  A more rigorous treatment of the Schwarzschild metric by making use of the
energy-momentum tensor of a single point particle as source term shows that
  $g_{00}=-\{1-\frac{2GM}{c^2r}-\frac{8G^2 M^2}{c^4 r^2}(\theta (r)-1)\}\exp
[2(\theta (r)-1)]$
  $g_{rr}=\{1-\frac{2GM}{c^2r}-\frac{8G^2 M^2}{c^4 r^2}(\theta (r)-1)\}^{-1}$
  The existence of a discontinuity at r=0 leads to an infinite repulsive force
that will change the ultimate fate of a free fall test particle to a bouncing
state.
