Talk:Friedmann–Lemaître–Robertson–Walker metric

Change name to "FRW model"?
I think this article should be renamed FRW model. One reason is that I need to refer to it a lot and can never remember this crazy name. Another reason (probably a much better one) is that noone would ever type in this phrase to search for information on FRW models. Comments? ---CH (talk) 23:46, 17 September 2005 (UTC)


 * I agree; this seems to be the most common usage. I've never heard it referred to as the FLRW model. Maybe it's better to spell it out (as in Friedmann-Robertson-Walker model) for the actual title though. Either is fine with me. -- Fropuff 02:15, 18 September 2005 (UTC)


 * I agree. The word 'model' in the title is more informative as to the nature of the article than four hyphenated names. --Eddie | Talk 07:53, 19 September 2005 (UTC)


 * I'm confused; I would have expected this to be the FRW metric, or maybe the FLRW metric - it's talking purely about the metric, which while it does represent a model it is not the entire model, more a subset of the GR model. The current name ('Robertson-Walker coordinates') makes no sense to me. As such, I would propose a name change to FRW metric. The names don't need to be spelled out, methinks. When any name change is made, remember to change the links from other pages (especially the cosmology template). Mike Peel 17:01, 25 February 2006 (UTC)


 * I do not agree. I am a physicist, I have taken many courses on Relativity and FRW is a metric, not a model. it's a mathematical object, not a model of behavior for a system. Astrobayes 23:56, 3 June 2006 (UTC)


 * Based on the above comments, I think that Friedmann-Robertson-Walker metric is probably the best candidate for a name change - reasons: (1) clearly, 'coordinates' should not be in the title, as it's a reference to the metric (2) I think it's the most common reference to the metric ('FRW model', as Astrobayes suggests, is not technically correct, even though many people refer to the metric as the model). MP  (talk) 18:32, 17 June 2006 (UTC)


 * This is not supposed to be a USA-UK-centric encyclopedia, nor an EU-Russian-centric encyclopedia. Friedmann was Russian, Lemaitre was Belgian, Robertson was from the US, Walker was from the UK. Check out French authors e.g. Lachieze-Rey & Luminet 1995 http://arxiv.org/abs/gr-qc/9605010 who use only Friedmann and Lemaitre. For FLRW, just a few examples of usage by practising scientists are e.g.
 * A non-linear Schrodinger type formulation of FLRW scalar field cosmology, Jennie D'Ambroise, Floyd L. Williams, http://arxiv.org/abs/hep-th/0609125
 * Deviation of geodesics in FLRW spacetime geometries, George F R Ellis, Henk van Elst http://arxiv.org/abs/gr-qc/9709060
 * Distance-Redshift in Inhomogeneous FLRW, R. Kantowski, J.K. Kao, R. C. Thomas http://arxiv.org/abs/astro-ph/0002334
 * NPOV surely favours FLRW. So i'm shifting back to FLRW. FLRW metric is fine rather than FLRW model. Boud 00:33, 31 December 2006 (UTC)

FLRW is very common in professional litterature. Strictly speaking it would be better to say "FLRW universe" or "FLRW" Model. The Metric is reazlly due to RW and the dynamical equations were derived independantly by F and L. ABlanchard 24 November 2009


 * Perhaps it would be a good idea to encourage people to refer to the "FLRW metric" when citing the derivation from GR and to the "FLRW model" when referring to the commonly used practical solution having additional particular assumptions. Otherwise it's just distracting semantics.

Kentgen1 (talk) 12:54, 3 November 2010 (UTC)

Derivation
I was wondering whether a derivation of the FRW metric might be handy? I've got a fairly nice (read 'hand waving') derivation from a Part II Experimental and Theoretical Physics course at Cambridge (3rd year Astrophysics) that does it without the need for tensor calc... I can have a look at whacking it up if people think it is worthy? MikeMorley 18:48, 11 June 2006 (UTC)


 * Fear not, when I get to it I plan a complete revision/reorganization of all articles pertaining to coordinate charts in gtr or exact solutions in gtr (this article fits into both categories, obviously), which will include derivations of such things. ---CH 01:20, 12 June 2006 (UTC)


 * Well, scratch that; see next section. ---CH 03:42, 1 July 2006 (UTC)


 * I'd like to have a look at it too if you come by it. Has Friedmann's paper ever been translated from German? JDoolin 02:41, 5 September 2006 (UTC)

I want to add, that the metric is not a SOLUTION to the einstein equations. It is a metric imposed by the symmetries homogenities etc...


 * In my opinion putting derivations in an encyclopedia is a Bad Idea. Wikipedia is, above all and foremost an encyclopedia, not a collection of student coursebooks where derivations are appropriate. In an encyclopedia you publish results, not intermediaries, unless it has an explicit bearing upon the subject. And that is not the case here. Instead I would focus more on interpretation of the result. 82.72.112.188 23:00, 15 December 2006 (UTC)


 * It needn't be a full derivation, but it could explain qualitatively where the metric comes from. To write, "Einstein's field equations are not used in deriving the general form for the metric: it follows from the geometric properties of homogeneity and isotropy" doesn't help the reader to understand what homogeneity and isotropy have to do with the thing. In my mind you need at least some explanation of where metric comes from if you're going to claim a serious article. JKeck (talk) 19:40, 16 April 2011 (UTC)

To the above two entries: I think it should be a full derivation, which would be very appropriate for this encyclopedia, since many physicists misinterpret the FRW metric. The only way to understand what it means is to see a full derivation. This derivation should include the fact that what is called "t" is really tau (proper time on comoving clocks), so that tau is automatically orthogonal to the comoving position coordinate "r" (ie, tau is a worldline parameter). The derivation should also specify how one knows what the energy-momentum tensor components are in these rather unnatural coordinates. However, in all the textbooks I've seen, including Weinberg's, and even in Friedman's and Lemaitre's original articles, no derivation is given that is complete. My hypothesis is that the FRW metric is not rigorous, and that that fact is the source of the confusion. Kate Rosser May 21, 2013 — Preceding unsigned comment added by Kate rosser (talk • contribs) 21:10, 21 May 2013 (UTC)


 * To JKeck: See the article on n-sphere which explains the structure of a positively-curved simply connected space which is homogeneous and isotropic. This can be extended to negatively curved spaces by replacing the radius of curvature by an imaginary number. JRSpriggs (talk) 21:04, 17 April 2011 (UTC)

That may be so in general but in this case I think it is important to be clear about the derivation. Many textbooks are not clear on this and often leave critical steps as an exercise to the reader. 94.66.41.69 (talk) 06:59, 23 August 2010 (UTC) For example: in d'Inverno's book, which is cited as giving a good account, an equation (22.30) is transformed by introducing a rescaled radial coordinate r*= sqrt|K|.r to derive the customary RW formula. The transformation is left as an exercise but it is not trivial as r* involves time (through K) and well as distance and the result does not follow as differentials dr,dt get mixed up. 94.66.3.25 (talk) 07:25, 24 August 2010 (UTC)94.66.90.209 (talk) 07:20, 25 August 2010 (UTC) Further remark: it is not generally acknowledged that K must depend on time - when we talk of a space of constant curvature it is constant in the space variables not time. A space of nonzero constant curvature can only expand if K varies with time - think of the ant on the expanding balloon analogy. K is inverse radius squared i.e. 1/a(t)^2. 94.66.90.203 (talk) 14:25, 25 August 2010 (UTC)


 * You misinterpret k. It is only the spatial curvature at time t0 when a(t0)=1. k is a constant independent of time. The spatial curvature at an arbitrary time t is k/(a(t))2. One should never get cross terms between dr and dt. JRSpriggs (talk) 03:19, 26 August 2010 (UTC)

You say I misinterpret k (which I call K). But I only use standard notation and definitions as per d'Inverno. You use a special interpretation of your own. You introduce time t0 when a(t0)=1 (defined, as I understand from the main article, 'for the present cosmological era') And you rescale the RW formula to be only valid for the present era. Is that what you are saying? But at other times (e.g. just after the Big Bang) when spatial curvature is 1/R(t)^2 (where R(t) = R(t0) a(t)) cross terms will appear - so no RW formula ! 94.66.22.0 (talk) 07:12, 27 August 2010 (UTC)


 * This article is not non-standard. You appear to be very confused. Either your source is wrong or you are misinterpreting it.
 * Essentially, we are taking a hypersphere for space and expanding or contracting it as needed while time passes. Quoting from some course notes I found on-line, "Even a homogeneous and isotropic universe only appears so to a special set of freely falling observers, called comoving observers. These observers are “going with the flow” of the expanding universe. They have constant values of r, &theta;, and &phi;, and the proper distance between them increases in proportion to a(t).". JRSpriggs (talk) 07:41, 27 August 2010 (UTC)

The book of d'Inverno is recommended in the bibliography of the main article. Why dont you read it and if it is wrong, suggest that it be removed, saying why it is wrong. But I can assure you it is completely standard and in accordance with the historical development. What you say is true - it is the ant on the balloon model. But the only relevance it has to the RW formula is to illustrate that expansion is not possible without time change in the 'constant' curvature. This fact invalidates the customary rescaling. (If the ant has a friend ant on the balloon then they move apart - they are comoving observers)94.66.48.65 (talk) 13:23, 27 August 2010 (UTC)

CONCLUSION: the book of d'Inverno (and others) mistakenly rescales distance to proper distance within the RW formula. It cant be done. The formula must be stated directly in comoving coordinates. The Wiki article makes a similar error and needs changing. It says there are two possible conventions: one with Gaussian curvature etc and one with k=-1,0,1 etc. These conventions are inconsistent. Avoiding mixed up notation, let r' be distance, r be comoving distance so that r'=r R (omitting zero curvature case). In the 1st convention there is a term (dr')^2 corresponding to R^2 (dr)^2 in the 2nd convention. These are unequal since dr' = dr R + r dR. Remaining terms agree. Therefore unequal inconsistent formulae. In fact the first convention is incorrect, the second correct. I propose the text be changed. Any objections? Wikipedia should be accurate on such an important topic.94.66.75.99 (talk) 18:02, 3 September 2010 (UTC)


 * NO! The fact that you disagree with all the sources should tell you something. You are wrong. The article is correct presently; and your changes would make it incorrect.
 * Do not make any changes unless you can provide a reliable source (see WP:RS) which agrees with your version!
 * Of course, the two conventions contradict each other &mdash; that is what it means to be two different conventions. However, there is a simple way to convert between them which shows that the underlying physics is the same. In either convention, the spatial curvature at time t is given by $$\frac{k}{{a(t)}^2} \,.$$ This fact can be used to derive a conversion between the two sets of values for k and a(t). This conversion involves multiplying k and a by constants which cancel out in the spatial curvature. The differential of these constants is zero.
 * Comoving distance is not represented by any single variable in these formulas. The comoving distance of a point from the origin at time t is r&times;a(t), if in hyper-spherical coordinates. However, the differential of the comoving distance does not appear in the metric, contrary to what you said. Your assumption that it does so appear may be your underlying error. JRSpriggs (talk) 04:50, 4 September 2010 (UTC)

(1) You say I disagree with all the sources and have my own version. But I was mainly pointing out inconsistencies in the versions of d'Inverno and the Wiki article. Apart from that I said that the RW formula should be expressed directly in terms of dimensionless comoving r. On this see Walker (1935) quoted in the bibliography, especially p.121 on 'Milne's hypothesis' (2) I do not see why it is OK for different conventions to contradict one another and find your 'simple way for conversion' totally incomprehensible. Can a simple way be explained simply? 94.66.81.185 (talk) 14:42, 6 September 2010 (UTC)


 * I will just respond to (2). In International System of Units, a distance given as "5" means five meters, but in the Imperial units it means five yards. Thus these two conventions contradict each other.
 * For simplicity, I will just consider the case when the spatial curvature is positive, and assume that $$t_0 \,$$ refers to the present time. Let $$k_1 \,$$ and $$a_1 (t) \,$$ be according to the first convention mentioned in the article. And let $$k_2 \,$$ and $$a_2 (t) \,$$ be according to the second convention mentioned in the article. Then $$a_1 (t_0) = 1 \,$$ and $$k_2 = +1 \,$$ and $$\frac{k_1}{(a_1 (t))^2} = \frac{k_2}{(a_2 (t))^2} \,.$$ Thus to convert from the second convention to the first: $$k_1 = \frac{1}{(a_2 (t_0))^2} \,$$ and $$a_1 (t) = \frac{a_2 (t)}{a_2 (t_0)} \,.$$ While to convert from the first convention to the second: $$k_2 = 1 \,$$ and $$a_2 (t) = a_1 (t) \cdot a_2 (t_0) = \frac{a_1 (t)}{\sqrt{k_1}} \,.$$ So all the conversion factors are just various powers of $$a_2 (t_0) \,$$ which are constants independent of time $$t \,$$ as I said. JRSpriggs (talk) 18:34, 6 September 2010 (UTC)

Thank you, I agree. (you make algebraic mistakes which do not affect the argument writing k/(a(t))^2 instead of k (a(t))^2). But you deal only with k and a(t), what about dr^2 which appears in the RW formula?94.66.36.63 (talk) 06:58, 8 September 2010 (UTC)


 * No, I did not make any algebraic error. As the universe gets larger over time, $$a (t) \,$$ increases which means that $$\frac{k}{(a (t))^2} \,$$ gets smaller, that is, the spatial curvature decreases as it should.
 * For any event in spacetime, let its value of $$r_1 \,$$ be as in the first convention and $$r_2 \,$$ be as in the second convention. Then the comoving proper distance (if in hyper-spherical coordinates) is $$r_1 a_1 (t) = r_2 a_2 (t) \,.$$ Thus $$r_1 \sqrt{k_1} = r_2 \sqrt{k_2} = r_2 \,$$ (again restricting to the case when curvature is positive, i.e. $$k_2 = +1 \,$$). Notice that the conversion does not introduce any dependence of r on t. JRSpriggs (talk) 10:05, 8 September 2010 (UTC)

Ok you did not make an algebraic error. Your final result is that r1 √k1 = r2 where k1 is Gaussian curvature which is dependent on t as I was saying above. So your equation is the same as r'/R(t) = r or r'=r R(t) which I previously wrote - and we are back to square one.94.66.31.109 (talk) 19:51, 9 September 2010 (UTC)


 * NO, YOU ARE MISSING THE POINT. k1 is a constant, independent of time. The spatial curvature is NOT $$k_1 \,,$$ the spatial curvature is $$\frac{k_1}{(a_1 (t))^2} = \frac{k_2}{(a_2 (t))^2} \,.$$ JRSpriggs (talk) 09:13, 10 September 2010 (UTC)

Now I see what you are intending, but if I am not mistaken you are still wrong because of your switch to hyperspherical coordinates.The definition of these is clear in the second convention but not in the first because the integrand defining X (=chi) then contains the Gaussian curvature which is a function of time. X is then a function of time which invalidates your argument. Your notation is truly very confusing with two meanings for k and three for r. Just think what it must be for a person who has not studied the subject. Added to this your description is very non-standard in spite of your protestations. You have this a(t) which changes meaning and equals 1 at an undefined time t0. Do you have any source reference or is it your own version? I am used to the version using only R(t) which occurs in the literature cited in the article (Robertson, Walker, d'Inverno) as well as in everything else I have read on the subject.94.66.12.93 (talk) 18:43, 11 September 2010 (UTC)


 * No. I am tired of trying to explain this very simple idea to you. If you still need help, go to the Reference desk. JRSpriggs (talk) 11:46, 12 September 2010 (UTC)

THE ERROR: Nevertheless I have finally managed to work out what you do and locate the error. To show consistency of the two conventions you switch to hyper-spherical coordinates and, to avoid time dependence in the first convention, you use for Gaussian curvature the constant value k1 evaluated at time t0 when a(t0)=1. But then the metric is only valid at t0 and so is useless. You could have allowed k to be time varying but this would give a metric not in RW form. Surely it is best to use only the second convention when all is OK.94.66.17.3 (talk) 07:49, 14 September 2010 (UTC)

Please tell me where I'm going wrong, but I have a problem with the dimensions (units) in the formulae for $$S_k(r) $$ in the section on hyperspherical coordinates. In the case where k is dimensionless, then r must be dimensionless also, to make the arguments of sin, sinh dimensionless, as they must be. What does r then represent? Dendropithecus (talk) 05:23, 17 July 2012 (UTC)


 * In the second convention using hyperspherical coordinates, r(t) is the proper distance (at time t) of the object in question from us divided by the scale factor a2(t). So it is the ratio of two distances and thus dimensionless. In this case, the scale factor is the radius of curvature of the universe which is the proper distance to the antipode divided by &pi; (assuming there is an antipode, i.e. that the spatial slice of the universe is S3 topologically). JRSpriggs (talk) 07:08, 17 July 2012 (UTC)


 * Thanks, JR. I wasn't expecting such a prompt reply! So I wasn't wrong, just a little confused! In the spherical model (k>0). this scaled version of r(t) appears to be the angle subtended at the "centre" by the arc, i.e. the angle between the normal to the surface and the axis of symmetry. It looks like being an angle in the hyperbolic model (k<0), but does this angle have any physical interpretation?


 * Would I be right in inferring that, if the distance coordinate is scaled, then the time coordinate must also be scaled, to render it dimensionless? What would be the scale factor for time? Would it be c times the spatial scale factor? Dendropithecus (talk) 12:03, 17 July 2012 (UTC)


 * Sorry, the article does say "Note that when k = +1, r is essentially a third angle along with θ and φ." Also time need not be scaled if a(t ) has the dimensions of space, i.e. a(t ) is multiplied by a2(t ). Is this correct?


 * It seems it's all there, and correct, of course, but it does tend to confuse anyone not familiar with the topic. Could it be improved for the benefit of us lesser mortals? Dendropithecus (talk) 12:39, 17 July 2012 (UTC)


 * I hope you don't mind, but I've made an attempt to resolve the issue that caused my confusion, which is that, when reading the section on hyperspheric coords, we were expected to remember, or to refer back to, the previous section. This meant we had to weed out the bits that were relevant to both types of coords from the bits that only applied to the reduced polar case. Please could you check this (please don't just reverse it) and edit any errors I may have introduced. Ta. Dendropithecus (talk) 03:42, 18 July 2012 (UTC)


 * It's just occurred to me that, to reduce the duplicity, the parts I've copied, that apply to both types, may be removed from the section on reduced polar coords and the phrase "As before" replaced by "In both of the above cases" or something similar, possibly with a new heading "Unit Conventions". What do you think? Dendropithecus (talk) 04:17, 18 July 2012 (UTC)


 * Since there is nothing outside of our spacetime (as far as we know now), the angle has no physical meaning. It just plays that role in the mathematics.
 * No, time is handled separately. It is not scaled or modified in any way. It does not even appear in $$\mathrm{d}\mathbf{\Sigma}^2 \,.$$
 * Your addition to the article is OK. I see no need to worry about a little duplication. Trying to avoid the duplication may cause more problems than it solves. JRSpriggs (talk) 13:19, 18 July 2012 (UTC)


 * why does that meet the conditions? not explained, not sourcedJuror1 (talk) 07:11, 4 December 2017 (UTC)
 * To Juror1: You are responding to something that happened over five years ago and almost a hundred edits back. So you need to provide more context for your statement or withdraw it. JRSpriggs (talk) 04:48, 5 December 2017 (UTC)

Students beware
I edited the July 2005 version of this article, planned to greatly improve it, and had been monitoring it for bad edits, but I am leaving the WP and am now abandoning this article to its fate.

Just wanted to provide notice that I am only responsible (in part) for the last version I edited; see User:Hillman/Archive. I emphatically do not vouch for anything you might see in more recent versions. I hope for the best, but gtr attracts many cranks, and cosmology has been "hijacked" by political action groups (creation science (sic) and all that) to further their agenda of suppressing scientific belief which they feel somehow threatens their particular religious dogmas, so the reader should beware; it seems likely that at least some future versions of this article will contain slanted information, misinformation, or disinformation.

Good luck to all students in your search for information, regardless!---CH 03:42, 1 July 2006 (UTC)


 * > "I am only responsible for the last version I edited"
 * No offense, and I know this was 12 years ago, but so what? WP is edited by a lot of people. Yes, kooks, crackpots, and religious nuts haunt all WP science articles. What's your point?
 * > " I emphatically do not vouch for anything you might see in more recent versions. "
 * Do you think someone will blame you for edits made by people after you leave WP?
 * Do you think you "own" this article and now you're abandoning ownership? What makes you different from other editors?
 * Again, I do not mean offense, and the guy will probably never read this, but I don't understand the... peculiar message.
 * I'm probably missing someone's point again because I'm autistic. In fact, I'm sure of it. Verdana ♥ '''Bold 05:02, 18 October 2018 (UTC)

Friedmann-Lemaître-Robertson-Walker metric
(Comments moved here from the newly-created talk page, which will be deleted shortly as I attempt to fix the mangled move. Mike Peel 11:10, 7 January 2007 (UTC))

Depending on geographical/historical preferences, this may be referred to under the names of a preferred subset of the four scientists Alexander Friedmann, Georges Lemaître, Howard Percy Robertson and Arthur Geoffrey Walker, e.g. Friedmann-Robertson-Walker (FRW) or Robertson-Walker (RW) or Friedmann-Lemaître (FL).

Boud 00:42, 31 December 2006 (UTC)

While most US/UK based cosmologists tend to choose the US-UK scientists and write RW or FRW (the latter including the Russian), European cosmologists e.g. Lachieze-Rey & Luminet 1995 http://arxiv.org/abs/gr-qc/9605010 sometimes use only Friedmann and Lemaitre.

A few examples of NPOV usage - FLRW - among practising scientists are e.g.
 * A non-linear Schrodinger type formulation of FLRW scalar field cosmology, Jennie D'Ambroise, Floyd L. Williams, http://arxiv.org/abs/hep-th/0609125
 * Deviation of geodesics in FLRW spacetime geometries, George F. R. Ellis, Henk van Elst http://arxiv.org/abs/gr-qc/9709060
 * Distance-Redshift in Inhomogeneous FLRW, R. Kantowski, J.K. Kao, R. C. Thomas http://arxiv.org/abs/astro-ph/0002334

Boud 00:48, 31 December 2006 (UTC)

You may want to mention that in writing the Friedmann equations, you are setting various constants (h, G, c, 8pi) equal to 1. These are common units for cosmologists, but not to incoming students. 76.23.174.125 23:54, 3 February 2007 (UTC)A. D.
 * How can you set 8pi = 1? Verdana ♥ '''Bold 05:06, 18 October 2018 (UTC)

It's called exclusively Friedmann in Russian cosmology literature including the acclaimed and universally used Landau-Lifshitz course. Therefore, the use of Friedmann alone must be included in the list of geographical variations. (There is no need to endorse or oppose it -- but it's a fact that this is in use -- and Friedmann is universally acknowledged to be the one who wrote the solutions first.) — Preceding unsigned comment added by 161.200.117.216 (talk) 07:20, 9 February 2018 (UTC)

Merge with Einstein's radius of the universe?
There's been a merge tag on Einstein's radius of the universe to merge it here. They don't seem related but I'm not a physicist. Does anyone know if they should be merged, or should the tag be removed? Tocharianne 16:23, 14 July 2007 (UTC)

Whoever put the "citation needed" tag around the $$R_\text{E}=c/\sqrt {4\pi G\rho}$$: that is easy to verify, just plug in that number into the Friedmann equations and the Hubble parameter becomes zero, so the static universe with matter, curvature and cosmological constant is accomplished, see here for the equations and here for the solution. --91.141.68.128 (talk) 22:35, 14 August 2023 (UTC)

Some possible merges
It looks to me as though Lemaitre metric and Lemaitre-Tolman metric ought to be merged here. Or if it's not appropriate to merge them, then at least those two articles should discuss the relationship between the various models, along the lines of the discussion in section 4 of this article. Gdr 14:39, 28 November 2007 (UTC)

Two Naviagtion templates
Aren't two Naviagational templates too much in an article ? I propose to remove the GR template, as I did already one month ago !(Sheliak (talk) 15:35, 7 February 2008 (UTC))

Criticisms
Why is there no criticisms section in this article? That seems horribly peculiar. 66.69.194.16 (talk) 15:18, 3 June 2008 (UTC)
 * The article is entirely about the mathematical nature of the metric itself and not about its physical interpretations. It mentions that it is the basis for the big bang cosmology only in passing; were this in fact an article about the FLRW cosmologies with physical interpretation, then one would expect opinions about interpretation, while this page gives none. When you assume homogeneity and isotropicity, you always get the FLRW metric - it also occurs as the solution to a collapsing sphere of relativistic dust; I am not sure if that is mentioned on the page. Atonita (talk) 17:04, 1 December 2008 (UTC)

I agree with 3 June 2008. As the article is written, it violates Wikipedia's own NPOV policy because it tacitly approves or endorses the views propounded in the article. The article barely acknowledges that the FLRW metric is useless without the Friedmann equations, which do assume a lot more than just the Cosmological Principle. The "FLRW Metric" needs a critique as does the "Friedmann Equations".

For a critique of the FLRW and the Friedmann Equations with their solutions, see my talk page under the heading "Critique of the Universe":

Kentgen1 (talk) 13:31, 28 October 2010 (UTC)


 * In response to this concern raised by 3 June 2008, I have written an abstract referring to a contribution under construction that is meant to be a new free standing article or else a segment to be appended to the FLRW Metric article or maybe to the Friedman Eqautions or perhaps to both. This proposed contribution has the working title “Critique of the Universe”.

In the full text contribution, also available on my Talk Page, the references are numbered in “belt and suspenders” duplicate style because, at present, some of them contain a comment as well as a reference or else the reference is incomplete – being only a reference to Wikipedia itself – so the citation would need to be edited and upgraded. I hope I can attract some editors who might like to do this.

As 3 June 2008 said of the FLRW Metric, both of the articles are defective in that they contain no criticisms, no caveats, no cautionary dicta. They lack a critique. This violates Wikipedia’s own NPOV policy because, to the reader, it appears as if Wikipedia sanctions or tacitly accepts the statements and conclusions given in these articles as scientific fact, when they are both just hypotheses.

As an editor pointed out, the FLRW metric is indeed just a mathematical object but, there are alternate mathematical expressions that do not require the same assumptions and do not lead to the same derived set of equations as a model. So, a caveat or two would be in order. Just because it is itself mathematically internally logical, does not mean that the math must actually make sense, particularly with regard to the required assumptions.

See my full text proposed contribution and help edit it at:

Kentgen1 (talk) 22:16, 29 October 2010 (UTC)

The sinc function
To : You revised several parts of the article and in doing so put in a reference to the Sinc function with which I have only a slight familiarity. Could you please explain how you get that
 * $$r \; \mathrm{sinc} \, (r \sqrt{k}) = \sqrt{|k|}^{-1} \sinh (r \sqrt{|k|}) \,$$

when k < 0. Thank you. JRSpriggs (talk) 05:48, 29 November 2008 (UTC)
 * FWIW this seems to me to be a non-standard extension of the meaning of "sinc". I like the rest of the changes made by BenRG but I think we should remove reference to sinc. PaddyLeahy (talk) 16:46, 1 December 2008 (UTC)
 * It's completely standard mathematically (sinc is defined over the whole complex plane), but I'll admit I've never seen anyone write the FLRW metric using sinc. Is the rewrite okay? -- BenRG (talk) 20:14, 1 December 2008 (UTC)
 * Here's one way to derive it: $$\frac{\sinh (r \sqrt{|k|})}{\sqrt{|k|}} = \frac{e^{r \sqrt{|k|}} - e^{-r \sqrt{|k|}}}{2 \sqrt{|k|}} = r \frac{e^{i(ir \sqrt{|k|})} - e^{-i(ir \sqrt{|k|})}}{2i(ir \sqrt{|k|})} = r \frac{\sin (ir \sqrt{|k|})}{ir \sqrt{|k|}} = r \; \mathrm{sinc} \, (ir \sqrt{|k|}) = r \; \mathrm{sinc} \, (r \sqrt{-|k|})$$. -- BenRG (talk) 20:14, 1 December 2008 (UTC)
 * To BenRG: Thank you for your explanation and for improving the article. JRSpriggs (talk) 07:53, 2 December 2008 (UTC)
 * Ditto. I do prefer it this way even though you are right about sinc...writing r sinc(r) for sin(r) is not very transparent for those of us of little mathematical brain and the extension to implicitly imaginary arguments was just plain confusing, as my comment above showed! PaddyLeahy (talk) 20:22, 2 December 2008 (UTC)

Note: homogeneity/isotropy applies to space, not spacetime
I have just replaced the false statement "the FLRW metric is the only one on a Lorentzian manifold that is both homogeneous and isotropic" with "the FLRW metric is the only one on a spacetime that is spatially homogeneous and isotropic." Metrics on homogenous/isotropic (maximally symmetric) 4D Lorentzian manifolds are the metrics of de Sitter space, anti-de Sitter space, and Minkowski space. In the FLRW model, only space is maximally symmetric, not spacetime. —Preceding unsigned comment added by 128.36.149.21 (talk • contribs) 19:57, 1 December 2008 (UTC)

INCOMPLETE EXPLANATIONS: The equations need work
The equations use undefined terms: sigma, omega, etc. These require explanation. If we're going to assume that all of the readers of the article are already familiar with the topic then there's no point in having the article in the first place. It also needs work from someone who can provide a physical description. I don't know the topic well enough to fix this page but I know enough to know that this page is seriously incomplete. Michael McGinnis (talk) 17:58, 9 September 2010 (UTC)


 * They are defined. Just read the article straight through. JRSpriggs (talk) 09:15, 10 September 2010 (UTC)

Incompatibility of FRW metric with general relativity
It is an awkward fact, conveniently overlooked, that the FRW is incompatible with GR. General relativity is based on a unified space-time continuum where all determinations of space and time variables (together) are permissible. The FRW separates time and space variables with time as essentially unique and giving a cosmic clock ticking away from time zero at the big-bang. This is related to the last-but-one comment.JFB80 (talk) 16:19, 8 February 2011 (UTC)


 * The point about FLRW metric separating time and space, giving co-moving (privileged) observers with a standard clock, sorely needs to be made in the article. That it is lacking is a definite shortcoming.


 * But FLRW is only incompatible with your philosophical interpretation/extension of GR. FLRW is simply a specification, a further constraint, of the GR equations—rather like the atoms in your body obey all the laws of physics PLUS the laws of biology, and comparable to specifying boundary conditions. Or perhaps you can cite a standard GR textbook that says it's incompatible? JKeck (talk) 01:33, 9 July 2011 (UTC)
 * No I cannot cite a standard GR textbook which says that GR and RW are incompatible and that exactly illustrates my point - they all avoid this awkwardness. But can you cite any textbook which discusses a standard clock ( = absolute time) within GR for comoving observers? If you can why did you not give a reference or describe it yourself? That would be interesting (and new). My point was made in a lecture some years ago by Prof Keswani of Delhi University who said it had been raised by Robertson 1933. Keswani quoted V.Illingworth (ed.): MacMillan Dictionary of Astronomy 1985 pp.101, 304 which put it There is a widespread belief that RW metric of an expanding  universe is based on relativity and the concept of a unified space-time continuum.  So you cannot say it is just my philosophical interpretation of GR.JFB80 (talk) 19:04, 29 March 2012 (UTC)


 * GR formulates the theory of gravity in a way that works for all coordinate systems and is thus independent of any particular choice of coordinate system. However, for us to solve the equations of GR, it is necessary to specify a particular coordinate system. Thus we must use a coordinate condition. In this case, a version of synchronous coordinates is used in which the spatial part of the metric is the product of a scalar function of time and tensor function of space. JRSpriggs (talk) 04:11, 30 March 2012 (UTC)
 * Definition of time by synchronous coordinates appears to be only local and not unique. How can this be identified with the absolute time of cosmology?JFB80 (talk) 19:43, 30 March 2012 (UTC)


 * I do not understand why you think that this definition of time is ambiguous. Imagine a particle of dust emerging from the Big Bang. It is affected only by the symmetrical gravitational force. A clock carried on that dust particle (and set to zero at the Big Bang for uniqueness) would measure this time coordinate, that is, proper time along a geodesic (free-falling trajectory). What is your problem with that? JRSpriggs (talk) 06:51, 31 March 2012 (UTC)
 * Regarding ambiguity I was just quoting from your link to 'synchronous coordinates'. Your idea may be correct in principle but needs to be clearly thought out and incorporated into the article as requested by contributor JKeck above. The connexion with RW metric has to be explained. (You need an initial metric form to be talking about geodesics and proper time - how do you go from that to RW?)JFB80 (talk) 05:06, 1 April 2012 (UTC)

I am sorry, but nothing you are saying makes any sense. So I cannot respond to it. JRSpriggs (talk) 12:02, 1 April 2012 (UTC)

Error in Field Eqns?
I think there's an error in the second equation in the "Solutions" section of the article. I was working this metric (k, lambda=0) with Mathematica and it tells me the second equation should be $$-2a \ddot a -\dot a^2=8\pi G_N p$$ which differs by an overall factor of $a^2$ on the left-hand side. — Preceding unsigned comment added by 76.20.12.36 (talk) 08:43, 9 June 2011 (UTC)


 * Could you explain that in plain mathematics and English without using the crutch of Mathematica? JRSpriggs (talk) 10:04, 9 June 2011 (UTC)


 * Apparently not! 67.162.165.126 (talk) 05:10, 20 May 2013 (UTC)

need to clarify
In the first equation, someone needs to explain the difference between tau and t. Merely seeing them ion opposite sides of an equation is enough in theory, but meaningless to someone struggling to understand this metric. 67.162.165.126 (talk) 05:22, 20 May 2013 (UTC)


 * I will explain it here. You can put it into the article in such fashion as you think will make it clear to naive readers.
 * Imagine a particle moving along a trajectory in FLRW spacetime. Consider two events along that trajectory which have an infinitesimal separation. In terms of the coordinate system being used, the separation can be described as infinitesimal changes in the four coordinates of the events: dt, dr, d&theta;, and d&phi;. The metric can be understood as a quadratic formula which allows one to calculate the change in the proper time (time measured by a clock carried by the particle), d&tau;, from those changes in coordinates. The coefficients (on the right side) in the formula (which are functions of the coordinates themselves: t, r, &theta;, and &phi;) are the components of the metric tensor. This is standard for metrics generally, not specific to FLRW.
 * Is that clear? JRSpriggs (talk) 08:09, 20 May 2013 (UTC)

Regarding the first equation, the problem is really notation. Instead of "tau", the correct symbol would be "s", the symbol for distance in 4-space. The symbol "tau" should be reserved for proper time, which is the time measured on a comoving clock. Ironically, and to compound the confusion even further, the symbol "t" in the FRW metric actually stands for proper time tau. Rigorously, the equation should be written ds**2 = d(tau)**2 - a(tau)**2 dr**2. — Preceding unsigned comment added by Kate rosser (talk • contribs) 21:30, 21 May 2013 (UTC)

Equations of the solution
In the section entitled "Solutions", p is not defined.

— Preceding unsigned comment added by 190.46.106.200 (talk) 20:10, 26 May 2014 (UTC)


 * p is the pressure, a part of the Stress–energy tensor, as one could discover by following any of several links in the article. JRSpriggs (talk) 02:12, 27 May 2014 (UTC)

Tau?
One

Someone needs to define tau in the first equation (on the left side of the equals sign). How is it different than t (on the right side)?

It must be a length of time, since it is multiplied by c to get spatial units. I suspect that tau is the age of the universe.

Two

I suppose the left side is the familiar 4D interval with the spatial distance adjusted by the scale factor, but why is it negated?

Three

Also, why isn't ct (on the right side, the length of time expressed as space) multiplied by the scale factor too, like space is? It's just another direction in spacetime like x, y, and z.

Thank you.

Verdana ♥ Bøld 16:21, 3 January 2015 (UTC)


 * Tau refers to the proper time along a particle's trajectory. "t" refers to the (maximum [added later]) hypothetical proper time along a free-falling trajectory beginning at the Big Bang and ending with the event in question, i.e. the age of the universe at that event. Multiplying the time term by a2 would just introduce unnecessary complications. JRSpriggs (talk) 18:52, 4 January 2015 (UTC)
 * Proper time is an idea coming from Minkowski space-time in SR. Here we are presumably talking about GR. How is it defined from the GR metric? Will it not depend on position so that it will not be the same for all freely falling particles? JFB80 (talk) 14:08, 9 February 2015 (UTC)


 * To JFB80: Proper time is not dependent on the metric (or SR or GR). Proper time is the time measured by a reliable clock as it travels along a given trajectory . Rather, the equation
 * $$- c^2 \mathrm{d}\tau^2 = - c^2 \mathrm{d}t^2 + {a(t)}^2 \mathrm{d}\mathbf{\Sigma}^2$$
 * can be thought of as defining the metric of spacetime in terms of proper time. In defining t here I am presuming that the symmetry of spacetime will ensure that there is only one free-falling trajectory connecting the Big Bang and another given event. Obviously that would not be the case if the symmetry were imperfect as it is in the real world, but FLRW is merely an approximation. JRSpriggs (talk) 17:21, 9 February 2015 (UTC)
 * By saying "connecting the Big Bang and another given event" you assume the Big Bang originated from a point like an explosion. But this is not so as then the cosmological principle fails. The Big Bang has to be everwhere at once. What you assume is essentially the same as Weyl's principle that the nebulae lie on a pencil of geodesics diverging from a common event in the past. JFB80 (talk) 21:03, 12 February 2015 (UTC)


 * The Big Bang is both everywhere at once and at a single point. When a(0)=0, the entire spatial slice (or instant of time) is reduced to a single event &mdash; everything at that time is at the same place since there is no distance between them. There is no other space to expand into. JRSpriggs (talk) 07:21, 13 February 2015 (UTC)


 * Yes the maths is understandable but not the physical situation which defies comprehension. So immediately after the Big Bang, when it is everywhere, where is our particle (=reliable clock) to be found? JFB80 (talk) 11:35, 16 February 2015 (UTC)


 * To JFB80: I made a mistake. I should have said that "t" refers to the maximum hypothetical proper time along a free-falling trajectory beginning at the Big Bang and ending with the event in question. Forget what I said about symmetry ensuring only one free-falling trajectory; that is not true.
 * The trajectory with the maximum proper time should always stay at the same location as where the event in question occurs. JRSpriggs (talk) 07:41, 17 February 2015 (UTC)
 * I am not understanding the situation clearly but it seems to me you probably need the minimum, not maximum, because if you think of a shower of particles arriving you are interested in knowing when the first arrives - something like a wave-front.JFB80 (talk) 17:11, 17 February 2015 (UTC)

If you remember from special relativity, moving objects experience less time passing (less proper time) than objects at rest. Because the sign of the spatial part of the metric is opposite of the sign of the temporal part, the most direct (straightest) path consumes the most proper time. This is opposite to what you would expect for the length of a spatial line where the most direct path is the shortest. JRSpriggs (talk) 22:07, 17 February 2015 (UTC)


 * You know when the first particle arrives but not when the last arrives because there could be one more. But I cannot really comment because I don't understand the situation well enough.JFB80 (talk) 07:56, 20 February 2015 (UTC)

Standard model
In the intro it says: "This model is sometimes called the Standard Model of modern cosmology", yet the Lambda-CDM model article say it is "...frequently referred to as the standard model of Big Bang cosmology". I imagine that this is just a matter of some small clarification required.Snori (talk)


 * This is not a contradiction, because the Lambda-CDM model is a special case of the Friedmann–Lemaître–Robertson–Walker metric. Everything said here about FLRW also applies to &Lambda;-CDM. The difference is that &Lambda;-CDM gives a more specific relationship between the pressure and the density. So there are versions of FLRW which are not versions of &Lambda;-CDM, but they are non-standard. JRSpriggs (talk) 17:31, 10 July 2015 (UTC)

wrong units

 * $$\mathrm{d}\mathbf{\Sigma}^2 = \frac{\mathrm{d}r^2}{1-k r^2} + r^2 \mathrm{d}\mathbf{\Omega}^2$$

it is not possible to add or subtract 1 and kr²[m²] Ra-raisch (talk) 00:17, 1 January 2016 (UTC)


 * Read the paragraph below the equation. It explains what the units of the various constants and variables are. You will see that they are consistent. JRSpriggs (talk) 14:23, 1 January 2016 (UTC)

Sign of Ricci scalar
made a correction to Friedmann–Lemaître–Robertson–Walker metric by changing the sign from negative to positive. He was right, but the reason he gave was wrong. The negative sign was not due to an erroneous calculation in the source; rather it is due to the fact that we have chosen the (-+++) signature for the metric while the source was using the (+---) signature. Specifically, the change of the sign of the metric does not affect the Christoffel symbols nor the Ricci tensor, but it does change the sign of the Ricci scalar. JRSpriggs (talk) 02:32, 24 December 2016 (UTC)

Possible mistake in section "Curvature" under "Cartesian coordinates"
I'm pretty sure there's a factor of $$1/a$$ missing in the spatial components of the Ricci tensor. What currently reads
 * $$ R_{xx}= R_{yy} = R_{zz} = c^{-2} (a \ddot{a} + 2 \dot{a}^2) $$

should probably be
 * $$ R_{xx}= R_{yy} = R_{zz} = c^{-2} \frac{a \ddot{a} + 2 \dot{a}^2}{a} \ . $$

Otherwise, the Ricci scalar does not come out correctly. Julian.eicher (talk) 21:05, 29 December 2021 (UTC)

How is a universal time defined?
It is stated that the FLRW metric assumes homogeneity and isotropy of space but is time dependent. The time is afterwards identified as the universal cosmic time of the Big Bang. But how can this be combined with General Relativity because both the Special and the General Theory of relativity assume time to be special to each observer? This is something to be explained isn't it? JFB80 (talk) 09:13, 9 July 2022 (UTC) I see that I previously raised this point in 2012 and there was a lot of discussion but in the end no explanation was actually made in the article. It still needs one. It is not enough to say that all the books say so. JFB80 (talk) 09:31, 9 July 2022 (UTC)


 * It is neither necessary nor possible to correct every misconception someone might have about relativity in this article. The problem is not in this article, nor in relativity, the problem is in your head. You are assuming that relativity says something which it does not say. When you say "time is special to each observer", you are using a vague expression to conflate different things.
 * A clock which I carry with me does not generally measure the same time interval between two events which a clock carried by a different observer would measure. However, if I take the locations of the events and the motion of the other observer into consideration, I should be able to calculate what interval he would measure.
 * In FLRW, the time coordinate is the time measured by a special observer -- one who has been free-falling since the Big Bang (when he set his clock to zero) and is currently located at the place where the event in question is occurring. It should not be necessary to say this in words because it is implied by the metric equation. JRSpriggs (talk) 00:48, 10 July 2022 (UTC)
 * Thank you for your explanation. But I don't think you can say "all the books say so" because I didn't read this anywhere in a book. Do you know a book that says it? It is only accepted in Wikipedia if in a book. It sounds a bit circular to me as the FLRW metric is defined based on the existence of the Big Bang and is then used to produce cosmological models showing the existence of the Big Bang.JFB80 (talk) 05:27, 10 July 2022 (UTC)


 * We may need a reliable secondary source to put a fact into Wikipedia, but we do not need any source to leave anything out of Wikipedia. I never said that "all the books say so". I gave you my interpretation of what the metric equation implies because you seemed to be unable to figure it out for yourself.
 * The mere existence of the FLRW theory is not being used as evidence for the Big Bang. But evidence which tends to confirm FLRW could reasonably be construed as evidence for the Big Bang. Just as evidence that the planets' orbits have the minor deviations from Kepler's laws predicted by Newton can be used as evidence for Newton's law of gravity. JRSpriggs (talk) 17:28, 10 July 2022 (UTC)
 * Some time you might like to check the 'Weyl postulate' which Hermann Weyl thought necessary to justify the existence of a universal time. It is not generally mentioned but is in J Narlikar The Structure of the Universe"
 * Oxford U P 1977 p.119 (Old but very readable) JFB80 (talk) 10:16, 16 July 2022 (UTC)
 * The original reference is H. Weyl Phys Zeit 1923 JFB80 (talk) 08:32, 29 July 2022 (UTC)