Dynamics of tilted cylindrical geometry

Abstract

In this paper, we study the dynamics of tilted cylindrical model with imperfect matter distribution. We formulate the field equations and develop relations between tilted and non-tilted variables. We evaluate kinematical as well as dynamical quantities and discuss the inhomogeneity factor. We also obtain the Raychaudhuri equation to study evolution of expansion scalar. The solutions of field equations are also investigated for static cylinder under isotropy and conformally flat condition. Finally, we analyze some thermoinertial aspects of the system.

Appendix

The non-vanishing components of the Weyl tensor are

$$\begin{aligned} &{C_{0101}=-\frac{A^{2}}{6} \biggl[2\frac{\dot{B}\dot{C}}{BC}- 2 \frac{A''}{A}+\frac{B''}{B}+\frac{C''}{C}+2 \biggl( \frac{A'}{A} \biggr)^{2} } \\ &{\phantom{C_{0101}=}{}-2\frac{B'C'}{BC}+2 \frac{\ddot{A}}{A}-\frac{\ddot{C}}{C}-\frac{\ddot{B}}{B}- 2 \biggl(\frac{\dot{A}}{A} \biggr)^{2} \biggr]} \\ &{\phantom{C_{0101}}= - \biggl(\frac{A^{2}}{BC} \biggr)^{2}C_{2323},} \\ &{C_{0202}=\frac{B^{2}}{6} \biggl[\frac{\ddot{A}}{A}- 2 \frac{\ddot{B}}{B}+\frac{\ddot{C}}{C}- \biggl(\frac{\dot{A}}{A} \biggr)^{2}+\frac{3\dot{A}}{A} \biggl(\frac{\dot{B}}{B}- \frac{\dot{C}}{C} \biggr) } \\ &{\phantom{C_{0202}=}{}+\frac{\dot{B}\dot{C}}{BC}-\frac{A''}{A}- \frac{B''}{B}+2\frac{C''}{C}+\frac{3A'}{A} \biggl( \frac{B'}{B}-\frac{C'}{C} \biggr)} \\ &{\phantom{C_{0202}=}{}-\frac{B'C'}{BC} \biggr] =- \biggl(\frac{B}{C} \biggr)^{2}C_{1313},} \\ &{C_{0303}=\frac{C^{2}}{6} \biggl[\frac{\ddot{A}}{A}- 2 \frac{\ddot{C}}{C}+\frac{\ddot{B}}{B}- \biggl(\frac{\dot{A}}{A} \biggr)^{2}-\frac{3\dot{A}}{A} \biggl(\frac{\dot{B}}{B}- \frac{\dot{C}}{C} \biggr)} \\ &{\phantom{C_{0303}=}{}+ \frac{\dot{B}\dot{C}}{BC}-\frac{A''}{A}+2 \frac{B''}{B}- \frac{3A'}{A} \biggl(\frac{B'}{B}- \frac{C'}{C} \biggr)-\frac{C''}{C} } \\ &{\phantom{C_{0303}=}{}+ \biggl(\frac{A'}{A} \biggr)^{2} -\frac{B'C'}{BC} \biggr]=- \biggl(\frac{C}{B} \biggr)^{2}C_{1212},} \\ &{C_{0212}=-\frac{B^{2}}{2} \biggl[\frac{\dot{B'}}{B} - \frac{\dot{C'}}{C}-\frac{\dot{A}}{A} \biggl(\frac{B'}{B}-\frac{C'}{C} \biggr)} \\ &{\phantom{C_{0212}=}{}-\frac{A'}{A} \biggl(\frac{\dot{B}}{B}-\frac{\dot{C}}{C} \biggr) \biggr]=- \biggl(\frac{B}{C} \biggr)^{2}C_{0313}.} \end{aligned}$$