9.3 False spikes

One way of constructing a spike is to start with a solution (Q, P ) with asymptotic expansions of the form of Equations (37View Equation) – (38View Equation), where 0 < va < 1 (due to [68Jump To The Next Citation Point], we know that such solutions exist and that we are free to specify the functions v ,ϕ, q,ψ a, assuming 0 < v < 1 a). Assume that q has the properties that q(𝜃0) = 0 and that q ⁄= 0 in the punctured neighborhood of 𝜃0. Applying an inversion to (Q, P), we obtain
(Q1, P1) = Inv (Q, P ). (42 )
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Figure 1: Q1 in the neighborhood of a false spike for a fixed τ.
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Figure 2: P1 in the neighborhood of a false spike for a fixed τ.

By combining the expansions (37View Equation) – (38View Equation) and (40View Equation), it is possible to conclude that if q(𝜃) ⁄= 0, there is the neighborhood of 𝜃 such that there are expansions for (Q1,P1 ) of the same form as the ones for (Q, P ) in that neighborhood. However, at the point 𝜃0, the situation is different. In fact, Equation (40View Equation) implies that

P (τ,𝜃 ) = − P (τ,𝜃 ) + ln[1 + e2P(τ,𝜃0)Q2 (τ,𝜃 )]. 1 0 0 0

Furthermore, the last term converges to zero exponentially due to Equation (37View Equation), Equation (38View Equation) and the fact that q(𝜃0) = 0. In particular,

lτi→m∞ P1 τ(τ, 𝜃0) = − va(𝜃0). (43 )
However, for 𝜃 in the punctured neighborhood of 𝜃0, we have
lτ→im∞ P1τ(τ,𝜃) = va(𝜃). (44 )
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Figure 3: P1τ in the neighborhood of a false spike for a fixed τ.

Due to the fact that the limit of P1τ has a discontinuity at 𝜃0, the point 𝜃0 is called a spike for the solution (Q1,P1 ).

In order to understand this phenomenon better, it is of interest to consider the asymptotic behavior of (Q, P ) and (Q ,P ) 1 1 in the upper half plane model. Since v (𝜃 ) > 0 a 0 and q(𝜃 ) = 0 0, we have

− P(τ,𝜃0) τli→m∞ ϕRH [Q (τ,𝜃0),P (τ, 𝜃0)] = τli→m∞[Q (τ, 𝜃0),e ] = (0,0).

In other words, the solution, at 𝜃0, converges to the origin in the upper half plane. On the other hand, in the punctured neighborhood of 𝜃 0, the solution converges a to point on the real line different from the origin. The inversion Inv 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,

2 lτim→∞ 𝒦1 (τ,𝜃) = va(𝜃)

for all 𝜃, where 𝒦 1 is the kinetic energy density associated with the solution (Q ,P ) 1 1. In other words, this limit is smooth even though the limit of P1τ is discontinuous. As a consequence of the nongeometric nature of the above spikes, we shall refer to them as “false spikes”.

In Figures 1View Image and 2View Image, we have plotted Q1 and P1, respectively, in the neighborhood of a false spike. The figures have been obtained by making a specific choice of va,ϕ, q,ψ, ignoring u and w in Equations (37View Equation) – (38View Equation) and applying an inversion. They represent the solution at a fixed point in time. We have also plotted P1τ in the neighborhood of a spike in Figure 3View Image.

9.3.1 True spikes

In order to give an example of a true spike, let (Q,P ) be a solution to (16View Equation) – (17View Equation) with asymptotic expansions of the form of Equations (37View Equation) – (38View Equation), where 0 < va < 1, q(𝜃0) = 0 and q(𝜃) ⁄= 0 in some punctured neighborhood of 𝜃0. Applying an inversion, we obtain (Q1,P1 ) according to Equation (42View Equation). Applying the Gowdy to Ernst transformation produces a true spike:

(Q2,P2 ) = GEq ,τ,𝜃(Q1, P1). 0 0 0

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Figure 4: P2 in the neighborhood of a true spike for a fixed τ.

In Figure 4View Image, we have plotted P2 in the neighborhood of a true spike; we have not plotted Q2, since it is regular in the neighborhood of the spike, even in the limit. The particular values of the constants (q0,τ0,𝜃0) are not of importance. Since P2 τ = 1 − P1τ, we conclude that

lim P (τ,𝜃 ) = 1 + v (𝜃 ), τ→ ∞ 2τ 0 a 0

due to Equation (43View Equation), and

lim P2τ(τ,𝜃) = 1 − va(𝜃) τ→ ∞

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Figure 5: P2τ in the neighborhood of a true spike for a fixed τ.

for 𝜃 in the punctured neighborhood of 𝜃0, due to Equation (44View Equation). In Figure 5View Image, we have plotted P2 τ in the neighborhood of a true spike. In the case of a true spike, the limit of 𝒦2, the kinetic energy density associated with the solution (Q2,P2 ), is discontinuous at 𝜃0. 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 [71Jump To The Next Citation Point, p. 2966]. For these reasons, we shall refer to 𝜃0 as a true spike associated with the solution (Q2, P2).


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