Inside charged black holes. I. Baryons

An extensive investigation is made of the interior structure of self-similar accreting charged black holes. In this, the first of two papers, the black hole is assumed to accrete a charged, electrically conducting, relativistic baryonic fluid. The mass and charge of the black hole are generated self-consistently by the accreted material. The accreted baryonic fluid undergoes one of two possible fates: either it plunges directly to the spacelike singularity at zero radius, or else it drops through the Cauchy horizon. The baryons fall directly to the singularity if the conductivity either exceeds a certain continuum threshold ${\kappa }_{\infty }$, or else equals one of an infinite spectrum ${\kappa }_n$ of discrete values. Between the discrete values ${\kappa }_n$, the solution is characterized by the number of times that the baryonic fluid cycles between ingoing and outgoing. If the conductivity is at the continuum threshold ${\kappa }_{\infty }$, then the solution cycles repeatedly between ingoing and outgoing, displaying a discrete self-similarity reminiscent of that observed in critical collapse. Below the continuum threshold ${\kappa }_{\infty }$, and except at the discrete values ${\kappa }_n$, the baryonic fluid drops through the Cauchy horizon, and in this case undergoes a shock, downstream of which the solution terminates at an irregular sonic point where the proper acceleration diverges, and there is no consistent self-similar continuation to zero radius. As far as the solution can be followed inside the Cauchy horizon, the radial direction is timelike. If the radial direction remains timelike to zero radius (which cannot be confirmed because the self-similar solutions terminate), then there is presumably a spacelike singularity at zero radius inside the Cauchy horizon, which is distinctly different from the vacuum (Reissner-Nordström) solution for a charged black hole.

Figure 1

Charge-to-mass $Q/M_c$ of the black hole as a function of the charge-to-mass density $q/{\rho }_b$ of baryons at the outer sonic point boundary, for an accretion rate ${\eta }_s=0.1$. Two cases are shown, one with zero conductivity, $\kappa =0$, the other with finite conductivity, $\kappa =1$. The charge-to-mass $Q/M_c$ of the black hole generally increases as the charge-to-mass density $q/{\rho }_b$ at the outer boundary is increased, although in the zero conductivity case the charge-to-mass $Q/M_c$ does turn down slightly at the largest charge-to-mass densities $q/{\rho }_b$. The maximum charge-to-mass density $q/{\rho }_b$ is set by the condition that ${\gamma }^2-{\beta }^2>0$.

Figure 2

The simplest model considered in this paper: a black hole that accretes uncharged baryons, at rate ${\eta }_s=0.1$. Quantities are plotted against radius $r$ in units where the charge-augmented mass at the outer sonic point is unity, $M_c=1$. Lines are dashed where quantities are negative. Disks mark the outer sonic point, where $V=\sqrt{w}\approx 0.577$, at which the boundary conditions are set. Short horizontal bars mark the horizon, where $V=1$. (Upper panel) Proper velocity $V$ of the similarity frame relative to the baryonic frame. (Middle panel) Dimensionless proper baryonic mass density $z\equiv 4\pi r^2{\rho }_b$. (Bottom panel) Interior mass $M$, proper acceleration $g$ experienced by the baryonic fluid, and the homothetic scalar $H$, Eq. (58).

Figure 3

Penrose diagram for the uncharged black hole of Fig. 2. The arrowed line represents schematically the world line of a parcel of baryonic fluid. The thin gray lines are lines of constant self-similar coordinate. The self-similar solution extends some way outside the sonic point outside the outer horizon, but it does not continue to $r=\infty$; the Penrose diagram as drawn presumes that the spacetime joins to an asymptotically flat space as $r\to \infty$.

Figure 4

Similar to Fig. 2, but for a black hole with charge-to-mass $Q/M_c=0.8$, accreting baryons with zero conductivity, $\kappa =0$. Lines are dashed where quantities are negative. Disks mark sonic points, where $V=\pm \sqrt{w}$. Short horizontal bars mark horizons, where $V=\pm 1$. (Upper panel) Proper velocity $V$ of the similarity frame relative to the baryonic frame. The velocity passes from $V=+\infty$ to $-\infty$ as the charged baryonic fluid passes from ingoing to outgoing. The baryonic fluid drops through the Cauchy horizon, beyond which the solution terminates at an irregular sonic point. (Middle panel) Dimensionless proper baryonic mass and charge densities $z\equiv 4\pi r^2{\rho }_b$ and $z_q\equiv 4\pi r^{2}q$. (Bottom panel) Interior mass $M$, proper acceleration $g$, and the homothetic scalar $H$.

Figure 5

Putative continuations of the similarity solution inside the Cauchy horizon for the model shown in Fig. 4, which has $Q/M_c=0.8$ and zero conductivity. Three cases are shown, ranging the gamut from an extremely weak shock to an extremely strong shock. In all cases the shocked fluid continues downstream for a short distance before terminating at an irregular sonic point, marked by a disk, where the proper acceleration diverges and the similarity solution cannot be continued. Dotted lines connect preshock and postshock quantities. (Top panel) Proper velocity $V$ of the similarity frame relative to the baryonic frame. The Cauchy horizon, where $V=-1$, is marked by a short horizontal bar. A shock decelerates the baryonic fluid from supersonic to subsonic, through the sound speed where $V=-\sqrt{w}\approx 0.577$. (Bottom panel) Proper acceleration $g$ experienced by the baryonic fluid in the vicinity of each of the three putative shocks. The acceleration changes from outward (negative, dashed lines) in the preshock fluid to inward (positive, continuous lines) in the postshock fluid.

Figure 6

Tentative Penrose diagram for a charged black hole, such as that shown in Figs. 4, 5, in which the baryons drop through a Cauchy horizon and undergo a shock before falling to a central spacelike singularity. The arrowed line represents schematically the world line of a parcel of baryonic fluid. The thin gray lines are lines of constant self-similar coordinate. The self-similar solution extends some way outside the sonic point outside the outer horizon, but it does not continue to $r=\infty$. The Penrose diagram as drawn presumes that the spacetime joins to an asymptotically flat space as $r\to \infty$. Similarly, the self-similar solution extends some way inside the shock inside the Cauchy horizon, but it does not continue to a singularity, but rather terminates at an irregular sonic point. The Penrose diagram as drawn presumes that the spacetime terminates at a spacelike singularity at $r=0$. A question mark emphasizes the fact that the similarity solution leaves undetermined how the spacetime continues (or not) to the singularity.

Figure 7

Velocity $V$ versus radius $r$ for pure baryonic models with various conductivities $\kappa$, increasing from zero at top left to a large value somewhat less than the maximum conductivity ${\kappa }_{\max }\approx 67.0$ at bottom right. At conductivities less than the continuum threshold ${\kappa }_{\infty }$, the top six panels, the baryonic fluid drops through the Cauchy horizon unless the conductivity equals one of an infinite spectrum of discrete values ${\kappa }_n$, in which case the fluid falls to a singularity at zero radius. The first two cases of the discrete spectrum, ${\kappa }_1$ and ${\kappa }_2$, are shown. The velocity $V$ passes through $\pm \infty$ where the baryonic fluid transitions between ingoing ($V>0$, continuous lines) and outgoing ($V<0$, dashed lines). Solutions between values of the discrete spectrum ${\kappa }_n$ are characterized by the number of times that the baryonic fluid cycles between ingoing and outgoing. At the continuum threshold ${\kappa }_{\infty }$, the bottom left panel, the solution exhibits discrete self-similarity, the baryonic fluid cycling repeatedly between ingoing and outgoing. At and above the continuum threshold ${\kappa }_{\infty }$, the bottom three panels, all solutions fall to a central singularity. Disks mark sonic points, where $V=\pm \sqrt{w}\approx \pm 0.577$. Horizons occur where $V=\pm 1$.

Figure 8

Velocity times radius, $Vr$, for two models with conductivities $\kappa ={\kappa }_1\pm ϵ$ just either side of the value ${\kappa }_1\approx 0.998$ that separates solutions that cycle once versus twice between ingoing ($V>0$, continuous lines) and outgoing ($V<0$, dashed lines). The inset shows a detail of how the velocities in the two models peel off from the critical solution, which itself would continue on the outgoing path as a horizontal line to zero radius.

Figure 9

Similar to Figs. 2, 4. Model with conductivity $\kappa =0.8$, less than the first critical conductivity ${\kappa }_1$. The baryonic fluid drops through the Cauchy horizon.

Figure 10

Similar to Figs. 2, 4. Model with conductivity at the continuum threshold conductivity $\kappa ={\kappa }_{\infty }$. The model displays discrete self-similarity.

Figure 11

Discrete self-similarity at the continuum threshold conductivity $\kappa ={\kappa }_{\infty }$. The horizontal axis is the radius $r$ folded over two periods, a period being 3.9463 decades. The vertical axis is the velocity $V$ multiplied by a power of radius, $r^{1.7081}$. The solution as shown passes through 31 periods, covering a total of about 120 decades of radius.

Figure 12

Angular size ${\chi }_{\mathrm{ph}}$ of the black hole, Eq. (73), and the blueshift of photons at the edge of the black hole, Eq. (74), perceived by an observer in either the baryonic frame or a radial free-fall frame, for three models. The models shown are (left panels) the uncharged baryonic model from Fig. 2; (middle panels) the charged nonconducting baryonic model from Fig. 4; (right panels) the discretely self-similar charged baryonic model with $\kappa ={\kappa }_{\infty }$, from Fig. 10. Disks mark the outer sonic point, where $V=\sqrt{w}\approx 0.577$. Horizontal bars mark the outer horizon, where $V=1$. Light from the outside universe is visible only from outside the Cauchy horizon, so lines terminate infinitesimally outside the Cauchy horizon even in models in which the baryons drop inside the Cauchy horizon (middle model).