🚀 go-pugleaf

In article <1992Aug8.072223.11413@nuscc.nus.sg> scip1061@nuscc.nus.sg (Marc Paul Jozef) writes:
>
> I saw some people suggesting that the existence of
>matter in the universe would cause the expansion-rate
>to slow down _because of mutual gravitational attraction_
>
>Of course, if one reads `gravitational attraction' as
>an interpretation of the mathematical solutions of
>Einstein's global scale equations (where the stress-
>energy tensor is idealized as that of an ideal fluid),
>then the assertion is correct, albeit somewhat
>meaningless (just giving _names- ).
>
>If one however reads this `mutual gravitational attraction'
>in an intuitive `Newtonian' sense, then the assertion
>is false, of course: the supposed homogeneity of the
>universe implies namely precisely that the mutual attraction
>is zero (in one point, the net contribution of gravitation
>of other points is zero due to isotropy).

This argument, while extremely appealing, is not necessarily right.  
If you try to compute the Newtonian gravitational field at a point 
in an infinite space filled with matter of uniform density, you 
have to do an integral which diverges.  If the integral converged, 
then it would have to converge to zero as you say, because of 
isotropy.  But since it diverges, it's probably better
to say that Newtonian gravity doesn't make a definite prediction 
at all about this situation.

Lest you think I'm splitting hairs here, let me give you an 
argument which "proves" that an infinite uniform-density space 
_can't_ have zero Newtonian gravitational field everywhere.

Remember Poisson's equation:  Newton's law of gravity implies that

   Laplacian of Phi = 4 Pi G rho

where Phi is the gravitational potential and rho is the density 
field.  If there were no gravitational field anywhere, then Phi 
would be constant everywhere.  But then its Laplacian would be 
zero, and that's not a solution to Poisson's equation.  

The point is just that Newtonian gravity is not well-equipped 
to handle this case.  It's a good thing that Einstein came along.

-Ted