# Why is Newtonian cosmology correct for curved space?

The Newtonian model of an expanding Universe gives Friedmann's equation exactly for non-zero spatial curvature $k$ (see http://hyperphysics.phy-astr.gsu.edu/hbase/astro/expuni.html). Instead of using the concept of spatial curvature the Newtonian model introduces $k$ as a constant that is proportional to the total energy of a co-moving shell of matter.

But the derivation assumes that the total mass inside a shell of radius $r$, $M_r$, is given by:

$$M_r = \frac{4}{3}\pi r^3 \rho.$$ I assume that this expression is only true for flat space with $k=0$.

Is it just luck that the derivation gives the correct result for curved space as well?

• Seems to me to be the same type of coincidence as the Schwarzschild radii of GR and of a "Newtonian black hole" agreeing as in Can a black hole be explained by newtonian gravity? – ACuriousMind Mar 23 '16 at 13:35
• Wow, yes, the equation derived using the Newtonian approach is exactly the same as the first Friedmann equation whether $k$ is zero or non-zero. Whether there's any great insight to be had here is debatable, but it's certainly an interesting result. – John Rennie Mar 23 '16 at 17:01
• @John Rennie : or wow, maybe it's the other way around. – John Duffield Mar 23 '16 at 21:22
• Ok, the method used isn't valid for an infinite mass distribution and the calculation has been deliberately manipulated to reproduce the Friedmann equation. The energy calculation should work taking any fixed point as an origin, not necessarily the centre of the shell, but it obviously doesn't so the deriver has manipulated the working to obtain the result they wanted. A shame, as for a moment I thought it might be fun to look into. – John Rennie Mar 24 '16 at 6:44

Why is Newtonian cosmology correct for curved space?

Actually, it isn't. The expansion of space is increasing, and space isn't curved. See this from the NASA WMAP article from : "We now know (as of 2013) that the universe is flat with only a 0.4% margin of error".

The Newtonian model of an expanding Universe gives Friedmann's equation exactly for non-zero spatial curvature $k$ (see http://hyperphysics.phy-astr.gsu.edu/hbase/astro/expuni.html).

The hyperphysics page says "a spherical shell of mass m which is expanding can be described in terms of its kinetic energy and gravitational potential energy". That's fine as far as it goes, but it's nothing like the universe. Space expands. It never ever collapsed due to its own gravity, because a gravitational field is a place where space is "neither homogeneous nor isotropic", not a place where space is falling towards some centre. And do note that the expansion is speeding up, not slowing down.

Instead of using the concept of spatial curvature the Newtonian model introduces $k$ as a constant that is proportional to the total energy of a co-moving shell of matter.

The concept of spatial curvature is suspect anyway, since only one of the three shape of the universe solutions was ever going to be right. In addition note this in this Baez article: "Similarly, in general relativity gravity is not really a 'force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial".

But the derivation assumes that the total mass inside a shell of radius $r$, $M_r$, is given by: $M_r = \frac{4}{3}\pi r^3 \rho$. I assume that this expression is only true for flat space with $k=0$. Is it just luck that the derivation gives the correct result for curved space as well?

IMHO it isn't luck, it's judgement. Imagine if I came up with an equation that covered all the bases for curvature and matched some Newtonian expression for a universe where the expansion is slowing down, such that it's "asymptotically approaching a rest condition at infinite time". If you then discovered that the universe is flat and always was flat, and that the expansion isn't slowing down, you wouldn't be overly impressed. Particularly if you knew that the "coordinate" speed of light increases with elevation. As Einstein said, the speed of light is spatially variable. What this means is that the ascending photon speeds up. I know this is counterintuitive, but it's true, check that with the PhysicFAQ editor Don Koks. When you replace that shell of matter with light, you get something that's much more like the universe. This is neither Newtonian cosmology nor curved space, but IMHO the universe evolving over time is akin to pulling away from a gravitating body in space. I suspect this is the underlying answer to your question.