# Cosmological Proper Distance

1. May 17, 2012

### Ich

This is based on a side discussion in the balloon analogy thread, see #49 and #53.

How is the definition of cosmological proper distance ("CPD" from now on) different from the usual definition of distance? Here, I want to discuss the respective defintions, what these definitions "really" mean (i.e. their operational implementation), and how you should interpret the numbers that you calculate. I'm concentrating on the distance of an event/object to an free falling observer in FRW spacetimes.

We define the distance of an object "now" in flat spacetime. In that case, you can compare the usual definition (standard coordinates in an inertial system) directly with CPD in an empty FRW universe, as both can be transformed easily into each other.

The usual definition is
- formally the length of the spacelike geodesic connecting observer and object that is normal to the worldline of the observer at a given event.
- operationally the length of a ruler stationary wrt the observer that connects both.

The CPD definition is
- formally the length of a curve of constant cosmological time connecting both at a given time. (The curve is a geodesic of the subspace defined by fixing the cosmological time.)
- operationally the total length of a chain of many small 'comoving' rulers, each in relative motion wrt its immediate neighbour according to the Hubble law, connecting both. Each ruler is supposed to lie end to end with its neighbour at the time you take the measurement.

The CPD definition is clearly different from the normal one. It also looks more complicated, but we can make the normal definition also more complicated if we generalize it to curved spacetimes. There it becomes
- formally the same as in flat spacetime, but now the geodesic is not simply a staight line.
- operationally the total length of a chain of many small 'static' rulers, each at rest wrt its immediate neighbour according to the Hubble law, connecting both. Each ruler is supposed to lie end to end with its neighbour all the time.
This definition is compatible to "Normal Coordinates", which represent arguably the next best thing to standard SR coordinates in flat spacetime.

Consequences: "Recession Velocity" d(distance)/d(time)

Velocity in both definitions is not limited to the speed of light. But there is an important difference: in the absence of spacetime curvature, the normal definition reduces to - you guessed it - the normal SR definition, which is always subluminal.
The FRW definition on the other hand is intrinsically different: it is actually no a velocity, but a rapidity, which becomes "superluminal" quite naturally. You don't have to invoke "motion through space" vs "motion of space" to explain that.

Operationally: FRW CPD/proper time are locally always the same as the normal coordinates of the fundamental (comoving) observers.
Your neighbour moves away from you at dv, and his neighbour moves away from him at dv, too.
In normal coordinates, you would use relativistic velocity addition w= dv+dv/(1+dv*dv) to calculate the total velocity. In FRW coordinates, you simply add the velocities, w=dv+dv, the same way you added the lengths of the rulers, even if they were in relative motion. This is not an exotic GR effect due to some miraculous "expansion of space", it is rather very straightforward: Add the dv, you get relative rapidity. Add the dv according to relativistic velocity addition, you get relative velocity.

So, first conclusion: what is always called "recession velocity" is actually not a velocity. You don't have to invoke subtle spacetime mechanisms to explain superluminal recession velocities. It's a rapidity, and rapidities go easily "superluminal".
Additional information: The different methods of adding small velocities along a curve of constant proper time are addressed in this paper. The "correct" application of the relativistic velocity addition law corresponds to parallel transport of the velocity vector, the algebraic addition to the recession velocity. Both are connected by the tanh function, as the definition of a rapidity requires.

I'll have more to say about distances and space curvature, but I'm too tired right now. I hope there's already enough to start a discussion in this post.

2. May 18, 2012

### Chalnoth

Well, no, they're exactly the same. It's just that the cosmological proper distance is specific to a FRW universe. This is because in FRW coordinates, observers on different galaxies in the universe are approximately stationary. If they are stationary, then a geodesic that is normal to their worldlines is constant in the FRW time coordinate.

3. May 18, 2012

### Ich

Sorry, I can't follow you here. Are you claiming that geodesics of FRW space are spacetime geodesics?
Also, the concept of "stationarity" you're using strikes me as odd. A general FRW spacetime is definitely not stationary, and even in the cases that are (empty and de Sitter, let's ignore the Einstein Universe), FRW space does generally not contain any spacetime geodesics.

4. May 18, 2012

### Ich

Ok, here's more on the difference between both.

Here, I use again the empty universe - the Milne model - as the best toy model. If you're not familiar with it: it's basically the empty "expanding" universe expressed in Minkowski coordinates. There are good places to get familiar with it, Ned Wright's website not the least.
Assume there's an inertial observer at coordinate position zero. We pick a spherical shell of events that happen all at cosmological time 1 (arbitrary units) in a cosmological distance 0.88 (there's some meaning to that number). We first drop y and z and concentrate on a t,x spacetime diagram.
So, what's the difference between normal and CPD definitions? Here's a graph in Minkowski coordinates: (ok sorry, it seems the graph is visible in the attachment only)
https://www.physicsforums.com/attachment.php?attachmentid=47411&stc=1&d=1337373065
The normal definition (blue line) is of course a straight line connecting the events at observer time 1.41. Their mutual distance in this system is 2, their distance to the observer is 1.
So in normal coordinates the events happened at time 1.41 in a distance of 1.

The FRW definition is a line of constant cosmological time = constant proper time of objects since the Big Bang. Such a line is obviously a hyperbola here, as $t^2 = tau^2+x^2.$ Hyperbolae are not geodesics of empty space, therefore the distance to the origin is different: 0.88, as said before.
In FRW coordinates, the events happened (as stated) at observer time 1 in a distance of 0.88.
Operationally, the difference is due to the comoving - not stationary - rulers used here: they see all distances Lorentz-contracted, therefore yielding a lower total distance.

There's an interesting consequence: a circle at Minkowski radius 1 has obviously a diameter of 2π. As the FRW rulers that measure circumference are oriented perpendicular to their motion, they'll also measure a circumference of 2π - but here with a radius of 0.88!
This imbalance between diameter and circumference translates directly to a negative curvature of space in comoving coordinates. With the help of Wikipedia, one can (relatively) easily calculate the curvature: it is -H².
So in FRW coordinates, we'd have to expect space curvature by the matter content of the universe, and additionaly a negative contribution from the moving rulers we're using. Which is in fact the case, here's the respective Friedmann equation rearranged:
$$\frac{k^2}{a^2} = \frac{8 \pi G \rho}{3} - H^2$$
We see: energy content curves space positively, while expressing it in comoving coordinates curves it negatively. An equilibrium of both is called "flat space" in the cosmological models, which is not at all a flat spacetime.

#### Attached Files:

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5. May 18, 2012

### Chalnoth

What is stationary is a particular set of observers with respect to the coordinate system. Those observers roughly correspond to real observers on real galaxies (not exactly, since real galaxies tend to move at a few hundred to a couple thousand km/s with respect to the coordinate system).

The critical point to be made here, however, is that the choice of proper distance is always going to be arbitrary: it depends entirely upon your choice of observer worldlines. With your Milne example, for instance, you haven't picked the same set of observers between the Milne space-time and the flat space-time: in the flat space-time, the stationary observers in the Milne space-time are no longer stationary, so you have to take their motions into account.

6. May 19, 2012

### RUTA

I like your comparison of "velocities" in the OP and post #5. I assume that the reason we use recession velocity = d(proper distance)/d(proper time) in all other FRW cases is because we don't have a global SR frame when spacetime is curved, so that definition is just not available to us.

7. May 19, 2012

### Chalnoth

Well, when actually doing cosmology, recession velocity is hardly ever mentioned, because it just isn't very meaningful. It's potentially interesting how it behaves, but it really doesn't affect anything.

8. May 19, 2012

### RUTA

It is in all the textbooks though ... it's the v in Hubble's law ... that's why people on this forum ask, I assume.

9. May 19, 2012

### Chalnoth

I think it's because it's one of those things that people feel like gives them an idea of what is happening. But it really isn't useful for doing science.

10. May 19, 2012

### Ich

I think you're confusing some things here.
"Stationary observers" can be connected by an orthogonal spacelike geodesic. The worldlines of those observers are integral lines of the timelike Killing field in a stationary spacetime.
Observers at a constant spatial coordinate in a time-dependent, non-stationary coordinate system (like FRW coordinates) are guaranteed not to meet that definition. The orthogonal line connecting them is never a spacetime geodesic.
Which is exactly my point. Choosing a different set of canonical observers, namely comoving instead of static ones, produces a different distance measure. As shown by the red and blue lines in the diagram. I didn't understand your "Well, no, they're exactly the same." in that context, as they clearly are not.
The Milne spacetime is flat, independent of its coordinate representation - Minkowski or FRW. If you wanted to say "canonical FRW observers are not at rest in Minkowski coordinates", that's true. It's also true that I chose different sets of observers, according to the respective coordinate representation. That's exactly what I have written, so I don't understand your former statement when you're now obviously agreeing.

So it's safe to say that you no longer claim that both definitions are the same?

11. May 19, 2012

### Chalnoth

The thing is, stationary is a matter of perspective.

Perhaps you are simply using "proper distance" as the term is used in Special Relativity. But that definition clearly doesn't work in a curved space-time. But the generalization of this definition to the cosmological definition is easy to understand: if we define a set of observers at every point, then the proper distance is a space-time geodesic which is orthogonal to those observers' word lines at every point.

But since this definition allows us to define a global time coordinate, a geodesic defined in this manner is always going to be independent of time, and therefore just a spatial geodesic. That fact is due to our specific coordinate choice, but the proper distance we obtain would be the same in any coordinate system provided we make the same choice of a set of observers.

Sorry, I misspoke. I did mean Minkowski coordinates vs. FRW coordinates.

12. May 21, 2012

### Ich

Hi Chalnoth,

as your post seems to be rather loosely connected to what I've written in this thread, please let me repeat this important question:

Are you aware of the fact that the curve along which FRW proper distance is measured is not a spacetime geodesic?

13. May 21, 2012

### Chalnoth

Sure it is. It's a constrained spacetime geodesic, but still a spacetime geodesic. In fact, it's constrained in precisely the way that makes it align well with the special relativity definition of proper distance: it's a geodesic which is orthogonal to the worldlines of a specific ensemble of observers.

14. May 22, 2012

### Ich

Sure.

Can we then, for the sake of clarity, agree on the following terminology in the context of this thread:
- "geodesic" means "geodesic", i.e. an example of the set of naturally privileged curves in a manifold with a connection that obeys the geodesic equation
- "spacetime geodesic" means a geodesic where the respective manifold is all of spacetime
- "geodesic of a subspace" means a geodesic where the respective manifold is a subspace of said spacetime. Such curves are examples of "constrained spacetime geodesics" such that the orthogonality constraint applies to the worldlines of the canonical observers that span said subspace.

It has the advantage of being compatible with the general usage of "spacetime geodesic" and my previous terminology in this thread.

With this terminology, I trust you are aware that geodesics of expanding FRW-space are not geodesics of spacetime?

15. May 22, 2012

### Chalnoth

If you want to take those constraints on what can or cannot be a geodesic, then the proper distance in special relativity isn't a geodesic either.

16. May 22, 2012

### Ich

Straight lines are not geodesics of flat spacetime? Is this another terminology problem?

17. May 22, 2012

### Chalnoth

You're making the constraint that they are orthogonal to the worldlines. You can't make the same sort of constraint in General Relativity so easily.

18. May 22, 2012

### Ich

Orthogonal to which worldlines? You're obviously talking about something different than I again, so please, before jumping to conclusions, answer:
If the answer is "yes", I'll restate what I've already written in my OP until there is a consensus. If it is "no", we'll have to work on that instead.

19. May 22, 2012

### Chalnoth

I just see the fact that the cosmological proper distance is given by a geodesic orthogonal to an ensemble of observers is a reasonable generalization of the SR result.

20. May 25, 2012

### Ich

So you're preferring a different definition of distance. IMO, your definition is too general to be useful and, more importantly, misses the point of this thread. Please read the links in the OP to see that I'm trying to explain here why FRW distance is not the same as the distance measured by a ruler.

Besides, I'd very much appreciate if you'd care to correct your errorneous Post #2. You explicitly referred to my definitions, not yours, and I feel that your mistaken claim that both are the same contributed to the failure of this thread to discuss and clear the issue. Maybe there is something to salvage.
Thanks in andvance.

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