What is a singular black hole beyond general relativity?

Exploring the characterization of singular black hole spacetimes, we study the relation between energy density, curvature invariants, and geodesic completeness using a quadratic $f(R)$ gravity theory coupled to an anisotropic fluid. Working in a metric-affine approach, our models and solutions represent minimal extensions of general relativity (GR) in the sense that they rapidly recover the usual Reissner-Nordström solution from near the inner horizon outwards. The anisotropic fluid helps modify only the innermost geometry. Depending on the values and signs of two parameters on the gravitational and matter sectors, a breakdown of the correlations between the finiteness/divergence of the energy density, the behavior of curvature invariants, and the (in)completeness of geodesics is obtained. We find a variety of configurations with and without wormholes, a case with a de Sitter interior, solutions that mimic nonlinear models of electrodynamics coupled to GR, and configurations with up to four horizons. Our results raise questions regarding what infinities, if any, a quantum version of these theories should regularize.

Figure 1

Representation of $z(x)$ as a function of the radial coordinate $x$ (in units of $r_{\star }={\tilde{\beta }}^{1/4}{r}_0$), for $\gamma =1.1$ (solid, red), $\gamma =1.5$ (dashed, orange) and $\gamma =2$ (dotted, green). Note that far from the bouncing region ($z=z_c$, $x=0$), where the wormhole throat is located, we have $z^2\simeq x^2$, which restores the GR behavior there.

Figure 2

Affine parameter $\lambda (z)$ (measured in units of $r_{\star }$) for radial null geodesics (with $E=1$), in the GR case ($\gamma =0$, dashed) and three $\gamma >1$ cases ($=1.1$, 1.5, 3, solid), for ingoing/outgoing trajectories. Far from $z=z_c(\gamma )$ (where the wormhole throat is located) all curves converge to the GR solution $\lambda (z)=z$, while for $z=z_c(\gamma )$, the curves $\lambda (z)$ (for $\gamma >1$) diverge to $\pm \infty$ which means that they are complete, as opposed to the GR case.

Figure 3

Representation of the affine parameter $\lambda (z)$ as a function of the radial coordinate $z$ (in units of $r_{\star }={\tilde{\beta }}^{1/4}{r}_0$), for $\gamma =1$. The dashed curves represent radial null geodesics in GR, while the solid ones are those of our gravity model. The upper/lower curve is the ingoing/outgoing light ray. Radial timelike geodesics with ${\delta }_1<{\delta }_c^{(1)}$ (black dotted curves) lie within the light cone and hit the origin in a finite affine time.

Figure 4

Representation of $x(z)$ as a function of the radial coordinate $z$ (in units of $r_{\star }={\tilde{\beta }}^{1/4}{r}_0$), for $\gamma =1$ (solid, red). The curve $x=z$ (dashed) is shown for comparison. We identify two regions where $x(z)$ is monotonic: one which tends to the GR solution $x=z$ and has a minimum, and another one close to the origin representing a nonasymptotically flat solution which may be disconnected from the exterior one.

Figure 5

Representation of $g_{tt}$ in the $\{\sigma >0,\beta >0\}$ case, as a function of the radial coordinate $z$, for $r_S/r_{\star }=10$, ${\delta }_1=1.3(r_S/4r_{\star })$ and $z_{\min }=2$ (solid curve). The dotted line represents the GR solution with the same parameters. Note that $g_{tt}$ cuts the $z$ axis four times, which implies four horizons. The two external horizons are almost coincident with the GR prediction. The other two are a result of the modified dynamics. The vertical grey line represents the location where $x(z)$ reaches its minimum, which is (close but) unrelated to the minima of $g_{tt}$. Note that the horizontal axis begins at $z=1$.

Figure 6

Representation of the affine parameter $\lambda (z)$ in the $\{\sigma >0,\beta >0\}$ case, corresponding to radially ($L=0$) infalling geodesics with $E=1.1$ in the geometry of Fig. 5 (solid curves) as compared to their GR counterparts (dashed curves). The upper pair represents null geodesics while the lower one is timelike. Deviations from GR (dashed curve) only arise near $z_{\min }$. In GR timelike observers bounce before reaching the center, while light rays get there in a finite affine time. In the quadratic $f(R)$ theory, both null and timelike geodesics reach the surface $z=1$ in a finite affine time. The vertical grid line represents the location where $x(z)$ reaches its minimum, which sets a saddle point for $\lambda (z)$. Note that the horizontal axis begins at $z=1$.

Figure 7

Representation of $g_{tt}$ extended into the $z<1$ region for the same parameters as in Fig. 5. The geometry now exhibits up to five horizons. The metric diverges as $1/z$ as $z\to 0$ (Schwarzschild like). Note that the energy density in the region $z<1$ is always negative, with a divergence at $z=1$. In our view, only the region $z>1$ should be regarded as physical.

Figure 8

Extension of the geodesics shown in Fig. 6 down to $z=0$. Both timelike and radial null geodesics take a finite affine time to reach the origin. Since the metric in this (unphysical) region diverges as $g_{tt}\sim 1/z$, the behavior is analogous to that found in the Schwarzschild black hole.

Figure 9

Representation of $x(z)$ for different values of $\gamma$ in the model with $s_{\sigma }=-1=s_{\beta }$.

Figure 10

Representation of $z(x)$ for $\gamma =1$ (solid green curve) and $\gamma =50$ (dashed blue curve) as compared to the GR behavior $z^2=x^2$ (dashed red line) in the model with $s_{\sigma }=-1$ and $s_{\beta }=1$. This shows that the physical metric describes a wormhole structure. Far from the central region the GR behavior is recovered $z^2(x)\approx x^2$.