Electromagnetic effects on the evolution of LTB geometry in modified gravity

Abstract

We study the influence of Palatini $f(R)$ gravity and tilted observer on the dynamics of Lemaître–Tolman–Bondi space-time in the presence of electromagnetic field. The imperfect charged fluid seen by tilted observer is considered in comparison with charged dust fluid seen by nontilted observer. We develop the relations between tilted and nontilted variables by including electric charge in Palatini $f(R)$ gravity. In this framework, we explore the evolution of energy density inhomogeneities for tilted and nontilted observers by calculating the energy conservation laws for charged fluid. Finally, we evaluate a constraint on the electric charge in the collapse of stellar objects, which leads to the instability of nontilted congruence.

Appendix

The parts of Eqs. (42) and (43) are

$$\begin{aligned} &{\mathcal{D}_{0} =-\dot{\mathcal{T}}_{00}+ \biggl( \frac{\mathcal{T}_{10}}{B ^{2}} \biggr) ' -\mathcal{T}_{00} \biggl( \frac{\dot{B}}{B}+\frac{3 \dot{f_{R}}}{2f_{R}} +\frac{2\dot{C}}{C} \biggr) }\\ &{\phantom{\mathcal{D}_{0} =}{}+ \frac{\mathcal{T}_{01}}{B ^{2}} \biggl( \frac{B'}{B} +\frac{2f'_{R}}{f_{R}}+ \frac{2C'}{C} \biggr)-\frac{\mathcal{T}_{11}}{B^{2}} \biggl( \frac{\dot{B}}{B}+\frac{ \dot{f_{R}}}{2f_{R}} \biggr)} \\ &{\phantom{\mathcal{D}_{0} =}{} -\frac{2\mathcal{T}_{22}}{C^{2}} \biggl( \frac{ \dot{C}}{C} +\frac{\dot{f_{R}}}{2f_{R}} \biggr) ,} \\ &{\mathcal{D}_{1} =-\dot{\mathcal{T}}_{10}+ \biggl( \frac{\mathcal{T}_{11}}{B ^{2}} \biggr) ' +\mathcal{T}_{00} \frac{f_{R}'}{2f_{R}}-\mathcal{T}_{10} \biggl( \frac{\dot{B}}{B}+ \frac{2\dot{f}_{R}}{f_{R}}+\frac{2\dot {C}}{C} \biggr) } \\ &{\phantom{\mathcal{D}_{1} =}{}+\frac{ \mathcal{T}_{11}}{B^{2}} \biggl( 2 \frac{C'}{C}+\frac{3f_{R}'}{2f_{R}} \biggr)-\frac{2\mathcal{T}_{22}}{C^{2}} \biggl( \frac{C'}{C}+\frac{f'_{R}}{2f _{R}} \biggr) .} \end{aligned}$$