Someone moves to the top of the World Trade Center in New York and moves back to the ground floor after one year according to a clock located at the ground floor. By how much did this person age?
In this problem, we can use that for weak gravitational fields the metric only slightly deviates from the Lorentzian metric, according to:
ds^2 = (1 + 2 V) dt^2 - (dx^2 + dy^2 + dz^2)
where V is the gravitational potential (denoted as phi in your notes) and we're using c = 1 units. The meaning of this is as follows. The value for the coordinates t, x, y, and z are arbitrary, they are just labels denoting space-time points that you are free to change, but ds^2 evaluated between two neighboring points will always tell you the distance between the points, no matter how you have decided to relabel the space-time coordinates. Here the distance refers to a distance defined in the four dimensional space time: ds^2 = time difference^2 - spatial ...
This solution provides a detailed, accessible solution fro people without knowledge of the theory of relativity.