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In the derivation of Newtonian Cosmology people simply replace for the mass inside a sphere as [see Liddle or Roos textbooks or APJ 1965 and 1996 papers]

$$M(r)=\frac{4π}{3} ρ\, r^3​​$$

​But this formula is based on volume of sphere in Euclidean geometry. If Universe would be curved the volume is different in say Spherical or Hyperbolic geometry. As a two dimensional example, the area of a circle of radius $r$ on a 2-sphere of radius $R$ is given by

$$A(r)=2\pi R^2\left(1-\cos\frac{r}{R}\right)$$

One may say that the $ρ$​​ is the effective one in Curved geometry, but then it should depend on $r$​​, violating homogeneity (as is evident by the 2d example in above).

Please give your answers or comments.

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  • $\begingroup$ Newtonian mechanics and gravitation exist in Euclidean space by construction. A curved space is not Newtonian. $\endgroup$
    – anna v
    Mar 9 '18 at 8:20
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Just remember this is just an ad-hoc argument to approach the dynamics of space-time. The notion of curved space-time is not present in the Newtonian formalism, if you actually want to be formal and include this from the beginning of the treatment, you'd need to use Einstein's field equations which will lead you to the Friedmann Equations.

It just turns out that if you follow the argument of the sphere in a Newtonian fashion, you will end up with something that looks very similar, but make no mistake, the actual dynamics of the space-time cannot be encapsulated by Newtonian dynamics

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  • $\begingroup$ But is there a bad habit in PSE of voting negatively and hiding? See my recent questions. Hiiiiii....from TeX.SE. $\endgroup$
    – Sebastiano
    Mar 18 at 23:23

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