Structure scalars in dissipative axial system in f(R) gravity

Abstract

This paper investigates the effects of $f{(}R{)}=R+\alpha R^2$ model on the dissipative anisotropic non-static rotating axial stellar system using structure scalars. By orthogonal splitting of the Riemann tensor, we obtain structure scalars and explore their roles for dissipative relativistic fluid configuration. We investigate the contribution of structure scalars in the evolution of various kinematical variables (e.g. expansion, shear and vorticity). Finally, we discuss thermodynamical aspects and the effect of dissipation on inertial mass density through transport equation. It is interesting to mention here that all our results reduce to the previous known results in the limiting case.

Appendix

The propagation equation for vorticity tensor (Eq. (26)) is

$$\begin{aligned} \frac{2}{3} \Theta \Omega _{\gamma \nu }+h^\alpha _\gamma h^\beta _\nu \Omega _{\alpha \beta ;\delta } V^\delta +2\sigma _{\alpha [\gamma }\Omega ^\alpha _{\nu ]}-h^\alpha _{[\gamma }h^\beta _{\nu ]}a_{\alpha ;\beta }=0. \end{aligned}$$

(66)

The propagation equation for shear is

$$\begin{aligned}&E_{\gamma \nu }-\frac{4\pi \Pi _{\gamma \nu }}{(1+2\alpha R)}+\frac{h_{\gamma \nu }}{6(1+2\alpha R)}\left( V^\pi V^\rho \nabla _\pi \nabla _\rho (1+2\alpha R)+\Box (1+2\alpha R)\right) \nonumber \\&\quad -\frac{h^\pi _\gamma h^\lambda _\nu }{2(1+2\alpha R)}\nabla _\pi \nabla _\lambda (1+2\alpha R)=h^\alpha _{(\gamma } h^\beta _{\nu )}a_{\beta ;\alpha }+a_\gamma a_\nu +\frac{h_{\gamma \nu }}{3}\left( \Omega ^2+2\sigma ^2-a^\beta _{;\beta }\right) \nonumber \\&\quad -\sigma ^\alpha _\gamma \sigma _{\nu \alpha }-\omega _\gamma \omega _\nu -h^\alpha _\gamma h^\beta _\nu \sigma _{\alpha \beta ;\delta } V^\delta -\frac{2}{3} \Theta \sigma _{\gamma \nu }. \end{aligned}$$

(67)

Here $\omega _\gamma$ is the vorticity vector given as

$$\begin{aligned} \omega _\gamma =\frac{1}{2}\eta _{\gamma \nu \lambda \delta }\Omega ^{\nu \lambda } V^\delta , \end{aligned}$$

or $\omega _\gamma =-\Omega S_\gamma$. Contracting Eq. (67) with KK, LL and KL, we obtain

$$\begin{aligned} Y_I&=\left( \Omega ^2+2\sigma ^2-a^\lambda _{;\lambda }\right) -\frac{\sigma _I^2}{3}-\sigma _{I;\lambda } V^\lambda -\frac{2 \Theta \sigma _I}{3}+3\left( a_I^2+a_{\lambda ;\nu } K^\lambda K^\nu \right) \nonumber \\&\quad +\frac{1}{1+2{\alpha }R}\left[ \frac{3\alpha }{B^2}\nabla _r \nabla _r R-\alpha \Box R-\frac{\alpha }{A^2}\nabla _t \nabla _t R\right] , \end{aligned}$$

(68)

$$\begin{aligned} Y_{II}&=\left( \Omega ^2+2\sigma ^2-a^\lambda _{;\lambda }\right) -\frac{\sigma _{II}^2}{3}-\sigma _{II;\lambda } V^\lambda -\frac{2 \Theta \sigma _{II}}{3}+3\left( a_{II}^2+a_{\lambda ;\nu } L^\lambda L^\nu \right) \nonumber \\&\quad +\frac{1}{1+2{\alpha }R}\left[ \frac{3\alpha }{r^2B^2 }\nabla _\theta \nabla _\theta R-\alpha \Box R-\frac{\alpha }{A^2}\nabla _t \nabla _t R\right] , \end{aligned}$$

(69)

$$\begin{aligned} Y_{KL}&=K^{(\lambda }L^{\nu )}a_{\nu ;\lambda }+\frac{\Omega \left( \sigma _{II}-\sigma _I\right) }{3}+a_I a_{II}+\frac{\alpha }{B^2 r (1+2\alpha R)}\nabla _r \nabla _\theta R. \end{aligned}$$

(70)

The time propagation equation for expansion scalar is given by

$$\begin{aligned} Y_T=-\Theta _{;\gamma } V^\gamma + a^\gamma _{;\gamma }-\frac{\Theta ^2}{3} +2(\Omega ^2-\sigma ^2)+\frac{2\alpha }{(1+2\alpha R)}\left( \Box R- 2 V^\lambda V^\delta \nabla _\lambda \nabla _\delta R\right) . \end{aligned}$$

(71)

The expressions for the scalars corresponding to electric and magnetic part of the Weyl tensor are

$$\begin{aligned} \varepsilon _I\!&=\!\frac{3}{A^2B^2}C_{1010},\quad \varepsilon _{II}\!=\!\frac{3}{A^2B^2r^2+G^2}C_{2020},\quad \varepsilon _{KL}\!=\!\frac{1}{AB\sqrt{A^2B^2r^2+G^2}}C_{1020},\\ H_1&=\frac{1}{2ABC^3(A^2B^2r^2+G^2)}\left( C_{0323}(G-A^2) -C_{0303}(G+B^2r^2)\right) ,\\ H_2&=\frac{1}{2C^3\sqrt{A^2B^2r^2+G^2}}\left( \frac{G}{A^2B^2r^2+G^2}C_{0323}\right. \\&\quad \left. -\frac{B^2r^2}{A^2B^2r^2+G^2}C_{0303}-\frac{1}{B^2}C_{0313}\right) , \end{aligned}$$

where the non-zero components of the Weyl tensor (like $C_{0303}$) can be obtained in a straight forward way.