[syhprum (OP)]

It is well understood that increasing the verlocity of a body to near c leads to i an increase in its apparent mass also that if two bodies start out a being the same age and one is accelerated to near c when it returns it is apparently younger.
this leads to the question how much energy must be expended to make a mass of 1Kg one second younger ?

[Halc]

It is well understood that increasing the verlocity of a body to near c leads to i an increase in its apparent mass also that if two bodies start out a being the same age and one is accelerated to near c when it returns it is apparently younger.
this leads to the question how much energy must be expended to make a mass of 1Kg one second younger ?

This doesn't lead to that question actually since it isn't a function of mass or especially energy.

So for instance, I can put two 1kg clocks on either end of a 10m stick that pivots in the middle and re-orient the whole thing (negligible energy required if it's balanced) so one clock is 10m above the other. Wait a while (a long while) and the clocks will be 1 second out of sync.  Easy.
You can do it with linear motion as well (SR instead of GR).  Take any small amount of energy and get one of the clocks moving, in a circle say, so it stays close by.  After some time, it will fall 1 second behind the clock that isn't on the roundabout ride.

[Bored chemist]

So for instance, I can put two 1kg clocks on either end of a 10m stick that pivots in the middle and re-orient the whole thing (negligible energy required if it's balanced) so one clock is 10m above the other. Wait a while (a long while) and the clocks will be 1 second out of sync.  Easy.

Am I right in thinking that , if you did this near the surface of a neutron star you would get a much bigger rate of change between the clocks- so the rate of divergence of the clocks is not a well defined function of energy or time?

[Halc]

Am I right in thinking that , if you did this near the surface of a neutron star you would get a much bigger rate of change between the clocks- so the rate of divergence of the clocks is not a well defined function of energy or time?

The rate of divergence indeed isn't a function of energy. The rate of divergence (having 'rate' in the description) is divergence/time, so I don't see how it would not be a function of time.
The OP didn't ask about a rate. He just asked what energy it took to get a second of dilation difference between a pair of clocks and specified no maximum time in which to complete the task.

Near a neutron star the tidal forces are significant and it would take an absolute stupid amount of energy to get our rod with the two clocks back to the horizontal orientation, but we were never asked to do that, so it's not a problem.

[Bored chemist]

so I don't see how it would not be a function of time

Good point.

My point was that there's no energetic "penalty" for making the divergence quickly.