The archimedes principle of time-dilation: If you go against energy your clocks will slowdown and if your clocks take energy pills they will run faster.
All types of energy slows down clocks, not just gravitational energy and there is nothing Einsteinian about it. Gravity just slows down in a particular way than any other forms of energy and Einstein gave its exact form. I will derive time dilation of gravity from
Time Dilation and Blackholes:
Energy slows down clocks because in the first place its energy that runs the clocks. If you do not have energy you will drop down dead. If a clock does not have energy it will drop down slow and then dead. If time does not have energy time will slow and eventually drop down dead. Time is dead in a blackhole, therefore time and space and energy they do not run in a blackhole. In-fact a blackhole is an intense form of gravity and has intensely stolen all space and time in the region. The fact is it has stolen all energy which is how space and time etc are used to create energy. [Energy is how space and time etc are used to create "energy"]. In-fact a blackhole is not black in color it might be an immense cosmos of colorful lights all decorated in a beautiful fashion but for each element that are there, there is no space or time available to them in a way space-time is not available to them, in a way actually energy is available to them. This is a overall region where this is occuring not a point like region. In the total region if you gain some energy then in a while or in a spacewhile you lost your energy again and this sports ensues. Eventually some elements/members may gain some energy towards the complete end of the region [called ergosphere] from where it escapes and reaches us. We see a blackedout space but we do not have eneough escapedos to tell us what it really looks like within: an osmosis of color or not. The combination of space, time and their derivatives making Energy is mathematics but exactly how and their evidence in nature is Physics.
All forms of energy dilate or contract time [depending on whether you go from more energy to less or the reverse]. A lesser to more energy means a gain in energy. So if time units are not adjusted this would mean energy is not conserved, compared to a situation where there is no such energy and the space-time is flat. [I described all this in a long intuitive article "Time dilation"]. So more energy or surplus would be accommodated by contracting the units of time which means time runs faster which should be called time contraction. Which is why in a lesser Gravity field time has to run faster because lesser Gravity [away from say earth] is a more energy compared to more Gravity [closer to earth] thats why you will see “Gravity slows down clocks” statements and receding objects are redshifted [redshifted=time runs faster because the time that was lost by going into making a more-curved region is now released as the region is flat.
If instead energy were not in the form 1/r [coming from inverse square central forces] the time dilation/contraction would be different in magnitude. Always remember your time slows down if your energy is less. Your energy is less if you are in a well. And energy slows down clocks because its energy that runs the clocks time in the first place [not the clock but the time] In-fact it take time and space to make energy therefore wherever there is energy there is space-time or lack thereof. These are called space-dilation, space-contraction, time-dilation and time-contraction effects. The literature just described length-contraction and time dilation for convenience and everyone remembered that, they never thought the other possibilities.
What will follow here is a very simple maths that will show why its a classical mechanics effect [gravitational time dilation and all time dilation are classical mechanics effects whose exact form is Relativity Special+General and which are significant only in the realm of the later. But here is a derivation why one could have got the formula for time dilation simply from classical mechanics. I do this for gravity case, one can perhaps apply a kinematic case which would give the dilation/contraction for special-relativistic or kinematic time dilation. One could also try other force forms and derive his equations for time dilation and be famous]
The maths has been derived it will be typeset later. (Edited MD, 25-02-2013) Well here is the maths which describes Newtonian Gravity showing signs of Relativistic dilation. Something I had mentioned (27-06-2012, this article) 6-12 months ago but hadn’t shown the results. There are some reservations in my mind since I do not exactly see the implications, but clearly the fractional increments in frequency, energy, length, time are related and the factor beta (Lorentz factor) is showing up.
If you check the last line you see that fractional shifts in frequency, energy, length and time are all beta which is the Relativistic time dilation/contraction factor given to the gravitational deflection r (also length) being comparable to the increments in time you have defined for the problem. That means howsoever small for gross objects there is a time dilation. It becomes apparent for objects that are so small that they fly off at tremendous speeds.