  The gravitational energy shift for photons is extended to all mass-equivalent
energies $E = mc^2$, obeying the quantum condition $E = h\nu$.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.
