On the stability of electrostatics stars with modified non-gauge invariant Einstein-Maxwell gravity

Abstract

We use a modified Einstein-Maxwell gravity to study stability of an electrostatic spherical star. Correction terms in this model are scalars which are made from contraction of Ricci tensor and electromagnetic vector potential. Our motivation to use this kind of exotic Einstein-Maxwell gravity is inevitable influence of cosmic magnetic field in inflation of the universe which is observed now but its intensity suppresses in the usual gauge invariant Einstein-Maxwell gravity. In this work, we use approach of dynamical systems to obtain stability conditions of such a star and we investigate interaction parts effect of the model on the stability.

Appendices

Appendix I

By looking at the Eqs. (2.4) and (2.8) we define

$$\begin{aligned} G_{\mu \nu }=T_{\mu \nu }^{total}=\frac{2\tau _{\mu \nu }}{\alpha A^2} \end{aligned}$$

(8.1)

in which

$$\begin{aligned} \tau _{\mu \nu }=T_{\mu \nu }-\frac{T^{em}_{\mu \nu }}{2}+\Sigma _{\mu \nu } \end{aligned}$$

(8.2)

with

$$\begin{aligned} \Sigma _{\mu \nu }= & {} -\frac{\alpha }{2} A_{\mu }A_{\nu } R+ \frac{\beta }{4} g_{\mu \nu } R_{\kappa \lambda }A^{\kappa }A^{\lambda }-\frac{1}{4}( \nabla _\lambda \nabla _\nu \theta _\mu ^\lambda +\nabla _\lambda \nabla _\mu \theta _\nu ^\lambda )\\{} & {} +\frac{1}{4} g_{\mu \nu } \partial _\kappa \big (\nabla _\lambda \theta ^{\kappa \lambda }\big )+\frac{1}{4} \Box \theta _{\mu \nu }. \end{aligned}$$

By using the Bianchi‘s identity $\nabla ^{\mu }G_{\mu \nu }=0,$ the Eq. (8.1) gives the following identity for general form of matter stress tensor $T_{\mu \nu }$ which we considered in this work such that

$$\begin{aligned} \nabla ^\mu T_{\mu \nu }=-\nabla ^\mu \Sigma _{\mu \nu }+\tau _{\mu \nu }\nabla ^\mu \ln A^2 \end{aligned}$$

(8.3)

because $\nabla ^\mu T_{\mu \nu }^{em}=0.$ In other words, unknown matter stress tensor $T_{\mu \nu }$ which we considered here should interact with other fields called as gravity and Maxwell fields to support covariant conservation condition for the total stress tensor $T_{\mu \nu }^{total}$ which we assumed behaves as an anisotropic imperfect fluid with the stress tensor form (2.8).

Appendix II

$$\begin{aligned} \Sigma =&\,8\alpha ^3\beta +4\alpha ^2\beta ^2-4\alpha \beta \nonumber \\&r[(8\alpha ^2-8\alpha ^4-8\alpha ^3\beta +2\alpha ^2\beta ^2+2\alpha \beta ^3)f\nonumber \\&+(2\alpha \beta -8\alpha ^4-8\alpha ^3\beta -2\alpha ^2\beta ^2+V\beta ^2+2\alpha ^2\beta +\alpha \beta ^2+8\alpha ^2)\psi ] \end{aligned}$$

(9.1)

$$\begin{aligned} \Sigma _V=&\,(4\alpha ^3\beta +2\alpha ^2\beta ^2-\alpha \beta )V-4\alpha ^3\beta -2\alpha ^2\beta ^2+\alpha \beta \nonumber \\&r[(16\alpha ^4+16\alpha ^3\beta +4\alpha ^2\beta ^2+4\alpha ^2\beta -16\alpha ^2+2\alpha \beta )\psi \nonumber \\&+(16\alpha ^4+16\alpha ^3\beta +4\alpha ^2\beta ^2-6\alpha ^2 \beta +\alpha \beta ^2-16\alpha ^2+10\alpha \beta -\beta ^2)f] \nonumber \\&r^2[(-4\alpha ^3-2\alpha ^2\beta -4\alpha \beta -2\beta ^2+4\alpha )\psi ^2+(-8\alpha ^4-24\alpha ^3\beta -18\alpha ^2\beta ^2\nonumber \\&-4\alpha \beta ^3-11\alpha ^2\beta - 4\alpha \beta ^2+8\alpha ^2+15\alpha \beta -2\beta ^2)f\psi \nonumber \\&+((8\alpha ^4+8\alpha ^3\beta +2\alpha ^2\beta ^2-8\alpha ^2+\alpha \beta )p_r-\alpha \beta \rho )V\nonumber \\&+(-8\alpha ^4-16\alpha ^3\beta -18\alpha ^2\beta ^2-6\alpha \beta ^3+8\alpha ^2+17\alpha \beta -6\beta ^2)f^2] \end{aligned}$$

(9.2)

$$\begin{aligned} \Sigma _f=&\,\psi [(8\alpha ^4+4\alpha ^3\beta -2\alpha ^2)V+24\alpha ^4+12\alpha ^3\beta -14\alpha ^2]\nonumber \\&+f[(8\alpha ^4-2\alpha ^2\beta ^2-2\alpha ^2+\alpha \beta )V+24\alpha ^4-6\alpha ^2\beta ^2-14\alpha ^2+7\alpha \beta ]\nonumber \\&+r[f^2(-16\alpha ^4-32\alpha ^3\beta -20\alpha ^2\beta ^2-4\alpha \beta ^3 -12\alpha ^3+8\alpha ^2\beta -\alpha \beta ^2\nonumber \\&+28\alpha ^2+8\alpha \beta +\beta ^2 )+\psi f(-16\alpha ^4-24\alpha ^3\beta -8\alpha ^2\beta ^2+4V\alpha \beta -2V\beta ^2\nonumber \\&+4\alpha ^3-2\alpha ^2\beta -2\alpha \beta ^2+40\alpha ^2+8\alpha \beta ) +(16\alpha ^4+8\alpha ^3\beta -8\alpha ^2)Vp_r\nonumber \\&+4V\alpha \beta \psi ^2+(8\alpha ^3+12\alpha ^2+4\alpha )\psi ^2]+r^2[f^3(32\alpha ^4+16\alpha ^3\beta -8\alpha ^2\beta ^2\nonumber \\&-4\alpha \beta ^3-14\alpha ^2-29\alpha \beta +6\beta ^2) +f^2\psi (32\alpha ^4+24\alpha ^3\beta -4\alpha ^2\beta ^2\nonumber \\&-4\alpha \beta ^3-2V\alpha \beta -5V\beta ^2-26\alpha ^3-9\alpha ^2\beta -\alpha \beta ^2-20\alpha ^2-41\alpha \beta +2\beta ^2)\nonumber \\&f\psi ^2(-2V\alpha \beta -4V\beta ^2-18\alpha ^3-18\alpha ^2\beta -6\alpha \beta ^2-14\alpha ^2-12\alpha \beta +2\beta ^2\nonumber \\&-4\alpha )+fV\rho (-2\alpha ^2+\alpha \beta )+fVp_r(-8\alpha ^4+2\alpha ^2\beta ^2+10\alpha ^2-\alpha \beta )] \end{aligned}$$

(9.3)

$$\begin{aligned} \Sigma _\psi =&\,4\alpha ^2\beta (1-V)+r[(8\alpha ^2\beta ^2+20\alpha ^3-26\alpha ^2\beta -9\alpha \beta ) \psi +\psi V(12\alpha ^3+2\alpha ^2\beta \nonumber \\&-2\alpha \beta ^2+\alpha \beta )+2\psi V^3\alpha \beta ^2 +f\psi (24\alpha ^3\beta -4\alpha ^2\beta ^2+20\alpha ^3-54\alpha ^2\beta +2\alpha \beta ^2)\nonumber \\&+fV\psi (12\alpha ^3-2\alpha ^2\beta +2\alpha \beta ^2)]+r^2[(16\alpha ^3-12\alpha ^2\beta )Vp_r +(-24\alpha ^4-8\alpha ^3\beta \nonumber \\&+14\alpha ^2\beta ^2 -2\alpha \beta ^3+24\alpha ^3+4\alpha ^2\beta +26\alpha \beta ^2+2\beta ^3)f^2 +fV\psi (8\alpha ^2\beta +4\alpha \beta ^2)\nonumber \\&+(-24\alpha ^4+20\alpha ^3\beta +10\alpha ^2\beta ^2+4\alpha \beta ^3+48\alpha ^3 +46\alpha ^2\beta +17 \alpha \beta ^2+32\alpha ^2\nonumber \\&-6\alpha \beta +\beta ^2)\psi f+ (8\alpha ^2\beta +4\alpha \beta ^2+2\beta ^2)\psi ^2V+4\alpha ^2\beta \rho V+(-8\alpha ^3\beta -4\alpha ^2\beta ^2\nonumber \\&+24\alpha ^3+28\alpha ^2\beta +10\alpha \beta ^2+24\alpha ^2+2\alpha \beta )\psi ^2]+ r^3[(4\alpha ^2\beta +2\alpha \beta ^2)V^2\psi p_r\nonumber \\&+(12\alpha ^3+6\alpha ^2\beta +8\alpha ^2-\alpha \beta )V\psi p_r+(4\alpha ^3+18\alpha ^2\beta -2\alpha \beta ^2)Vfp_r\nonumber \\&+(-2V\alpha \beta -16\alpha ^3-16\alpha ^2\beta -4\alpha \beta ^2+4\alpha ^2+4\alpha \beta +2\beta ^2-4\alpha )\psi ^3\nonumber \\&+(-36\alpha ^4-30\alpha ^3\beta -6\alpha ^2\beta ^2-28\alpha ^3 -37\alpha ^2\beta -5\alpha \beta ^2+4\beta ^3-8\alpha ^2\nonumber \\&-25\alpha \beta +2\beta ^2)f\psi ^2+ (-4\alpha ^2\beta -10\alpha \beta ^2-4\beta ^3-\beta ^2)Vf\psi ^2\nonumber \\&+(-36\alpha ^4-30\alpha ^3\beta +12\alpha ^2\beta ^2+6\alpha \beta ^3+24\alpha ^3 -76\alpha ^2\beta -58\alpha \beta ^2\nonumber \\&+4\beta ^3-16\alpha ^2-17\alpha \beta +6\beta ^2 )f^2\psi +(-4\alpha ^2\beta -6\alpha \beta ^2-6\beta ^3)f^2\psi V\nonumber \\&+(-4\alpha ^3-2\alpha ^2\beta +\alpha \beta )\rho \psi V+(36\alpha ^3-54\alpha ^2\beta -46\alpha \beta ^2+12\beta ^3)f^3\nonumber \\&+(-4\alpha ^3-2\alpha ^2\beta +2\alpha \beta ^2)\rho Vf] \end{aligned}$$

(9.4)

and

$$\begin{aligned} \Sigma _{p_t}=&\,(16\alpha ^5+16\alpha ^4\beta +4\alpha ^3\beta ^2-4\alpha ^3-2\alpha ^2\beta )V\psi \nonumber \\&+(48\alpha ^5-12\alpha ^3\beta ^2-28\alpha ^3+10\alpha ^2\beta )\psi \nonumber \\ {}&+(16\alpha ^5+20\alpha ^4\beta +6\alpha ^3\beta ^2-4\alpha ^3-3\alpha ^2\beta )Vf\nonumber \\&+ (48\alpha ^5-36\alpha ^4\beta -30\alpha ^3\beta ^2-28\alpha ^3+27\alpha ^2\beta )f\nonumber \\&+r[(64\alpha ^5+64\alpha ^4\beta +16\alpha ^3\beta ^2-24\alpha ^4-14\alpha ^3\beta +3\alpha ^2\beta ^2\nonumber \\&-40\alpha ^3+50\alpha ^2\beta -15\alpha \beta ^2)f^2+(8\alpha ^2\beta -6\alpha \beta ^2)V\psi f\nonumber \\&+((8\alpha ^2\beta -6\alpha \beta ^2)V+10\alpha ^2\beta -10\alpha \beta ^2+160\alpha ^4\beta +60\alpha ^3\beta ^2\nonumber \\ {}&+4\alpha ^2\beta ^3 -4\alpha ^2\beta ^2+112\alpha ^5+8\alpha ^4-64\alpha ^3)\psi f\nonumber \\&+(8\alpha ^2\beta -2\alpha \beta ^2)V\psi ^2+(48\alpha ^5+80\alpha ^4\beta +28\alpha ^3\beta ^2 +16\alpha ^4\nonumber \\ {}&+12\alpha ^3\beta +2\alpha ^2\beta ^2-24\alpha ^3-16\alpha ^2\beta +8\alpha ^2-4\alpha \beta )\psi ^2\nonumber \\ {}&+(32\alpha ^5+8\alpha ^4\beta -4\alpha ^3\beta ^2-16\alpha ^3+4\alpha ^2\beta )Vp_r] \nonumber \\ {}&+r^2[(-4\alpha ^3-2\alpha ^2\beta )\psi V\rho +(-4\alpha ^3-3\alpha ^2\beta )fV\rho +(4\alpha ^2\beta -\alpha \beta ^2)\psi V^2p_r\nonumber \\ {}&+(24\alpha ^5+24\alpha ^4\beta +6\alpha ^3\beta ^2+8\alpha ^4+2\alpha ^3\beta -\alpha ^2\beta ^2-12\alpha ^3)\psi Vp_r\nonumber \\ {}&+(32\alpha ^5+48\alpha ^4\beta +12\alpha ^3\beta ^2-2\alpha ^2\beta ^3-28\alpha ^3+3\alpha ^2\beta )fVp_r\nonumber \\ {}&+(4\alpha \beta ^2-2\alpha \beta +\beta ^2)\psi ^3V+(-4\alpha ^2\beta +2\beta ^3)f\psi ^2 V\nonumber \\ {}&+(-4\alpha ^2\beta -6\alpha \beta ^2+3\beta ^3)f^2\psi V\nonumber \\ {}&+(-32\alpha ^5-32\alpha ^4\beta -8\alpha ^3\beta ^2-8\alpha ^4+2\alpha ^2\beta ^2+12\alpha ^3\nonumber \\ {}&-8\alpha ^2\beta -3\alpha \beta ^2+12\alpha ^2+2\alpha \beta )\psi ^3\nonumber \\ {}&+(-112\alpha ^5-160\alpha ^4\beta -60\alpha ^3\beta ^2-4\alpha ^2\beta ^3-64\alpha ^4\nonumber \\ {}&-60\alpha ^3\beta -6\alpha ^2\beta ^2+4\alpha \beta ^3+84\alpha ^3 +18\alpha ^2\beta -6\alpha \beta ^2+20\alpha ^2 )f\psi ^2\nonumber \\ {}&+(-88\alpha ^5-240\alpha ^4\beta -166\alpha ^3\beta ^2-26\alpha ^2\beta ^3+4\alpha \beta ^4-52\alpha ^4\nonumber \\ {}&-65\alpha ^3\beta -12\alpha ^2\beta ^2+3\alpha \beta ^3 +112\alpha ^3+75\alpha ^2\beta -12\alpha \beta ^2 )f^2\psi \nonumber \\ {}&+(-8\alpha ^5-104\alpha ^4\beta -122\alpha ^3\beta ^2-24\alpha ^2\beta ^3\nonumber \\ {}&+6\alpha \beta ^4+44\alpha ^3+51\alpha ^2\beta -18\alpha \beta ^2 )f^3]. \end{aligned}$$

(9.5)