By combining the expansions (37) – (38) and (40), it is possible to conclude that if , there is the neighborhood of such that there are expansions for of the same form as the ones for in that neighborhood. However, at the point , the situation is different. In fact, Equation (40) implies that
Furthermore, the last term converges to zero exponentially due to Equation (37), Equation (38) and the fact that . In particular,
Due to the fact that the limit of has a discontinuity at , the point is called a spike for the solution .
In order to understand this phenomenon better, it is of interest to consider the asymptotic behavior of and in the upper half plane model. Since and , we have
In other words, the solution, at , converges to the origin in the upper half plane. On the other hand, in the punctured neighborhood of , the solution converges a to point on the real line different from the origin. The inversion maps the origin to infinity, but it maps the points on the real line different from the origin to points on the real line (different from the origin). In other words, the appearance of the “spike” described above is a result of the fact that the upper half plane model of hyperbolic space has a preferred boundary point, namely infinity. In the disc model of hyperbolic space, it is not meaningful to speak of spikes of the above form. In particular,
for all , where is the kinetic energy density associated with the solution . In other words, this limit is smooth even though the limit of is discontinuous. As a consequence of the nongeometric nature of the above spikes, we shall refer to them as “false spikes”.
In Figures 1 and 2, we have plotted and , respectively, in the neighborhood of a false spike. The figures have been obtained by making a specific choice of , ignoring and in Equations (37) – (38) and applying an inversion. They represent the solution at a fixed point in time. We have also plotted in the neighborhood of a spike in Figure 3.
In order to give an example of a true spike, let be a solution to (16) – (17) with asymptotic expansions of the form of Equations (37) – (38), where , and in some punctured neighborhood of . Applying an inversion, we obtain according to Equation (42). Applying the Gowdy to Ernst transformation produces a true spike:
In Figure 4, we have plotted in the neighborhood of a true spike; we have not plotted , since it is regular in the neighborhood of the spike, even in the limit. The particular values of the constants are not of importance. Since , we conclude that
due to Equation (43), and
for in the punctured neighborhood of , due to Equation (44). In Figure 5, we have plotted in the neighborhood of a true spike. In the case of a true spike, the limit of , the kinetic energy density associated with the solution , is discontinuous at . Consequently, the discontinuity is a geometric feature of the solution to the wave-map equations. Furthermore, the Kretschmann scalar blows up at different rates at the tip of the spike than in the punctured neighborhood; see the bottom of [71, p. 2966]. For these reasons, we shall refer to as a true spike associated with the solution .
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