Novel black-bounce spacetimes: Wormholes, regularity, energy conditions, and causal structure

We develop a number of novel “black-bounce” spacetimes. These are specific regular black holes where the area radius always remains nonzero, thereby leading to a “throat” that is either timelike (corresponding to a traversable wormhole), spacelike (corresponding to a “bounce” into a future universe), or null (corresponding to a “one-way wormhole”). We first perform a general analysis of the regularity conditions for such a spacetime and then consider a number of specific examples. The examples are constructed using a mass function similar to that of Fan-Wang and fall into several particular cases, such as the original Simpson-Visser model, a Bardeen-type model, and other generalizations thereof. We analyze the regularity, energy conditions, and causal structure of these models. The main results are several new geometries, more complex than before, with two or more horizons, with the possibility of an extremal case. We derive a general theorem regarding static spacetime regularity and another general theorem regarding (non)satisfaction of the classical energy conditions.

Figure 1

Carter-Penrose diagram for the case when we have a traditional two-way traversable wormhole in the sense of Morris and Thorne.

Figure 2

Carter-Penrose diagram for the case when we have a one-way wormhole geometry with an extremal null throat; for all relevant examples this corresponds to fixing $a=a_{\mathrm{ext}}$.

Figure 3

Carter-Penrose diagram for the maximally extended case where we have one horizon location in each universe. One may propagate through the event horizon, at $r=r_{+}$, to reach the spacelike bounce hypersurface at $r=0$, before bouncing into a future reincarnation of our own Universe. Infinitely many future copies of the Universe exist if we extrapolate the time coordinate “up” the page; only two are displayed here for tractability.

Figure 4

Graphical representation of the possibilities of $f(r)$, given by (72). For $a<a_{\mathrm{ext}}$, the signature changes four times, which translates as four horizons: two event and two Cauchy horizons. For $a=a_{\mathrm{ext}}$, we have a black bounce with two symmetric degenerate horizons. For $a>a_{\mathrm{ext}}$, no horizon exists. See the text for more details.

Figure 5

In this example the horizon location is extremal. Mathematically we therefore have repeated roots $r_{+}=r_C$ of $f(r)=0$. Since the extremal horizon ($H_{+}=C_{+}=r_{+}$) is as usual an infinite proper distance from any point not on the extremal horizon, the Carter-Penrose diagram is somewhat misleading in that it would be infeasible to propagate through the extremal horizon to reach the hypersurface at $r=0$. The bounce surface is now timelike, given $f(r)$ does not switch sign through the extremal horizon.

Figure 6

Graphical representation of the possibilities of $f(r)$, given by (86) with $r_0=2\pi a$.

Figure 7

Graphical representation of the energy density $\rho (r)$, to the model (86) with $r_0=2\pi a$.

Figure 8

Graphical representation of $f(r)$ for (90). In the left side $n=1$ and right side $n=2$. For $n=2$, the extreme value is approximately $a_{\mathrm{ext}}\approx 5.16315560586775m/{\pi }^2$.

Figure 9

Carter-Penrose diagram for the case where we have an extremal horizon in our Universe, followed by a timelike bounce hypersurface at $r=0$, bouncing into a separate universe without horizons. Since the extremal horizon $(H+=C+=r+)$ is as usual an infinite proper distance from any point not on the extremal horizon, the Carter-Penrose diagram is somewhat misleading in that it would be infeasible to propagate through the extremal horizon to reach the hypersurface at $r=0$.

Figure 10

Carter-Penrose diagram for the case where we have an inner and outer horizon in our Universe, followed by a timelike bounce hypersurface at $r=0$, bouncing into a separate universe without horizons.

Figure 11

Graphical representation of the energy conditions for (90) with $n=1$, $a=a_{\mathrm{ext}}=4m/\pi$, ${\kappa }^2=8\pi$, in the left plot, and $n=2$, $a=a_{\mathrm{ext}}\approx 5.16m$, ${\kappa }^2=8\pi$ in the right plot.

Figure 12

Graphical representation of $f(r)$ for (92), with $n=1$ (left) and $n=2$ (right) for different values of $a$.

Figure 13

Energy density for (92) with $n=1$, ${\kappa }^2=8\pi$ (left) and $n=2$, ${\kappa }^2=8\pi$ (right).