# Distances in cosmology (relativistic) [closed]

so, my problem is that distance measures in cosmology seem to ignore General Relativity GR (or even STR). There, the proper distance is the distance w.r.t. a specific rest frame!

This means that, in order not to violate relativity, we always need to give a resframe to which we are referring to when speaking about distances. In other words: We need to give the frame of reference from which we are measuring that distance.

Now, my question is: Other distance measures than the proper distance from GR, like for example the "Light-travel distance" (or "lookback distance), do not seem to respect STR (or GR) in its definitions: "The distance light traveled from A (source) to B (observer) taking into account the expansion of the universe."

But from which restframe?! From the restframe of the source (e.g. distant galaxy) or of the observer (e.g. on earth)? The frame we are talking about also has implications for the "loockback time" which results from that distance!

Other distance measures like the "comoving" distance makes this clear: It is the frame in which the expansion of the universe is zero.

• I presume you've looked at this? physics.stackexchange.com/q/643610 and en.wikipedia.org/wiki/Distance_measures_(cosmology) These other distances don't claim to be "proper distances" so I'm not sure what the problem is? Commented Jun 8, 2021 at 14:15
• Yes, I did, thanks! Here the author says: "At the beginning of our considerations S and D do not move with respect to each other in the comoving coordinates." Well okay that makes sense, but after a certain time they are not in rest which respect to each other anymore (due to the expansion of the universe). That is actually what confuses me. I am missing the following information in those definitons: From which restframe are these distances given? Commented Jun 8, 2021 at 14:17

You are correct that distances and times are relative, but everyone writing textbooks and research papers in astronomy and cosmology knows this, so you can be sure that, apart from mistakes and oversights, people are using well-defined quantities, and it is up to you to find out what the standard definitions are. I will not list all of them but I will provide a few pointers.

If the discussion concerns cosmology, then if we first adopt comoving coordinates then one can define a distance between fundamental observers (roughly speaking, galaxies or galaxy clusters) at any given instant of cosmic time. This is often called the proper distance. You consider the spacelike surface at some instant of cosmic time, and find the line of minimum proper length, in that surface, between the given galaxies $$A$$ and $$B$$. The proper distance between them is then $$\int_{(A)}^{(B)} ( g_{ij} dx^i dx^j )^{1/2}$$ where $$g_{ij}$$ is the metric within the hypersurface and the integral is along the line of minimal length. It may seem amazing that you thus get an invariant distance measure, but this is because by specifying a spatial hypersurface you have adopted a reference frame in a certain sense, because you have specified which set of events you are deeming "simultaneous". This is not a local inertial frame at either galaxy $$A$$ or galaxy $$B$$, it is the comoving frame which extends everywhere, but this frame does manage to coincide (locally) with local inertial rest frames at both $$A$$ and $$B$$.

Most of the other measures are concerned with the separation of events at different cosmic times. If the events are also at different comoving spatial coordinates then the definition of distance becomes more tricky. One way is simply to provide the comoving coordinate values at both events, and leave the reader to figure out what they wish from that (the reader will also need to know the expansion history to do this). Another interesting quantity is the amount of elapsed cosmic time (i.e. proper time for the fundamental observers) between the events. If light traveled from one of the events to the other, then there is null geodesic between them and then the elapsed cosmic time between the two spatial hypersurfaces tells you something about how far apart the events are. Again, one needs to know $$a(t)$$ or equivalent information to get quantitative information about other matters, such as the proper distance now between receiver and the comoving location of the source. But simply multiplying the time by $$c$$ gives some sort of distance measure, and perhaps this is the one your text is using. However you are right to say that that distance measure, on its own, does not provide much in the way of useful information, so astronomers mostly do not use it.

In practice, rather than mentioning cosmic time, it is very common to mention cosmic redshift $$z$$. One can develop formulae relating $$z$$ to other measures, and these formulae typically involve an integral over the expansion history.

Further standard measures go by names such as luminosity distance and angular diameter distance.

It is not possible to say "this event at redshift $$z$$ happened at distance $$X$$ from the Milky Way" because, as your question correctly states, there is no single $$X$$ which is "the distance" (unless $$z$$ is small). In a research paper one therefore speaks of "luminosity distance" or whatever. A distance measure that is often used in popular presentations is to take some event $$A$$ which may have happened long ago, and then consider how far it is now (i.e. proper distance) from ourselves to the fundamental observer at the comoving coordinate location of $$A$$. This is what is going on when people speak of "the size of the observable universe", and it is usually this measure which is being referred to when someone speaks of "the distance to the surface of last scattering" from which our cosmic background radiation comes.

• Firstly: Thank you very much. Secondly: Two short follow-up questions: 1. This is my most crucial confusion: In your definition of proper distance, you speak of two galaxies and their distance in an instant of time. From where are we measuring this distance: Galaxy A, B or from a third system (the comoving one?)? This will have an effect on the distance. And 2. Do I see it rigth that the usage of "proper distance" is a different one than in GR or is it the same here, because here we seem to use it for the distance between fundamental observers, contrary to distance in the respective restframe Commented Jun 9, 2021 at 23:01
• @SimonFischer "proper distance" makes sense when you have a spatial hypersurface. I edited the answer to spell it out a bit more. Commented Jun 10, 2021 at 0:18