I'm having some trouble with Exercise 5.1 in Shapiro's BH,WD&NS book, which goes as follows:
Consider two particles of mass $m$ at distance $r$ and $r+h$, such that $h\ll r$, on the same vertical line from the center of earth. The particles fall freely from rest a the same time $t=0$ towards the earth surface. Show that an observer falling with one particle will see the separation between the particles gradually increase. Translate this into a quantitative statement about the observer's local inertial frame.
This is the first exercise in this chapter, right after the first principle of GR. The author has not introduced curvature and other definitions yet. How can I understand this problem both quantitatively and intuitively?
Here is how I think about it:
I should solve this problem in two reference frames: the Earth, which is an inertial reference frame, with Minkowski metric, and the free falling particle frame. The two events are the other particle starting falling and after Earth time $dt$ the time and location of it. Compare in free falling frame if the particle is approaching to get the result. Am I conceptually correct?
Also I am confused because even if I treat the Earth as a stationary object it has gravitational field so how is distance $r$ and $r+h$ defined? I don't think the Earth frame is inertial.