Gravitational Collapse (Simple model)

I've been plodding through Weinberg's 'Gravitation & Cosmology' and have a
specific question about section 11.9 which discusses the Oppenheimer Snyder
model for gravitational collapse - the simplest conceivable treatment of a
black hole:

There's an interior solution inside a uniform collapsing ball of dust and an
exterior (Schwarzchild) solution applying to the empty space outside the
ball. He does a coordinate conversion on the interior solution and shows
that it matches the exterior solution on the boundary.

I was hoping the first derivatives of g_mu nu would also turn out to be
continuous across the boundary (in the bar coordinates) but they're not. For
example along a curve of the form: r bar, theta bar, phi bar = Constants , t
bar = Varying, the function A is constant in the exterior region, but the
interior version of A (eqn. 11.9.32) has a nonzero derivative since it's a
function of r^2/R(t) = (r bar)^2/R(t)^3 and t can't be constant along this
curve.

Two questions:

(1) How does the continuity only of g prove the legitimacy of the (interior)
solution ?

(2) Does a global coordinate system exist in which the components of g are
differentiable even on the boundary ?

Something similar occurs in Newtonian theory: For a point fixed in space at
a radius r, the gravitational potential increases while more dust is falling
past radius r, but as soon as the boundary surface passes it, the potential
suddenly becomes constant. This leads to an infinite second time derivative
of the field at the spacetime boundary but it's not an issue since time
derivatives don't enter the field equation.

*****

To Bill Hobba, Tom Roberts, & Ken Tucker: Thanks much for your replies to my
previous post.




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