Self-gravitating spherically symmetric fluid models in Brans–Dicke gravity

Abstract

This paper is devoted to study self-gravitating spherically symmetric fluid models in Brans–Dicke gravity. We formulate a set of equations which govern the dynamics of evolving gravitating fluids through Weyl tensor, shear tensor, expansion scalar, anisotropy, energy inhomogeneity, dissipation as well as scalar field. We also discuss some particular cases according to different dynamical conditions. It is concluded that fluid models for regular distribution of scalar field are consistent with general relativity and models due to irregular distribution of scalar field deviate from theory of general relativity.

Appendix

The scalar field terms of Eqs. (28) and (29) are given by

$$\begin{aligned} H^{\phi }_{1}= & {} -A\left( \phi ^{,\mu ;\nu }+\frac{\omega _{BD} \phi ^{,\mu }\phi ^{,\nu }}{\phi }\right) _{;\nu }+\left( \frac{2\phi \square \phi +\omega _{BD}\phi ^{,\alpha }\phi _{,\alpha }-\phi V(\phi )}{2A\phi }\right) _{;\nu } +\frac{\dot{\phi }}{\phi } \\&\quad \times \,\left( -A\tilde{\rho } +\phi ^{,t;t}+\frac{\omega _{BD}(\phi ^{,t})^{2}}{\phi } -\frac{\omega _{BD\phi ^{,\alpha }\phi _{,\alpha }}}{2\phi A}\right) \\&\quad +\frac{\phi '}{\phi }\left( \tilde{q}AB+ \phi ^{,t;r}+\frac{\omega _{BD}\phi ^{,t}\phi ^{,r}}{\phi }\right) ,\\ H^{\phi }_{2}= & {} B\left( \phi ^{,\mu ;\nu }+\frac{\omega _{BD} \phi ^{,\mu }\phi ^{,\nu }}{\phi }\right) _{;\nu }+\left( \frac{2\phi \square \phi +\omega _{BD}\phi ^{,\alpha }\phi _{,\alpha }-\phi V(\phi )}{2B\phi }\right) _{;\nu } +\frac{\dot{\phi }}{\phi }\\&\quad \times \,\left( \tilde{q}AB +\phi ^{,t;r}+\frac{\omega _{BD}\phi ^{,t}\phi ^{,r}}{\phi }\right) \\&\quad +\,\frac{\phi '}{\phi }\left( \tilde{p}_{r} +\phi ^{,r;r}+\frac{\omega _{BD}(\phi ^{,r})^{2}}{\phi } -\frac{\omega _{BD\phi ^{,\alpha }\phi _{,\alpha }}}{2\phi B}\right) . \end{aligned}$$

The scalar energy terms $H^{\phi }_{3}$ and $H^{\phi }_{4}$ in geodesic fluid are as follows

$$\begin{aligned} H^{\phi }_{3}= & {} -A\left( \phi ^{,\mu ;\nu }+\frac{\omega _{BD} \phi ^{,\mu }\phi ^{,\nu }}{\phi }\right) _{;\nu }+\left( \frac{2\phi \square \phi +\omega _{BD}\phi ^{,\alpha }\phi _{,\alpha }-\phi V(\phi )}{2A\phi }\right) _{;\nu } +\frac{\dot{\phi }}{\phi } \\&\quad \left( -A\tilde{\rho } +\phi ^{,t;t}+\frac{\omega _{BD}(\phi ^{,t})^{2}}{\phi } -\frac{\omega _{BD\phi ^{,\alpha }\phi ^{,\alpha }}}{2\phi A}\right) + \frac{\phi '}{\phi }\left( \phi ^{,t;r} +\frac{\omega _{BD}\phi ^{,t}\phi ^{,r}}{\phi }\right) , \\ H^{\phi }_{4}= & {} B\left( \phi ^{,\mu ;\nu }+\frac{\omega _{BD} \phi ^{,\mu }\phi ^{,\nu }}{\phi }\right) _{;\nu }+\left( \frac{2\phi \square \phi +\omega _{BD}\phi ^{,\alpha }\phi _{,\alpha }-\phi V(\phi )}{2B\phi }\right) _{;\nu } +\frac{\dot{\phi }}{\phi } \\&\quad \left( \phi ^{,t;r}+\frac{\omega _{BD}\phi ^{,t}\phi ^{,r}}{\phi }\right) +\frac{\phi '}{\phi }\left( \tilde{p}_{r} +\phi ^{,r;r}+\frac{\omega _{BD}(\phi ^{,r})^{2}}{\phi } -\frac{\omega _{BD\phi ^{,\alpha }\phi _{,\alpha }}}{2\phi B}\right) . \end{aligned}$$