# Cosmic rest frame, comoving distance, and proper distance

## Main Question or Discussion Point

Hi
I am trying to understand more about these terms. I am currently studying a course about relativity and cosmology, but I am finding the textbook (Open University) difficult to follow. Can anyone help me untangle and make some simple sense of these different terms? Thanks (I accidentally posted this earlier in the relativity forum)

## Answers and Replies

Bandersnatch
Cosmic rest frame, which I think is meant to mean the same as CMBR-rest frame (aka comoving coordinates), will be the reference frame in which the universe looks isotropic - specifically, it means that you'd see the cosmic microwave background radiation as uniform across the whole sky. In any other reference frame, some part of the CMBR will be blue-shifted in the direction of motion and red-shifted in the opposite patch of the sky.

Proper distance is what you'd measure if you could stop the expansion, take a measuring stick, and count how far a given object is.

Comoving distance is the distance that you'd get if you were to "factor out" the influence of expansion.
Imagine a grid of coordinates, and various groups of galaxies as objects on that grid. If you multiply each and every distance on that grid by the same number every now and then, you get proper distance growth (expansion). But despite proper distances growing, the relative positions of objects on the grid remain unchanged. Comoving distance lets you focus on those relative positions, without bothering how far actually (i.e., in proper distance terms) objects are. It doesn't have as tangible a meaning as proper distance, but is useful in calculations.

Equation-wise, it's $d(t)=a(t)d_0$
where
$d(t)$ is the proper distance; it is a function of time - i.e., grows as the universe expands
$d_0$ is the comoving distance; it is a constant set at some arbitrarily-chosen value (normally as the proper distance at present time)
$a(t)$ is the scale factor; it tells you by what factor has the $d_0$ changed; it's a function of time

In our universe, you can take the current proper distance to some galaxy, call it comoving distance, and say that 10 billion years ago it was X light years away in terms of proper distance, or that it was Y times closer than the comoving distance.

marcus
Gold Member
Dearly Missed
Hi Resurgance, I subscribe to everything Bandersnatch said--which was basically the perfect concise complete answer. I even hesitated to post because it would cover up post #2. So everybody please go back and read post #2!
But I wanted to say hello and glad you are wondering about cosmic rest frame, one of the nicest things in cosmology,maybe even in all of science. The ancient light (the "cmb") is approximately uniform, the same temperature in all directions. As it should be because at one time space was filled more or less evenly with a cloud of hot gas which then cooled just enough to become transparent. So the ancient hot gas and the ancient light that remains from it DEFINES what it means to be not moving. Not moving with respect to the ancient light.

If we were at rest then the ancient light would be EXACTLY the same average temperature in all directions. But it was around 1977 using a U-2 Spy Plane with a little microwave temperature receiver in a blister on top of it, that they discovered a WARM SPOT in the microwave sky. A warm spot in the ancient light, that was around the constellation of LEO (which you can see overhead in the Springtime of the year). A patch of the microwave sky that was just about 1/8 of one percent warmer!

That meant that the solar system was moving at about 1/8 of one percent of the speed of light, in the direction marked in the sky by the constellation Leo, and that it was a DOPPLER effect of our moving through the approximately uniform soup of ancient light. And it meant there should be a doppler COLD spot in the opposite direction in the sky from Leo, 180 degrees reversed. (around the constellation Aquarius.) Where the temperature was 1/8 of one percent cooler because of our moving away from the light coming from that direction. Here are some pictures.
http://aether.lbl.gov/www/projects/u2/

1/8 percent of 300,000 km per second is about 380 km per second, that is about what speed the solar system is going, relative to CMB-rest.
So we can correct our redshift observations for that motion and record the data AS IF the galaxies were observed from a reference frame at rest relative to the ancient light (or one could say at rest with respect to the hot gas cloud of ancient matter (before it began to fall together and coagulate into wisps and clumps all over the place and the chunks acquired their separate motions, that is to say) back when it was more more even and stationary.

380/300,000 the ratio of speeds (same as the doppler shift in temperature)
about 1/8 of a percent.

Last edited:
marcus
Gold Member
Dearly Missed
Again, be sure to scroll back to post #2 by Bandersnatch for a concise complete reply to the thread questions.
Continuing a more loquacious discussion, the idea of being at cosmic REST (ancient light roughly same temp in all directions, no doppler hotspot) gives you a standard cosmic TIME. This is the time as measured by observers all over the universe who are at rest.

Because they are all at rest they could in principle synchronize their clocks, they all see the same average temperature of the CMB and they see it cooling as distances expand and they all estimate the same age of the universe. The relativity effects of moving observers don't come in to mess things up. We'll ignore gravitational time dilation and assume they are all reasonably far from intense gravitational fields that might slow their clocks. So the criterion of rest gives us a "universe standard time" which the equation model of the cosmos can run on and which we can use to define PROPER DISTANCE, the distance between two things at a given moment.

If you didn't have an idea of universe time you couldn't talk about pausing expansion and freezing things are at a given moment to allow you to measure distance.
That's what proper distance is, the conventional distance you would measure (with string, or radar, or however) if you could pause expansion long enough to do it.

ChrisVer
Gold Member
marcus
Gold Member
Dearly Missed
Resurgance, both Bandersnatch and ChrisVer have given concise clear reply, let me continue being loquacious in case that's also useful. You asked about cosmic rest frame, proper distance, comoving distance.
I responded so far about cosmic rest frame, cosmic STANDARD TIME (needed to define proper distance and the quantities appearing in Hubble law, which has a time-depended rate) and proper distance.

One more topic is comoving distance. In the large picture of things, bits of matter don't move around very much compared with the spreading out of distances.
So to a first approximation one can IGNORE the motion of galaxies in their surrounding space---which tends to be limited to only a few hundred km/second.
Whereas distances between CMB stationary observers may be increasing at several time speed of light.

that is not like ordinary motion we are used to, nobody GETS anywhere by the mere growth of distances, everybody just becomes farther apart.

So, because collections of matter, e.g. galaxies are NOT MOVING very much we can treat them as CMB stationary observers and TAG THEM WITH THE DISTANCE THEY HAVE FROM US AT THIS MOMENT, and that distance tag will not change over time. We imagine they have born that tag for the whole expansion history. It is their "now distance" identifier. That is their comoving distance, and it is a very useful tag.

Because we might want to talk about how much matter is currently in our observable region, and how has that increased over time, as light from more and more matter has come in and we "hear from" a wider and wider circle of stuff. this is not EXPANSION, this is the growing amount of matter of all forms that we have observational data for.
How do we describe how much we expect that to grow in the future? We need a measure of radius which is not affected by expansion and then we can say "that much" matter: out to such and such comoving radius.

Cosmic rest frame, which I think is meant to mean the same as CMBR-rest frame (aka comoving coordinates), will be the reference frame in which the universe looks isotropic - specifically, it means that you'd see the cosmic microwave background radiation as uniform across the whole sky. In any other reference frame, some part of the CMBR will be blue-shifted in the direction of motion and red-shifted in the opposite patch of the sky.

Proper distance is what you'd measure if you could stop the expansion, take a measuring stick, and count how far a given object is.

Comoving distance is the distance that you'd get if you were to "factor out" the influence of expansion.
Imagine a grid of coordinates, and various groups of galaxies as objects on that grid. If you multiply each and every distance on that grid by the same number every now and then, you get proper distance growth (expansion). But despite proper distances growing, the relative positions of objects on the grid remain unchanged. Comoving distance lets you focus on those relative positions, without bothering how far actually (i.e., in proper distance terms) objects are. It doesn't have as tangible a meaning as proper distance, but is useful in calculations.

Equation-wise, it's $d(t)=a(t)d_0$
where
$d(t)$ is the proper distance; it is a function of time - i.e., grows as the universe expands
$d_0$ is the comoving distance; it is a constant set at some arbitrarily-chosen value (normally as the proper distance at present time)
$a(t)$ is the scale factor; it tells you by what factor has the $d_0$ changed; it's a function of time

In our universe, you can take the current proper distance to some galaxy, call it comoving distance, and say that 10 billion years ago it was X light years away in terms of proper distance, or that it was Y times closer than the comoving distance.
Hi Bandersnatch. Thanks for your reply. It is a great help. Can I just ask how you include equations so clearly in your post? Thanks again.

ChrisVer
Gold Member
Using LaTeX....
you can either use writing the equation between double dollar symbols (\$)
or write them within [it_ex] [/it_ex] (without the underscore).
The rest is whether you know how to write the equation in that environment, e.g. subscripts are written within the brackets after an underscore A_{a} will give $A_{a}$, for superscripts you change _ into ^ etc...if it's just one you dont have to contain the brackets, eg you could as well write A_a

Son
Again, be sure to scroll back to post #2 by Bandersnatch for a concise complete reply to the thread questions.
Continuing a more loquacious discussion, the idea of being at cosmic REST (ancient light roughly same temp in all directions, no doppler hotspot) gives you a standard cosmic TIME. This is the time as measured by observers all over the universe who are at rest.

Because they are all at rest they could in principle synchronize their clocks, they all see the same average temperature of the CMB and they see it cooling as distances expand and they all estimate the same age of the universe. The relativity effects of moving observers don't come in to mess things up. We'll ignore gravitational time dilation and assume they are all reasonably far from intense gravitational fields that might slow their clocks. So the criterion of rest gives us a "universe standard time" which the equation model of the cosmos can run on and which we can use to define PROPER DISTANCE, the distance between two things at a given moment.

If you didn't have an idea of universe time you couldn't talk about pausing expansion and freezing things are at a given moment to allow you to measure distance.
That's what proper distance is, the conventional distance you would measure (with string, or radar, or however) if you could pause expansion long enough to do it.
. So proper distance in GR is a bit similar to proper distance in SR - yes? In SR the proper distance between two events, is the distance measured by an observer, in whose reference frame the events are simultaneous. I think that I am beginning to get this. If a(t) = 1 for our (present) time, then the comoving distance would be equal to the proper distance that we measure. At earlier times a(t) < 1 and the proper distances would be smaller - yes?
Thanks - lots of interesting stuff in your post.

Bandersnatch
Good question Resurgance, good answer Bandersnach, good follow up Marcus. I have liked the the flow of concepts. I thought about a reply to the original question then I saw the answer given and the follow up. I realized then what a disruption and discourtesy it would have been for me to answer part of the first question. I apologize for my past behavior in this manner. I will not interject my thinking into other threads in the future. I will just start a topic and go from there.

A primer on LaTeX can be found here:
https://www.physicsforums.com/help/latexhelp/
Thank you. Sorry it has taken me a couple of weeks to get back here, but thanks again, for the information and help in understanding about comoving distances etc.

Hi Resurgance, I subscribe to everything Bandersnatch said--which was basically the perfect concise complete answer. I even hesitated to post because it would cover up post #2. So everybody please go back and read post #2!
But I wanted to say hello and glad you are wondering about cosmic rest frame, one of the nicest things in cosmology,maybe even in all of science. The ancient light (the "cmb") is approximately uniform, the same temperature in all directions. As it should be because at one time space was filled more or less evenly with a cloud of hot gas which then cooled just enough to become transparent. So the ancient hot gas and the ancient light that remains from it DEFINES what it means to be not moving. Not moving with respect to the ancient light.

If we were at rest then the ancient light would be EXACTLY the same average temperature in all directions. But it was around 1977 using a U-2 Spy Plane with a little microwave temperature receiver in a blister on top of it, that they discovered a WARM SPOT in the microwave sky. A warm spot in the ancient light, that was around the constellation of LEO (which you can see overhead in the Springtime of the year). A patch of the microwave sky that was just about 1/8 of one percent warmer!

That meant that the solar system was moving at about 1/8 of one percent of the speed of light, in the direction marked in the sky by the constellation Leo, and that it was a DOPPLER effect of our moving through the approximately uniform soup of ancient light. And it meant there should be a doppler COLD spot in the opposite direction in the sky from Leo, 180 degrees reversed. (around the constellation Aquarius.) Where the temperature was 1/8 of one percent cooler because of our moving away from the light coming from that direction. Here are some pictures.
http://aether.lbl.gov/www/projects/u2/

1/8 percent of 300,000 km per second is about 380 km per second, that is about what speed the solar system is going, relative to CMB-rest.
So we can correct our redshift observations for that motion and record the data AS IF the galaxies were observed from a reference frame at rest relative to the ancient light (or one could say at rest with respect to the hot gas cloud of ancient matter (before it began to fall together and coagulate into wisps and clumps all over the place and the chunks acquired their separate motions, that is to say) back when it was more more even and stationary.

380/300,000 the ratio of speeds (same as the doppler shift in temperature)
about 1/8 of a percent.
Thanks Marcus. I am really struggling with this course at the moment, and your comments here, are very helpful. Cheers