BIRS Workshop Lecture Videos

Banff International Research Station Logo

BIRS Workshop Lecture Videos

Vacuum Boundary Problem of the Metric Produced by Perfect Fluid Mass Distributions Makino, Tetu


The mathematical study of the metric produced by compactly supported perfect fluid mass distributions governed by the Einstein-Euler equations is now developing. The main difficulty of the study comes from the situation that the boundary between the mass and the vacuum requires mathematically delicate treatise. This so called `physical vacuum boundary' is a kind of free boundary. The spherically symmetric evolution of the metric near spherically symmetric equilibria, say, solutions of the Tolman-Oppenheimer-Volkoff equation, is studied in [1] by application of the Nash-Moser theorem. The metric can be matched to the exterior Schwarrzschild metric in the vacuum region, but it turns out to be the boundary is of class C^2 if and only if the metric is static. The existence of the axially symmetric metric of slowly rotating mass under the weak gravitational field has been established in [2] in a bonded region containing the support of the density. But the global continuation of the metric beyond the bounded region has not yet been done. Actually we are not sure whether the matching can be done using the Kerr metric, and, what on earth, we are not sure whether the metric can be continued as asymptotically flat one or not. Much less the problem of time evolution near this rotating stationary metric is completely open. [1] T. Makino, Kyoto J. Math., 56 (2016), 243-282. DOI 10.1215/21562261-3478880. [2] T. Makino, J. Math. Physics., 59(2018), 102502. DOI 10.1063/1.5026133.

Item Media

Item Citations and Data


Attribution-NonCommercial-NoDerivatives 4.0 International