  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.
