On a Wave Map Equation Arising in General Relativity

We consider a class of spacetimes for which the essential part of Einstein’s equations can be written as a wave map equation. The domain is not the standard one, but the target is hyperbolic space. One ends up with a 1+1 non-linear wave equation, where the space variable belongs to the circle and the time variable belongs to the positive real numbers. The main objective of this paper is to analyze the asymptotics of solutions to these equations as t → ∞. For each point in time, the solution defines a loop in hyperbolic space, and the first result is that the length of this loop tends to zero as t−1/2 as t → ∞. In other words, the solution in some sense becomes spatially homogeneous. However, the asymptotic behaviour need not be similar to that of spatially homogeneous solutions to the equations. The orbits of such solutions are either a point or a geodesic in the hyperbolic plane. In the non-homogeneous case, one gets the following asymptotic behaviour in the upper half plane (after applying an isometry of hyperbolic space if necessary). i) The solution converges to a point. ii) The solution converges to the origin on the boundary along a straight line (which need not be perpendicular to the boundary). iii) The solution goes to infinity along a curve y = const. iv) The solution oscillates around a circle inside the upper half plane. Thus, even though the solutions become spatially homogeneous in the sense that the spatial variations die out, the asymptotic behaviour may be radically different from anything observed for spatially homogeneous solutions of the equations. This analysis can then be applied to draw conclusions concerning the associated class of spacetimes. For instance, one obtains the leading order behaviour of the functions appearing in the metric, and one can conclude future causal geodesic completeness.