There are many sets of coordinates that provide different and complementary physical viewpoints of Schwarzschild spacetime. Here is another one.
(a) Consider again a material particle that radially free falls from rest at infinity. Show that the Schwarzschild coordinate time t is related to the proper time t_ff of the particle [we just called this tau in lecture] according to
dt = dt - (2Mr)¹/² / r - 2M dr.
Use this to transform the spacetime metric from Schwarzschild coordinates (t, r, theta, phi) to new coordinates (t r, θ, ∲).
