Inside charged black holes. II. Baryons plus dark matter

This is the second of two companion papers on the interior structure of self-similar accreting charged black holes. In the first paper, the black hole was allowed to accrete only a single fluid of charged baryons. In this second paper, the black hole is allowed to accrete in addition a neutral fluid of almost noninteracting dark matter. Relativistic streaming between outgoing baryons and ingoing dark matter leads to mass inflation near the inner horizon. When enough dark matter has been accreted that the center-of-mass frame near the inner horizon is ingoing, then mass inflation ceases and the fluid collapses to a central singularity. A null singularity does not form on the Cauchy horizon. Although the simultaneous presence of ingoing and outgoing fluids near the inner horizon is essential to mass inflation, reducing one or the other of the ingoing dark matter or outgoing baryonic streams to a trace relative to the other stream makes mass inflation more extreme, not the other way around as one might naively have expected. Consequently, if the dark matter has a finite cross section for being absorbed into the baryonic fluid, then the reduction of the amount of ingoing dark matter merely makes inflation more extreme, the interior mass exponentiating more rapidly and to a larger value before mass inflation ceases. However, if the dark matter absorption cross section is effectively infinite at high collision energy, so that the ingoing dark matter stream disappears completely, then the outgoing baryonic fluid can drop through the Cauchy horizon. In all cases, as the baryons and the dark matter voyage to their diverse fates inside the black hole, they only ever see a finite amount of time pass by in the outside universe. Thus the solutions do not depend on what happens in the infinite past or future. We discuss in some detail the physical mechanism that drives mass inflation. Although the gravitational force is inward, inward means opposite direction for ingoing and outgoing fluids near the inner horizon. Mass inflation is driven by a feedback loop in which the general relativistic contribution to the gravitational force sourced by the radial pressure accelerates the ingoing and outgoing fluids through each other, which increases the radial pressure, which increases the gravitational force.

Figure 1

A black hole with the same parameters, an accretion rate ${\eta }_s=0.1$ and a charge-to-mass $Q/M_c=0.8$, as that shown in Fig. 4 of Paper 1, except that here the black hole accretes dark matter in addition to baryons, with density ratio ${\rho }_d/{\rho }_b=0.1$ at the outer sonic point. Quantities are plotted against radius $r$ in units where the charge-augmented mass at the outer sonic point is unity, $M_c=1$. To reveal more detail, the radial axis is split into two regimes with different horizontal and vertical scales. 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 and charge densities $z\equiv 4\pi r^2{\rho }_b$ and $z_q\equiv 4\pi r^{2}q$, and dimensionless proper dark matter mass density $z_d\equiv 4\pi r^2{\rho }_d$. (Bottom panel) Interior mass $M$, proper acceleration $g$ experienced by the baryonic fluid, and the homothetic scalar $H$. The exponential increase of mass $M$ over a modest range of radii above the inner horizon, $r\sim 0.4–0.1$, is the signature of mass inflation.

Figure 2

Dimensionless exponential inflationary scale length $l$ as a function of the ratio ${\rho }_d/{\rho }_b$ of dark matter to baryonic proper mass densities at the outer sonic point, for black holes accreting at rate ${\eta }_s=0.1$ and with charge-to-mass $Q/M_c=0.8$. Shorter scale lengths $l$ signify more extreme mass inflation.

Figure 3

Similar to Fig. 1, but for a black hole in which the absorption rate of dark matter by baryons is effectively infinite above a certain large collision energy. Disks mark sonic points, where $V=\pm \sqrt{w}$. Short horizontal bars mark horizons, where $V=\pm 1$. Mass inflation begins, but is cut short as soon as all the dark matter is absorbed. With the dark matter gone, the baryons almost immediately drop through the Cauchy horizon. The similarity solution terminates inside the Cauchy horizon at an irregular sonic point.

Figure 4

Angular size ${\chi }_{\mathrm{ph}}$ of the black hole and the blueshift of photons at the edge of the black hole perceived by observers in either the baryonic frame or the free-fall dark matter frame, for (left) the model of Fig. 1, where the black hole accretes noninteracting dark matter, and (right) the model of Fig. 3, where the black hole accretes dark matter whose cross section for absorption by baryons is effectively infinite at high energy. Light from the outside universe is visible only from outside the Cauchy horizon, so lines terminate infinitesimally outside the Cauchy horizon even in the model at right, in which the baryons drop inside the Cauchy horizon.

Figure 5

Ingoing dark matter and outgoing baryons streaming through each other at radius $r$ inside the black hole were accreted by the black hole at different times. The graph shows the ratio $t_d/t$ of ages of the black hole when the dark matter versus baryons were accreted. For example, a ratio $t_d/t=2$ means that the black hole was twice as old when it accreted the dark matter as it was when it accreted the baryons. The solid line is for the model with noninteracting dark matter shown in Fig. 1, while the dashed line almost coincident with the solid line is for the model shown in Fig. 1, in which the dark matter has an effectively infinite cross section for absorption by baryons at high energy.