From the viewpoint of an observer in an orbiting rocket, how does time behave as another rocket falls toward a black hole?

From the perspective of an observer in an orbiting rocket, time behaves differently for the rocket that is falling toward a black hole due to the effects of gravitational time dilation as described by Einstein's theory of General Relativity. As the falling rocket approaches the black hole, the intense gravitational field around the black hole affects the passage of time.

For the observer in the orbiting rocket, time will appear to run slower for the falling rocket relative to their own experience of time. This means that while time seems normal in the orbiting rocket, the falling rocket experiences time at a much slower rate as it gets closer to the black hole. Consequently, the events on the falling rocket unfold more slowly compared to those on the observer's rocket. This phenomenon is a consequence of the way gravity influences spacetime, where stronger gravitational fields (like those near a black hole) lead to slower passage of time.

This understanding of gravitational time dilation is essential in the context of black holes, illustrating how time does not behave uniformly across different gravitational conditions.