### 12.1 Asymptotic behavior

That the polarized vacuum T^{3}-Gowdy spacetimes are future causally geodesically complete was
announced in [21]; see [21, Step 3, p. 1677]. However, we shall here follow the presentation of [52]. The
relevant equation to study is Equation (29). However, in the study of the expanding direction, it is
convenient to change the time coordinate to the original areal time . The equation then becomes
Since the equation is linear and the coefficients do not depend on the spatial coordinate, it is clear that the
spatial average of ,
solves the same equation. Furthermore, it is clear that there are constants and such
that

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

with zero average, i.e.,

for all , and a function , the average of which is also zero, such that

Furthermore, and its first derivatives decay as and the division of Equation (52) is
unique; see [52, Corollary 11, p. 183] (note that the statement of this result is also to be found
in [21, (7a), p. 1675]). In short, given a solution , there are , and as above (and
then is uniquely determined). It is of interest to ask if it is possible to go in the other
direction. In other words, given , and 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, , and , 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 T^{3}-Gowdy spacetimes are future causally-geodesically complete; see [52, Corollary 21,
p. 190].