The most obvious candidate for such a carrier is the set of wave functions of a particle. They take over the role that the particle world lines occupy in classical mechanics. Indeed, in the classical limit, any world line can be recovered from the wave functions via the principle of constructive interference. We shall find that the imposition of the Eotvos property on the wave functions is not only extremely restrictive, but also forces us into viewing spacetime from a perspective which is very different from the customary one.
First recall the de Broglie relations. They relate the wavelength and the frequency of a particle's wave function to its momentum and energy, and to its mass. Next recall that, quite generally, the set of dynamical constants of motion of particles not only characterizes the particles, but also indicates the nature of the reference frame within which they move. In fact, there is a one-to-one correspondence between the constants of motion (``complete set of commuting observables'') and the type of reference frame in which they are observed. Thus conservation of momentum and energy for all particles implies that they are observed in a free-float (inertial) reference frame. Consequently, if one chooses to describe the quantum dynamics of the particles in the momentum representation, then, by default, one has chosen to observe these particles relative to a free-float frame. This means that the choice of the representation characterized by a complete set of commuting observables (here the momentum and the energy) goes hand in hand with the customary view of observing spacetime from the perspective of free float frames.
In the momentum representation the wave function depends explicitly on the mass of the particle. Consequently, this wave function does not qualify as a carrier of the imprints of gravitation.
We shall now remedy this deficiency by exhibiting the quantum dynamics of the particle in a new representation relative to which the wave function does have the Eotvos property. The spacetime perspective of this representation consists of the familiar four Rindler sectors induced by two (noninertial) frames accelerating into opposite directions.