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and all symbols other than "r" are defined. Generally an equation in physics should work no matter what system of units is chosen but this not the case here and "r" and "dr" do not appear to be related in the way one would expect.
Elsewhere I found that "k" can only take values 1, 0, and -1 (dimensionless). also has to be dimensionless so this should be written as where Rmax is some maximum value of r (which has to be the case if k = 1). Alternatively, if "r" is by convention a dimensionless fraction of Rmax (which should be explained) then one assumes "dr" is also dimensionless and something needs to be added to the expression to make that dimensionless.
Redshift as an unsuitable measure of distance
An essential requirement for any measure of distance is additivity for collinear points. i.e. if A, B, C are on a straight line (or geodesic) then dist AB + dist BC = dist AC. It is easy to verify that redshift does not have this property. As a result redshift measurements are dependent on the point from which they are measured. Usually the point of measurement is the Earth leading to a geocentric view of the Universe not satisfying the cosmological principle. JFB80 (talk) 02:39, 25 March 2016 (UTC)
To address your precise point: You've defined "measure of distance" as a linear quantity. Redshift, however, is in general a nonlinear function of distance. For example, using the Friedmann-Robertson-Walker metric, redshift is the ratio of the scale factor a(t) at the time of observation to a(t) at the time of emission, where a(t) is nonlinear. So redshift is an indicator of distance, but not a "distance measure" per se.Karl pomeroy (talk) 21:04, 3 July 2016 (UTC)
Indeed. I don't get your point though, because it's possible to derive distance from redshift, it's just sensitive to other parameters. And while additivity does not hold, other relations do, so given the redshifts to B and C from A, one can work out the redshift to C from B. Banedon (talk) 02:47, 25 March 2016 (UTC)
Thank you for your reply. I have been trying to understand how cosmologists in practice estimate distances from redshifts. I don't find this clearly explained, perhaps you can. It seems to me you can get either (a) a (possibly rough) estimate of time the light takes to travel or (b) make some assumption about the model and so use theoretical formulae to find a(t) and then hopefully the distance. My remarks were prompted by the thought that, most often, redshift is used to characterize distance and with the commonly used definition (which seems to be incorrect), a distorted picture arises.JFB80 (talk) 06:34, 28 March 2016 (UTC)
Well part of the problem is that 'distance' has multiple meanings in cosmology, and furthermore is sensitive to other parameters such as the density of matter and / or dark energy. In other words, the "distance" (using this loosely, since there are different meanings of distance) to a faraway body is not the same in a universe with zero dark energy compared to one in which dark energy makes up ~70% of the universe and matter the remaining ~30%. There is a formula for it, which as far as I recall is an integral equation that has to be evaluated numerically. Light travel time is much easier, since the speed of light in a vacuum is a constant so simply dividing the "distance" by the speed of light grants the time. This might help: . Banedon (talk) 06:48, 28 March 2016 (UTC)
To clarify further: redshift is the observable quantity. Distance is the derived quantity. Since we can't directly measure distances, we might as well carry around the thing we can measure. Astronomers refer to redshift as it is independent of a particular set of cosmological parameters, so that as those parameters are refined (via e.g. CMB, BAO, or Supernova measurements), we can trivially update our distances. As Banedon alluded to, you can derive many different useful "distances" from the redshift and the cosmological parameters, so the thing to hang on to is the base quantity. To compute your distance of choice from redshift, you have to (almost always numerically) integrate the Friedmann_equation. Ned Wright's Cosmological Calculator is a handy way to play with this.
This could probably be made more clear in the article. - Parejkoj (talk) 16:33, 28 March 2016 (UTC)
Yes I think the algorithm should be explained in simple terms. To find light-travel time and distance for nearer objects seems fairly simple using Hubble's original law. But in general Friedmann's equation must be integrated numerically to find H(t). Then how is distance found? Is it obvious as I don't see that explained anywhere in Wikipedia? Presumably it is programmed in Ned Wright's calculator but that needs a lot of deciphering. JFB80 (talk) 21:07, 29 March 2016 (UTC). I think the distance must be found by integrating incremental distances using the time-variable Hubble Law but as I said it is not made clear anywhere.JFB80 (talk) 21:46, 29 March 2016 (UTC)
No, it's not obvious how redshift is transformed into "distance" (again, using the word loosely). This is material typically covered in an advanced undergraduate course in cosmology. A quick search on Wikipedia does show some coverage though: Distance measures (cosmology). I'm woefully short on time though, and can't add it to this article. Banedon (talk) 09:33, 31 March 2016 (UTC)
Thank you for the comment. The most direct method seems to be the Mattig formula. JFB80 (talk) 17:07, 1 April 2016 (UTC)
┌────────────────────────────────────────────────────────────────────────────────────────────────────┘That formula is not valid for the universe we live in: we have dark energy, while it assumes Omega_lambda=0. There really is no closed form solution for our Universe. The best discussion of this on Wikipedia is section 1 of Distance measures (cosmology), which was clearly written based on Hogg (1999), which it references. I've added a couple sentences to it to try to clarify things. If you think this helps, we should like to that article from section 3.2. It would be nice to reference Barbara Ryden's Cosmology book as well, in that section.. - Parejkoj (talk) 19:06, 2 April 2016 (UTC)
The mathematical derivation of cosmic redshift (section 3.2.1) based on the Friedmann-Robertson-Walker metric looks absolutely rigorous. However, when I apply this step-by-step procedure (breaking up and recombinging the integrals) to the Schwarzschild metric, I get a blueshift rather than a redshift, ie. the wrong answer. I have tried several variations on the integrand and the limits of the integral, and have even tried varying the speed of light. All attempts produced a blueshift or no shift at all. Obviously, this could be my own systematic errors. Does anyone know whether this procedure is truly rigorous? Or is there some hidden assumption that makes it work for the FRW metric but not other metrics? It should work for every metric. Karl pomeroy (talk) 20:22, 3 July 2016 (UTC)
Sounds like a sign error. Make sure you're using the right signature. jps (talk) 19:20, 16 March 2017 (UTC)