Gravitational domain walls and the dynamics of $G$

From the point of view of elementary particle physics the gravitational constant $G$ is extraordinarily small. This has led to ask whether it could have decayed to its present value from an initial one commensurate with microscopical units. A mechanism that leads to such a decay is proposed herein. It is based on assuming that $G$ may take different values within regions of the universe separated by a novel kind of domain wall, a "G-wall". The idea is implemented by introducing a gauge potential $A_{\mu\nu\rho}$, and its conjugate $D$, which determines the value of $G$ as an integration constant rather than a fundamental constant. The value of $G$ jumps when one goes through a $G$-wall. The procedure extends one previously developed for the cosmological constant, but the generalization is far from straightforward: (i) The intrinsic geometry of a $G$-wall is not the same as seen from its two sides, because the second law of black hole thermodynamics mandates that the jump in $G$ must cause a discontinuity in the scale of length. (ii) The size of the decay step in $G$ is controlled by a function $G(D)$ which may be chosen so as to diminish the value of $G$ towards the asymptote $G=0$, without fine tuning. It is shown that: (i) The dynamics of the gravitational field with $G$ treated as a dynamical variable, coupled to $G$-walls and matter, follows from an action principle, which is given. (ii) A particle that impinges on a $G$-wall may be refracted or reflected. (iii) The various forces between two particles change when a $G$-wall is inserted in between them. (iv) $G$-walls may be nucleated trough tunneling and thermal effects. The semiclassical probabilities are evaluated. (v)~If the action principle is constructed properly, the entropy of a black hole increases when the value of the gravitational constant is changed through the absorption of a G-wall by the hole.