As I understand, it is considered the weakness of gravity in relation to the strong, weak and electromagnetic that those interaction dominate at quantum scales while any gravitational interaction at quantum scales is imperceptible.
Gravitational effects are described in established theory as proportional to local mass.
The standard answer is the equivalence principle. Particles follow geodesics that are extremal paths in spacetime. These paths are the same for any mass m so long as this test mass is small m > 8πGm^2. If this test mass is sufficiently large it has its own appreciable curvature and this results in a highly complex problem similar to the 3-body problem in Newtonian mechanics.
I have though been thinking about this a bit differently. The equivalence principle states that the freely falling frame is equivalent to a frame in flat spacetime far removed from any gravity field. I have been thinking of this according to quantum field theory. A freely falling frame is one which has a vacuum equivalent to a quantum vacuum in flat spacetime. The geodesic the frame falls under is one which preserves this vacuum.
This may be extended to massive particles and massless particles. A massless particle is on a null geodesic and is then on a “null frame.” These frames are then projective varieties over massless quantum fields in that vacuum. These massless fields all have transverse modes, but no longitudinal modes. A massive particle has a longitudinal mode, and the vacuum modes have dispersion as a result. This is due to the presence of the Higgs field.
An accelerated frame is one which has a vacuum equivalent to a particular vacuum near a black hole. Depending on the acceleration parameter this can be a different vacuum. In this way aspects of spacetime physics can be thought according to quantum mechanical or quantum field properties. Spacetime is probably an emergent phenomenon of quantum physics.
Gravity as understood in a Newtonian sense can be seen as an illusion. According to the General Theory of Relativity, it is the curvature of spacetime which forces matter to move towards other matter (i.e. experience gravitational acceleration). The 'force' of the attraction is related to the gradient of the curvature. As James mentions in his comment above, gravitational interaction at quantum scales would be imperceptible, but would nevertheless exist as even at the small quantum scale, spacetime is occupied. You might not be able to measure it, but you could calculate it. Therefore all matter, at all scales are affected by spacetime curvature.
Ludwig, guess t you want to emphasise the duality between geometry and accelerations; one may interpret gravity as a geometric property, i.e., as the curvature of space.
I would say this view is a different, or a dual, way –although it is in some branches of physics a necessity– to look at one and the same interaction of gravity. What is a good view depends on the premise and goal. In classical physics one starts from geometry and tries to work out models for observations. In cosmology it goes the other way around; one starts from the observations (of light) and tries to work out the geometry of space from this.
Yes, quite right, I sometimes do both, for instance I am developing a measurement system based on a 12.5 cm refractor to measure the deflection of light around the Sun (Eddington type experiment), this should provide information of the gradient of spacetime curvature close to the Sun, and if accurate enough, could even provide details on the oblateness of the Sun. That is, I will be going from observations to the geometry.
To build upon the comments by Lawrence, Ludwig,......., equivalence principle from which ensues a geometrical formulation of gravitation (first realized by Einstein) holds the key to the answer. The observed equivalence principle according to which acceleration of an object is independent of its mass leads to the general theory of relativity in which it is the space-time geometry that masquerades as gravity. Then, in the absence of other forces, any test particle will simply be following a geometrical path (namely, the geodesic) regardless of its mass. This does not hold good for any other force, e.g. for charge particles in an electric field, acceleration depends on charge to mass ratio. That's why no other force can be `geometrified'.
If the question is taken to refer to equivalence principals, there's an interesting discussion of measurement methods in http://arxiv.org/pdf/gr-qc/0103067v1.pdf.
If taken in reference to the simplest, most fundamental observations of nature that originated during the Renaissance period, the answer seems painfully obvious: cannonballs and ball bearings in freefall are accelerated to the same velocity not by any property of their own internal structure and composition - but because of the Earth's internal structure and composition of its mass. The difference in the mass, and resulting gravitational effects, of a cannonball and a ball bearing is insignificant in comparison to the sum of their individual mass and the Earth's mass. Of course, the effect is much different when comparing the motion of a cannonball dropped from a tower on the Earth to one that is dropped from a tower on the moon.
Again, this is all obvious, except that it does not seem to have been properly appreciated by those who considered the results to be so profoundly significant...
The cosmic background gravitational potential plays a more important role than the gravitational force. I have proposed a cosmological model with several fundamental constants turned into functions of the gravitational potential.