Talk:Comoving and proper distances

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)

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 [https://relativity.yukterez.net/28.html#1 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 [https://umwelt-wissenschaft.de/forum/naturwissenschaftliche-themen/138-mein-bb-inflation?start=60#7261 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)

File:Cosmos-animation Lambda-CDM.gif

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)

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 $\backslash chi$ for the comoving distance."
• Roos{{cite book

|title=Introduction to Cosmology

|edition=4th

|first1=Matts

|last1=Roos

|publisher=John Wiley & Sons

|year=2015

|isbn=978-1-118-92329-0

|page=37

|url=https://books.google.com/books?id=RkgZBwAAQBAJ}} [https://books.google.com/books?id=RkgZBwAAQBAJ&pg=PA37 Extract of page 37 (see equation 2.39)] says "Another convenient comoving coordinate is $\backslash chi$." Pg 36. Page 37 does imply that $\backslash chi$ is a comoving distance when we are a geodesic between two points observed by light.

• Webb{{cite book

|title=Measuring the Universe: The Cosmological Distance Ladder

|edition=illustrated

|first1=Stephen

|last1=Webb

|publisher=Springer Science & Business Media

|year=1999

|isbn=978-1-85233-106-1

|page=263

|url=https://books.google.com/books?id=ntZwxttZF-sC}} [https://books.google.com/books?id=ntZwxttZF-sC&pg=PA263 Extract of page 263] does not even use a symbol. Pg 261.

• Lachièze-Rey{{cite book

|title=The Cosmological Background Radiation

|edition=illustrated

|first1=Marc

|last1=Lachièze-Rey

|first2=Edgard

|last2=Gunzig

|publisher=Cambridge University Press

|year=1999

|isbn=978-0-521-57437-2

|pages=9–12

|url=https://books.google.com/books?id=3LO75VmI9BMC}} [https://books.google.com/books?id=3LO75VmI9BMC&pg=PA11 Extract of page 11] says $\backslash chi(r)$ (not function of t like Roos) is the inverse of $S\_k$ and that the "comoving proper distance" is $d(r)\; =\; d\_\backslash text\{proper\}(r,t\_0)\; =\; R(t)\backslash chi(r)$ Pg 12

• Dodelson{{Cite book |last=Dodelson |first=Scott |title=Modern cosmology |date=2003 |publisher=Academic Press |isbn=978-0-12-219141-1 |location=San Diego, Calif}}{{rp|34}} defines $\backslash eta\; =\; \backslash int\_\{0\}^\{t\}\; \backslash frac\{dt\text{'}\}\{a(t\text{'})\}$ 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 $\backslash chi(a)\; =\; \backslash int\_\{t(a)\}^\{t\_0\}\; \backslash frac\{dt\text{'}\}\{a(t\text{'})\}$
• 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, $S\_k(r)\; \backslash rightarrow\; r$" explicitly saying the literature is ambiguous. Peacock defines conformal time as $\backslash eta\; =\; \backslash int\_\{0\}^\{t\}\; c\backslash frac\{dt\text{'}\}\{R(t\text{'})\}$ and the dimensionless scale factor as $a(t)\backslash equiv\; \backslash frac\{R(t)\}\{R\_0\}$, so Peacock's conformal time is Dodelson's divided by $R\_0$.
• Zee{{Cite book|url=https://www.worldcat.org/oclc/820123453|title=Einstein gravity in a nutshell|last=A.|first=Zee|date=5 May 2013|isbn=9780691145587|location=Princeton|oclc=820123453}} calls Dodelson's conformal time value "coordinate distance". Pg 532
• Huterer{{cite book |last1=Huterer |first1=Dragan |title=A Course in Cosmology |date=2023 |publisher=Cambridge University Press |isbn=978-1-316-51359-0}} 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 $\backslash chi$ 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.

{{reflisttalk}} Johnjbarton (talk) 00:00, 27 May 2025 (UTC)