to the comoving ones. For example the Earth is not comoving which is shown by the dipole anisotropy we observe in CMB. The anisotropy is used to calculate our speed with respect to the comoving frame also called the rest frame of CMB. Yet, the Earth doesn't have 'engines' to propel itself so it is in a free fall.

 

[MeJennifer]

There are free fall (inertial) observers that are not comoving. They have a nonzero speed with respect to the comoving ones. For example the Earth is not comoving which is shown by the dipole anisotropy we observe in CMB. The anisotropy is used to calculate our speed with respect to the comoving frame also called the rest frame of CMB. Yet, the Earth doesn't have 'engines' to propel itself so it is in a free fall.

Smallphi, how familiar are you with the FLRW solution?

All mass-energy objects co-move in a FLRW solution. You mention the Earth, the Earth cannot possibly be described in a FLRW metric as an inertially moving object.

It seems you mix up the FLRW solution with a cosmological model based on such a solution. You must realize that the FLRW solution, like most other solutions, is extremely simple and idealized.

If our universe would be an exact FLRW solution it would be really booring and for sure we would not be able to have this conversation, in fact we would not even exist.

 

[smallphi]

How well the smoothed out FRW describes the actual universe and observers in it is currently being debated and nobody knows the answer.

In the idealized homogeneous and isotropic FRW, as well as in any spacetime, there are gazillion of geodesics passing through a spacetime point and all of them are world lines of free fall observers. Only one of them is the worldline of a comoving observer.

Idealized observers are assumed to not contribute significantly to the energy momentum tensor and just because all the matter that drives the FRW metric is comoving in the adapted coordinate system doesn't mean a free fall observer has to be comoving.

 

[MeJennifer]

A few comments here:

A FLRW solution is not the same as the ΛCDM model! These two should not be mixed up!

Be careful about the physical interpretation of non-co-moving mass-energy objects in the ΛCDM model since this model based on the FLRW solution.

While you can consider an infinite number of classes of observers or test-particles in a solution you cannot, unpunished, introduce arbitrary objects that have mass or energy and them interpret them physically with respect to the metric.

Anyway, this whole discussion is quite irrelevant as to whether something is a coordinate dependent property or not.