### 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 to
Equations (16) – (17), let , where is defined in Equation (24). Then the
limit
always exists due to [79, Lemma 6.17, p. 1011] (as an aside, it is worth noting that is
a real analytic function from the open unit disc to the real numbers if is the hyperbolic
distance from the origin of the unit disc to ; see [79, Remark 6.18, p. 1011]). It would be
reasonable to call this limit, which we shall refer to as , 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.