10.4 Two dimensional version of the asymptotic velocity

In the proof of strong cosmic censorship, it is of importance to note that it is meaningful to consider the velocity to be a two-dimensional object. The two-dimensional character is most easily seen by considering the solution in the disc model. Given a solution x = (Q, P ) to Equations (16View Equation) – (17View Equation), let z = ϕRD ∘ x, where ϕRD is defined in Equation (24View Equation). Then the limit
[ z ρ ] lim ----- (τ,𝜃) τ→ ∞ |z|τ

always exists due to [79Jump To The Next Citation Point, Lemma 6.17, p. 1011] (as an aside, it is worth noting that ρ∕|z| is a real analytic function from the open unit disc to the real numbers if ρ is the hyperbolic distance from the origin of the unit disc to z; see [79Jump To The Next Citation Point, Remark 6.18, p. 1011]). It would be reasonable to call this limit, which we shall refer to as v(𝜃), the velocity, since it contains information not only concerning size, but also concerning direction. However, the most important aspect of this construction is that it is two-dimensional. An essential step of the proof of strong cosmic censorship is to perturb away from zero velocity. In the case of polarized Gowdy, this is not possible. However, if the velocity is a two-dimensional object, it is at least potentially possible.


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