Self-similar cosmological solutions with dark energy. II. Black holes, naked singularities, and wormholes

We use a combination of numerical and analytical methods, exploiting the equations derived in a preceding paper, to classify all spherically symmetric self-similar solutions which are asymptotically Friedmann at large distances and contain a perfect fluid with equation of state $p=(\gamma -1)\mu$ with $0<\gamma <2/3$. The expansion of the Friedmann universe is accelerated in this case. We find a one-parameter family of self-similar solutions representing a black hole embedded in a Friedmann background. This suggests that, in contrast to the positive pressure case, black holes in a universe with dark energy can grow as fast as the Hubble horizon if they are not too large. There are also self-similar solutions which contain a central naked singularity with negative mass and solutions which represent a Friedmann universe connected to either another Friedmann universe or some other cosmological model. The latter are interpreted as self-similar cosmological white hole or wormhole solutions. The throats of these wormholes are defined as two-dimensional spheres with minimal area on a spacelike hypersurface and they are all nontraversable because of the absence of a past null infinity.

Figure 1

$V$ is plotted against $-1/z$ for $\gamma =1/3$. The thick line corresponds to the exact Friedmann solution ($A_0=0$), which has one similarity horizon.

Figure 2

(a) $W=8\pi G\mu r^2$ and (b) $W/z^2=8\pi G\mu t^2$ are plotted against $-1/z$ for $\gamma =1/3$. The thick line $W/z^2=12$ corresponds to the exact Friedmann solution ($A_0=0$).

Figure 3

(a) $S=R/r$ and (b) $zS=R/t$ are plotted for $\gamma =1/3$. The thick line $z^{2}S=1$ corresponds to the exact Friedmann solution ($A_0=0$).

Figure 4

(a) $M=2Gm/r$ and (b) $zM=2Gm/t$ are plotted for $\gamma =1/3$. The thick line $z^{4}M=4$ corresponds to the exact Friedmann solution ($A_0=0$).

Figure 5

$M/S=2Gm/R$ is plotted for $\gamma =1/3$. The thick line corresponds to the exact Friedmann solution ($A_0=0$).

Figure 6

The conformal diagram of the exact flat Friedmann solution. The thin curves and lines denote similarity surfaces, i.e., orbits of $z=\mathrm{\text{constant}}$. There is a similarity horizon at $z=z_1(>0)$.

Figure 7

The $t\mathrm{\text{−}}r$ plane of the exact flat Friedmann solution. A fluid element moves from the bottom to the top. $r=0$ and $r\to +\infty$ correspond to spacelike infinity and the physical center ($R=0$), respectively. $t=0$ corresponds to the big-bang singularity, which is represented by a zigzag line. There is a similarity horizon at $z=z_1(>0)$, corresponding to a cosmological event horizon.

Figure 8

The conformal diagram of the cosmological naked singularity solution. There is a similarity horizon at $z=z_1(>0)$. $z=z_{*}(>0)$ with $0<t<\infty$ corresponds to a timelike singularity at which the mass is negative.

Figure 9

The $t\mathrm{\text{−}}r$ plane of the cosmological naked singularity solution. $r=0$ corresponds to spacelike infinity. There is a similarity horizon at $z=z_1(>0)$, corresponding to a cosmological event horizon. $z=z_{*}(>0)$ with $0<t<\infty$ corresponds to a timelike singularity of negative mass and is represented by a zigzag line.

Figure 10

(a) $V$ and (b) $M/S$ as functions of $zS(=R/t)$ for the cosmological black hole solutions.

Figure 11

The value of $zS=R/t$ for various horizons is plotted as a function of the parameter $A_0$ for the cosmological black hole solutions with $\gamma =1/3$.

Figure 12

The conformal diagram of the cosmological black hole solution. There are two similarity horizons at $z=z_1$ and $z=z_2(z_2<0<z_1)$. $z=z_1$ with $0<t<\infty$ is the cosmological event horizon, while $z=z_2$ with $0<t<\infty$ is the black hole event horizon. $z=z_{*}(<0)$ with $0<t<\infty$ gives a spacelike singularity with positive mass.

Figure 13

The $t\mathrm{\text{−}}r$ plane of the cosmological black hole solution. $r=0^{+}$ corresponds to spacelike infinity.

Figure 14

(a) $W/z^2$ and (b) $z^{2}S$ are plotted for $\gamma =1/3$.

Figure 15

The conformal diagram of the Friedmann-quasi-Friedmann wormhole solution. There are two similarity horizons, $z=z_1$ and $z=z_2(z_2<0<z_1)$, both corresponding to cosmological event horizons. $z=0^{+}$ and $z=0^{-}$ give two distinct null infinities, described by Friedmann and quasi-Friedmann asymptotics, respectively. $z=\pm \infty$, however, is described by the Kantowski-Sachs asymptotic as $t$ increases from 0 to $\infty$.

Figure 16

The $t\mathrm{\text{−}}r$ plane of the Friedmann-quasi-Friedmann wormhole solution. We show the solution with a wormhole throat at $z=z_{\mathrm{t}}$ in the positive $z$ region. $r=0^{+}$ and $0^{-}$ correspond to the Friedmann and quasi-Friedmann infinities, respectively. $t=0$ corresponds to the big-bang singularity, represented by the quasi-Kantowski-Sachs solution.

Figure 17

The conformal diagrams of the cosmological white hole solutions with: (a) no similarity horizon; (b) one degenerate similarity horizon $z=z_1(>0)$; and (c) two distinct similarity horizons $z=z_1$ and $z=z_2(0<z_1<z_2<\infty )$. For each case, $z=z_{*}(<0)$ is a spacelike singularity with positive mass. In (b), $z=z_1$ with $t=\infty$ corresponds to future null infinity. In (c), $z=z_2$ with $t=\infty$ corresponds to future null infinity.

Figure 18

The $t\mathrm{\text{−}}r$ plane of the cosmological white hole solution in the case without a similarity horizon. $r=0^{+}$ and $+\infty$ correspond to the Friedmann and quasistatic infinities, respectively. There is a wormhole throat at $z=z_{\mathrm{t}}$ in the positive $z$ region. $z=z_{*}(<0)$ with $0<t<\infty$ corresponds to a spacelike singularity with positive mass, represented by a zigzag line.

Figure 19

The fictitious conformal diagram of the Friedmann-constant-velocity wormhole solution. There is a similarity horizon at $z=z_1(>0)$, corresponding to the cosmological event horizon. $z=\infty$ with $0<t<\infty$ corresponds to a timelike boundary of spacetime at infinity.