Ring wormholes and time machines

In the present paper we discuss properties of a model of a ring wormhole, recently proposed by Gibbons and Volkov [Phys. Rev. D 96, 024053 (2017); J. Cosmol. Astropart. Phys. 05 (2017) 039; Phys. Lett. B 760, 324 (2016)]. Such a wormhole connects two flat spacetimes which are glued through disks of the radius $a$ bounded by the string with negative angle deficit $-2\pi$. The presence of the string’s matter violating the null-energy condition makes the wormhole static and traversable. We study gravitational field of static sources in such a spacetime in the weak-field approximation. In particular, we discuss how a field of an oblate thin massive shell surrounding one of the wormhole’s mouths is modified by its presence. We also obtain a solution of a similar problem when both mouths of the wormhole are located in the same space. This approximate solution is found for the case when the distance $L$ between these mouths is much larger than the radius $a$ of the ring. We demonstrate that the corresponding locally static gravitational field in such a multiply connected space is nonpotential. As a result of this, the proper time gap for the clock’s synchronization linearly grows with time and closed timelike curves are formed. This process inevitably transforms such a traversable ring wormhole into a time machine. We estimate the timescale of this process.

Figure 1

A ring wormhole connecting two flat spaces $R_{+}$ and $R_{-}$.

Figure 2

Ring wormhole in space $\tilde{R}$ with two asymptotic regions $R_{-}$ and $R_{+}$. Each of the domains $R_{\pm }$ is covered by cylindrical coordinates ${\rho }_{\pm },{\phi }_{\pm },z_{\pm }$. Directions related to angular coordinate ${\phi }_{\pm }$ are not shown. They can be obtained by rotations around axes ${\rho }_{\pm }=0$. The whole space $\tilde{R}$ is covered by one copy of the oblate spheroidal coordinates $\chi \in (-\infty ,\infty )$, $\vartheta \in (0,\pi )$ and $\phi \equiv {\phi }_{\pm }\in (-\pi ,\pi )$. Dashed lines corresponds to $\chi =\text{const}$, dotted lines to $\vartheta =\text{const}$. Double lines indicate the disks which are identified as indicated by arrows. They represent the mouths of the wormhole.

Figure 3

Ring wormhole space $R_{\mathrm{wh}}$ with one asymptotic region. We have global cylindrical coordinates $\rho ,\phi ,z$ and two copies of the oblate spheroidal coordinates ${\chi }_{\pm },{\vartheta }_{\pm },{\phi }_{\pm }$. The coordinate $\phi \equiv {\phi }_{\pm }$ is not shown. The disks $D_{\pm }$ representing the wormhole’s mouths are indicated by double lines located at $z=\pm \ell /2$. The dashed (${\chi }_{\pm }=\text{const}$) and dash-dotted (${\vartheta }_{\pm }=\text{const}$) lines correspond to the domains ${\overline{V}}_{\pm }$ described in the text and Fig. 4. The coordinates in ${\overline{V}}_{\pm }$ are related to the opposite coordinates in the leaked domains ${\widehat{V}}_{\mp }$ by ${\chi }_{+}=-{\chi }_{-}$ and ${\vartheta }_{+}=\pi -{\vartheta }_{-}$. Compare also Fig. 5. In the further discussion we put a massive shell at a constant value of ${\chi }_{-}={\chi }_{\mathrm{o}}$.

Figure 4

Domains $V_{-}$ (top) and $V_{+}$ (middle) are covering together the whole space. They are composed by the “primary” parts ${\overline{V}}_{\pm }$ and leaked parts ${\widehat{V}}_{\pm }$. The bottom diagram shows two intersections $\overline{V}$ and $\widehat{V}$ of these two domains. Horizontal direction corresponds to the $z$ axis, vertical to the $\rho$ direction. The mouths $D_{\pm }$ of the wormhole $W$ are represented by the thick vertical lines. The distance between mouths is $\ell$.

Figure 5

Coordinate maps $({\chi }_{-},{\vartheta }_{-})$ in the domain $V_{-}$ (top half) and $({\chi }_{+},{\vartheta }_{+})$ in $V_{+}$ (bottom half).

Figure 6

Functions $\mathcal{Z}(\chi )$.

Figure 7

Potential $U_{\mathrm{zm}}$ (top) and field strength $w_{\mathrm{zm}}^j$ (bottom) of the zero mode. The left diagram shows the field in space $R_{-}$, and the right one in $R_{+}$. Thick segments on the ${\rho }_{\pm }$ axes represent the mouths of the wormhole. Angular coordinate $\phi$ is ignored and can be recovered by a rotation around the axes ${\rho }_{\pm }=0$.

Figure 8

Potential $U_{\mathrm{sh}}$ (top) and field strength $w_{\mathrm{sh}}^j$ (bottom) of a thin massive spheroidal shell located around the wormhole in $R_{+}$. The left diagram shows the field in space $R_{-}$, and the right one in $R_{+}$. Thick segments on the ${\rho }_{\pm }$ axes represent mouths of the wormhole. The shell (depicted by the thick ellipse) is confocal with the wormhole mouth. Angular coordinate $\phi$ is ignored and can be recovered by a rotation around the axes ${\rho }_{\pm }=0$.

Figure 9

Functions ${\mathcal{Z}}_{-}(\chi )$ and ${\mathcal{Z}}_{+}(\chi )$.

Figure 10

Potential $U_{\mathrm{hom}}$ (top) and field strength $w_{\mathrm{hom}}^j$ (bottom) of an asymptotically homogeneous field in $R_{+}$. The left diagram shows the induced field in space $R_{-}$, and the right one the modified homogeneous field in $R_{+}$. Thick segments on the ${\rho }_{\pm }$ axes represent mouths of the wormhole. Angular coordinate $\phi$ is ignored and can be recovered by a rotation around the axes ${\rho }_{\pm }=0$.

Figure 11

Coordinate domains around the wormhole in the presence of a gravitational field of a thin massive shell near the left wormhole mouth. Domains $V_{-}$ (top) and $V_{+}$ (middle) are covering together the whole space. The division into main part ${\overline{V}}_{\pm }$ and leaked part ${\widehat{V}}_{\pm }$ is indicated. The bottom diagram shows two intersections $\overline{V}$ and $\widehat{V}$ of these two domains. The wormhole mouths $D_{\pm }$ are represented by the thick vertical lines. The real size of both mouths is the same. However, since the gravitational field of the shell affects the geometry, the global Cartesian coordinates used for mapping to the diagram do not represent the geometry properly. The coordinate size of both mouths is thus different, as can be observed in the diagram. The distance between mouths should be large—diagrams do not reflect this feature accurately.

Figure 12

Diagram shows the zeroth-order approximation of potentials $U_{\pm }$ drawn along the axis $z$. The potentials $U_{-}$ and $U_{+}$ are continuous on the domains $V_{-}$ and $V_{+}$, respectively. They coincide on the intersection domain $\overline{V}$ but differ by a constant $\mathrm{\Delta }U$ on the intersection domain $\widehat{V}$. The falloff of the potential is exaggerated to compensate a small distance between mouths. The top part of the diagram indicates the domains $V_{-}$ and $V_{+}$ and the position of the wormhole and of the massive shell.

Figure 13

Finite wormhole with distant mouths. The diagram shows coordinates $x_{-}^j$ and $x_{+}^j$ defined on the domains $U_{-}$ and $U_{+}$, respectively. Since the potential modifies the geometry, the coordinates continuously extended through the wormhole does not match the original coordinates.

Figure 14

Potential $U$ [top] and field strength [bottom] of the thin massive shell localized around the wormhole mouth $D_{-}$. The potential depicted here is smooth outside the wormhole throat, but it is discontinuous at the throat. The zeroth-order value $U_{\text{sht}}$ of the potential corresponds to the shell in space without a wormhole. The correction given by induced field $U_{\text{shi}}$ makes the field strength smooth through the wormhole throat.

Figure 15

The diagram shows the first-order approximation of potentials $U_{\pm }$ drawn along the axis $z$. The potentials $U_{-}$ and $U_{+}$ are continuous on the domains $V_{-}$ and $V_{+}$, respectively. The potentials include the induced field correcting the zeroth-order contribution on the wormhole throat. Both potentials coincide on the intersection domain $\overline{V}$ but differ by a constant $\mathrm{\Delta }U$ on the intersection domain $\widehat{V}$. The top part of the diagram indicates the domains $V_{-}$ and $V_{+}$ and the position of the wormhole and of the massive shell.

Figure 16

Contour $C$.

Figure 17

Red dashed lines mark the connection of events via wormhole. Before the massive shell was turned on around $D_{-}$ the events are globally synchronized. After an adiabatic formation of a massive shell the pace of time at $D_{-}$ slows down as seen by an observer at infinity and the mouths of the wormhole become desynchronized from his or her point of view. The pace of clocks at the mouths is symbolically presented by oscillating lines. When the time difference accedes $\ell /c$ the chronology horizon forms. After that the time machine appears.

Figure 18

Projection of $M$ onto the space of Killing trajectories $S$.

Figure 19

Time gaps.