Re: [Phys-l] Gauss' Law

[John Denker <jsd@av8n.com>]

On 03/04/2008 11:09 AM, Robert Carlson wrote:
>  It's not clear to me, using symmetrical
> arguments, how one can establish that the mass outside
> the Gaussian surface in these situations contributes
> zero gravitational field on the Gaussian surface. 
> Perhaps I'm missing something.

As Joel R. said, this is the key point.  One way to do
business is:
  1) Start at R=0 and argue that the field is zero by
   symmetry /at that point/.  See proof below.
  2) Work your way out, using successive Gaussian surfaces,
   all centered on R=0.

We agree that there are innumerable other starting points
and innumerable other surfaces for which the symmetry
argument would not work.

The proof is not entirely trivial.  One classic method is
proof by contradiction:

   Assume by way of contradiction that the field is nonzero
   at R=0.  Suppose we have two observers, viewing the scene
   from two different directions.  The mass distribution
   is the same (rotational invariance) the field vector 
   must be different in the two reference frames (since it's 
   a vector, and is nonzero by hypothesis).   It's impossible 
   for the same source to produce two different fields vectors 
   at the same point.  In contrast, the situation makes sense
   if the field is zero at R=0.  QED.  

   FWIW this logic gets used a lot, in simple problems and 
   in not-so-simple problems.  One famous embodiment is the 
   Wigner-Eckart theorem.

   More generally, symmetry arguments are the heart and soul
   of physics.

For students who have never seen this type of reasoning, it
is pretty mysterious.  But with practice it becomes as easy
as breathing.


> One can of course start with a spherical or infinite
> cylindrical shell of mass and integrate using Newton's
> Law of Gravitation as is done in Tipler for a
> spherical shell. 

That's certainly doable, but not worth the trouble.