Consider two objects presented in the figure below. Objects have equal masses and are separated by a distance of 60 light seconds.
Assume that we move left object by 3 light seconds to the left in 30 seconds. This requires energy input, lets say it's equal to X. Change in potential energy is also equal to X.
Now, the right object does not 'know' yet that the left object was moved. Left object was moved 30 seconds ago and this information requires 60 seconds to reach the right object.
We have 2 options:
A: We can move the right object by 3 light seconds to the right with the speed of 0.1c now.
B: We can move it with the same speed and by the same distance later - let's say after 60 seconds.
Clearly case A requires the same energy input as when moving left object = X. This is because right object still 'thinks' that the left mass in the same place.
In the case B the right object 'knows' that the left mass was move so we will need less energy to move it. Energy needed < X.
The end state in A and B is the same. Why it is possible to achieve it with 2 different energy inputs? If we choose method A the increase in potential energy is smaller than energy input X. Where is the missing energy?
Maybe the 'speed of gravity' is infinite?