Jason,
I'm not sure whether you speak partial differential equations or not, so I'll try to express the final equation in my essay in English.
The equation reads phonetically: partial-sub-rho(time) = partial-sub-x(mass)
What is shown in the derivation, but not explicitly shown in the final equation is that the units are inverse Planck's constant, that is, the right hand side is "per unit of action".
Now partial-sub-rho(time), where rho stands for volume, means "the change of time in a region of space".
and partial-sub-x(mass), where x stands for distance, means "the change of mass with distance" (across the region of space).
The result is a simple equation that represents space, time, distance, and mass in quantum units of action.
Now this probably won't make much sense if you think of solid mass, like a chunk of lead, but if you think that a gravitational field (in a volume of space) has energy (proportional to the field squared, like all fields, according to Maxwell) and use Einstein's E=mc**2, then we can think of the change in the gravity (across the region) where the distance x is the 'width' of the region in the direction of maximum gravitational change. It usually helps to draw a picture at this point where each side is represented.
So we have a change in time (time dilation) in a region where we have a change in the gravitational field energy/mass and the two are related. This simple (and beautiful) equation fell right out of my generalized Heisenberg quantum relation, which fell right out of my Master equation that claims that if we start with one field, and nothing else in the universe, the field can only evolve by interacting with itself.
Because you are very interested in time dilation, I thought that I would try to turn you on to this (quantum) way of looking at it.
Forgive me if you are an expert in partial differential equations, and simply let my explanation remain for non-experts who are interested in time dilation in a simpler way that it is typically explained.
Regards,
Edwin Eugene Klingman