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?© BrainMass Inc. brainmass.com September 26, 2018, 10:33 am ad1c9bdddf - https://brainmass.com/physics/gravity/gravitational-time-dilation-602449
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.