Possible Quantum Gravity Effects in a Charged Bose Condensate Under a Variable e.m. Field

Abstract

We prove that in Euclidean quantum gravity, in the weak field approximation, a “local” positive cosmological term μ2(x) induces localized gravitational instabilities. Such a term can be produced by the coupling to an external Bose condensate. We study the static classical limit of the functional integral in the presence of the (regulated) instabilities. ∗e-mail: modanese@science.unitn.it This model is applied to a phenomenological analysis of recent experimental results. A new demonstration experiment is described. 04.20.-q Classical general relativity. 04.60.-m Quantum gravity. 74.72.-h High-Tc cuprates. The behavior of a Bose condensate – or more specifically of a superconductor – in an external gravitational field has been the subject of some study in the past [1]. The presence in a superconductor of currents flowing without any measurable resistance suggests that it could be used as a sensitive detection system, in particular for gravitational fields. The possible back-reaction of induced supercurrents on the gravitational field itself has been studied too, in analogy with the familiar treatment of the Meissner effect. As one can easily foresee, it turns out that the “gravitational Meissner effect” is extremely weak: it was computed for instance that in a neutron star with density of the order of 10 kg/m the London penetration depth is ca. 12 km [2]. The reason for the extremely weak coupling of the supercurrents to the classical gravitational field is very general and originates of course from the smallness of the coupling between gravity and the energy-momentum tensor of matter Tμν . One might wonder whether in a quantum theory of gravity – or at least in an approximation of the theory for weak fields – the Bose condensate of the Cooper pairs, due to its macroscopic quantum character, can play a more subtle role than a simple contribution to the energy-momentum tensor. In a quantum-field representation the condensate is described by a field with non-vanishing vacuum expectation value φ0, possibly depending on the spacetime coordinate x. It is interesting to insert the action of this field, suitably “covariantized”, into the functional integral of gravity, expand the metric tensor gμν in weak field approximation and check if the only effect of φ0(x) is to produce gravity/condensate interaction vertices proportional to powers of κ = √ 16πG. One finds that in general this is not the case; the quadratic part of the gravitational action is modified too, by receiving a negative definite contribution. It can thus be expected that the condensate induces localized instabilities of the gravitational field, in a sense