Symmetries of spacetimes embedded with an Electromagnetic String Fluid

Abstract

The electromagnetic string fluid (EMSF) is an anisotropic charged string fluid interacting with a strong magnetic field. In this fluid we consider the double congruence defined by the 4-velocity of the fluid $u^a$ and the unit vector $n^a$ along the magnetic field. Using the standard 1+3 decomposition defined by the vector $u^a$ and the 1+1+2 decomposition defined by the double congruence ${u^a,n^a}$ we determine the kinematic and the dynamic quantities of an EM string fluid in both decompositions. In order to solve the resulting field equations we consider simplifying assumptions in the form of collineations. We decompose the generic quantity $L_{X}g_{ab}$ in a trace $\psi$ and and a traceless part $H_{ab}$. Because all collineations are expressible in terms of the quantity $L_{X}g_{ab}$ it is possible to compute the Lie derivative of all tensors defined by the metric i.e. the Ricci tensor, the Weyl tensor etc. This makes possible the effects of any assumed collineation on the gravitational field equations. This is done as follows. Using relevant identities of Differential Geometry we express the quantity $L_{X}R_{ab}$ where $R_{ab}$ is the Ricci tensor in terms of the two irreducible parts $\psi,H_{ab}$. Subsequently using the gravitational field equations we compute the same quantity $L_{X}R_{ab}$ in terms of the Lie derivative of the dynamic variables. We equate the two results and find the field equations in the form $L_{X}{\text{Dynamic variable}}=F(\psi,H_{ab},{\text{ dynamic variables}})$. This result is general and holds for all gravitational systems and in particular for the EMSF. Subsequently we specialize our study at two levels. We consider the case of a Conformal Killing Vector (CKV) parallel to $u^a$ and a CKV parallel to $n^a$.