Talk:Comoving and proper distances

Uniform image size

Who was the elephant in the china shop that changed the image sizes so that images that belong together were scaled differently? Looks like crap, if you have to change one size also change the others to the same size, at least those which belong together. --Yukterez (talk) 23:46, 1 December 2024 (UTC)

Proper distace uses comoving rulers

The definition for the proper distance was wrong since it implied the proper distance was measured with a static ruler, while in reality it is measured with a set of comoving rulers each moving with the Hubble flow. I changed that to the correct definition and hope nobody changes it back.

That is shorter than the distance measured with a static ruler, which would break at the Hubble radius. Such rulers are used in diagonal coordinates without crossterms and where gᵣᵣ=-1/gₜₜ, most famous example is the classic De Sitter metric in Droste style coordinates which have a coordinate singularity at the Hubble radius like Schwarzschild in Droste coordinates has one at the Schwarzschild radius.

The FLRW in proper distace coordinates has the same structure as the Schwarzschild metric in raindrop coordinates, meaning gₜₜ=1-v², gᵗᵗ=1, gₜᵣ=gᵗʳ=v, gᵣᵣ=-1, gʳʳ=-1+v² where v=Hr for the FLRW and v=-c√(rₛ/r) for Schwarzschild. In both cases the local clocks and rulers are moving with v. --Yukterez (talk) 15:01, 19 December 2024 (UTC)

Why this image?

comoving coordinates

Comoving coordinates separate the exactly proportional expansion in a Friedmannian universe in spatial comoving coordinates from the scale factor $a(t).$ This example is for the ΛCDM model.

The article has an image of some dots moving apart. In my opinion it has no illustrative value for this topic or any thing else. The caption makes no connection to the image, does not explain what the dots are and what the frame means. As far as I can tell it just a distraction. Other opinions? Johnjbarton (talk) 22:25, 26 May 2025 (UTC)

Definitions

As far as I can tell the "Definitions" is confused and mixed up. To be fair, different textbooks use different symbols/terminology sometimes multiple ones in the same text.

• "Many textbooks use the symbol $\chi$ for the comoving distance."
  • Roos says "Another convenient comoving coordinate is $\chi$." Pg 36. Page 37 does imply that $\chi$ is a comoving distance when we are a geodesic between two points observed by light.
  • Webb does not even use a symbol. Pg 261.
  • Lachièze-Rey says $\chi (r)$ (not function of t like Roos) is the inverse of $S_k$ and that the "comoving proper distance" is $d(r)=d_{\text{proper}}(r,t_0)=R(t)\chi (r)$ Pg 12
  • DodelsonTemplate:Rp defines $\eta =\underset{0}{\overset{t}{\int }}\frac{d{t}^{\prime }}{a(t^{\prime })}$ as the "total comoving distance" light could travel in t and is also called "conformal time". Dodelson then defines "another important comoving distance is that between a distance emitter and us" as $\chi (a)=\underset{t(a)}{\overset{t_0}{\int }}\frac{d{t}^{\prime }}{a(t^{\prime })}$
  • Peacock says the radial increment of comoving distance is $dr$ then says that there are different ways that the Robertson Walker metric can be defined with the two common ones using different definitions of "comoving distance, $S_k(r)\to r$" explicitly saying the literature is ambiguous. Peacock defines conformal time as $\eta =\underset{0}{\overset{t}{\int }}c\frac{d{t}^{\prime }}{R(t^{\prime })}$ and the dimensionless scale factor as $a(t)\equiv \frac{R(t)}{R_0}$, so Peacock's conformal time is Dodelson's divided by $R_0$.
  • Zee calls Dodelson's conformal time value "coordinate distance". Pg 532
  • Huterer defines conformal time the same way as Dodelson and call the Dodelson "distance emitter and us" the coordinate distance. Huterer says coordinate distance is also called conformal distance or radial distance and is used by experimental cosmologists while conformal time is more often used in theory. The proper distance $d_p$ is a length of a geodesic at a fixed time or fixed scale factor; freeze the universe and measure with a meter stick. Pg 25.

So "many" textbooks don't use $\chi$ for comoving distance. Partly this issue is a consequence of expanding space time: distance is intrinsically complex and one needs to specify what your distance means. Template:Reflisttalk Johnjbarton (talk) 00:00, 27 May 2025 (UTC)