Nonsingular black holes, wormholes, and de Sitter cores from anisotropic fluids

We study Born-Infeld gravity coupled to an anisotropic fluid in a static, spherically symmetric background. The free function characterizing the fluid is selected on the following grounds: i) recovery of the Reissner-Nordström solution of General Relativity at large distances, ii) fulfillment of classical energy conditions, and iii) inclusion of models of nonlinear electrodynamics as particular examples. Four branches of solutions are obtained, depending on the signs of two parameters on the gravity and matter sectors. On each branch, we discuss in detail the modifications on the innermost region of the corresponding solutions, which provides a plethora of configurations, including nonsingular black holes and naked objects, wormholes, and de Sitter cores. The regular character of these configurations is discussed according to the completeness of geodesics and the behavior of curvature scalars.

Figure 1

Radial function $z(x)$ for case I, as follows from the integration of Eq. (37) in this case. From bottom to top, the solid curves represent $\xi =1$ (blue), $\xi =5$ (orange), and $\xi =10$ (green), with the dashed red line corresponding to $|x|$. The wormhole throat is located at $z_c=({\xi }^2+1{)}^{1/4}$.

Figure 2

Metric component $g_{tt}(z)$ of case I taking $\xi =1$, for which the wormhole throat is located at $z_c=2^{1/4}$ (represented by the vertical dashed black line). We find i) Reissner-Nordström-like solutions with two (blue solid, ${\delta }_1=1/10$, $r_{\star }/r_S=1/6$), a single degenerate (blue dotted, ${\delta }_1=3$, $r_{\star }/r_S=1/6$), or zero (blue dashed, ${\delta }_1=3/4$, $r_{\star }/r_S=1/6$) horizons; ii) Schwarzschild-like solutions with a single horizon (red solid, ${\delta }_1=9/6$, $r_{\star }/r_S=1/6$); and iii) Minkowski-like solutions with a single horizon (orange solid, ${\delta }_1={\delta }_c\approx 0.464$, ${\delta }_2=5$) or none (orange dashed, ${\delta }_1={\delta }_c\approx 0.464$, $r_{\star }/r_S=1/6$). All solutions are asymptotically flat.

Figure 3

The affine parameter $\tilde{u}(x)$ for null radial geodesics of case I, as given by Eq. (54). Here, we take values $\xi =1$ (blue), $\xi =5$ (orange), and $\xi =10$ (green), with the dashed red curve representing $\tilde{u}(x)=x$ and corresponding to the GR behavior. As it is obvious from this plot, null radial geodesics in this case are complete.

Figure 4

Radial function $z(x)$ for case II. From bottom to top, the curves represent $\xi =3/4$ (solid blue), $\xi =1$ (dotted black), $\xi =3/2$ (solid orange), $\xi =4$ (solid green), and $\xi =10$ (solid brown), with the two dashed straight red lines corresponding to $|x|$. The wormhole throat is located at $z_c=({\xi }^2-1{)}^{1/4}$, provided that ${\xi }^2>1$; otherwise, the radial function extends to $z=0$ (dashed black, $\xi =1$, and solid blue, $\xi =3/4$).

Figure 5

The affine parameter $\tilde{u}(x)$ for null radial geodesics of case II, where we verify the reliability of the approximation (74) around the wormhole throat $z=z_c=({\xi }^2-1{)}^{1/2}$ (corresponding to $x=0$ in this plot). Here, we take values $\xi =3/2$ (orange), $\xi =3$ (green), and $\xi =10$ (brown), with the dashed red curve representing $\tilde{u}(x)=x$ and corresponding to the GR behavior. These (non-GR) geodesics are complete. Moreover, we also depict the limit configuration with ${\xi }^2=1$, for which the wormhole throat lies at $z=x=0$, which cannot be reached in finite affine time by null radial geodesics (see Sec. 5c3 for details). In the GR region ($z\to \infty$), all curves converge to the GR behavior, $\tilde{u}(x)\approx x$.

Figure 6

Representation of $x(z)$ of case III (corresponding to $\{s_{\epsilon }=+1,s_{\beta }=-1\}$) for $\xi =0$ (solid black), $\xi =1$ (dotted red), $\xi =2$ (dashed blue), and $\xi =4$ (solid orange).

Figure 7

Representation of $x(z)$ of case IV for $\xi =1$ (solid blue), $\xi =3$ (solid red), and $\xi =5$ (solid orange), as compared to $|x|=z$ (dashed black). The vertical dotted lines set the minimum radius available for the radial function, $z=1$, where the energy density diverges.