Freely Falling Finite Frames Near a Black Hole

Abstract

It is well-known that the Riemann curvature tensor has no discontinuity at the black hole horizon. It is also well-known that a freely falling observer takes finite time to reach the horizon from an outside point. However, the usual assumption is that such an observer resides in a frame of reference (spaceship) of infinitesimal size. This assumption is justified as long as the coordinates are continuous enough to assume that the observer's frame is small compared to the variations of the metric from a local flat metric. Such an assumption may be invalid when the coordinate system has not only a discontinuity but a singularity like the one at the horizon. Hence, here, the characteristics of a finite frame (a spaceship) near a black hole horizon is discussed. It is shown that clocks placed at the front and rear ends have different time scales even in the limit when they reach the horizon at the same time. This renders such a frame physically meaningless. It is also argued that the forces that are expected to keep a realistic frame (like a spaceship) in one piece tend to zero near the horizon. So, a physical spaceship is expected to fall apart near the horizon.