3.3 Global existence for small initial data

In [142Jump To The Next Citation Point] the authors also consider the problem of global existence in Schwarzschild coordinates for small initial data for massive particles. They show that for such data the v-support is bounded on [0,T ). Hence, the continuation criterion implies that T = ∞. The resulting spacetime in [142Jump To The Next Citation Point] is geodesically complete, and the components of the energy-momentum tensor as well as the metric quantities decay with certain algebraic rates in t. The mathematical method used by Rein and Rendall is inspired by the analogous small data result for the Vlasov–Poisson equation by Bardos and Degond [32]. This should not be too surprising since for small data the gravitational fields are expected to be small and a Newtonian spacetime should be a fair approximation. In this context we point out that in [143] it is proven that the Vlasov–Poisson system is indeed the non-relativistic limit of the spherically-symmetric Einstein–Vlasov system, i.e., the limit when the speed of light c → ∞. In [150] this result is shown without symmetry assumptions.

As mentioned above the local and global existence problem has been studied using other time gauges, in particular Rendall has shown global existence for small initial data in maximal-isotropic coordinates in [156Jump To The Next Citation Point].

The previous results refer to massive particles but they do not immediately carry over to massless particles. This case is treated by Dafermos in [63Jump To The Next Citation Point] where global existence for small initial data is shown in double null coordinates. The spacetimes obtained in the studies [142, 156Jump To The Next Citation Point, 63] are all causally geodesically complete and appropriate decay rates of the metric and the matter quantities are given.


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