Gravitational collapse of a magnetized fermion gas with finite temperature

Abstract

We examine the dynamics of a self-gravitating magnetized fermion gas at finite temperature near the collapsing singularity of a Bianchi-I spacetime. Considering a general set of appropriate and physically motivated initial conditions, we transform Einstein–Maxwell field equations into a complete and self-consistent dynamical system amenable for numerical work. The resulting numerical solutions reveal the gas collapsing into both, isotropic (“point-like”) and anisotropic (“cigar-like”), singularities, depending on the initial intensity of the magnetic field. We provide a thorough study of the near collapse behavior and interplay of all relevant state and kinematic variables: temperature, expansion scalar, shear scalar, magnetic field, magnetization, and energy density. A significant qualitative difference in the behavior of the gas emerges in the temperature range T/m _f ∼10⁻⁶ and T/m _f ∼10⁻³.

Appendix A: Contributions to EOS

The thermodynamical potential of a fermion gas has two contributions, see Eq. (9), and thus we can express the magnetization as $M_{f}=M_{f}^{\mathrm{SQFT}}+M_{f}^{\mathrm{QFT}}$. Taking it into account we can rewrite the EOS (42)–(47) in the following form:

$$\begin{aligned} P_{\|} =&-\varOmega_f^\mathrm{SQFT} \underbrace{-\varOmega_f^\mathrm{QFT}-\frac{B^{2}}{2}} \\ \equiv& \overline{P}_{\|} (B,\mu,T ) +\Delta P_{\|} (B ), \end{aligned}$$

(51)

$$\begin{aligned} P_{\bot} =& -\varOmega_f^\mathrm{SQFT}-BM_f^\mathrm{SQFT} \underbrace{-\varOmega_f^\mathrm{QFT}-BM_f^\mathrm{QFT}+ \frac{B^{2}}{2}} \\ \equiv& \overline{P}_{\bot} (B,\mu,T ) +\Delta P_{\bot} (B ), \end{aligned}$$

(52)

$$\begin{aligned} U =&-\varOmega_f^\mathrm{SQFT}+TS+\mu N \underbrace{- \varOmega_f^\mathrm{QFT}+\frac{B^{2}}{2}} \\ \equiv& \overline{U} (B,\mu,T ) +\Delta U (B ), \end{aligned}$$

(53)

where we identify by $\overline{P}_{\|}$, $\overline{P}_{\bot}$ and $\overline{U}$ the terms corresponding to our approximate EOS, while ΔP _∥, ΔP _⊥, and ΔU represent those that have been neglected. Note that these last terms are only field dependent.

In order to analyze the contribution of ΔP _∥, ΔP _⊥, and ΔU to its respective EOS, we evaluate the expressions $\Delta P_{\|} / \overline{P}_{\|}$, $\Delta P_{\bot} / \overline{P}_{\bot}$, and $\Delta U / \overline{U}$ in the sets of solutions obtained previously. In all the cases the numerical results show that these terms are not significant for values of magnetic field less than or equal to the critical magnetic field.

To exemplify that mentioned above, in Fig. 7 we show the fractions of the EOS neglected during the evolution of the gas starting from ϕ(0)=10⁻³, b(0)=5×10⁻⁵, $\tilde{\mu}(0)=2$, S ₂(0)=0, and S ₃(0)=1. As we can observe these values remain close to zero for B≲B _c , specifically $\Delta P_{\|} / \overline {P}_{\|}$, $\Delta P_{\bot} / \overline{P}_{\bot}$, and $\Delta U / \overline{U}$ remain below the line fraction=0.135. The graph of magnetic field is included for comparison.

Fig. 7

In the upper (lower) part are shown the fraction of contribution to EOS of the neglected terms (values of magnetic field) versus proper time