In the previous Chap. 7 , we have shown that spacelike momentum field \(p^{\mu }\) so far ignored as non-physical ghost field is an inevitable constituent element of the spacelike part of physical space-time. If so, then, a natural question we will ask next would be “what about the timelike part of physical space-time ?” In order to answer this intriguing question, we first show that the Hamiltonian structure we have touched upon in Chap. 6 again plays an important role in leading the discussion. Except for a couple of towering accomplishments in modern physics, namely, quantum physics and Einstein’s theory of (special & general) relativity, so-called Nonlinear Science emerged in 1980s with catch-phrases of “chaos”, “soliton” and “fractal” provided a lot of important new concepts in modern physics. In our present context, we think that the concept of self-similarity in “fractal” becomes especially important. In Chap. 7 , we have investigated scale-free dynamics on electromagnetic field. In geophysical fluid dynamics, the most important dynamical element is known as Ertel’s potential vorticity (PV) closely related to the Casimir in the generalized Hamiltonian we discussed in Chap. 6 . Owing to the pioneering study of Penrose, we know that space-time can be represented as a certain kind of spin-network, so that we conjecture that, by applying the notion of self-similarity, the concept of PV may be extended to the timelike part of physical space-time. The main theme of this chapter is to show that it is actually the case.