We put the question, how gravitational fields are communicated to space, and how they are recognized in space by moving objects. The general answer given to this question in these days is that field quanta of the gravitational field, called gravitons, are responsible for the communication of such fields. This answer is taken serious here, and then consequently is put into serious doubts when studying several physically coherent consequences. We start by investigating the gravitational influence of cosmic mass associations on moving massive objects when hereby the effect of propagating gravity field quanta, i.e. the gravitons, communicating the position of gravity sources, is taken into account. It is shown that moving objects are affected by gravity sources from a relativistically displaced position, like stars are seen under an aberration angle compared to the true source position. The astonishing effect of that mislocation is that a moving object at its passage through the center of a centrally symmetric galactic mass association undergoes a permanent gravitational deceleration of its relative motion. Furtheron it is shown that under this new view the problem of a planetary object, orbiting a central mass like the sun, takes a non-Keplerian solution, because the orbiting planet permanently should lose orbital angular momentum, since it permanently experiences a gravitational force component antiparallel to its orbital motion. From that an orbital decay time can be derived which predicts that a terrestrial planet should have a spiralling-in period of only a few 104 years which, compared to the expected age of the Earth of about 4.5 Billion years, represents a big problem of understanding. In addition, in view of the presumed action of gravitons from an aberrated source position, this article studies the propagation of cosmic photons over the photospheric limbs of stars. We show that when the center of the gravity field acts from a relativistically displaced position, then no energy change of limb photons when again propagating to larger distances should occur - which does confirm Einstein‘s view - , but also no deflection from the original propagation direction of the photon should occur - which contradicts Einstein‘s derivations. In this article we do not yet offer a rational solution of these newly pinpointed problems, but we simply end with the recommendation to perhaps reinvestigate the theoretical concept of gravitons, thought as to be the quantum messengers of gravitational fields.