The Concept of Time in 2 D Gravity

We show that the " time " t s defined via spin clusters in the Ising model coupled to 2d gravity leads to a fractal dimension d h (s) = 6 of space-time at the critical point, as advocated by Ishibashi and Kawai. In the unmagnetized phase, however, this definition of Hausdorff dimension breaks down. Numerical measurements are consistent with these results. The same definition leads to d h (s) = 16 at the critical point when applied to flat space. The fractal dimension d h (s) is in disagreement with both analytical prediction and numerical determination of the fractal dimension d h (g), which is based on the use of the geodesic distance t g as " proper time ". There seems to be no simple relation of the kind t s = t d h (g)/d h (s) g , as expected by dimensional reasons.