Wheeler-dewitt Metric and the Attractivity of Gravity

We investigate the class of ultralocal metrics on the configuration space of canonical gravity. It is described by a parameter α, where α = 0.5 corresponds to general relativity. For α less than a critical value the signature is positive definite, while for all other values it is indefinite. We show that in the positive definite case gravity becomes repulsive. From the primordial helium abundance we find that α must lie between 0.4 and 0.55. To appear in Phy. Lett. A In observations of phenomena both within and outside the solar system, the general theory of relativity has passed many viable tests in the last decades. This success, together with the conceptual simplicity and beauty of the theory, is the reason why it is commonly regarded as the fundamental theory of gravity. And yet, it is desirable to study alternative theories of gravity, for the following reasons. First, by recovering general relativity from a more general, alternative, theory in an appropriate limit one can quantify its success by excluding parameter values referring to the alternative theory. The perhaps best known example is the Jordan-Brans-Dicke theory which in addition to the metric contains a scalar field playing the role of a dynamical gravitational “constant.” The coupling of this additional field to matter is specified by a parameter ω which must satisfy, from Viking radar measurements, the observational constraint ω > 600 (ω → ∞ recovers general relativity). The second reason is quantum theory. Since all other known interactions are successfully described in a quantum framework, one expects that a quantum theory of gravity is also more fundamental than its classical counterpart. Such a theory could then in principle lead to classical limit different from general relativity. One example is superstring theory with its prediction of dilaton fields (similar to the Jordan-Brans-Dicke field) and antisymmetric tensor fields in addition to the metric. While most alternative theories, such as scalar-tensor theories or higher derivative theories, are given in a Lagrangean prescription, the viewpoint put forward here will be the opposite one. The motivation is provided by canonical quantum gravity in its geometrodynamical version, where the central kinematical concept is a wave functional on superspace, the configuration space of all three-geometries. By varying the metric on this configuration space, we obtain a class of alternative theories even in the classical limit (to which the discussion will be restricted). The study of these alternative theories will provide an interesting new viewpoint on general relativity itself. The central role in canonical gravity is played by the Hamiltonian constraint (with c = 1)