Structure scalars and cylindrical systems in Brans-Dicke gravity

Abstract

This paper explores structure scalars of cylindrically symmetric spacetime in Brans-Dicke gravity. We construct twelve scalar factors using orthogonal splitting of the Reimann tensor and study their distinct dynamical interpretations. These structure scalars are used to derive a set of governing equations of the evolving system. It is concluded that cylindrical systems are necessarily inhomogeneous. Finally, we show that all possible static inhomogeneous cylindrical solutions can be obtained through structure scalars.

Appendix

The non-zero components of BD equations are

$$\begin{aligned}& G^{0}_{0} =\frac{1}{\phi} \bigl(T^{0(m)}_{0}+T^{0(\phi)}_{0}\bigr)= \rho^{(\mathit{eff})} \\& \phantom{G^{0}_{0}}=\frac{1}{\phi} \biggl({\rho}+ \phi^{,t}_{;t}+ \Box\phi+\frac{\omega_{BD}}{\phi} \biggl[\phi^{,t}\phi_{;t} +\frac{1}{2}\phi_{,\alpha} \phi^{,\alpha} \biggr] \biggr) \\& \phantom{G^{0}_{0}=}{} +\frac{V(\phi)}{2\phi}, \end{aligned}$$

(58)

$$\begin{aligned}& G^{0}_{1}=\frac{1}{\phi} \bigl(T^{0(m)}_{1}+T^{0(\phi)}_{1}\bigr) = \frac{\omega_{BD}}{\phi^{2}}\bigl(\phi^{,t}\phi_{,r}\bigr) + \frac{\phi^{,t}_{;r}}{\phi}, \end{aligned}$$

(59)

$$\begin{aligned}& G^{1}_{1}=\frac{1}{\phi} \bigl(T^{1(m)}_{1}+T^{1(\phi)}_{1} \bigr)=p^{(\mathit{eff})}_{r} \\& \phantom{G^{1}_{1}} = \frac{1}{\phi} \biggl(p_{r}+ \phi^{,r}_{;r}-\square\phi +\frac{\omega_{BD}}{\phi} \biggl[ \phi^{,r}\phi_{,r} -\frac{1}{2}\phi^{,\alpha} \phi_{,\alpha} \biggr] \biggr) \\& \phantom{G^{1}_{1}=}{} -\frac{V(\phi)}{2\phi}, \end{aligned}$$

(60)

$$\begin{aligned}& G^{2}_{2}=\frac{1}{\phi} \bigl(T^{2(m)}_{2}+T^{2(\phi)}_{2} \bigr)=p^{(\mathit{eff})}_{z} \\& \phantom{G^{2}_{2}} =\frac{1}{\phi} \biggl(p_{z}- \square\phi -\frac{\omega_{BD}}{2\phi}\phi^{,\alpha}\phi_{,\alpha} - \frac{V(\phi)}{2\phi}\biggr) , \end{aligned}$$

(61)

$$\begin{aligned}& G^{3}_{3}=\frac{1}{\phi} \bigl(T^{3(m)}_{3}+T^{3(\phi)}_{3} \bigr) =p^{(\mathit{eff})}_{\phi} \\& \phantom{G^{3}_{3}}=\frac{1}{\phi} \biggl(p_{\perp}-\square \phi -\frac{\omega_{BD}}{2\phi}\phi^{,\alpha}\phi_{,\alpha} -\frac{V(\phi)}{2\phi} \biggr) , \end{aligned}$$

(62)

and the wave equation is

$$\begin{aligned} \square\phi =&\frac{1}{3+2\omega_{BD}} \biggl[-\rho^{m}+p^{m}_{r} +p^{m}_{\perp}+p^{m}_{z} \\ &{}+\phi \frac{dV(\phi)}{d\phi}-2V(\phi) \biggr]. \end{aligned}$$

(63)

The non-zero components of the Weyl tensor in Eqs. (8) and (9) are

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