Assuming the minimal time to send a bit of information in the Einstein clock synchronization of the two clocks located at different positions, we introduce the extended metric to the information space. This modification of relativity changes the red shift formula keeping the geodesic equation intact. Extending the gauge symmetry hidden in the metric to the 5-dimensional general invariance, we start with the Einstein-Hilbert action in the 5-dimensional space-time. After the 4+1 decomposition, we obtain the effective action which includes the Einstein-Hilbert action for gravity, the Maxwell-like action for the velocity field and the Lagrange multiplier term which ensures the normalization of the time-like velocity field. As an application, we investigate a solution of the field equations in the case that a 4-dimensional part of the extended metric is spherically symmetric, which exhibits Schwarzschild-like space-time but with the minimal radius. As a discussion we present a possible informational model of synchronization process which is inherently stochastic. The model enables us to interpret the information quantity as a new spatial coordinate.

34 pages, 7 figures