Einstein-Vlasov system with equal-angular momenta in AdS5

Abstract

We investigate solutions of the five-dimensional rotating Einstein-Vlasov system with an $R\ifmmode\times\else\texttimes\fi{}SU(2)\ifmmode\times\else\texttimes\fi{}U(1)$ isometry group. In a five-dimensional spacetime, there are two independent planes of rotation; thus, considering $U(1)$ symmetry on each rotation plane, we may impose an $R\ifmmode\times\else\texttimes\fi{}U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ isometry to a stationary spacetime. Furthermore, when the values of the two angular momenta are equal to each other, the spatial symmetry gets enhanced to $R\ifmmode\times\else\texttimes\fi{}SU(2)\ifmmode\times\else\texttimes\fi{}U(1)$ symmetry, and the spacetime has a cohomogeneity-1 structure. Imposing the same symmetry to the distribution function of the particles of which the Vlasov system consists, the distribution function can be dependent on three mutually independent and commutative conserved charges for particle motion [energy, total angular momentum on $SU(2)$ and $U(1)$ angular momentum]. We consider the distribution function which exponentially depends on the $U(1)$ angular momentum and reduces to the thermal equilibrium state in spherical symmetry. Then, in this paper, we numerically construct solutions of the asymptotically anti--de Sitter Einstein-Vlasov system.