If I do not interpret equivalence principle wrongly, it says that we could always free fall ourselves to achieve local inertial frame. The metric would be $ds^2=-dt^2+dx^2+dy^2+dz^2$.
Suppose an astronaut am nearby an object described by Schwarzschild metric, say a black hole. He free falls himself, subject to gravity only and follow the geodesic. What metric would he see? Is it still the ordinary Lorentz frame's metric or Schwarzschild metric?
And as an observer observing the tragedy, I am using Schwarzschild metric to view the event. Am I understanding it correctly?