Thought Experiment on Dark Matter

[notknowing]

Thinking about the hypothetical dark matter, I came to the following (primitive) thought experiment. Consider a large sphere (say radius R) filled uniformily with matter and a small test mass located at radius R1. This test mass will experience a force to the centre of the sphere and the strenght of this force is proportional to the mass content of the sphere limited to R1 (the effect of the outer mass cancel outs, as is well known). Now, increase the radius R to a very large value. Again, the force should remain the same. In fact, one can make R arbitrarily large.
So, now consider an infinite space, filled uniformily with matter. The test mass will now not experience a force towards a specific direction, because of symmetry.
So, one gets the strange thing that there is a difference between R being arbitrary large and R being infinite. What does this mean ?


Rudi Van Nieuwenhove

 

[Garth]

What does this mean ?

In the infinite universe there is no centre of mass. Every point is the centre of mass.

If you have a line with a sequence of numbers from x to y there is a number that is 'equidistant' from the end points x and y. In a line with an infinite sequence from - $\infty$ to +$\infty$ every point on the line is equidistant from the 'end points'.

Garth

 

[chronon]

So, one gets the strange thing that there is a difference between R being arbitrary large and R being infinite. What does this mean ?

You're right, it is a paradox. If you think of an infinite universe in terms of a large sphere with radius tending to infinity then you deduce that a static universe is impossible - everything would collapse together. But if you think of the forces on an arbitrary particle then you find that everything can stay where it is.

Einstein's solution to the problem was to accept the first view, but to introduce a cosmological constant to prevent it. There's some evidence that Newton also considered this, but in the end went for the second point of view, and so insisted upon 'absolute' space.

Neither solution really allows a static universe, since small perturbations would always grow.

The real universe is not static, so the problem doesn't arise.