Unfortunately, the maths is beyond me. Does it translate into saying that the Earth holds the moon in orbit without expending any energy?
Hehe you must be one of those people who're put off by ugly looking formulas :P It's very simple:
E = mc^2
m = m_0 / sqrt(1-v^2/c^2)
If you put v=0 you see that both types of mass are the same.
If you have v=c then relativistic mass m and the total energy E become infinite.
Regarding the moon you can use classical mechanics. If the orbit was circular:
F = ma
work = F * (distance traveled in the direction of F).
There's acceleration because of the curved orbit, so there's a force F. However that force is always perpendicular to the direction of motion, so the 2nd equation says no work is done.
It's not really circular but where there's a component of F in the direction of motion, kinetic energy is given to the moon. That's cancelled out by kinetic energy the moon loses in another part of the orbit when there's a component of F opposite to the direction of motion.
