Tuesday, January 7, 2025

A large uniform universe necessarily has the spatial metric flat in Minkowski-newtonian model

UPDATE January 12, 2025: The flatness problem really is not about the flat spatial metric in the universe, but the question: why the mass density of the universe is quite close to the "critical density"?

The critical density means that the velocity of a galaxy at a distance R from us is close to the escape velocity from the mass M contained within the distance R from us.

For example, the universe might have a mass density which is only 1/100 of the current one. Then the velocity of a distant galaxy would be 10X the escape velocity.

Our Minkowski & newtonian model does not explain why the velocity is close to the escape velocity.

A possible explanation: a bounce-back model in which an initially almost static cloud of dust collapses, and then bounces back through some unknown physical mechanism. Then the dust cloud has roughly the critical density.


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In the Friedmann equations, the mass density of the universe has to be set very carefully to the "critical density", in order to ensure that the spatial metric is flat, and will stay flat for a long time. We know that the spatial metric in the observable universe is roughly flat on the large scale.

In our own, Minkowski-newtonian gravity model, the spatial metric is determined by a different inertia of a test mass m to different directions. For example, around a neutron star, the radial metric is stretched because the inertia of a test mass is larger in the radial direction: there is "energy shipping" to the test mass m if it moves radially, which adds extra inertia. The spatial metric bulges at the neutron star, and is not flat.

If the universe is spatially very large (much larger than 15 billion light-years), and almost uniform, then the inertia of a test mass m cannot vary much in any direction, within the observable universe whose radius is only 15 billion light-years. This implies that the spatial metric is almost flat.

We do not need any fine-tuning of the mass density of the universe to a "critical" value. It is enough to demand that the universe is large and almost uniform.

This observation solves the flatness problem of cosmology. We do not need to assume inflation to fine-tune the mass density to the critical one at the beginning.

It may also explain why the expansion looks similar to all directions. If we are dealing with a huge expanding dust ball, the expansion may locally look rather uniform. The uniformity of the CMB would be explained if the dust ball is really large.

Still, we have to assume uniformity of the mass density in a very large dust ball. Why did the uniformity arise?

Also, we do not understand how Minkowski-newtonian handles a dust ball which is much inside its own Schwarzschild radius. Why is it not frozen, like a black hole is?

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