I have been trying to figure out how Gauss' law of gravitation implies that the gravitational field within a spherically symmetrical shell is zero. I have spent a very long time thinking about this and trying to find an answer online, but no luck. I would appreciate hints rather than a complete answer.
So far, some of my thoughts have been:
The net flux of the gravitational field of any mass not contained within some bounded surface is zero through the surface, even though the gravitational field inside the surface due to this single mass is non-zeo
I have thought about constructing arbitrarily small closed surfaces within the shell, but that hasn't gotten me anywhere.
In a nutshell, I realise that the difficulty I am having is that all Gauss' law tells us about is what happens at some bounding surface. It doest seem to tell us anything at all about the field within some surface. How can one extrapolate this information from Gauss' law?