
Abstract 
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 “Gwall”. The idea is implemented by introducing a gauge potential A, 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 Gwall. The procedure extends one previously developed for the cosmological constant, but the generalization is far from straightforward: (i) The intrinsic geometry of a Gwall 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 Gwalls and matter, follows from an action principle, which is given. (ii) A particle that impinges on a Gwall may be refracted or reflected. (iii) The various forces between two particles change when a Gwall is inserted in between them. (iv) Gwalls 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 Gwall by the hole. 
