El Naschie’s Cantorian gravity and Einstein’s quantum gravity

A geometric formulation of the gravitation theory on a fractal space–time is given. Then, a connection is introduced on a tangent bundle, the connection coefficients, the Riemann curvature tensor and the Einstein field equations are calculated. We can characterize the “relative resolution parameter” as a ‘state scale’ of the coordinate system, as compared with Einstein’s formulation of the relativity principle, in which the relative speed characterize the movement state of the frame. In such a context, it results, by means of a dilation operator, the equivalence of this model with Einstein’s quantum gravity. Through the gravitational instantons synchronization, the correspondence with El Naschie’s gravity is established.