My question relates to page 118 of these notes, which is reproduced below for your convenience. Specifically, I have difficulty understanding the line "This shows that time slows in a region of stronger gravity (smaller $\Phi$)". On the contrary, if we define $\Phi_A = g(h_0+h)$ and $\Phi_B = gh_0$, then equation (A.1.9) seems to show that the time interval measured by Bob (at the bottom of the rocket) between signals is smaller than that measured by Alice. This is because in this scenario $\Phi_B - \Phi_A < 0$. Surely this isn't right. What is going wrong here?
If Bob says the time interval is smaller than Alice says, then time is going slower for Bob. For example, perhaps Bob says one second has passed while Alice says two second have passed. In this case Alice will think "come on Bob, your clock is ticking too slowly" and Bob will think "wow! Alice is aging rapidly."
This is correct. The fact that they're inside a non-accelerating rocketship doesn't matter, Bob is simply in stronger gravity because he's closer to the surface of the earth. Gravity gets weaker as you go away from the earth's surface. Obviously the effect on these scales is miniscule.