The gravitational energy shift for photons is extended to all mass-equivalent energies $E = mc^2$, obeying the quantum condition $E = h u$.On an example of a relativistic binary system, it was shown that the gravitational energy shift would imply,in contrast to Newtonian gravity, the gravitational attraction between full mass-equivalent energies. The corresponding space-time metric becomes exponential. A good agreement was found with all results of weak field tests of General relativity. The strong field effects in a binary system can be easily studied. A long standing problems of Pioneer and other flyby anomalies were also discussed in connection with the violation of total energy conservation. It was shown that relatively small energy non-conservation during the change of the orbit type could explain these persistent anomalies.
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