Is the observation of an observer local in general relativity? Taking Schwarzschild spacetime as an example, an observer at infinity can observe events happened in his neighbourhood at infinity and measure the corresponding physical quantities. I want to know whether the observer at infinity can observe events happened at finite $r$.
 A: No, the Schwarzschild observer is not a local observer. The observer at infinity is an idealised observer. An observer actually at an infinite distance would be useless because it would take infinite time for them to receive information from any finite $r$. To explain this observer, we need to invoke the concept of limits from calculus. We put the observer at a finite $r$ which is sufficiently far from $r=0$ that spacetime is almost flat, and then in the limit as $r\to\infty$ spacetime curvature approaches zero, and our finite observer approaches the observer at infinity.
However, the Schwarzschild observer does not directly observe events. Instead, they correlate events observed by local observers. A good description of this procedure is given in the Wikipedia article on Gullstrand–Painlevé coordinates. This article first explains Schwarzschild coordinates so that it can then describe how Gullstrand–Painlevé coordinates differ from them.

Schwarzschild coordinates
A Schwarzschild observer is a far observer or a bookkeeper. He does not directly make measurements of events that occur in different places. Instead, he is far away from the black hole and the events. Observers local to the events are enlisted to make measurements and send the results to him. The bookkeeper gathers and combines the reports from various places. The numbers in the reports are translated into data in Schwarzschild coordinates, which provide a systematic means of evaluating and describing the events globally. Thus, the physicist can compare and interpret the data intelligently.

