I understand that the coordinate system (CS) for a distant observer Od is different than that for an observer Of who is falling radially toward the event horizon of a non-rotating black hole (BH). Using the Schwarzschild metric, I would like to understand the transformation equations that calculate the change in observations of time, t, and radial position, r, as seen by both Od and Of. Here is the scenario I have in mind. Each observer has a one of two identical clocks, syncrhonized at time t0. Starting at a given radius, r0, which is the distance from the center of the BH (in Od's CS at time t0), Od observes as Of falls towards the BH. Od is maintained stationary at r0 by a suitable repulsion engine. t0 is the time a which Of begins to fall radially toward the BH. Using repulsion engines, there are stationary marker objects that can be observed by both Od and Of. As Of passes the marker mr at radius r (in Od's CS), both observers can measure the value of t at Od's radius r (no doubt getting different values. Od and Of can communicate with each other with light speed messages. The messages they exchange enable each observer to know: a) when the other (using the other's clock) saw Of passing mr, and b) when a message sent by one observer was received by the other.In this scenario, both Od and Of measure t(r) for the falling Of and both can calculate the two respective r(t) functions: td(r) and tf(r). Questions: 1) What are the functions rd(r) and rf(t)? 2) Are either/both of these two functions different as measured/calculated by Od and Of? If so, in what way? I am looking forward to a discussion answering these two questions, as well as a description of how the answers were developed.