13 Expanding Direction, The General Case

Clearly, the situation is more complicated in the nonpolarized case. Consequently, it is natural to start by asking questions concerning the rough behavior of solutions. It is also natural to carry out numerical simulations. This was done in [8Jump To The Next Citation Point], a paper, which played an important role in the further development of the subject. In the polarized case, the difference between the solution and its spatial average converges to zero. Does the same phenomenon occur in the general case? This is perhaps too complicated a question to start with, but it is of interest to know how the spatial variation of the solution evolves with time. In order to define what is meant by the spatial variation, recall that Equations (11View Equation) – (12View Equation) can be viewed as wave-map equations in which the metric of the target is given by Equation (20View Equation). Furthermore, at each point in time, the solution defines a loop in hyperbolic space. The natural definition of the spatial variation of the solution at one point in time is the length of this loop with respect to the hyperbolic metric (20View Equation). In other words, it is natural to ask how

∫ āˆ˜ ------------ ā„“ = Pšœƒ2+ e2PQ2šœƒdšœƒ S1

evolves with time. In the case of polarized Gowdy, we know that

−1āˆ•2 −1 Pšœƒ = t νšœƒ + O (t )

asymptotically; see Equation (52View Equation) and the adjacent text. Consequently, ā„“ = O(t−1āˆ•2) in that case, but there is no better estimate.

 13.1 Energy decay
 13.2 Proof of decay of the energy
  13.2.1 Toy model
  13.2.2 Polarized case
  13.2.3 General case
 13.3 Asymptotic ODE behavior
  13.3.1 Conserved quantities
  13.3.2 Interpreting the conserved quantities as ODEs for the averages
 13.4 Geometric interpretation of the asymptotics
 13.5 Concluding remarks
 13.6 Geodesic completeness

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