The Wider Perspective

We comprehend gravitation in two stages. First we identify the agent which carries the imprints of gravitation. In Newton's formulation this agent is the set of particle trajectories, in Einstein's formulation the set of particle world lines, and in the quantum formulation the set of correlations between the wave amplitudes in a pair of oppositely accelerating Rindler frames.

In Newton's formulation the imprints consist of the bending of the particle trajectories, in Einstein's formulation they consist of the deviations of the geodesic world lines, and in the quantum formulation they consist of the deviations of the correlations away from the ``Planckian'' and the ``fluctuation'' spectral values given by Eq.(13).

The second stage of our comprehension consists of relating these imprints to the source of gravitation. In Newton's formulation this relation is the Poisson equation, in Einstein's formulation the field equations of general relativity, and in the quantum formulation we do not know the answer as yet.

The equations for gravitation are an expression of the relation between the properties inside a box and the resulting geometrical imprints of gravitation on the surface of the box which surrounds the matter source of gravitation. In the Newtonian theory the mass inside the box is proportional to the total amount of gravitational force flux through the surface of the box. In Einstein's theory the amount of energy and momentum in the box is proportional to the total amount of moment of rotation on the surface of the box[16,17].

Alternatively, if the box is swept out by a coplanar collimated set of moving null particles (e.g. neutrino test particles), then, upon moving from one face of the box to the other, the neutrino pulse gets focussed by an amount which is proportional to the amount of matter inside the swept-out box. This proportionality, when combined with energy-momentum conservation, is expressed by the Einstein field equations[18]. In this formulation the directed neutrino pulse is a classical (i.e. non-quantum mechanical) carrier of the imprints of gravitation, and the amount by which the pulse area gets focussed is the gravitational imprint which is also classical. By letting the neutrino pulse go into various directions, one obtains the various components of the Einstein field equations.

It is interesting that the logical path from the classical imprints to these field equations consists of a temporary excursion into the quantum physics relative to an accelerated frame. This excursion starts with the demand that one describe the state of the matter inside the box relative to the frame of a uniformly accelerated observer. The acceleration of this frame is to be collinear with the motion of the neutrino pulse. If one complies with this demand, then the matter passing through the neutrino pulse (event horizon) area is a flow of heat energy relative to the accelerated frame. The temperature is the acceleration temperature given by the Davies-Unruh formula. This permits one to assign an entropy to the matter inside the box swept out by the neutrino pulse. Consequently, the Rindler (boost) heat energy in the box is the product of the matter entropy times the acceleration temperature.

Finally, the Einstein field equations follow from the Bekenstein hypothesis[19] that the entropy be proportinal to the change of the area of the neutrino pulse area as it passes through the box. It is obvious that this deduction of the field equations from the Bekenstein hypothesis is only a temporary excursion into the quantum physics relative to an accelerated frame. Indeed, even though the proportionality between entropy and area consists of the squared Planck-Wheeler length, reference to Planck's constant $\hbar$ gets cancelled out by the Davies-Unruh temperature in the product which makes up the Rindler heat flux, the source for the Einstein field equations. Consequently, the disappearance $\hbar$ from the heat flux guarantees that $\hbar$ will not appear in the Einstein field equations.

It seems evident that a quantum mechanical comprehension of gravitation must start with a purely quantum mechanical carrier of its imprints.