The space-times symmetries and equilibrium distributions of the charged fluids. The equations of the state

R. A. Daishev

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, Tome 148 (2006) no. 3, pp. 49-64

Résumé

Space-times $V_4$, admittings a groups of homothetic motions $H_r$ with charged fluid as its source are discussed. It is assumed that the vector of macroscopic velocity of the fluid is collinear to the time-like vector $\mathbf Y=\xi^i \partial_i$ of the group's Lie algebra. We prove that if $(\rho+p)\neq 0$, the vector $\mathbf Y$ is the vector of the Lie algebra, corresponding to isometric transformations of the group $H_r$ and giving rise to time-like ideal of the Lie algebra of the group $H_r$. All space-times $V_4$, admitting a groups of homothetic transformations with indicated properties, are selected. Equations $(T^{ik}+E^{ik})_{|k}=0$ are integrated entirely, and all possible equations of state of investigated fluid are presented. It is founded that equation of state of the fluid practically uniquely fixed by the space-time's symmetry and pressure, energy density as well as electrical charges expressed solely through “field” quantities: $A_k\xi^k $ and $\xi_k\xi^k$, where $A_k$ is the 4-potential of an electromagnetic field.