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 $\alpha$, where $\alpha =0.5$ corresponds to general relativity. For $\alpha$ 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 $\alpha$ must lie between $0.4$ and $0.55$.