Figure 1

Behaviors of the Lagrangian densities of admissible NEDs around the vacuum (cases B1, B2, B3 and IRD) and on the boundary of their domain of definition at $Y=0$ (cases A1, A2 and UVD). In between both domains, these functions must be strictly monotonic increasing in order for the energy to be positive definite. The parameters $p$, $\sigma$ and $q$ correspond to the exponents of $r$ in Eqs. (8, 10, 9), respectively, determining the central and asymptotic field behaviors of the ESS solutions.Reuse & Permissions

Figure 2

Qualitative behavior of the energy curves $y={\epsilon }_{\mathrm{in}}(r,Q)$ (for fixed $Q$) for the different admissible IRD models [the curve UVD corresponds to the primitive of $r^2{T}_0^0(r,Q)$, defined up to an arbitrary constant]. The slopes of these curves at $r=0$ diverge in the A1 and UVD cases and take finite values ($=8\pi Qa$) in the A2 cases. In all cases these curves are monotonic, convex and diverge asymptotically with vanishing slope. The cutting points between these curves and the beam of dashed lines $y=\frac{r+C}{2}$ define the horizons of the configurations. The tangency points correspond to extreme black hole and extreme black point configurations. Otherwise the configurations are naked singularities and one or two-horizons black holes. The cutting points between the energy curves and the straight line $C=0$ provide the radius of the critical configurations (which are meaningless for the UVD case).Reuse & Permissions

Figure 3

Behavior of $\lambda (r)$ for the IRD-A1 configurations. When $C>0$ there can be naked singularities [curve I, $C>C_{\mathrm{extr}}(Q)$], extreme black holes [curve II, $C=C_{\mathrm{extr}}(Q)$] or two-horizons black holes [curves III and IV, $0<C<C_{\mathrm{extr}}(Q)$]. Solutions with $C\le 0$ lead to single-horizon black holes (curve V, $C=0$, corresponding to the critical configuration and curve VI for $C<0$). The structure of the limit configurations obtained as $C\to 0^{\pm }$ can be easily visualized in this figure. All curves approach asymptotic flatness slower than the Schwarzschild solution. Curves I to IV describe also the typical qualitative behavior of the metric function in the IRD-UVD cases.Reuse & Permissions

Figure 4

Qualitative behavior of $\lambda (r)$ for the IRD-A2 models when $\sigma >1$. This behavior is similar in the three cases (b), (c), (a), ($16\pi qa⪌1$, respectively), excepting by the sign of the metric function of the critical configurations at the center, which is ${\lambda }_{\mathrm{crit}}(r=0)⪋0$, respectively. Then, by simplicity, we have plotted a unique set of curves for the different configurations and modified the relative position of the $\lambda =0$ axis for each case (continuous horizontal lines). The metric functions diverge at the center to $+\infty$ when $C>0$ and to $-\infty$ for $C<0$, whereas for the critical configuration ($C=0$), they attain a finite value, which is positive, null or negative, depending on the case. The slope of the critical configuration at the center diverges for this value of $\sigma$. Curve I corresponds to naked singularities in all cases. Curve II corresponds to extreme black hole configurations in the $16\pi qa>1$ case and to naked singularities otherwise. Curves III and IV correspond to two-horizons black holes in the $16\pi qa>1$ case and to naked singularities otherwise. The dashed curve V is the critical configuration and corresponds to single-horizon black holes in the $16\pi qa>1$ case, to extreme black points when $16\pi qa=1$ and to naked singularities in the $16\pi qa<1$ case. Curves VI and VII correspond to single-horizon black holes in all cases. Curves IV and VI show how the metric approaches the behavior of the critical case as $C\to 0$ from above and below, respectively.Reuse & Permissions

Figure 5

Same comments as in Fig. 4, but now for $\sigma <1$. The slope at the center of the metric function for the critical configuration ($C=0$) vanishes in this case (see also the discussion in the text).Reuse & Permissions

Figure 6

Same comments as in Fig. 4, but now for $\sigma =1$. The slope at the center of the metric function for the critical configuration ($C=0$) is finite in this case (see also the discussion in the text).Reuse & Permissions