Dynamical black hole in a bouncing universe

We analyze the causal structure of McVittie spacetime for a classical bouncing cosmological model. In particular, we compute the trapping horizons of the metric and integrate the trajectories of radial null geodesics before, during, and after the bounce takes place. In the contracting phase up to the occurrence of the bounce, a dynamical black hole is present. When the universe reaches a certain minimum scale, the trapping horizons disappear and the black hole ceases to exist. After the bounce, the central weak singularity becomes naked. In the expanding phase, for large positive values of the cosmic time, the behavior of null geodesics indicates that the solution contains a black hole. These results suggest that neither a contracting nor an expanding universe can accommodate a black hole at all times.

Figure 1

The black line indicates the location of the trapping horizons in McVittie spacetime for the $\mathrm{\Lambda }\mathrm{CDM}$ model. The white zone corresponds to the regular region while the light pink corresponds to the antitrapped region, respectively. The dot dashed line denotes the location of the singularity. Here, we fixed $m_0=0.479$ and $H_0=1/3$ as an example.

Figure 2

The black line indicates the location of the trapping horizons in McVittie spacetime for a dust background model. The white zone corresponds to the regular region while the light pink corresponds to the anti-trapped region, respectively. The dot dashed line denotes the location of the singularity. Here, we fixed $m_0=0.479$ as an example.

Figure 3

Radial outgoing null geodesics in McVittie spacetime for the $\mathrm{\Lambda }\mathrm{CDM}$ model. The black line indicates the location of the trapping horizons and the dot-dashed line denotes the singular surface $R=2m_0$. Here, we fixed $m_0=0.479$ and $H_0=1/3$ as an example.

Figure 4

Radial outgoing null geodesics in McVittie spacetime for a dust cosmological background. The black line indicates the location of the trapping horizons and the dot-dashed line denotes the singular surface $R=2m_0$. Here, we fixed $m_0=0.479$ as an example.

Figure 5

Radial ingoing null geodesics in McVittie spacetime for the $\mathrm{\Lambda }\mathrm{CDM}$ model. The black line indicates the location of the trapping horizons and the dot-dashed curve denotes the singular surface $R=2m_0$. Here, we fixed $m_0=0.479$ and $H_0=1/3$ as an example.

Figure 6

Radial ingoing null geodesics in McVittie spacetime for a dust cosmological background. The black line indicates the location of the trapping horizons and the dot-dashed curve denotes the singular surface $R=2m_0$. Here, we fixed $m_0=0.479$ as an example.

Figure 7

The black line indicates the condition $\sqrt{3}G{m}_0/c^3=t_0$. The light yellow zone corresponds to values of $m_0$ and $t_0$ such that $\sqrt{3}G{m}_0/c^3>t_0$ while the light red region satisfies the condition $\sqrt{3}G{m}_0/c^3<t_0$. Here, $m_0$ is in units of solar masses $M_{\odot }$.

Figure 8

The black lines indicate the location of the trapping horizons. The white zones correspond the regular regions, the light pink zone indicates the antitrapped region and the light blue zone denotes the trapped region. The dot dashed curve denotes the location of the singularity. The lower plot zooms into the region near the bounce. Here, $m_0=50M_{\odot }$, and $t_0=5\times {10}^{-5}\mathrm{s}$.

Figure 9

Radial outgoing geodesics in McVittie spacetime for a bouncing cosmological background. The black line indicates the location of the trapping horizons while the dot dashed curve denotes the location of the singularity. The lower plot zooms into the region near the bounce. Here, $m_0=50M_{\odot }$, and $t_0=5\times {10}^{-5}\mathrm{s}$.

Figure 10

Radial ingoing geodesics in McVittie spacetime for a bouncing cosmological background. The black line indicates the location of the trapping horizons while the dot dashed curve denotes the location of the singularity. The lower plot zooms into the region near the bounce. Here, $m_0=50M_{\odot }$, and $t_0=5\times {10}^{-5}\mathrm{s}$.

Figure 11

Radial outgoing geodesics in McVittie spacetime for a bouncing cosmological background in the region close to the bounce. The black line indicates the location of the trapping horizons while the dot dashed curve denotes the location of the singularity. Here, $m_0=50M_{\odot }$, and $t_0=5\times {10}^{-5}\mathrm{s}$.

Figure 12

Radial ingoing geodesics in McVittie spacetime for a bouncing cosmological background in the region close to the bounce. The black line indicates the location of the trapping horizons while the dot dashed curve denotes the location of the singularity. Here, $m_0=50M_{\odot }$, and $t_0=5\times {10}^{-5}\mathrm{s}$.

Figure 13

Trajectories of radial ingoing null geodesics in the $z–l$ plane for McVittie spacetime in a bouncing cosmological model. Here, $m_0=50M_{\odot }$, and $t_0=5\times {10}^{-5}\mathrm{s}$.

Figure 14

Closeup in the $z–l$ plane showing the trajectories of radial ingoing null geodesics approaching the surface $x=2$. Expression (31) is represented by the dotted black curve. Here, $m_0=50M_{\odot }$, and $t_0=5\times {10}^{-5}\mathrm{s}$.

Figure 15

Trajectory of a future-directed outgoing null geodesic that emerges from the singular point $(l,z)=(0,0)$ and reaches $z=1$ ($x\to \infty$) in a finite interval of time.

Figure 16

Light cone structure in McVittie spacetime for a bouncing cosmological model in the region $t<0$. The dotted curves represent the null ingoing geodesics while the dot dashed curves the null outgoing geodesics. The regions shaded in grey show some light cones and the black arrow indicates the future direction.

Figure 17

Light cone structure in McVittie spacetime for a bouncing cosmological model in the region $t>0$. The dotted curves represent the null ingoing geodesics while the dot dashed curves the null outgoing geodesics. The grey shadow regions show some light cones and the black arrow indicates the future direction.

Figure 18

Qualitative Penrose diagram of the McVittie spacetime for a bouncing cosmological model in the expanding region ($t>0$). The dotted lines represent null ingoing geodesics while the dashed lines null outgoing geodesics. Here, ${\mathcal{H}}^{+}=X_{-}\to 2$, $t\to \infty$, and ${\mathcal{J}}^{+}$ is the future null infinity.

Figure 19

Qualitative Penrose diagram of the McVittie spacetime for a bouncing cosmological model in the contracting region ($t<0$). The dotted lines represent null ingoing geodesics while the dashed lines null outgoing geodesics. Here, ${\mathcal{H}}^{-}=X_{-}\to 2$, $t\to -\infty$, and ${\mathcal{J}}^{-}$ is the past null infinity.