# GR exercise: falling particles on earth's surface

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.

• Here's a hint: the radial geodesic equation is identical to the Newtonian force law if the test particle has zero angular momentum. – Jerry Schirmer Aug 2 '14 at 20:21
• Don't have access to Shapiro's book, but would assume he wants the reader at that point to derive a tidal acceleration expression based on the Newtonian approximation? – Johannes Oct 11 '14 at 3:53

Possibly you are over-engineering this. The difference in the gravitational accelerations for the two particles is the second time derivative of the separation between the two particles. A binomial expansion then seems to give a simple result when $h \ll r$ ...
The quantitative statement? Not sure about that, but clearly you want the acceleration discussed above, to be below some tolerance threshold, which means $h$ must be smaller than some value involving the tolerance threshold, $r$ and $GM$.
However, you then also need to consider the fact that the separation grows with time. So what was an acceptable value of $h$ at $t=0$ will grow to become unacceptably large at some time later. This would require expressions for $h(t)$ and $r(t)$.