Electromagnetic field and spherically symmetric dissipative fluid models

Abstract

This paper studies a few properties of Lemaître–Tolman–Bondi (LTB) space–time to the dissipative cases that may lead to its extension in Maxwell-f(R, T) gravity, where R is the Ricci scalar and T is the trace of energy–momentum tensor. Using Misner and Sharp mass formalism, we have first established a relationship between the Weyl tensor and other physical variables. The role of electric charge in the development of the Bianchi identities was also investigated. The physical importance of the effective form of structure scalars was then analysed in view of some realistic backgrounds. We also discussed the generalisations of LTB and the extension of LTB based on structure scalars and a few symmetry properties under constant curvature conditions.

Appendices

Appendix A

The parts of the equations are given below:

$$\begin{aligned} \chi _{00}=&\bigg [\mu f_{T}-\bigg (f-\frac{R}{2}f_{{R}}\bigg )+\frac{f''_{R}}{G^2}\nonumber \\&-\dot{f_{R}} \big (\frac{{\dot{G}}}{G} +2\frac{{\dot{M}}}{M}\bigg )-f'_{R}\bigg (\frac{G'}{G^3}-\frac{2M'}{G^2M}\big )\bigg ], \end{aligned}$$

(A.1)

$$\begin{aligned} \chi _{01}=&\left[ \dot{f'_{R}}-\frac{{\dot{G}}}{G}\bigg (\dot{f_{R}}+f'_{R}\bigg ) \right] ,\end{aligned}$$

(A.2)

$$\begin{aligned} \chi _{11}=&\left[ -\mu G^2 f_{T}+\bigg (f-\frac{R}{2} f_{R}\bigg ) G^2-G {\dot{G}} \dot{f_{R}}\right. \nonumber \\&\left. -\frac{G'}{G} f'_{R}+G^2 \ddot{f_{R}}+\dot{f_{R}} G^2 \bigg (\frac{{\dot{G}}}{G}+2\frac{{\dot{M}}}{M}\bigg )\right. \nonumber \\&\left. +f'_{R}G^2\bigg (\frac{G'}{G^3}-\frac{2M'}{G^2 M}\bigg ) \right] , \end{aligned}$$

(A.3)

$$\begin{aligned} \chi _{22}=&\left[ -\mu M^2 f_{T}+\bigg (f-\frac{R}{2} f_{R}\bigg ) M^2+\frac{M M'}{G^2}f'_{R}\right. \nonumber \\&\left. -M {\dot{M}} \dot{f_{R}}+M^2 \ddot{f_{R}}-\frac{M^2}{G^2}f''_{R}+M^2\bigg (\frac{{\dot{G}}}{G}\right. \nonumber \\&\left. +2\frac{{\dot{M}}}{M}\bigg ) \dot{f_{R}}+f'_{R}\bigg (\frac{G'}{G^3}-\frac{2M'}{G^2 M}\bigg )M^2\right] . \end{aligned}$$

(A.4)

Some more terms are given below:

(A.5)

(A.6)

(A.7)

(A.8)

Appendix B

The charge terms are given as follows:

$$\begin{aligned} Q^{00}&=\frac{1}{f_{r}}\frac{s^2}{8\pi M^4},\quad Q^{11}=-\frac{1}{f_{r}}\frac{s^2}{8\pi G^{2}M^4},\nonumber \\ Q^{22}&=\frac{1}{R^4}\frac{1}{f_{r}}\frac{s^2}{8\pi M^2}, \end{aligned}$$

(B.1)

$$\begin{aligned} Q^{33}&=\frac{1}{R^4\sin ^2\theta }\left[ \left( \frac{1+f_{T}}{f_{r}}\right) \frac{s^2}{8\pi M^2}-f_{T}\frac{s^2}{2M^4}M^2\right] ,~\nonumber \\ Q_{00}&=\left[ \left( \frac{1+f_{T}}{f_{r}}\right) \frac{s^2}{8\pi M^4}-f_{T}\frac{s^2}{2M^4}\right] ,\end{aligned}$$

(B.2)

$$\begin{aligned} Q_{11}&=-\left( \frac{1+f_{T}}{f_{r}}\right) \frac{s^2 G}{8\pi G M^4}+f_{T}G^2\frac{M^2}{2M^4},~\nonumber \\ Q_{22}&=\left[ \left( \frac{1+f_{T}}{f_{r}}\right) \frac{s^2}{8\pi M^2}-f_{T}\frac{s^2}{2M^4}M^2\right] ,\end{aligned}$$

(B.3)

$$\begin{aligned} Q_{33}&=\left[ \left( \frac{1+f_{T}}{f_{r}}\right) \frac{s^2}{8\pi M^2}\sin ^2\theta -f_{T}\frac{s^2}{2M^4}M^2\sin ^2\theta \right] ,\end{aligned}$$

(B.4)

$$\begin{aligned} Q_{4}&={Q'_{10}}+\left( \frac{Q_{2}}{G^{2}}\right) '+\frac{3{M'}}{M}Q_{10} +\frac{3{M'}}{MG^2}Q_{2}\nonumber \\&\quad -\left( \frac{Q_{0}}{2}\right) ',\nonumber \\ ~Q_{7}&=4\pi (Q_{\gamma \zeta } V^{\zeta }V^{\gamma }+Q_{\gamma \delta }V^{\gamma }V^{\delta }),\end{aligned}$$

(B.5)

$$\begin{aligned} Q_{8}&=\frac{1}{\chi _{\eta }\chi _{\zeta }-\frac{h_{\eta \zeta }}{3}} \Bigg (4\pi \Bigg (Q_{\eta \zeta } +Q_{\gamma \zeta }V_{\eta }V^{\gamma }\nonumber \\&\quad +Q_{\gamma \delta }\delta _{\eta \zeta } V^{\gamma }V^{\delta }\Bigg )-\frac{1}{3}h_{\eta \zeta }Q_{7}\Bigg ),\end{aligned}$$

(B.6)

$$\begin{aligned} Q_{9}&=-\pi \epsilon ^{\rho \epsilon \zeta }\Big (\epsilon _{\pi \rho \zeta }Q^{\pi }_{\epsilon } \nonumber \\&\quad +\epsilon _{\pi \epsilon \zeta }Q^{\pi }_{\rho }+\epsilon _{\rho \theta \zeta } Q^{\theta }_{\epsilon } +\epsilon _{\epsilon \theta \zeta }Q^{\theta }_{\rho }\Big ),\ \end{aligned}$$

(B.7)

$$\begin{aligned} Q_{10}&=\frac{1}{\chi _{\eta }\chi _{\zeta }-\frac{h_{\eta \zeta }}{3}} \left( -\pi \epsilon ^{\rho \epsilon }_{\eta }(\epsilon _{\pi \rho \zeta }Q^{\pi }_{\epsilon }\right. \nonumber \\&\quad \left. +\epsilon _{\pi \epsilon \zeta }Q^{\pi }_{\rho }+\epsilon _{\rho \theta \zeta } Q^{\theta }_{\epsilon } +\epsilon _{\epsilon \theta \zeta }Q^{\theta }_{\rho })-\frac{1}{3}Q_{9}h_{\eta \zeta } \right) , \end{aligned}$$

(B.8)

$$\begin{aligned} Q_{A}&=\left( {\frac{1}{f_{r}}\frac{s^2}{8\pi M^4}}\right) -G{\dot{G}}\left( \frac{1}{f_{r}}\frac{s^2}{8\pi G^{2}M^4}\right) \nonumber \\&\quad +\left( \frac{{\dot{G}}}{G} +\frac{2{\dot{M}}}{M}\right) \frac{1}{f_{r}}\frac{s^2}{8\pi M^4}, \end{aligned}$$

(B.9)

$$\begin{aligned} Q_{B}&=-\frac{1}{8\pi f_{R}}\bigg (\frac{s^2}{G^2M^4}\bigg )'\nonumber \\&\quad -\frac{1}{8\pi f_{R}}\left( 2\frac{{G}'}{G}+\frac{{M'}}{M}\right) \frac{s^2}{G^2M^4}\nonumber \\&\quad -2\frac{{M'}M}{G^2}\frac{1}{8\pi f_{R}}\frac{s^2}{M^6}, \end{aligned}$$

(B.10)

$$\begin{aligned} Q_{C}&=\frac{3{M'}Q_{2}}{G^2 M}+\left( \frac{Q_{2}}{2G^2}\right) '-\left( \frac{Q_{0}}{2}\right) ', \end{aligned}$$

(B.11)

$$\begin{aligned} Q_{D}&=\frac{1}{8\pi f_{R}}\bigg (\frac{s^2}{G^2M^4}\bigg )'-\frac{1}{8\pi f_{R}}\left( 2\frac{{G'}}{G}+\frac{{M'}}{M}\right) \frac{s^2}{G^2M^4}\nonumber \\&-2\frac{{M'}}{G^2}\frac{1}{8\pi f_{R}}\frac{s^2}{M^5}. \end{aligned}$$

(B.12)