(a) One possible choice of slices of simultaneity for Schwarzschild spacetime is the set of 3-surfaces {t = const}, where t is the Schwarzschild time coordinate. Show that the unique family of observers for whom these are the simultaneities are the static observers, with world lines {(r, θ, ∅) = const, t varying}. Explain why these slices of simultaneity and families of observers exist only outside the horizon of a black hole and cannot be extended into the interior. Draw a picture of the world lines of these observers and their slices of simultaneity in an Eddington-Finkelstein spacetime diagram.

(b) A second possible choice of simultaneities is the set of 3-surfaces {t̃ = const}, where t̃ is the Eddington-Finkelstein time coordinate. What are the world lines of the observers for whom these are the simultaneities? Draw a picture of those world lines in an Eddington-Finkelstein spacetime diagram. Note that they and their simultaneities cover the interior of the hole as well as its exterior.