Unexpected distant values of m/r in a finite universe

[Jonathan Scott]

I've often wondered about what happens when you try to add up the potential due to everything in the universe in a Newtonian way, especially in the context of the "Sum for Inertia" which seems to suggest a connection between Mach's Principle and GR in the context of rotation.

Today I noticed an oddity with m/r values when one considers a finite closed universe, which can easily be visualised using the toy model of the universe as the surface of a ball.

How should we measure the distance r to an extremely distant source as seen by an observer? The obvious answer is by parallax. If we move our viewpoint slightly sideways, the direction of the distant object appears to change and we can triangulate the distance. An alternative way to see the same quantity is to consider waves being emitted by the object, and to measure the curve of the wave fronts as they pass the observer, and to calculate the distance from that. Even if we don't have sufficiently sensitive equipment to do this in practice, we can at least define distance as the quantity which would be measured by that process in theory.

But what happens in a finite universe, illustrated by the ball shape? Imagine an observer at the north pole. Wave fronts from objects which are further and further away will have less and less of a curve. An object at the equator will produce a wave front which is a straight line locally by the time it passes the north pole, suggesting infinite distance. And an object which is further away than the equator (half way round our toy universe) will actually produce wave fronts which are curved in a converging direction towards the observer when they reach the north pole, giving a negative calculated distance, approaching zero for an object near the opposite pole of the toy universe.

In Newtonian theory, a negative distance would make the potential repulsive, so anything more than half of the universe away in our toy ball universe would repel gravitationally rather than attracting, and the repulsion would be greater for the most distant objects (although perhaps in a more realistic universe m would be red-shifted too much to have much effect, and this toy model also fails to take into account the finite speed of light and the expansion of the universe).

I don't know whether this unexpected effect on m/r terms might have any relevance to the real universe, but it shows up an interesting and unexpected side-effect of attempting to apply Newtonian thinking on that scale.

 

[Jorrie]

I see two problems with your toy model that may possibly make your idea invalid. Firstly, the universe expands in a way that makes it (spatially) virtually flat at present. Even if it is spatially marginally closed, it must be so big that the cosmological horizon at the pole would have a tiny radius, as fraction of the circumference. So there is no hope of observing signals from your 'equator'.

I do not understand what you mean by "measure the curve of the wave fronts as they pass the observer". How is this different from parallax measurement? You would still need a very long baseline detection.

 

[Jonathan Scott]

I do not understand what you mean by "measure the curve of the wave fronts as they pass the observer". How is this different from parallax measurement? You would still need a very long baseline detection.

It is intended to be identical to parallax measurement (and equally difficult to detect in practice) but easier to visualise.

And I agree there would be many problems in relating it to reality. My main point is merely that any semi-Newtonian concept of adding up the gravitational potential due to distant galaxies contains a surprise if you evaluate it in the way that you would for nearby objects.