Bjarne said:
All astronomic objects are periodical deceleration and accelerating.
In the deceleration period kinetic energy most be lost, - and will be nowhere to be found, where is it really ?
Maybe you will say it is converted to potential gravitionel energy, but the problem is that we can point on it - can we?
The total effective energy of something freely falling in a gravitational field (subject to some constraints) is normally assumed to be constant in relativity as in Newtonian theory, so that for example the total gravitational effect of a system of bodies as seen from a distance is constant regardless of internal gravitational interactions (assuming that those interactions are not strong enough to produce significant gravitational radiation).
One simplified way of looking at it is that time dilation due to gravitational potential modifies the effective rest energy of a test object as seen from a distance, exactly cancelling out changes in the kinetic energy. (Of course the rest energy as seen locally is unaffected). The difference between the original rest energy and the time-dilated rest energy is effectively equivalent to the potential energy.
Unfortunately, that simplified idea has serious limitations, in that the time dilation back on the source due to the test object also modifies the effective rest energy of the source by exactly the same amount, so the system has actually lost twice the required potential energy when energy is added up in that simplified way.
I'm not aware of any widely accepted satisfactory way of explaining the location of potential energy in GR, and it can certainly be proved that the location of gravitational energy cannot be uniquely defined. However, if you look at field theory approaches instead, in a weak approximation, an alternative model is to assume a source density of energy in the field of g
2/(8 pi G) for field with acceleration g, in a similar way to the energy density of an electrostatic field. With this assumption, the total energy in the field is positive and mathematically exactly opposite to the potential energy. This means that the doubled potential energy lost by each pair of source objects due to mutual time dilation is corrected by the positive total energy of the field, giving the expected conserved total energy as in the Newtonian view. Although this works mathematically, there are no obvious grounds for assuming that it represents physical reality, and it is incompatible with the usual GR approach, so it's probably best to treat it just as an illustration.
Within GR itself there are approaches that use "pseudotensors" (in particular the Landau-Lifgarbagez pseudotensor) to describe a possible way of mapping gravitational energy in such a way that total energy is conserved from a given point of view, subject to certain restrictions. I believe that this approach also effectively assigns a sort of energy density to the field, but I don't know how that compares with the field approach described above.