Under special conditions (Meissner-effect levitation in a high frequency magnetic field and rapid rotation) a disk of high-$T_c$ superconducting material has recently been found to produce a weak shielding of the gravitational field. We show that this phenomenon has no explanation in the standard gravity theories, except possibly in the non-perturbative Euclidean quantum theory.

In 4D nonperturbative Regge calculus a positive value of the effective cosmological constant characterizes the collapsed phase of the system. If a local term of the form $S^{\prime }={\Sigma }_{h\epsilon \{h_1,h_2,\dots \}}{\lambda }_h{V}_h$ is added to the gravitational action, where $\{h_1, h_2, \dots \}$ is a subset of the hinges and $\left\{{\lambda }_h\right\}$ are positive constants, one expects that the volumes $V_{h_1}, V_{h_2}, \dots$ tend to collapse and that the excitations of the lattice propagating through the hinges $\{h_1, h_2, \dots \}$ are damped. We study the continuum analogue of this effect. The additional term $S^{\prime }$ may represent the coupling of the gravitational field to an external Bose condensate.

A Wilson loop is defined, in four-dimensional pure Einstein gravity, as the trace of the holonomy of the Christoffel connection or of the spin connection, and its invariance under the symmetry transformations of the action is shown (diffeomorphisms and local Lorentz transformations). We then compute the loop perturbatively, both on a flat background and in the presence of an external source; we also allow some modifications in the form of the action, and test the action of "stabilized" gravity. A geometrical analysis of the results in terms of the gauge group of the Euclidean theory, SO(4), leads us to the conclusion that the corresponding statistical system does not develop any configuration with localized curvature at low temperature. This "nonlocal" behavior of the quantized gravitational field strongly contrasts with that of usual gauge fields. Our results also provide an explanation for the absence of any invariant correlation of the curvature in the same approximation.