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Talk:Friedmann equations

Inconsistent use of small k
In F.E., as given in the article now (2019 Aug 29), the k is the inverse of the comoving radius of curvature squared (positive or negative), i.e. +-1/R^2 and has dimensions of L^-2. However, in paragraph below k is used as a SIGN of k. It should either be a new variable (usually designated kappa), or it should say k>0, k<0. ... — Preceding unsigned comment added by 2600:1702:1BD0:8CD0:855:E16C:9983:EF9F (talk) 07:41, 29 August 2019 (UTC)

Context
The author of this article says that the Friedmann equations relate certain cosmological parameters in the context of general relavity. I would like to modify this statement to say that these equations define certain cosmological models in general relativity, usually called the Friedmann dusts (or matter dominated Friedmann models) and Friedmann radiation fluids (or radiation dominated Friedmann models). The equations themselves arise in the course of deriving these models in a comoving coordinate chart.

 * Isn't "relating practical real parameters within a theoretical context" the same as a "model"? All anyone needs to do is be careful when referring to the "FLRW metric", as derived from GR assuming the Cosmological Principle, or when referring to the "FLRW model", when certain additional assumptions are made in order to arrive at practical solutions within the metric.

Kentgen1 (talk) 13:10, 3 November 2010 (UTC)

Omega
I just hunted for 20 minutes and didn't find an article specifically on Omega, the density parameter. I put in a blurb about it on the Omega page, and created two new redirect articles, density of the universe and density parameter. I feel Omega deserves its own page, but I can't do it. --zandperl 13:42, 21 March 2006 (UTC)

 * Or it could go under either Omega (cosmology) or Omega (astronomy). --zandperl 13:47, 21 March 2006 (UTC)

 * I agree that we need an Omega (cosmology) article, and density parameter should redirect to it instead of this page. I also can't do it now, but maybe at the end of this semester. --Keflavich 01:45, 16 April 2006 (UTC)

 * Shouldn't "Density of the Universe" have its own page with discussion of:

-Variation in density.

-large scale variation and structure ~ link to page.

-Talk about steady state universe's that call for creation of matter to maintain constant density of universe.

Development
There should be some more development of where the equations come from. I may add some straightforward derivation. Alejandr013 21:02, 4 August 2006 (UTC)

It would be nice to include the actual value of the cosmic density of matter here. I've been searching the web for days trying to find a value, it seems to be one of the values that everybody else knows but never bothers to write down. AJH —Preceding unsigned comment added by 210.98.27.91 (talk) 10:21, 11 April 2008 (UTC)

speed of light
I may be braindead, but I can't for the life of me see how ρ and $$p/c^2$$ have the same units. –Joke 02:27, 14 August 2006 (UTC)

ρ is density, Kilograms over Meters cubed ρ=:$$\frac{Kg}{m^3}$$

$$p/c^2$$ is Pressure over velocity squared, or Force over area over velocity squared so
 * $$\frac{p}{c^2} = \frac{F}{Ac^2}=\frac{F/A}{(m/s)^2}=\frac{\frac{Kg*(m/s^2)}{m^2}}{(m/s)^2}=\frac{kg}{m^3}$$

Differentiating ρ, P, and p in physics can be difficult (did you catch the pun?). I thought it was standard to put pressure P capitalized and p little as momentum, and rho as density. What are the conventions on Wikipedia? Alejandr013 20:49, 14 August 2006 (UTC)

 * I think you left out a $$c^2$$ factor at the beginning -- I've taken the liberty of correcting it. Yes, it all comes down to the naming conventions for GR.  Is $$\rho$$ the mass density or the energy density?  In non-GR physics $$\rho$$ is normally mass density, but since c=1 is normally set by Geometrized unit system it doesn't matter most of the time.  Of course a general audience will not necessarily know of these unit conventions -- and according to WP policy articles should written with the general public in mind as well as the experts.

 * There is an unoffical set of naming conventions in GR: User:Hillman/WikiProject GTR/Policies. --Michael C. Price talk 01:25, 15 August 2006 (UTC)

The $$\rho$$ which appears in the equations is mass-density, as noted above, but later in the article there is talk of energy density and vacuum energy. This is confusing, at least it was to me; the first time I read it I assumed $$\rho$$ was indeed energy density, but then noticed the dimensions of the equation wouldn't balance. I think it should be clarified by specifically saying 'mass density' when ρ is introduced. Or, perhaps by changing to $$\rho$$ being energy density which seems to be used on other pages (e.g. cosmological constant). Then the term becomes
 * $$\frac{4\pi G}{c^2}\left(\rho_e + 3p\right)$$

The subscript e for energy density would be nice, but I guess this is no place to introduce new notation? E4mmacro 04:17, 9 April 2007 (UTC)

A bit late in way of reply, but to clarify, $$\rho c^2$$ is the energy density; $$\rho $$ is the energy density divided by $$c^2$$; it is not the `mass density'. In `usual' units of $$c=1$$, $$\rho $$ is referred to as the energy density. See Coles and Lucchin, The Origin and Evolution of Cosmic Structure, for example. 129.78.56.231 (talk) 07:10, 29 October 2018 (UTC)

mistake
I think, there is a mistake in the equations from the beginning. Instead of a2 in the last term on the r.h.s. there should be R2, where R = R(t) = a(t).R0. The dimension of the first equation is then s-2- same for all terms on the left and right hand side of the equation...

Whoplaysdice 09:56, 29 May 2007 (UTC)

 * If you are speaking of this equation:
 * $$H^2 \equiv \left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G \rho + \Lambda}{3} - K\frac{c^2}{a^2}$$,
 * then the right-most term already has units of reciprocal seconds squared. The Gaussian curvature when a=1, K has units of reciprocal meters squared. c has units of meters per second. a has units of one (i.e. arbitrary units or unit-less). So it is: m-2·m2s-21 = s-2. Is that OK? JRSpriggs 10:26, 29 May 2007 (UTC)

Thanks for the explanation very much. I haven't noticed that K has units of reciprocal meters squared... I'm sorry. Whoplaysdice 10:32, 29 May 2007 (UTC)

 * As a matter of fact, if the curvature is constant within a polygon, the area of the polygon times that curvature is just the "spherical" excess (in radians) of the sum of the interior angles of the polygon relative to the value in Euclidean space. JRSpriggs 04:38, 30 May 2007 (UTC)

 * In the article and everywhere else, k ist dimensionless -1, 0 or 1. Using here K=1/R²=(H/c)² instead of k makes sense. Who dares to correct it? There are so many places with k in the article. Ra-raisch (talk) 18:15, 18 June 2019 (UTC)

Expansion lowers temperature
added a new section Friedmann equations. The equation of state for the perfect fluid appeared to be the ideal gas law. But then I realized that the "constant" w is proportional to the absolute temperature. However, the temperature of a gas decreases when it expands, see Adiabatic cooling. So the equation should probably be corrected to account for the decrease in temperature. JRSpriggs (talk) 14:10, 23 April 2008 (UTC)

 * Uh, no, w is not proportional to T. For simple models ("dust", "radiation", etc) w is constant and luckily these models are quite good approximations to the real universe for quite long periods of its history. Only when w is independent of T (or equivalently of a) do we get nice closed-form solutions. Of course the real universe is a mixture of dust, radiation, dark energy etc but usually one dominates. Only when there is a cross-over between cases does the effective value of w change significantly, which happens relatively quickly (especially on a log-log plot!). PaddyLeahy (talk) 12:12, 25 April 2008 (UTC)

 * To Paddy: Thanks for clarifying that. JRSpriggs (talk) 13:14, 25 April 2008 (UTC)

Simpler 2nd equation
The 2nd equation:
 * $$\dot{H} + H^2 \equiv \frac{\ddot{a}}{a} = -\frac{4 \pi G}{3}\left(\rho+\frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3},$$

could be replaced by:
 * $$\dot{\rho} = -3 H (\rho + \frac{p}{c^2}),$$

which eliminates $$\Lambda ,$$

The original 2nd equation can be recovered, when combined with the 1st equation. Any objections to replacing the 2nd equation with this? --Michael C. Price talk 10:13, 26 April 2008 (UTC)

 * That is equivalent, so we might want to mention it in the article. However, the original second equation has historical significance; and it is named the "Friedman's acceleration equation". So I think we must retain it. Remember that the equations given emerge more or less directly from Einstein field equations. JRSpriggs (talk) 11:46, 26 April 2008 (UTC)

Mistake
There appears to be a minor error near the beginning of the article where it states that k is the normalized curvature parameter at a=1. It is not defined at a=1 at all; k is the ratio of the modulus squared of a, divided by a squared. That is how it appears in my lecture notes from Imperial College. If my notes are wrong then someone should correct me before next Wednesday when I have a 'Particle Cosmology' exam there! In the meantime, I'll simply remove the "at a=1" bit of the sentence. Dazza79 (talk) 21:06, 23 May 2008 (UTC)

 * I reverted your edit because the text was correct as it stood. Obviously, I cannot read your notes, so I cannot comment on them. But you should bear in mind that different authors use different notations, different definitions for a, k, etc.. It is imperative that the notation in the article be consistent with itself, not that it be consistent with your notes. JRSpriggs (talk) 00:04, 24 May 2008 (UTC)

 * I agree with Dazza79; k = +/-1 or 0 is how k is normalised by rescaling a, according to whether the universe is closed, open or flat. --Michael C. Price talk 05:45, 24 May 2008 (UTC)

 * To MichaelCPrice: One can only choose k to be one of those three specific values by sacrificing the connection of a to the present which also means that R0 would not be the present day radius of curvature. This is not helpful, in my opinion. Adjusting the variables so that the present has a special status is appropriate because, for us who live in the present, it is special. JRSpriggs (talk) 21:30, 24 May 2008 (UTC)

I don't see how renormalising 'k' removes the connection of 'a' to the present? What exactly do you mean by this phrase? The FRW metric for example contains both a 'k' and an 'a(t)'. So surely I am free to choose any value of a, at any time t, including the present, irrespective of my choice of +/-1 or 0 for k. If I choose a=1 to be the present, and k happens to be -1 say, then at a=2 k will still be -1. So how does my renormalised choice of -1 for the value of k affect the value of a? It should be totally independent. Or put another way, If I live in an open universe today I will still be living in an open universe tomorrow. 'a' can be as small or large as you like and the rescaled value of k remains the same. Yet the way I'm reading the article I get the impression that, with the wording as it stands, 'k' has the value it has now only at the present time. So if it is -1 today, it might be 0 at some point in the future? Dazza79 (talk) 22:54, 25 May 2008 (UTC)

 * To Dazza79: Let the cosmological constant and the equation of state for the contents of the universe both be fixed; and assume that the universe is homogeneous and isotropic, i.e. it has a Friedmann-Lemaître-Robertson-Walker metric. Then the possible instantaneous states (time slices) of the universe form a two-parameter class. The first parameter amounts to choosing a time for the time-slice. You may use any of $$t, a, H\,$$ (among others) for it. The second parameter describes the spacetime continuum as a whole. When the spatial curvature is positive, the second parameter amounts to choosing the minimal spatial curvature, i.e. the spatial curvature when $$H = 0\,$$. Since the second parameter varies continuously, it cannot be represented correctly by k if k is restricted to have values in the discrete set {-1, 0, +1}.
 * You asked what can stop you from choosing a and k to be anything you want? Well, what if I want k to be a real number other than -1, 0, or +1; perhaps +4.37? JRSpriggs (talk) 05:14, 27 May 2008 (UTC)

 * OK. Then I think the word 'normalised' in the article is out of place. If 'k' can vary continuously, then it is not the normalised spatial curvature is it? Seeing the word normalised made me assume that we were taking the often used convention of putting k = +/-1 or 0. Let's simply call it the spatial curvature. I will edit the article accordingly. If you disagree, then my next question is how would you distinguish between the normalised spatial curvature as described above, and the k=+/-1, 0 spatial curvature, which I would certainly describe as normalised? Dazza79 (talk) 07:03, 27 May 2008 (UTC)

 * Looks good to me: now reads "k is the spatial curvature when a = 1 (i.e. today)". As currently written, k is clearly a dimensional quantity (inverse length squared), so it is only "normalised" in the sense that it applies now, which is already stated. Therefore 'normalised' was at least redundant if not wrong, and certainly misleading. PaddyLeahy (talk) 17:28, 27 May 2008 (UTC)
 * OK. I agree with dropping "normalised". JRSpriggs (talk) 05:38, 28 May 2008 (UTC)
 * Hawking & Ellis' Large Scale Structure of Space-Time, have k = +1, 0 or -1 and called this value "normalized".--Michael C. Price talk 07:16, 29 May 2008 (UTC)

Surely we can remove all references to R and R0 by simply stating that a is the hyper-radius of the model universe? To explain the origin of the k term in the equation we need simply note that for a static hypersphere the Riemann curvature term is $$6 k \frac{c^2}{a^2}$$, where k = +/-1 or 0 etc. This makes it very easy to see why k = 0 in the flat example, since a -> infinity when flat, and k = -1 is simply a -> ia for the hyperbolic case. --Michael C. Price talk 07:49, 28 May 2008 (UTC)

 * Why would we want to remove such references? IMHO the article would be incomplete without making clear what these terms mean, since very frequently (as in Dazza79's lecture notes) the Friedman equation is written in terms of R rather than a. PaddyLeahy (talk) 14:17, 28 May 2008 (UTC)
 * I personally find it clearer to just talk in terms of a and not R, but each according to their taste. There's no reason why we can't have both explanations.  BTW I think the current explanation of k is totally confusing (and non-standard -- see note below about MWT and HE).  The current explanation states that k is the spatial curvature today, leaving unanswered what it was yesterday and will be tomorrow.  (You may think that the next two sentences clears everything up, but I disagree.) Anyway, I leave that clarification to others.
 * Any objections to a statement that a can also be interpreted as the hyper-radius, in which case k is restricted to +/-1 or 0? --Michael C. Price talk 17:35, 28 May 2008 (UTC)

The standard references, Misner, Wheeler & Thorne's Gravitation and Hawking & Ellis' Large Scale Structure of Space-Time, have k = +1, 0 or -1; HE even called its value "normalized".--Michael C. Price talk 07:14, 29 May 2008 (UTC)

How nice of JRSpriggs to revert my changes, without explanation here. Charming as ever. I trust an explanation of quite how my changes were "mistaken" will be appearing at some point.--Michael C. Price talk 19:16, 31 May 2008 (UTC)
 * My GR notes by Ray d'Inverno include the comment in the Robertson-Walker metric section 6, subsection 2, "k=+1 ..... The space is closed .... This is why in this case R(t) is often referred to as the 'radius of the universe'." before going on to give the Friedman equations in terms of same k & R. --Michael C. Price talk 07:59, 1 June 2008 (UTC)

MichaelCPrice's mistakes
The scalar curvature, R, is a scalar and as such it has only one component which cannot reasonably be called "spatial". Even if the definitions which follow this are inter-convertible, they are not equivalent because they yield different values for k and a. JRSpriggs (talk) 13:38, 1 June 2008 (UTC) --Michael C. Price talk 15:59, 1 June 2008 (UTC)
 * "$$k \over a^2$$ is one-sixth of the spatial component of the scalar curvature"
 * "$$a$$ and $$k$$ have two possible equivalent definitions"
 * It is not reasonable to call a the radius of curvature of the universe when k=0 because in that case the radius of curvature is either infinite or, more correctly, undefined.
 * Well of course they yield different values for k and a, as explained in the text, but $$k \over a^2$$ is invariant and is what appears in the equation. Your point is what precisely?
 * For an isotropic homogeneous universe, modeled as a 3-sphere with radius a, we have :$$R = \frac{6}{a^2}(\ddot{a} a + \dot{a}^2 + kc^2)$$. k = 1 for sphere, -1 for a hyperboloid. The spatial component refers to the last term. Perhaps you can suggest a better description?
 * It is well understood, even by beginners, that Euclidean geometry corresponds to a surface of a sphere of infinite radius.

Ambiguous Spelling: Friedmann or Friedman
Throughout the article, his name is spelled two different ways. I think it would be better if we settled on a single spelling; ideally, the same way he did. Apparently, he spelled it with a single 'n', "Friedman". I've hunted around on the web, a bit, and found what I think could be a good reference. It appears to be an image of the title page of one of his papers: http://www.springerlink.com/content/l23864w241673530/fulltext.pdf?page=1 Michael McGinnis (talk) 19:25, 15 March 2009 (UTC)

 * Of his two famous papers, the earlier one (which you linked) spells it Friedman, and the later one spells it Friedmann. The "Friedmann" spelling is overwhelmingly more common in modern physics papers, and it's also the spelling used in every online encyclopedia I checked except Wikipedia, so I think we should stick with "Friedmann", and I think the article on the man himself should be moved as well. -- BenRG (talk) 13:13, 16 March 2009 (UTC)

 * Actually, his name is "Алекса́ндр Алекса́ндрович Фри́дман" in Russian. The final three characters "ман" transliterate as "man". However, we are supposed to use the name by which most English speaking readers would know him, according to Naming conventions. JRSpriggs (talk) 01:39, 17 March 2009 (UTC)

 * Thank you, BenRG and JRSpriggs, for clearing this up for me. I do feel that a consistent spelling is needed, if only to reduce the possibility of confusion for people who might think that "Friedman" and "Friedmann" are two different people. I'm going to change all references to "Friedmann".Michael McGinnis (talk) 19:58, 19 March 2009 (UTC)

Mistake on Copernican Principle
Hi all!

I suspect the statement "There are serious consequences if homogeneity and isotropy (the Copernican Principle) are not quite true, [...]" is incorrect. The Copernican Principle states that the Earth does not occupy any particular position in the Universe. What the author refers to is the Cosmological Principle, which states that the Universe is isotropic about all locations and homogeneous. For a reference, please see Peacock's "Cosmological Physics" on page 66:

"[...] most scientists believe that it is not reasonable to adopt a cosmological model in which the universe is simply a joke played for the benefit of mankind. This attitude is encapsulated in the Copernican principle, which states that humans are not privileged observers."

This definition is much more compatible with what Copernicus did actually discover, namely that the Sun and not the Earth was at the center of the Solar System.

There is a certain confusion about this topic in the scientific community. Me too I was confused about the various definitions, until I read pages 65-66 of Peacock's book. They are an eye-opener.

Cheers,

Guido —Preceding unsigned comment added by Coccoinomane (talk • contribs) 08:26, 24 May 2009 (UTC)

 * I implemented the above proposed change. I also added the following clarifying line:
 * "Notice, however, that the mainstream of Cosmological research endorses a homogeneous and isotropic Universe on scales larger than ~100 Mpc." I think it is correct to inform the reader of this.

 * Cheers,

 * Guido
 * Coccoinomane (talk) 08:13, 3 June 2009 (UTC)

Kentgen1's additions
To : See Help:Section. Heading for sections, subsections, subsubsections, etc. should be created using equals signs on either side of the title. For example, " ==Your title here== " generates a section title "Your title here" when left-justified on a line. Thus your use of bold-face text to create a title (e.g. "Caveats to the Use of the Friedmann Equations to Model the Universe") is inappropriate. Also, it is preferred to use tripled single quotation marks to delimit bold-face rather than  and .

We do not assume that the matter in the universe is an ideal gas. I do not know how you reached the conclusion that we so assume.

If the Earth were at the center of a large spherical void within a denser surrounding, that would not cause the matter near the edge of the void to accelerate away from us. Thus this is not an alternative explanation for the acceleration due to the cosmological constant.

The deviation of gravitational potential field from its background value is proportional to 1/r. The resulting gravitational force field (gradient of the potential) is proportional to 1/r2. This holds for ordinary celestial bodies as well as super-massive black-holes (outside the near field effect); there is no distinction between types of bodies in this respect. I do not see how you get that this would cause a violation of the cosmological principle which has to do with average behavior. That is, the homogeneity and isotropy are understood to be approximations which hold best at large scales.

Yes, the universe is adiabatic because there is no external system to exchange heat with or do work upon. JRSpriggs (talk) 07:16, 21 June 2010 (UTC)

 * Third opinion: Edits like this are unacceptable because of their disruptive nature. There is simply no reason to put Wikipedia links inline like that. But even beyond that, there's a lack of reliable sourcing overall. Consider the text:
 * "The equations also rely on the simplification that the substance of the universe may be treated like an ideal fluid or gas. (http://en.wikipedia.org/wiki/FLRW_metric, http://en.wikipedia.org/wiki/Adiabatic_process , http://en.wikipedia.org/wiki/Ideal_gas_law )"
 * Wiki articles are not considered reliable, so how is the conclusion that the equations rely on that "simplification" drawn? If we could find some actual sources to back that up then at least the issue could be discussed, but until then it seems to be Kentgen's original research on the topic. —  Hello Annyong  (say whaaat?!) 19:14, 1 July 2010 (UTC)
 * Fourth opinion : the ideal gas law is not assumed. If it was then the pressure would not appear as a free variable to be solved for. Kentgen's original research is incorrect. --Michael C. Price talk 22:57, 24 July 2010 (UTC)

Derivation of simplified Friedmann equations
Since issues have repeatedly arisen concerning the origin of the equations and the correct values of the constants in them, I think it would be helpful to derive them here. For simplicity, I will focus on the special case when k2≈0 and Λ=0. Any components of tensors or pseudo-tensors omitted below are zero. Assume that
 * $$ ds^2 = - c^2 dt^2 + {a(t)}^2 ( dx^2 + dy^2 + dz^2 + dw^2 ) \,$$

with
 * $$ w = \sqrt{ k^{-1} - ( x^2 + y^2 + z^2 ) } \,.$$

Then
 * $$ dw = \frac{ -2 x dx -2 y dy -2 z dz }{ 2 w } \,.$$
 * $$ dw^2 = \frac{ ( x dx + y dy + z dz )^2 }{ w^2 } = \frac{ k ( x dx + y dy + z dz )^2 }{ 1 - k ( x^2 + y^2 + z^2 ) } \approx k ( x dx + y dy + z dz )^2 \,.$$

So the metric tensor (general relativity) is given by
 * $$ g_{0 0} = - c^2 \, ; \quad g_{i j} \approx a^2 ( \delta_{i j} + k x^i x^j ) \,$$

where it is understood that a is a function of time only. Thus
 * $$ g^{0 0} = - c^{-2} \, ; \quad g^{i j} \approx a^{-2} ( \delta_{i j} - k x^i x^j ) \,.$$

The partial derivatives of the metric are
 * $$ g_{i j, 0} \approx 2 a \dot{a} ( \delta_{i j} + k x^i x^j ) \,$$
 * $$ g_{i j, m} \approx k a^2 ( \delta_{i m} x^j + x^i \delta_{j m} ) \,.$$

The Christoffel symbol is defined by
 * $$ \Gamma^\alpha_{\beta \gamma} = {1 \over 2} g^{\alpha \sigma} ( g_{\sigma \beta, \gamma} + g_{\gamma \sigma, \beta} - g_{\beta \gamma, \sigma} ) \,.$$

So
 * $$ \Gamma^0_{i j} \approx c^{-2} a \dot{a} ( \delta_{i j} + k x^i x^j ) \,.$$
 * $$ \Gamma^i_{0 j} = \Gamma^i_{j 0} \approx a^{-1} \dot{a} \delta_{i j} \,.$$
 * $$ \Gamma^i_{j m} \approx k x^i \delta_{j m} \,.$$
 * $$ \Gamma^\rho_{\rho 0} \approx 3 a^{-1} \dot{a} \,.$$
 * $$ \Gamma^\rho_{\rho i} \approx k x^i \,.$$

The Ricci curvature tensor is defined by
 * $$ R_{\alpha\beta} = \partial_{\rho}{\Gamma^\rho_{\beta\alpha}} - \partial_{\beta}\Gamma^\rho_{\rho\alpha} + \Gamma^\rho_{\rho\lambda} \Gamma^\lambda_{\beta\alpha} - \Gamma^\rho_{\beta\lambda}\Gamma^\lambda_{\rho\alpha} \,.$$

So
 * $$ R_{0 0} \approx 0 - 3 ( a^{-1} \ddot{a} - a^{-2} \dot{a}^2 ) + 0 - 3 a^{-2} \dot{a}^2 = - 3 a^{-1} \ddot{a} \,.$$
 * $$ R_{0 i} = R_{i 0} \approx 0 - 0 + k a^{-1} \dot{a} x^i - k x^i a^{-1} \dot{a} = 0 \,.$$
 * $$ R_{i j} \approx [ c^{-2} ( \dot{a}^2 + a \ddot{a} ) ( \delta_{i j} + k x^i x^j ) + 3 k \delta_{i j} ] - k \delta_{i j} + 3 c^{-2} \dot{a}^2 ( \delta_{i j} + k x^i x^j ) - 2 c^{-2} \dot{a}^2 ( \delta_{i j} + k x^i x^j ) = c^{-2} ( 2 \dot{a}^2 + a \ddot{a} ) ( \delta_{i j} + k x^i x^j ) + 2 k \delta_{i j} \,.$$
 * $$ R = g^{\alpha \beta} R_{\alpha \beta} \approx 3 c^{-2} a^{-1} \ddot{a} + 3 c^{-2} ( 2 a^{-2} \dot{a}^2 + a^{-1} \ddot{a} ) + 6 a^{-2} k = 6 c^{-2} ( a^{-2} \dot{a}^2 + a^{-1} \ddot{a} ) + 6 a^{-2} k \,.$$

The Einstein field equations (without Λ) are
 * $$R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R = {8 \pi G \over c^4} T_{\mu \nu} \,$$

which give
 * 0 0: $$ - 3 a^{-1} \ddot{a} - {1 \over 2} ( - c^2 ) [ 6 c^{-2} ( a^{-2} \dot{a}^2 + a^{-1} \ddot{a} ) + 6 a^{-2} k ] = 3 a^{-2} \dot{a}^2 + 3 c^2 a^{-2} k \approx {8 \pi G \over c^4} \rho c^4 \,.$$
 * 0 i or i 0: 0 ≈ 0.
 * i j: $$ c^{-2} ( 2 \dot{a}^2 + a \ddot{a} ) ( \delta_{i j} + k x^i x^j ) + 2 k \delta_{i j} - {1 \over 2} ( a^2 ( \delta_{i j} + k x^i x^j ) ) [ 6 c^{-2} ( a^{-2} \dot{a}^2 + a^{-1} \ddot{a} ) + 6 a^{-2} k ] = c^{-2} ( - \dot{a}^2 - 2 a \ddot{a} ) ( \delta_{i j} + k x^i x^j ) - k \delta_{i j} \approx {8 \pi G \over c^4} p a^2 ( \delta_{i j} + k x^i x^j ) \,.$$

These simplify to
 * $$ a^{-2} \dot{a}^2 + c^2 a^{-2} k \approx { 8 \pi G \over 3 } \rho \,$$ ( which is the first Friedmann equation )
 * $$ ( - a^{-2} \dot{a}^2 - 2 a^{-1} \ddot{a} ) - c^2 a^{-2} k \approx {8 \pi G \over c^2} p \,.$$

If we add these together, we get
 * $$ - 2 \frac{\ddot{a}}{a} \approx {8 \pi G \over 3} \left( \rho + \frac{3 p}{c^2} \right) \,$$

that is
 * $$ \frac{\ddot{a}}{a} \approx {- 4 \pi G \over 3} \left( \rho + \frac{3 p}{c^2} \right) \,$$

which is the second Friedmann equation. JRSpriggs (talk) 08:42, 22 March 2013 (UTC)

 * thank for writing this new section. but how does dw arise in metric? spherical coordinates transfom to Cartesian coordinates?Earthandmoon (talk) 14:16, 21 March 2013 (UTC)

 * The assumption of Friedmann–Lemaître–Robertson–Walker metric is that our three dimensional space is a hyper-sphere in a four dimensional space (with time added later). w is the extra dimension in that four dimensional space and it is characterized by $$k^{-1} = (\text{radius of curvature of spatial slice when a=1})^2 = x^2 + y^2 + z^2 + w^2 \,.$$ Which coordinate system we use for the three-space does not affect the result, so I just chose what I thought would be the most convenient one. The coordinates I chose are the Cartesian coordinates t,x,y,z associated with the reduced-circumference polar coordinates. JRSpriggs (talk) 14:31, 22 March 2013 (UTC)
 * Greek indices vary over 0,1,2,3. Roman indices vary over 1,2,3. x1,x2,x3 refer to x,y,z respectively.
 * If you doubt any step or do not understand it, please indicate the first such questionable step and I will attempt to explain it or fix it. JRSpriggs (talk) 08:04, 26 March 2013 (UTC)
 * thank you, i got it. Your derivation is nice. :)Earthandmoon (talk) 10:57, 26 March 2013 (UTC)

The trace of the Einstein field equations (which is independent of the choice of the spatial coordinates) is
 * $$ - R = {8 \pi G \over c^4} T \,$$
 * $$ - [ 6 c^{-2} ( a^{-2} \dot{a}^2 + a^{-1} \ddot{a} ) + 6 a^{-2} k ] = {8 \pi G \over c^4} ( - c^2 \rho + 3 p ) \,$$
 * $$ \frac{ \dot{a}^2 }{ a^2 } + \frac{ \ddot{a} }{ a } + \frac { k c^2 }{ a^2 } = { - 4 \pi G \over 3 } \left( - \rho + \frac{ 3 p }{ c^2} \right) \,$$

which can be combined with the first Friedmann equation to get the second. JRSpriggs (talk) 08:06, 28 March 2013 (UTC)

Deriving the continuity equation (conservation of energy)
The time derivative of (00 component)
 * $$ \frac{\dot{a}^2}{a^2} + \frac{kc^2}{a^2} = \frac{8 \pi G}{3} \rho \,$$

is
 * $$ \frac{2\dot{a}\ddot{a}}{a^2} - \frac{2\dot{a}^3}{a^3} - \frac{2kc^2\dot{a}}{a^3} = \frac{8 \pi G}{3} \dot{\rho} \,.$$

Substituting in
 * $$ \frac{\ddot{a}}{a} = -\frac{4 \pi G}{3}\left(\rho+\frac{3p}{c^2}\right) \,$$

gives
 * $$ - H \frac{8 \pi G}{3}\left(\rho+\frac{3p}{c^2}\right) - 2 H \frac{8 \pi G}{3} \rho = \frac{8 \pi G}{3} \dot{\rho} \,.$$

Throwing out the common factor gives
 * $$ - H \left(\rho+\frac{3p}{c^2}\right) - 2 H \rho = \dot{\rho} \,,$$

that is,
 * $$ \dot{\rho} = - 3 H \left(\rho+\frac{p}{c^2}\right) \,$$

which is the continuity equation. This says that the reduction in the mass in a volume is equal to the amount that moves out its sides due to the expansion of the universe plus the mass equivalent of the energy expended by pressure pushing that mass out. JRSpriggs (talk) 08:44, 28 March 2013 (UTC)

As a check, I will derive the continuity equation by another method. In general, it takes the form
 * $$0 = T^{\mu \nu}{}_{;\nu} = T^{\mu \nu}{}_{,\nu} + T^{\sigma \nu} \Gamma^{\mu}{}_{\sigma \nu} + T^{\mu \sigma} \Gamma^{\nu}{}_{\sigma \nu} \,$$

as indicated at stress–energy tensor. In our case, the stress-energy of a fluid at rest is
 * $$ T^{0 0} = \rho \,;\quad T^{i j} = p \, g^{i j} \approx p \, a^{-2} ( \delta_{i j} - k x^i x^j ) \,.$$

Thus the time component is
 * $$0 = T^{0 \nu}{}_{;\nu} = T^{0 \nu}{}_{,\nu} + T^{\sigma \nu} \Gamma^{0}{}_{\sigma \nu} + T^{0 \sigma} \Gamma^{\nu}{}_{\sigma \nu} \,$$
 * $$0 \approx \dot{\rho} + p \, a^{-2} ( \delta_{i j} - k x^i x^j ) \, c^{-2} a \dot{a} ( \delta_{i j} + k x^i x^j ) + \rho \, 3 a^{-1} \dot{a} \,$$
 * $$0 \approx \dot{\rho} + 3 p c^{-2} a^{-1} \dot{a} + 3 \rho a^{-1} \dot{a} \,$$
 * $$ \dot{\rho} \approx - 3 H ( p c^{-2} + \rho ) \,$$

which agrees with the continuity equation found above. Let us also examine the i component which is
 * $$0 = T^{i \nu}{}_{;\nu} = T^{i \nu}{}_{,\nu} + T^{\sigma \nu} \Gamma^{i}{}_{\sigma \nu} + T^{i \sigma} \Gamma^{\nu}{}_{\sigma \nu} \,$$
 * $$0 \approx - 4 k p a^{-2} x^i + 3 k p a^{-2} x^i + k p a^{-2} x^i \,$$
 * $$0 \approx 0 \,.$$

This identity means that conservation of linear momentum is automatic in FLRW. JRSpriggs (talk) 06:32, 2 April 2013 (UTC)

Can the Friedmann equations be solved?
The Friedmann equations and the conserved energy equation are differential equations, can we solve them for a(t) function? How does the Friedmann equations describe the evolution of Universe in time? It describes the expansion of universe, right? but How about initial conditions (t=0)? Could you tell more about Cosmological principle, and why from it we can derive FLRW metric? Thank you for your attention. Earthandmoon (talk) 12:47, 28 March 2013 (UTC)
 * i read the article and found some useful information about a(t) in case k=0. how does a(t) look with k=1 or -1, and Λ Earthandmoon (talk) 12:52, 28 March 2013 (UTC)
 * Given an additional formula for the pressure, a numerical solution of the Friedmann equations is possible. You may specify freely k and the values of a and ρ at an initial time. Then a can be updated with the first Friedmann equation; and ρ with the continuity equation. Pressure p must be provided by the added equation.
 * Exact solutions are possible in some special cases (see Friedmann equations). If we assume k=0 and
 * $$ p = w \rho c^2 \,$$
 * for some constant w ≠ −1, then
 * $$ \dot{\rho} = - 3 H \left(\rho+\frac{w \rho c^2}{c^2}\right) = - 3 H ( 1 + w ) \rho \,$$
 * $$ \frac{ d \ln \rho }{ d t } = - 3 ( 1 + w ) \frac { d \ln a }{ d t } \,$$
 * $$ \rho = \rho_0 \left( \frac{ a }{ a_0 } \right)^{- 3 ( 1 + w )} \,.$$
 * So the first Friedmann equation becomes
 * $$ \frac{\dot{a}^2}{a^2} = \frac{8 \pi G}{3} \rho_0 \left( \frac{ a }{ a_0 } \right)^{- 3 ( 1 + w )} \,$$
 * $$ \left( a^{\tfrac32 ( 1 + w ) - 1} \dot{a} \right)^2 = \frac{8 \pi G \rho_0}{3} \, a_0^{3 ( 1 + w )} \,$$
 * $$ a^{\tfrac32 ( 1 + w ) - 1} \dot{a} = \sqrt{ \frac{8 \pi G \rho_0}{3} } \,\, a_0^{\tfrac32 ( 1 + w )} .$$
 * Integrating with respect to t gives
 * $$ a^{\tfrac32 ( 1 + w )} = ( t - t_0 ) \, \tfrac32 ( 1 + w ) \sqrt{ \frac{8 \pi G \rho_0}{3} } \,\, a_0^{\tfrac32 ( 1 + w )} + a_0^{\tfrac32 ( 1 + w )} \,.$$
 * $$ a^{\tfrac32 ( 1 + w )} = \left[ ( t - t_0 ) \, ( 1 + w ) \sqrt{ 6 \pi G \rho_0 } + 1 \right] a_0^{\tfrac32 ( 1 + w )} \,.$$
 * $$ a = a_0 \left[ 1 + ( t - t_0 ) \, ( 1 + w ) \sqrt{ 6 \pi G \rho_0 } \, \right]^{ \frac{2}{3( 1 + w )} } .$$
 * $$ \rho = \rho_0 \left[ 1 + ( t - t_0 ) \, ( 1 + w ) \sqrt{ 6 \pi G \rho_0 } \, \right]^{-2} .$$
 * On the other hand, if w = −1 (i.e. p = −ρc2), then
 * $$ \rho = \rho_0 \,.$$
 * $$ a = a_0 \exp{ \left[ ( t - t_0 ) \, \sqrt{ \frac{8 \pi G \rho_0}{3} } \, \right] } .$$
 * JRSpriggs (talk) 10:15, 1 April 2013 (UTC)

 * The exact solutions above assume that the universe is expanding. So that one takes the positive square-root. If the universe were contracting, then just change the sign of the square-root.
 * For an expanding universe, w > −1 implies that the universe begins with a big bang at
 * $$ t_{\text{Big Bang}} = t_0 - \left[ ( 1 + w ) \sqrt{ 6 \pi G \rho_0 } \, \right]^{-1} \,.$$
 * If w < −1, then the universe ends with a big rip at
 * $$ t_{\text{Big Rip}} = t_0 + \left[ ( -1 - w ) \sqrt{ 6 \pi G \rho_0 } \, \right]^{-1} \,.$$
 * If w = −1, then neither occurs. JRSpriggs (talk) 10:56, 1 April 2013 (UTC)
 * What does t mean here? It is our local time, or of an infinite observer, or of the entire universe? Earthandmoon (talk) 12:50, 1 April 2013 (UTC)
 * Because in both FLRW metric and this article I have not found the definition of t, and $$\tau$$. Earthandmoon (talk) 12:59, 1 April 2013 (UTC)
 * The physical meaning of the coordinates is implied by the metric equation. In some articles, such as Schwarzschild metric, explanations are provided, but they are actually redundant because they follow from the metric equation. The proper time, τ, is the time measured by a clock moving along the same world line with the test particle. In FLRW, t is time measured by clocks which have been free-falling since the Big Bang and thus are at rest relative to the average motion of local matter (using the assumption of homogeneity and isotropy). This assumes they were synchronized at the Big Bang and thus remain synchronized locally. JRSpriggs (talk) 05:16, 2 April 2013 (UTC)
 * Thank you for all!Earthandmoon (talk) 15:00, 2 April 2013 (UTC)

The expansion of the universe will turn into a contraction if and when
 * $$ \dot{a} = 0 \,;\quad \ddot{a} < 0 ,$$

that is, if and when
 * $$ \rho \, a^2 = \frac{3 k c^2}{8 \pi G} \,$$

(which requires positive curvature of space) and
 * $$ p > - \tfrac13 \rho c^2 \,.$$

JRSpriggs (talk) 05:16, 2 April 2013 (UTC)

No Vandalism Intended 246 GeV Energy Contained in Higgs Field 3.71295775e-28 kg * c^2 * (pi *376.730313462) = 246.508252 GeV

(((246.508252 GeV) / (c^2)) / (3.71295775e-28 kg)) / pi = 376.730313

((6.6740800122e-11 * (newtons / (meters^2))) / (3.71295775e-28 * 2 * (kg / (meter^3))))^0.5 = 299792458 m / s

((2 * G * (1 kg)) / (c^2)) / 4 = 3.71295774e-28 meters

https://en.wikipedia.org/wiki/Friedmann_equations#Density_parameter

To date, the critical density is estimated to be approximately five atoms (of monatomic hydrogen) per cubic metre, whereas the average density of ordinary matter in the Universe is believed to be 0.2–0.25 atoms per cubic metre

(3.71295775e-28 kilograms) / 1.67377e-27 = 0.221832017 Atomic Mass Units

http://vixra.org/pdf/1310.0191vD.pdf Time and the Black-hole White-hole Universe

Riess and Perlmutter (notes 2) using Type 1a supernovae calculated the end of the universe tend ∼ 1.7e-121 ∼ 0.588e+121 units of Planck time; tend ∼ 0.588e+121 ∼ 0.2e+71yrs (19) The maximum temperature Tmax would be when tage = 1. What is of equal importance is the minimum possible temperature Tmin - that temperature 1 unit above absolute zero, for in the context of this expansion theory, this temperature would signify the limit of expansion (the black-hole could expand no further). For example, if we simply set the minimum temperature as the inverse of the maximum temperature; Tmin ∼ 1/Tmax ∼ 8π/TP ∼ 0.1773e−30 K (20) This would then give us a value ‘the end’ in units of Planck time (∼ 0.35e+73 yrs) which is close to Riess and Perlmutter; t-end = T^4 max ∼ 1.014e+123 (21) The mid way point (Tmid = 1K) becomes T^2 max ∼ 3.18e+61 ∼ 108.77 billion years.

http://vixra.org/pdf/1310.0191vD.pdf

(1 / ((((5.39116e-44 seconds) * 4.769959e+120 seconds)^0.5) / (4pi))) / G = 3.71295788e-28 kg/m^3

(4.769959e+120 seconds) / 0.588e+121 = 0.811217517 seconds — Preceding unsigned comment added by 2602:306:833C:5980:3CAC:1509:A79A:BD65 (talk) 05:17, 21 May 2017 (UTC)

Free fall in FLRW
Particles which have a proper motion in an expanding universe should decelerate relative to their immediate surroundings. To examine this effect, I use the formula for acceleration at User:JRSpriggs/Force in general relativity applied to a free-falling particle
 * $$ \ddot{x}^{\beta} = - \Gamma^{\beta}_{\alpha\mu} \dot{x}^{\alpha} \dot{x}^{\mu} + \Gamma^{0}_{\alpha\mu} \dot{x}^{\alpha} \dot{x}^{\mu} \dot{x}^{\beta} \,.$$
 * $$ \ddot{x}^{i} = -2 \Gamma^{i}_{0 j} \dot{x}^{j} - \Gamma^{i}_{j m} \dot{x}^{j} \dot{x}^{m} + \Gamma^{0}_{j m} \dot{x}^{j} \dot{x}^{m} \dot{x}^{i} \,.$$
 * $$ \ddot{x}^{i} \approx -2 a^{-1} \dot{a} \delta_{i j} \dot{x}^{j} - k x^i \delta_{j m} \dot{x}^{j} \dot{x}^{m} + c^{-2} a \dot{a} ( \delta_{j m} + k x^j x^m ) \dot{x}^{j} \dot{x}^{m} \dot{x}^{i} \,.$$
 * $$ \ddot{x}^{i} \approx -2 H \dot{x}^{i} - k x^i \dot{x}^{j} \dot{x}^{j} + c^{-2} a^2 H \dot{x}^{j} \dot{x}^{j} \dot{x}^{i} + c^{-2} a^2 H k x^j \dot{x}^{j} x^m \dot{x}^{m} \dot{x}^{i} \,.$$

JRSpriggs (talk) 11:44, 7 April 2013 (UTC)

Friedmann qquations from R
We can use $$ R = 6/a^2 (\ddot{a} a + \dot{a}^2 + k) $$ to make Lagrange function $$ L = \sqrt{-g}(R-2\Lambda) $$ to find Friedmann equations using variational principle. But we find only one equation, not both.95.55.17.46 (talk) 14:45, 30 January 2014 (UTC)

 * See Einstein–Hilbert action for the derivation of the Einstein field equations. Your argument assumes that variations in the metric δgμν which violate the symmetry of the Friedmann–Lemaître–Robertson–Walker metric can be safely ignored. That is false. JRSpriggs (talk) 10:38, 31 January 2014 (UTC)

Is Omega-k correctly defined?
I can't find any other source for Omega-k "spatial curvature density". It seems intuitive that the definition should be Omega-k = Omega - 1. Which is correct? Further, how is Omega-k related to k? Asgrrr (talk) 12:27, 5 June 2016 (UTC)

Critical value range and it's Big Bang overrun
The Universe permanently maintains an allowed critical value range of energy with five constituents, a. expansion, b. mean time flow, c.dark matter, d. dark energy, e. leptohadronic matter. All these constituents change proportionally but are extant inside an allowed critical value range. Big Bang launches when there isn't any other method to maintain that equilibrium. We know that the Big Bang event occurred in the past, at approximately 13.8 billion years ago, because it deleted most data of our predecessor universe (all data except very few under the noise level of the low energy range that cause some low energy anisotropy, but under the noise level, although even under the noise level officially aren't detectable we can still see some non random neither homogenous patterns at that range). The maximum anisotropy of the full spectrum occurred before the Big Bang, but due to inflation shifted all paleosympantic (concerning our predecessor universe) energy spectrum to the subnoise (under the threshold level of noise) of the ultra low energy part of our spectrum. A simplistic popular belief is that each universe exists among two critical value enents, the paleosympantic (concerning our predecessor universe) Big Bang that produced our universe, and the universal death meta-Big Bang that will generate our offspring universe. Others focus only on the critical value part of the overall range of possible analogies among the universal constituents. Ω isn't functional at any level. The critical density occurs from a critical value allowed range. Thus the Ω parameter is permanently close to 1, but due to quantum noise never one. At extreme analogies among the energetic contributing universal constituents Ω cannot be close to 1 except if the universe explodes; and exactly that is what it does in order to maintain the Ω critical density inside the allowed critical value. The homogeneity of the thorough universal energy range isn't something other than the Big Bang goal. The Ω critical value range of the critical density itself forces homogeneity by expanding more or less universal regions during the Big Bang to achieve it, but the homogenous full spectrum of energy result, buries the anisotropic noise of the previous universe underneath the ultra low energy part of the spectrum at subthreshold levels, officially non detectable - but due to quantum mechanical probabilistic diffusion of information, will be discovered in the future (with some uncertainty because it is a subthreshold semi-anisotropy, it isn't a complete quantum decohesion or a pure anisotropy, because the Big Bang prevented that from happening, and that makes our job more difficult, but not impossible).

I understand what you mean but your English isn't correct. If the absolute value of the fullrange spectrum of energy is lower than the critical value, the universe is no longer possible to maintain Ω and and thus Big Bang occurs. -- 05:59, 26 July 2016 (UTC)

Imaginary radius of curvature
I am not sure what the meaning of the following sentence is:

If k = −1, then (loosely speaking) one can say that i· a is the radius of curvature of the universe.

I would suggest to remove the sentence if there is no easy way to supplement it with a simple clarification. --Falktan (talk) 10:34, 25 September 2016 (UTC)

 * See Gaussian curvature. If the two principal curvatures have the same sign so that the Gaussian curvature is positive, then the radius of curvature, a, is the reciprocal of their geometric mean. What is meant here is that if the two principal curvatures have opposite signs (e.g. the Gaussian curvature is negative), then the reciprocal of the geometric mean of their absolute values is a. JRSpriggs (talk) 14:06, 25 September 2016 (UTC)

Confusing Text
The text following the last equation in the Density parameter section is confusing and misleading to a reader with little background knowledge. The problem is that the sign of Ω_0,k is backwards. That is the sign is positive for a universe with negative (hyperbolic) curvature, and negative for a universe with positive (hyperspherical) curvature. In particular, the variable k was previously introduced with a description that k=+1 corresponds to positive curvature, and k=-1 corresponds to negative curvature. I hope someone with the appropriate skills will edit this text to remove this confusion. BHBloom (talk) 18:01, 28 April 2017 (UTC)

Bulk Elastic Properties & Wave Speed & Friedmann_equations Density_parameter
((((x GHz) * (3.71295774e-28 (m^2))) / Planck's constant) * (1 kg)) = 299792458^2

((((160.390033967 GHz) * (3.71295774e-28 (m^2))) / Planck's constant) * (1 kg)) = c^2

((((CMBR Hz) * (Friedmann Mass Density (m^2))) / Planck's constant) * (1 kg)) = c^2

1/(160.390033967 GHz * 3.71295774e-28 kg  / Planck's constant  / c^2  )^0.5 = 1 m^2/s (square meter per second)

https://en.wikipedia.org/wiki/Viscosity#Kinematic_viscosity

The kinematic viscosity "momentum diffusivity" is the ratio of the dynamic viscosity μ to the density of the fluid ρ. It is usually denoted by the Greek letter nu (ν) and has units m^2/s   — Preceding unsigned comment added by 2602:306:833C:5980:EF:7508:2AC:611B (talk) 01:24, 20 September 2017 (UTC)

((0.224533978 atomic mass unit)) * (4pi/10^7 ) / (0.5^0.5 s/m^2) = (6.62607005e-34 Joule Second)

(3.72847436e-28 kilograms) / (3.71295775e-28 kilograms) = 1.00417904  — Preceding unsigned comment added by 2602:306:833C:5980:F885:B7DF:2E16:93A6 (talk) 14:58, 29 May 2017 (UTC)

((6.6740800122e-11 * (newtons / (meters^2))) / (3.71295775e-28 * 2 * (kg / (meter^3))))^0.5 = 299792458 m / s

(6.6740800122e-11 /299792458^2 /2) = 3.712957749840332194829297612558e-28 kg

"physical mechanism" is the ( Inertial & Elastic Properties of the Medium) times the Mass Loading per cubic Meter http://hyperphysics.phy-astr.gsu.edu/hbase/permot3.html http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/souspe2.html#c1

http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/imgsou/soundwater.gif

(((6.6740800122e-11)/2 (newtons / (meters^2))) / (3.712958e-28 (kg / (meter^3))))^0.5 = 299792448 m / s

https://en.wikipedia.org/wiki/Friedmann_equations#Density_parameter

To date, the critical density is estimated to be approximately five atoms (of monatomic hydrogen) per cubic metre, whereas the average density of ordinary matter in the Universe is believed to be 0.2–0.25 atoms per cubic metre

http://www.wolframalpha.com/input/?i=((+(6.6740800122e-11%2F2)+Newtons+%2F+meters%5E2)%2F(3.712958e-28+kg%2Fmeter%5E3))%5E0.5+%3D

(((6.6740800122e-11)/2 N/m^2 (newtons per square meter))/(3.712958e-28 kg/m^3 (kilograms per cubic meter)))^(1/2) = 299792448

1 / ((6.6740800122e-11 / 2) * c) = 99.958075 s / m

0.2235995768 AMU = 3.712958e-28 kg

((6.6740800122e-11 * (newtons / (meters^2))) / (3.71295775e-28 * 2 * (kg / (meter^3))))^0.5 = 299792458 m / s

http://www.wolframalpha.com/input/?i=((6.6740800122e-11+*+(newtons+%2F+(meters%5E2)))+%2F+(3.71295775e-28+*+2+*+(kg+%2F+(meter%5E3))))%5E0.5+%3D

((6.6740800122e-11 Pascals) / (3.71295775e-28 * 2 * (kg / (meter^3))))^0.5 = 299792458 m/s

(6.6740800122e-11 / 3.71295775e-28 /2)^0.5 = 299792458

(6.6740800122e-11 /299792458^2 /2) = 3.712957749840332194829297612558e-28 kg

http://www.wolframalpha.com/input/?i=((+6.6740800122e-11+Newtons+%2F+meters%5E2)%2F(2+*+3.712957749840332194829297612558e-28+kg%2Fmeter%5E3))%5E0.5+%3D

(3.712957749840332194829297612558e-28 kg * (c^2)) / (meter^3) = 3.33704001e-11 pascals = G/2 — Preceding unsigned comment added by 2602:306:833C:5980:45B6:FE90:C7B3:8D3F (talk) 07:11, 14 May 2017 (UTC)

David Fuller

Density of the universe
I suggest creating a separate article named Density of the universe because at this moment it redirects to this article (Friedmann equations), which is full of formulae. The people who want to read the article Density of the universe they actually just want to know how many atoms (or grams) of matter are in the average cubic meter of this Universe. —  Ark25  (talk) 00:36, 21 July 2017 (UTC)
 * I modified the redirect to go to the section "Density parameter" which contains that information. JRSpriggs (talk) 10:59, 21 July 2017 (UTC)

Query about the Density parameter
I am confused about something regarding the Density parameter.

Lets say hypothetically it is the case that the value of ρ is actually greater than ρc and that the universe is therefore finite. If the Universe is expanding, this surely means that ρ will be decreasing? What happens when ρ becomes equal to ρc? Does the universe suddenly switch from being finite to being infinite? How can this make sense?

I would be very grateful if someone could untangle my confusion here. 81.149.43.187 (talk) 17:11, 11 October 2017 (UTC)

 * First, you are assuming that the critical density, ρc, does not change as time passes. Since it depends on the Hubble constant H (which is not actually constant), it does change.
 * Second, in the scenario you envision, the expansion would stop and become a contraction before the density became equal to the critical density. So the value of the density would begin to rise again preventing the equality from occurring. JRSpriggs (talk) 19:12, 11 October 2017 (UTC)

Hi JRSpriggs Thank you very much for your reply. It has clarified things somewhat. 81.149.43.187 (talk) 14:19, 13 October 2017 (UTC)

rescaled
$$\Omega_c=-\frac{kc^2}{H_0^2 a_0^2}\;$$ If k is meant to be K=1/R²=(c/H°)², then this turns to Ωc = -1/a² = -1. But what the heck is Ωc supposed to be? It is not the critical density, I guess it is supposed to be the curvature densitiy which is everywhere else called Ωk. On the other hand the value of Ωc ≈ -1 points to the critical value, but the minus sign is wrong then. So i guess it is a mixture and should be Ωc = 1+Ωk Ra-raisch (talk) 19:14, 18 June 2019 (UTC)
 * As far as I understood now, there are two conventions: k using as curvature index (-1,0,1) or (usually K) as the curvature (1/m²) and also a using as the scaling factor or the scaling parameter (m) which should be A in order to distinguish them. Thus here it should be Ωk and not Ωc, leaving k and a as is, the ambiguity of which should be cleared and naming altered throughout the article. Ra-raisch (talk) 22:44, 23 June 2019 (UTC)

Friedmanns original equation could not possibly have included the term "a(t)"!
... because a(t) wasn't invented until four years (1929) after his death (1925)!

A presentation of the original equation would be quite interesting. Hilmer B (talk) 15:18, 19 August 2020 (UTC)

 * Do you have any reliable source to back up that statement? JRSpriggs (talk) 19:34, 19 August 2020 (UTC)

 * Do you have any reliable source to falsify that statement? Did really Friedmann invent and use "a(t)" as it is interpreted today? Hilmer B (talk) 19:46, 20 August 2020 (UTC)

 * To a mathematician using the radius of a hyper-sphere as a function of time when the hyper-sphere varies in size is obvious. If you need to see that Friedmann was aware of it, see his letter to Ehrenfest here where he says "a world, the space of which possesses a curvature radius varying with time;". JRSpriggs (talk) 20:44, 23 August 2020 (UTC)

I guess you're referring to II:1 in that letter: “From my second note in Russian that I have sent to you, you have seen that under certain assumptions common to those of Einstein and De Sitter it is possible to obtain the universe with the space of a (spatially!) constant curvature, the radius of curvature of which is varying with time.”

Friedmann does here not suggest that he shared the belief that the scale of space itself varies with time, or ?! It may rather suggest that he, himself, considered spatial variations in the curvature. No trace of an expanding “a(t)” universe i this statement.

The quality and the beauty of Friedmann's equation is rather the way he relates the curvature of spacetime to (the local) density. He may very well have allowed the curvature to vary in space. What Einstein-deSitter suggested was a variation also in the time dimension. My guess is that they were worried because their interpretation of ∇2Φ = 4πGρ did only allow positive curvature in a static universe. (∇2Φ > 0 ...)

So back again to the original question: I guess that we can both agree upon that 1/R2 should have appeared in Friedmann's original, and I guess only, version of this equation. But was it also 1/a2(t)R2? And in that case how was that a(t) defined? Exactly: what did Friedmann's original equation look like? Hilmer B (talk) 19:07, 24 August 2020 (UTC)

 * Unfortunately, the original papers of Friedmann are behind a wall: you have to pay to see them and then they are in Russian or German. This is a common problem, forcing us to rely on secondary sources. But the mere fact that those papers are not available to us is not evidence that Friedmann was unaware of what is now purported to be their contents. JRSpriggs (talk) 19:30, 24 August 2020 (UTC)

Weird - Friedman died 95 years ago. Not even Disney has managed to get such a prolonged protection ... So - the conclusion is that none of us knows. A case similar to the situation of Schrödinger's poor cat ... Well, thanks for your contribution. Hilmer B (talk) 20:39, 25 August 2020 (UTC)

Justification for eliminating the Cosmological Constant from the First Equation
You properly derive the first equation as:

$$\frac{\dot a^2+k c^2}{a^2}=\frac{8\pi G \rho+\Lambda c^2}{3}$$

But then you go on to state that:

These equations are sometimes simplified by replacing

$$\rho \rightarrow\rho-\frac{\Lambda c^2}{8\pi G}$$

to give:

$$\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^2}$$

What is the justification for this substitution? I might guess that you assume that the Cosmological Constant is so small with respect to the other terms that it is effectively zero, but you can't just drop a major variable without an explanation.

 * It is a question of how the cosmological term is interpreted. Is it a complication in the law of gravity (as originally conceived)? Or is it a mysterious "dark energy" which should be included in the stress-energy tensor on the other side of the equation? Lately, people have been leaning to the latter interpretation. JRSpriggs (talk) 03:31, 21 February 2021 (UTC)

title: Talk:Friedmann equations


text:
Hinduism in Thailand

Hinduism in Thailand is a minority religion followed by 84,400 (0.1%) of the population as of 2020. Despite being a Buddhist-majority nation, Thailand has a very strong Hindu influence. The majority of Thai Hindus reside in Bangkok, Chonburi, and Phuket.

History
Although Thailand has never been a majority-Hindu country, it has been influenced by Hinduism. Despite the fact that today Thailand is a Buddhist-majority country, many elements of Thai culture and symbolism demonstrate Hindu influences and heritage. Southeast Asia, including what is now Thailand, has been in contact with Hinduism through India for over 2000 years. Indian settlement in Southeast Asia has been ongoing since the 6th century BCE and has continued into the modern era, influenced by various socioeconomic and political factors. Tamil and Gujarati immigrants migrated to Thailand in the late 1800s, working in the gem and textile industries, followed by a larger migration of both Sikhs and Hindus from Punjab in the 1890s.

The popular Thai epic Ramakien is based on the Buddhist Dasaratha Jataka, which is a Thai variant of the Hindu epic Ramayana. The national emblem of Thailand depicts Garuda, the vahana (mount) of Vishnu.

The city Ayutthaya, near Bangkok, is named after Ayodhya, the birthplace of Rama. Numerous rituals derived from Hinduism are preserved in rituals, such as the use of holy strings and pouring of water from conch shells. Furthermore, Hindu-Buddhist deities are worshiped by many Thais, such as Brahma at the famous Erawan Shrine, and statues of Ganesha, Indra, and Shiva. Reliefs in temple walls, such as the 12th-century Prasat Sikhoraphum near Surin (Thailand), show a dancing Shiva, with smaller images of Parvati, Vishnu, Brahma, and Ganesha.

The Devasathan is a Hindu temple established in 1784 by King Rama I. The temple is the centre of Hindus in Thailand. The royal court Brahmins operate the temple and perform several royal ceremonies per year.

An annual Giant Swing ceremony known as Triyampavai-Tripavai was held in major cities of Thailand until 1935, when it was abolished for safety reasons. The name of the ceremony was derived from the names of two Tamil language Hindu chants: Tiruvempavai and Tiruppavai. It is known that Tamil verses from Tiruvempavai — poet pratu sivalai ("opening the portals of Shiva's home") — were recited at this ceremony, as well as the coronation ceremony of the Thai king. According to T.P. Meenakshisundaram, the name of the festival indicates that Thiruppavai might have been recited as well. The swinging ceremony depicted a legend about how the god created the world. Outside shops, particularly in towns and rural areas, statues of Nang Kwak as the deity of wealth, fortune and prosperity (a form of Lakshmi) are found.

The elite, and the royal household, often employ Brahmins to mark funerals and state ceremonies such as the Royal Ploughing Ceremony to ensure a good harvest. The importance of Hinduism cannot be denied, even though much of the rituals has been combined with Buddhism.

Earliest Vishnu Sculpture Discovered in Thailand
Southern and Central Thailand are possibly the earliest regions, where Vaishnavism flourished in the whole of Southeast Asia. Several sculptures of Vishnu are discovered from the earliest period. These smaller-than-human-sized sculptures are stylistically comparable to Mathura and Amaravati schools of Indian Art.

Second Phase of Vishnu Sculpture from Central Thailand, 7th-8th centuries CE
The human-sized sculpture of the second phase Vishnu with Pallava influence was discovered in Sri Mahosot in Central Thailand. Vishnu is attired in a plain cylindrical crown along with a transparent lower garment. The style of the sculpture shares a similarity with the ones discovered in Southern Thailand.

Sculptures of Vishnu and Krishna discovered from Sri Thep
Sri Thep is one of the most important Dvaravati cities in Central Thailand which yields several stone sculptures of Vaishnavism. Vishnu and Krishna are the most common finds. Vishnu stands in the posture of Tribhanga, Krishna is portrayed with the left band lifting Mount Govardhana.

Four-armed Vishnu, 14th century CE
After the 13th century CE, the Thais were converted to Theravada Buddhism, kings tried to claim himself as the incarnation of Vishnu. Vishnu bronze image was moulded for Sukhothai's royal private worship. The image smiles with serenity, similar to Sukhothai Buddha image.

Tracing Shaivism in Thailand
Shiva is the mighty god of destruction of the universe, and sins. Shaivism, which originated in South India, was transmitted to the southern and central parts of Thailand. This occurred during the 6th-10th century A.D. Lord Shiva is manifested in various forms. Among them the most popular one is that of the lingam, in the shape of men's genital organs. It depicts the unison of the male and female energies, which created the entire universe according to Hindu thought. Several forms of Shiva Lingam have been found in Southern Thailand. One has the Lord Shiva's face. This is known as the Mukha Lingam. Similarly, a gold lingam, belonging to a hermit's private collection, has been discovered at a cave in the region. A lingam shaped stone known as the Swayambhulingam has also been discovered.

Thai Brahmin community
Thailand has two ethnic Thai Brahmin communities - Brahm Luang (Royal Brahmins) and Brahm Chao Baan (folk Brahmins). All ethnic Thai Brahmins are Buddhist by religion, who still worship Hindu Gods. The Brahm Luang (Royal Brahmins) mainly perform royal ceremonies of the Thai monarch, including crowning of the king. They belong to the long family bloodline of Brahmins in Thailand, who originated from Tamil Nadu. The folk Brahmins are the category of Brahmins who are not from a lineage of priests. Generally, these Brahmins have less knowledge about the rituals and ceremonies. Apart from this there are also Indian Brahmins from India who migrated to Thailand more recently.

Brahmins once conducted the royal ceremony in other Southeast Asian countries as well. The rituals were reinstated in Cambodia after the overthrow of the Khmer Rouge. The Brahmins of Myanmar have lost their role due to the abolition of monarchy.

Historian Damrong Rajanubhab has mentioned about three kind of Brahmins, from Nakhòn Sī Thammarāt, from Phatthalung, and those who originated from Cambodia.

Indian Hindus
During the Sukhothai and Ayutthaya periods, evidence of the presence of sizable number of Indians in the Thai court is described by a number of western travelers. However most of the contemporary Indians came to Thailand after 1920, and during the first half of the 19th century.

The Mariamman Temple, Bangkok is the first temple built in the South Indian architectural style. It was built in 1879 by Vaithi Padayatchi, a Tamil Hindu immigrant.

Demographics
According to the 2005 Thai census, there are 52,631 Hindus living in Thailand, making up just 0.09% of the total population. In the 2010 census, that number decreased to 41,808 Hindus in Thailand, constituting 0.06% of the population. In the 2015 census, this population decreased to 22,110, or 0.03%.

However, the PEW Research data found that Hinduism constituted 0.1% of the Thai population in 2014 and is increasing in Thailand as a percentage of the country's population. The PEW Research data reports that the Hindu population is expected to increase from 0.1% in 2014 to 0.2% by 2050.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Hinduism in Thailand


text: User talk:69.23.252.110

title: User talk:69.23.252.110


text:
KroniK

KroniK (also stylized as Kroniꓘ) was an American professional wrestling tag team in World Championship Wrestling (WCW), the World Wrestling Federation (WWF) and All Japan Pro Wrestling (AJPW), composed of Brian Adams and Bryan Clark in the early 2000s.

World Championship Wrestling (2000–2001)
Adams and Clark first joined together as a duo in April 2000 in World Championship Wrestling (WCW). Upon joining together as a tag team, KroniK (originally spelled Kronic) assisted Vince Russo in his vision of a clean sweep for his New Blood alliance at the 2000 Spring Stampede pay-per-view. Adams and Clark interfered in the WCW World Tag Team Championship match, allowing the team of Buff Bagwell and Shane Douglas to win the title.

While in WCW, KroniK moved between babyface and heel roles several times - holding the WCW World Tag Team Championship twice. In the final months of WCW, KroniK became hired muscle, adopting the catchphrase "breakin' necks and cashin' checks", similar to the WWF's popular APA tag team. Their contracts were not picked up by the WWF when WCW was sold by AOL Time Warner.

World Wrestling Federation (2001)
In September 2001, Brian Adams and Bryan Clark each made their re-debut as a World Wrestling Federation (WWF) tag team. Recruited by Stevie Richards, KroniK was called in to settle past differences Richards had with The Undertaker, stemming from the disbanding of Richards' stable, "Right to Censor" earlier that year. KroniK wrestled their first tag team match defeating Kaientai on the September 20, 2001 episode of SmackDown! and their last tag team match for the WWF a few days later at Unforgiven, losing to then WCW Tag Team Champions The Brothers of Destruction (Kane and The Undertaker).

After two dark matches, four appearances on Raw and SmackDown! and a pay-per-view appearance, Clark was released from his contract for being unconditioned and unsafe during their PPV match, while Adams began working for the Heartland Wrestling Association, which served as WWF's developmental territory, for conditioning. Adams requested his release from the WWF in November 2001.

WWA, All Japan Pro Wrestling, and retirement (2002–2003)
Following their WWF departure, KroniK made appearances for World Wrestling All-Stars, their debut match with promotion taking place at Revolution, defeating the Navajo Warrior and Ghost Walker in less than five minutes.

KroniK then joined All Japan Pro Wrestling in the summer of 2002 during the Pro Wrestling Love era, making an immediate impact by defeating Keiji Mutoh and Taiyo Kea for the AJPW World Tag Team Championship on July 17, 2002.

On the July 20, 2002 edition of AJPW TV, KroniK participated in the Stan Hansen Cup Four Way initiative. KroniK would eliminate the team of George Hines and Johnny Smith, before being eliminated by "Dr. Death" Steve Williams and Mike Rotunda after Williams pinned Adams after delivering the Doctor Bomb. Mike Barton and Jim Steele would go onto win the tournament. Despite KroniK's failure in winning the Stan Hansen Cup, after the match they would cut a confident and rather cocky promo talking about how they were still the AJPW World Tag Team Champions, calling everyone in the initiative "losers", "stupid" and "not smart enough" to be the champions.

KroniK would only make one successful title defense against Barton and Steele on August 30, 2002 as part of a two day pay-per-view. On the second day of the pay-per-view on August 31, KroniK defeated Tomoaki Honma and Yuto Aijima in a non-title match. KroniK was later stripped of the titles in October as they canceled the rest of their appearances in 2002, as Adams suffered a shoulder injury while training for a boxing career.

They returned for the WRESTLE-1 show on January 19, 2003 in what would be KroniK's last match, losing to Keiji Muto and Goldberg. Along with Goldberg, Steve Williams and Mike Rotunda, KroniK left AJPW during a gaijin exodus, which occurred after Mrs. Baba's last appearance behind the scenes in AJPW. In that match, Adams suffered a spinal injury that forced him into retirement, and Clark decided to leave during the exodus. This was also when AJPW's sale finalized, the company having been in the process of being sold in late 2002.

Adams died on August 13, 2007, after mixing the painkiller buprenorphine with the muscle relaxant carisoprodol, as well as the sedatives chlordiazepoxide and alprazolam. Though the Hillsborough County medical examiners determined that each drug was found to be at therapeutic levels, the combination of the drugs together were enough to impede Adams' respiratory system and cause his death.

Championships and accomplishments

 * All Japan Pro Wrestling
 * World Tag Team Championship (1 time)
 * World Championship Wrestling
 * WCW World Tag Team Championship (2 times)
 * Wrestling Observer Newsletter
 * Worst Worked Match of the Year (2001) vs. The Brothers of Destruction at Unforgiven
 * Worst Tag Team (2000, 2001)

title: KroniK


text: Category:United Nations images

title: Category:United Nations images


text: File:Nelly - Nellyville - Album.jpg

title: File:Nelly - Nellyville - Album.jpg


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Talk:Bubs

Sounds Black
He sounds totally black.

So.

So maybe that should be added to the article. None of the other characters sound black.

Maybe it shouldn't be added to the article. —BazookaJoe 00:39, 17 September 2005 (UTC)

I agree that it should be added. 70.237.47.94 18:15, 18 February 2007 (UTC)

Someone changed my edit back. I think it's significant that he's the only character that sounds black, especially given the context of his character's actions. 70.237.47.94 20:02, 18 February 2007 (UTC)

Redd Foxx?
He sounds more like Scatman Crothers to me.

Racist caricature
I don't know about the sounding black discussion but the dialect, the chicken references, the obesity and being a character shown to "break dance," I'm just lead to the belief that this is the case. 24.187.77.211 04:02, 12 July 2007 (UTC)

title: Talk:Bubs


text:
Loch Lyon

Loch Lyon (Scottish Gaelic, Loch Lìobhann) is a freshwater loch in Glen Lyon, located in Perthshire, Scotland, which feeds the River Lyon, a tributary of the River Tay. Loch Lyon lies in Glen Lyon, that is the longest enclosed Glen in Scotland, stretching over 30 mi from Fortingall in the east to Loch Lyon in the west. Sir Walter Scott described Glen Lyon as the longest, loneliest and loveliest glen in Scotland.

Reservoir
The original natural loch was much expanded by a hydro-electric dam, part of the North of Scotland Hydro-Electric Board's Breadalbane scheme.

title: Loch Lyon


text:
User talk:Acmurray

The state of coral reefs in the Philippines and Indonesia are drastically reducing in size to help benefit many tourists and hobbyists in countries such as our own. In the Philippines 80% of coral mangroves have been lost since 1920. This decline can be attributed to mostly blast and cyanide fishing. Consiquently, the fish population has also suffered and declined. Many of the fish and coral one might see in Aquariums today are most likely from Southeast Asian reefs. Moreover, Indonesia has the most significant reef resource in Southeast Asia and unfortunately suffers as the Philippines do. One differnce however, is though the reefs on the West Coast of Indonesia are damaged, the ecology of the Northeast reefs remains in good shape. Overall, the decline of reefs around the world can be attributed to many forms of pollution, but most importantly they can be attributed to human population growth.

References: 1)Columbia Encyclopedia

2)Regional Perspectives: East Asian Seas. May 26, 2005. www.ncdc.noaa.gov/paleo/outreach/coral/sor/sor_asia.html

title: User talk:Acmurray


text:
User talk:Zantastik/archive2

Maps of Korea
I got them from Korean web sites. Various places. I doubt they are copyrighted. They are all from government sources. So, I do not think there is anybody behind to claim the copyrights.

Salut,
D'après ce que j'ai vu sur la page de MRW : tu est étudiant en histoire, première année. J'ai pour toi une bonne, et une moins bonne, mais bonne nouvelle quand même. Le wikipédia USA est beaucoup plus riche en ce qui concerne les biographies historique, même pour les biographies d'hommes français. Pour les autres fait historique, les 2 wikis se valent, et se complètent. Yug
 * La bonne : c'est qu'on essait de construire ceci : Histoire pour étudiants en Licence ; pour organiser en commun le travail des étudiant en histoire sur wikipedia.fr.
 * l'autre, c'est que tu trouvera plus d'info sur Villèle ici : Jean-Baptiste Guillaume Joseph, comte de Villèle.

Gay rights map
Bonjour, I do not know if I have the time to edit it, as someone change the colouring scheme after I made an edit to the image on my hard drive, but never bothered to upload it. - Earl Andrew 06:29, 22 Mar 2005 (UTC)

Adrian Rogers
Dozenist is planning to add some less controversial views held by Rogers (such as his support for close-knit families and strong parental involvement).Jersyko 06:21, Mar 23, 2005 (UTC)

Hardly a stranger
Actually, I feel I do know of you, as I often see your name popping up on my watchlist. Good work! Thanks for your contributions to Wikipedia. Cheers, -Willmcw 10:49, Mar 23, 2005 (UTC)

mmmm
Response: Something the casual observer would find insignificant enough to deserve laughter. The bitterness and entitlement is fading away. BTW...RE: TK-P, how are some of those articles going to be managed? Lotsofissues 09:12, 31 Mar 2005 (UTC)

 * I would prefer to rewrite but I know nothing about the subject - didn't stop me from rewriting Cosmo. :-)

Church of Christ edits and a question
Zan, thanks. I hope someone goes through my changes with a jaundiced eye.

Also, is there a list of articles that need to be translated? I can translate from Spanish to English. Danlovejoy 15:23, 5 Apr 2005 (UTC)

--Zantastik 08:29, 7 Apr 2005 (UTC)

Orange Mound
I've enjoyed reading some of the articles you've written/contributed too... Your Orange Mound project caught my eye. What prompted you to develop an article on the Orange Mound neighborhood (rather than say, Midtown or Cooper-Young)? I lived in Memphis while in college and I've often thought about developing more Memphis articles but I lack motivation. Good luck! --Chiacomo 07:29, 9 Apr 2005 (UTC)
 * ... "Orange Mound Jazz Messengers" --Chiacomo 04:51, 27 Apr 2005 (UTC)

Daylily
Zan I uploaded this cropped version of the daylily pic cause I do agree with the guy who said it looks good except for the wall in the background...do you think we should nominate this version or just stick with the original? -- user:JennaMarie83

Image Copyrights
Hello! Take your time with that Sweet Tea pic, not a big deal at all, seriously. As to your inquiry, my first thought would be if the photograph had artistic merit on its own you would probably be ok. If however you just took a dead on photograph of the logo and cropped it you might be treading the line. But that said, the man to ask is Quadell. He spearheads the Image Tagging Project and the Image Sleuthing Project, and as such knows a bunch about image copyrights. I think he just moved so he may be MIA for a couple days. Drop him a line! --MaxPower 13:07, 2005 Apr 11 (UTC)

Hungary stubs
I've been adding to most of them. john k 14:19, 11 Apr 2005 (UTC)

Image copyright
Hello Zantastic. This is a tricky area of the law. When you take a photo that includes a logo or other copyrighted work, you own the copyright to the image - but you might be violating the other copyright-holder's copyright. This is where fair use comes in. It is certainly a fair use of the Nike logo to take (and even publish) a photo of a guy wearing a Nike shirt. Some really copyright-paranoid companies won't publish a photo that includes a logo "just in case", but there's no way a suit like that would succeed. On the other hand, if you were to publish a photo of a Nike logo, where it was clear you were simply trying to reproduce the logo, a fair use claim wouldn't hold up. In the case you mentioned, with tea brands showcase, it would be best if you used a pic of the whole box, and not just the front of the box. You want it to be evident that the majority of the artistic content in your work is your own, and that the photo is not intended to be a reproduction. Then, if I were you, I'd tag the photo as GFDL, but leave a note on the image description page that the inclusion of any copyrighted logos is contended to be "incidental" and a fair use of the material. Hope this helps, – Quadell (talk) (sleuth) 17:22, Apr 11, 2005 (UTC)

 * I replied at Commons:Commons talk:Copyright tags. dbenbenn | talk 21:47, 12 Apr 2005 (UTC)

I got a response from my copyright professor. He analogized such photos to photos of buildings, which are specifically allowed so long as the buildings are visible from a public place. However, he was not *certain* that photos of branded products would not be fair use, adding that it might require a case-by-case analysis. I sent him another e-mail with specific examples, so hopefully I can post a more definitive answer soon. In any event, it seems the law is by no means clear cut in this area (so now I don't feel so bad about not knowing the answer :) ). - Jersyko 22:09, Apr 12, 2005 (UTC)
 * Even "public" buildings aren't always safe... I read a copyright case where the courts held that a single "copyrighted" building constituted too much of the infringing use for fair-use rules to apply. As an aside, the same 'relative proportions' fair-use test seems to be a more common way copyvios occur on wiki. Feco
 * You know, it's really a shame that we have to take time to deal with a question like this. If only the United States had sensible copyright laws . . . - Jersyko 22:24, Apr 12, 2005 (UTC)

Ancestry maps
Please see Talk:Ancestries of Americans for the explanation. Thank you. -- Stevey7788 02:54, 12 Apr 2005 (UTC)

Posting Copyvios
I came across the EdmundKean page that you copyvio'd. I couldn't find it cross listed on WP:CP, which is where all copyvio pages are supposed to be referenced. The current copyvio notice is unclear, so users don't always know that they must manually add pages to WP:CP. The squiggle-bracket isn't cool enough to do that automatically. I had been making the same mistake until another user told me about it. Apologies if you're aware of all this already. Feco 00:37, 14 Apr 2005 (UTC)

 * Per our conversation, please take a look at User:Feco/Templates/copyvioDRAFT. It's a re-worked copyvio template that addresses the problem we discussed (it also attempts to address some other things). Review Template talk:Copyvio for more. Feco 01:37, 14 Apr 2005 (UTC)

Tooth Enamel
Hey, i saw you nominated tooth enamel for featured article statuts. While i completely agree with you that it deserves to be featured, do you think we should submit it for peer review first (Peer_review)? I guess it could obtain featured status without it, but its chances might be better if it went through peer review first, especially since i think a lot of people might be turned off by the number of dead wikilinks in the article (even if Dozenist does go back and write stubs for all of them eventually ;) ). - Jersyko 15:31, Apr 30, 2005 (UTC)

NAS
Aloha. I believe the book you added to New anti-Semitism, La nouvelle judéophobie is the original version of the English translation that is already listed as Rising From the Muck : The New Anti-Semitism in Europe. Both of these listings should be merged, somehow. --Viriditas | Talk 23:59, 2 May 2005 (UTC)

audentis/audaces fortuna iuvat
Don't know about that "other user", but the sources, as I stated in my revert before, quite clearly have audentis, not audaces. I really think we should stick with the original and mention alternative versions in the body of the text, as I did (which addition does not make a lot of sense the way it is now ;-)) Regards T.a.k. 11:28, 14 May 2005 (UTC)

Bible Adventures
Awesome edits to the Bible Adventures page :). However, the picture seems to have disappeared with the addition of the (very cool) infobox.  I don't know how to fix it, but i'm hoping you do. - Jersyko 22:17, May 19, 2005 (UTC)
 * How odd. No matter how many times i refresh, i can't see it in mozilla, but it shows up in IE.  Oh well.  Yes, please feel free to add the infobox to the other articles.  We need to finish writing those articles too . . . - Jersyko 01:52, May 20, 2005 (UTC)

Schoolwatch.
Howdy,

Thanks for taking interest in User:GRider/Schoolwatch. May I suggest that in the future you place kept schools in User:GRider/Schoolwatch/Archive, using the existing format. This greatly simplifies maintaining schoolwatch, and gives it a more polished appearance. Also that archive will eventually be used to make a statistical argument that most school's survive VfD, so it's important that the archive be maintained. Again, thanks for editing.

Cheers, --Klonimus, 04:43, 21 May 2005 (UTC)

My talk page
If you don't like me deleting comments I dislike from my talk page, please discuss it with me. Reverting my talk page to a version I dislike is rude and confrontational IMO. Frankly, I'm not interested in anything FM has to say, nor in anyone else taking up his crusade of hysterical theatrics against me. I find it very telling that Mel Etitis, a user I have had near constant conflict with, chose to bring these groundless claims to my attention, rather than a neutral party. I wasn't interested in hearing it, and I deleted it. I will do so again if necessary. If you disapprove, discuss, rather than revert my user name space. Thanks, Sam Spade 08:54, 21 May 2005 (UTC)

What Sam characterizes as "hysterical theatrics against him," I call an attempt at rehabilitating him and educating others while seeking justice for his personal attack and subsequent attempts to cover it up. FeloniousMonk 15:12, 21 May 2005 (UTC)

Photos to be transferred to commons
Are you interested in any specific photo or it is just a general suggestion? I will transfer all of them, no problem. Cheers! --Viosan,

SS
Thanks for your comment, and for trying to keep Wikipedia open and above-board. Although it's generally considered bad manners to delete comments from your own Talk page (except for archiving), and often suggests that you have something to hide, it's not actually forbidden. SS is well known for deleting comments that show him in a bad light (a look through the History of his Talk page is illuminating), and it provokes comment occasionally, but although there's another User who's the subject of an RfC for similar behaviour (see Requests for comment/Neutrality (2)), I don't think that that's really justified. I'd let him try to hide the criticism — it isn't really hidden, and the attempt does nothing but show him up. Thanks anyway. Mel Etitis ( Μελ Ετητης ) 22:44, 21 May 2005 (UTC)

 * Out of interest, this is from Talk etiquette FAQ:

 * Can I blank my talk page whenever I want?

 * There's no hard-and-fast rule that says you can't, however many people will suspect that you are trying to hide something or ignore other contributors if you do it too often. Most users do archive their talk pages periodically to a personal subpage -- either when the page gets too large, on a regular schedule, or when they take a wikivacation.
 * In the past, attempts at reasoning with certain problematic users resulted in their use of this tactic. In that case, the removal of material on a talk page is viewed as a kind of vandalism and incivility, which combined with other issues may be grounds for a case before Arbitration.

 * So there you have it... Mel Etitis ( Μελ Ετητης ) 11:14, 22 May 2005 (UTC)

Removing info from talk pages
I noticed your enquiry. People remove content they don't like from their talk pages all the time. It never goes, as it's in the history. It is not something one should do on article talk pages, but your own is a different matter. Cheers, --Silversmith 01:00, 22 May 2005 (UTC)

title: User talk:Zantastik/archive2


text: User talk:202.79.179.225

title: User talk:202.79.179.225


text: User talk:202.79.179.226

title: User talk:202.79.179.226


text: User talk:210.50.132.29

title: User talk:210.50.132.29


text: File:The drowners.jpg

title: File:The drowners.jpg


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Bill Mosienko

William Mosienko (November 2, 1921 – July 9, 1994) was a Canadian professional ice hockey right winger who played 14 seasons in the National Hockey League (NHL) for the Chicago Black Hawks from 1942 to 1955. He is best noted for recording the fastest hat trick in NHL history. In a 1952 game against the New York Rangers, Mosienko scored three goals in 21 seconds.

In the NHL, Mosienko won the Lady Byng Memorial Trophy in 1945 as the most gentlemanly player in the league, played in five All-Star Games and was twice named to the second All-Star team. He left the league in 1955 to help bring professional hockey to his hometown of Winnipeg. He helped create the Winnipeg Warriors of the Western Hockey League and was a three-time All-Star in his four years of play in the league. He won the league championship in 1956.

Mosienko was inducted into the Hockey Hall of Fame in 1965 and into the Manitoba Sports Hall of Fame and Museum in 1980.

Early life
Mosienko was born on November 2, 1921, in Winnipeg, Manitoba. He was one of 13 children, having eight brothers and four sisters. His parents, Daniel and Natalia, were Ukrainian immigrants who came from Ekaterinoslav (modern Dnipro, Ukraine). In Canada Daniel worked as a boilermaker with the Canadian Pacific Railway. Mosienko grew up in Winnipeg's north end.

He developed a passion for hockey and began playing at the age of 10 with the Tobans and later Sherburn athletic clubs. At 17, he sought to try out with the St. James Canadians junior team in 1939 and after being told he was too young, instead joined the Winnipeg Monarchs. He played the 1939–40 season with them and scored 21 goals in 24 games. Though still eligible for junior hockey, Mosienko left the team as they did not cover the cost of transportation for him within Winnipeg, later noting that as he was from a poor family he needed that money to help him out.

Chicago Black Hawks
Chicago Black Hawks player Joe Cooper discovered Mosienko playing on outdoor rinks in Winnipeg and recommended that Hawks management sign him. He had also been signed by the New York Rangers, but as Chicago submitted their claim first, they retained Mosienko's rights. Mosienko signed with Chicago at the age of 18 and was assigned to their minor league team, the Providence Reds of the American Hockey League. He played 36 games with Providence during the 1940–41 season, scoring 33 points, before being reassigned to the Kansas City Americans of the American Hockey Association, whom Chicago had just purchased as an additional minor-league affiliate. He finished the AHA season with 4 points in 7 games. The Black Hawks first recalled Mosienko in 1942, replacing players who had left to fight World War II. He scored his first two NHL goals on February 9, 1942, 21 seconds apart. An ankle injury kept him out for several weeks, and he finished the season with 14 points in 12 NHL games.

Mosineko returned to Winnipeg for the summer of 1942 and worked in a defence plant. He tried to enlist in the military but was refused for medical reasons, likely related to his small size. He was also unable to cross into the United States as a result of travel restrictions and played the 1942–43 season primarily with the Quebec Aces of the Quebec Senior Hockey League, appearing in 8 games for them and scoring 5 points. He appeared in two games when the Hawks traveled to Toronto. He finally established himself as an NHL regular in 1943–44 and scored a career high 70 points. Mosienko and his linemates Clint Smith and Doug Bentley combined to score 219 points, at that time an NHL record. His 70 points set a team rookie record that stood for 37 years until broken by Denis Savard in 1980–81.

In 1944–45, Mosienko finished sixth in league scoring with 54 points and did not record a single penalty minute throughout the course of the season. He was awarded the Lady Byng Memorial Trophy as the NHL's most gentlemanly player and named a second-team All-Star. He was also voted the most popular player on the team by its fans.

The following season, Chicago coach Johnny Gottselig paired Mosienko with the Bentley brothers: Max and Doug. The trio formed what came to be known as the "Pony Line" due to their speed and small size and emerged as one of the top scoring trios in the NHL. Max Bentley led the league in scoring in 1945–46, and despite missing time with a knee injury, Mosienko finished sixth. He was again named a second-team All-Star. He played in the first National Hockey League All-Star Game prior to the start of the 1947–48 season, during which he suffered a broken leg and was initially feared lost for the year. He ultimately missed the first two months of play, while the Pony Line was broken up for good shortly after his injury when Max Bentley was traded to the Toronto Maple Leafs.

Mosienko remained a key contributor to the Black Hawks offence, appearing in five All-Star Games during his career. His most famous moment came on March 23, 1952, in a game against the New York Rangers on the final night of the regular season. He scored three goals in a 21-second span of the third period against New York's Lorne Anderson to set a new record for the fastest hat-trick by one player. Linemate Gus Bodnar assisted on all three goals, and he nearly had a fourth goal 45 seconds later on a shot that deflected off the goalpost. Mosienko's feat remains an NHL record.

Due to his small size, Mosienko was a frequent target of physical abuse from larger opponents, and suffered numerous injuries as a result. Despite this, he refrained from retaliation and was well regarded across the league for his gentlemanly conduct. In 14 NHL seasons, he totaled only 121 penalty minutes and had just one fight.

After the 1953–54 season Mosienko decided to retire from hockey, but the Black Hawks asked him to return for one more season, which he agreed to do. He played 64 games and had 27 points in the 1954–55 season, his last in the NHL. Feeling the Black Hawks lowballed him in regards to salary, Mosienko retired a second and final time from the NHL in 1955.

Winnipeg Warriors
Retired from the NHL, Mosienko returned to Winnipeg, where he was offered a contract with the newly founded Winnipeg Warriors of the Western Hockey League. He led the Warriors to the President's Cup championship in the team's first year of 1955–56 and helped the Warriors defeat the Quebec Hockey League's Montreal Royals for the Edinburgh Trophy, Canada's minor professional championship. He scored two goals in the deciding sixth game of a 3–1 victory to capture the trophy. He was named to the WHL All-Star team three times: 1957, 1958 and 1959. Though he scored 88 points in the 1958–59 season, Mosienko announced his retirement after 18 seasons of professional hockey. He coached the team for one season in 1959–60, but quit after one year as he found it took up too much time.

At the time of his retirement from the NHL, Mosienko was seventh all-time in league scoring with 540 points. He was inducted into the Hockey Hall of Fame in 1965, and to the Manitoba Sports Hall of Fame and Museum in 1980. Additionally, Mosienko is honoured by the Manitoba Hockey Hall of Fame and in 2000 was named to its provincial All-Century first All-Star team.

Off the ice
In 1947 Mosienko and Joe Cooper built a bowling alley in Winnipeg, which opened in 1948. Cooper ran the business during Mosienko's playing career, though Mosienko would actively take a role during summers when he was in Winnipeg. On retiring from hockey Mosienko became more involved, though his relationship with Cooper worsened, and Mosienko ultimately bought out Cooper's interest in the bowling alleys. Mosienko and his family owned and operated them until 2007. He was a supporter of Winnipeg minor hockey and Manitoba oldtimers associations. There is an arena named in Mosienko's honor in his hometown of Winnipeg.

Mosienko had known his wife, Wanda Swita, since they were both children; they were married on July 13, 1946. They had three children: two sons and one daughter. His grandson, Tyler, was also a professional hockey player. The younger Mosienko remembers his grandfather as a humble man who would join him on the family's backyard rink when he was learning to skate as a child. Diagnosed with stomach cancer in 1986, Mosienko initially recovered, but in 1991 a follow-up exam found the cancer had spread to his brain; he died on July 9, 1994. He was buried in Brookside Cemetery in Winnipeg. In October 2021, a book about Mosienko's life titled Mosienko: The Man Who Caught Lightning In A Bottle by Ty Dilello was published.

title: Bill Mosienko


text: User talk:202.7.183.130

title: User talk:202.7.183.130


text:
User talk:Aurbina

Image:Moai Rano raraku.jpg
Thank you for uploading Image:Moai Rano raraku.jpg, it is currently nominated to become a featured picture on en.wikipedia. Would you happen to have a higher-resolution version of this photograph? ˉˉanetode╞┬╡ 15:06, 12 June 2006 (UTC)

Featured Picture
Congratulations! --Fir0002 01:07, 26 June 2006 (UTC)

Do you..?
Do you live on Easter Island? That featured picture is really nice. And I was just curious to know if you were a native to the island? -24.92.41.95 15:33, 3 September 2006 (UTC)

Re: Do you..?
No I do not, I was only for a week there.. and only a single day at that location, but I was looking for that particular Moai all the time, so I took a sequence of 6 or 7 pics of it when I found it.

Use in textbook
HI, I'd like to use this picture in a textbook on sustainable development. Although you released it to public domain, it would only be fair to recognize your contribution. Should it just say "Courtesy of Aurbina, Wikimedia Commons"? If you want to get back to me by e-mail: efranc65_at_yahoo_com. Thanks! Efranc65 (talk) 16:49, 21 October 2008 (UTC)

Now Commons
File:Motu Nui, Easter Island.jpg, an image uploaded to Wikipedia from this account in February 2005, was transferred to commons with the new file name File:Motu Nui Easter Island.jpg (Commons:File:Motu Nui Easter Island.jpg) by JanKub. — Athaenara ✉  07:03, 16 July 2009 (UTC)

Orphaned non-free image File:Gits-megatechbody-picturevinyl.jpg
 Thanks for uploading File:Gits-megatechbody-picturevinyl.jpg. The image description page currently specifies that the image is non-free and may only be used on Wikipedia under a claim of fair use. However, the image is currently orphaned, meaning that it is not used in any articles on Wikipedia. If the image was previously in an article, please go to the article and see why it was removed. You may add it back if you think that that will be useful. However, please note that images for which a replacement could be created are not acceptable for use on Wikipedia (see our policy for non-free media).

PLEASE NOTE:

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Thank you. DASHBot (talk) 05:35, 12 May 2010 (UTC)

Notification: changes to "Mark my edits as minor by default" preference
Hello there. This is an automated message to tell you about the gradual phasing out of the preference entitled "Mark all edits minor by default", which you currently have (or very recently had) enabled.

On 13 March 2011, this preference was hidden from the user preferences screen as part of efforts to prevent its accidental misuse (consensus discussion). This had the effect of locking users in to their existing preference, which, in your case, was. To complete the process, your preference will automatically be changed to  in the next few days. This does not require any intervention on your part and you will still be able to manually mark your edits as being 'minor'. The only thing that's changed is that you will no longer have them marked as minor by default.

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Thank you for your understanding and happy editing :) Editing on behalf of User:Jarry1250, LivingBot (talk) 17:07, 15 March 2011 (UTC)

Thank You for Wikipedia Freeware Picture
Hello, My name is Duane Hurst and I recently made a free (non-commercial) English web site to share information with people. I added links to your Wikipedia/Wikimedia freeware picture (http://en.wikipedia.org/wiki/File:Moai_Rano_raraku.jpg). I also gave credit to you on my web pages for your work. Thank you for sharing with the public. My website is:

http://www.freeenglishsite.com/

I add pictures such as yours to one of the following major sections of my site: 1. World section - contains information and over 10,000 images of every world country and territory. Link at: http://www.freeenglishsite.com/world/index.htm

2. USA section - contains information and images of every USA state and territory. Link at: http://www.freeenglishsite.com/world/usa/index.htm

3. English section - "Mel and Wes" lessons in conversation format. Stories are located in various USA states and world countries such as China, England, Germany, Japan, Mexico and Thailand. Each lesson has many slang terms and idioms, which I link to my Slang Dictionary. This eventually will have over 5,000 terms. Currently, it has about 3,000 slang and idioms. I regularly add new lessons and slang terms. Link at: http://www.freeenglishsite.com/english/lessons/index.htm Slang Dictionary link at: http://www.freeenglishsite.com/english/slang/Eslang_a.htm

Prior to retirement, I taught English at several private and public universities in the United States.

Please share this free site with your friends. I hope all will enjoy the pictures and find the English information useful. Sincerely, Duane Hurst in Utah, USA

Email address: duanerhurst@freeenglishsite.com --75.162.209.2 (talk) 00:46, 17 December 2012 (UTC)

title: User talk:Aurbina


text: User talk:68.80.109.129

title: User talk:68.80.109.129