12.1 Asymptotic behavior

That the polarized vacuum T3-Gowdy spacetimes are future causally geodesically complete was announced in [21Jump To The Next Citation Point]; see [21Jump To The Next Citation Point, Step 3, p. 1677]. However, we shall here follow the presentation of [52Jump To The Next Citation Point]. The relevant equation to study is Equation (29View Equation). However, in the study of the expanding direction, it is convenient to change the time coordinate to the original areal time t. The equation then becomes
1- Ptt + tPt − Pšœƒšœƒ = 0. (51 )
Since the equation is linear and the coefficients do not depend on the spatial coordinate, it is clear that the spatial average of P,
1 ∫ āŸØP āŸ© = --- P (⋅,šœƒ)dšœƒ, 2π S1

solves the same equation. Furthermore, it is clear that there are constants a and b such that

āŸØP āŸ© = a lnt + b.

It is of interest to know what the asymptotic behavior of the remainder is. It turns out that there is a solution ν to the ordinary wave equation, i.e.,

νtt − ν šœƒšœƒ = 0,

with zero average, i.e.,

∫ ν (t,šœƒ) = 0 S1

for all t > 0, and a function ψ, the average of which is also zero, such that

−1āˆ•2 P(t,šœƒ) = a ln t + b + t ν(t,šœƒ) + ψ (t,šœƒ). (52 )
Furthermore, ψ and its first derivatives decay as t−3āˆ•2 and the division of Equation (52View Equation) is unique; see [52Jump To The Next Citation Point, Corollary 11, p. 183] (note that the statement of this result is also to be found in [21Jump To The Next Citation Point, (7a), p. 1675]). In short, given a solution P, there are a, b and ν as above (and then ψ is uniquely determined). It is of interest to ask if it is possible to go in the other direction. In other words, given a, b and ν as above, is there a solution with the above form of asymptotics. The answer to this question is yes; see [78, Proposition 1, p. 1649]. In other words, a, b and ν, with properties as above can be considered to be data at the moment of infinite expansion. Using the above information, it is possible to prove that polarized vacuum T3-Gowdy spacetimes are future causally-geodesically complete; see [52Jump To The Next Citation Point, Corollary 21, p. 190].
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