Inside the event horizon of a Schwarzschild black hole the radial coordinate becomes timelike and the time coordinate becomes spacelike (they change signatures).
I read that timelike coordinates are one-directional and spacelike ones are bidirectional because an object with positive mass can only move along a timelike path, and that that means that inside the event horizon the radial coordinate can only decrease as an observer's proper time increases, leading to an inevitable encounter with the singularity. Does that mean that coordinate time can decrease if it's spacelike in the current metric?
Could an observer equipped with a rocket engine manipulate their time coordinate, or otherwise enter a geodesic with a fluctuating time coordinate? What would it look like?