Rotating wormholes in five dimensions with equal angular momenta: large asymmetry regime

Abstract

We clarify the relationship between rotation and the energy condition for stationary rotating wormhole solutions of the Einstein equations coupled to a phantom field in five-dimensional spacetime with equal angular momenta, particularly with large asymmetry between the two sides. It was shown by Dzhunushaliev et al. that the violation of the null energy condition can become arbitrarily small due to rotation. We find that the degree of violation of the null energy condition is essentially determined by the angular momentum and shows little dependence on asymmetry, that is, the mass difference between the two asymptotic regions. We also discuss the relation between the wormhole spacetime and the Myers-Perry black hole. We find that the geometry asymptotes to the extremal Myers-Perry spacetime in the limit of large angular momentum, while the non-extremal black hole geometry cannot be reproduced in any limit.

Figures (7)

• Figure 1: The cross-sectional area $A$ as a function of $x$. The left panel shows the results for fixed $a_{-\infty}=0$ and $c_{\omega}=0, 0.1, 0.5, 1, 3$, while the right panel shows those for fixed $c_{\omega}=0.5$ and $a_{-\infty}=-2, -1, 0, 1, 2$. The solid and dashed curves describe the profiles in the region $l>l_{\rm th}$ and $l<l_{\rm th}$, respectively.
• Figure 2: Phase diagrams showing the dependence of the mass for $l \to +\infty$, $M_{+}$ (upper left), the mass for $l \to -\infty$, $M_{-}$ (upper right), and the scalar charge $Q$ (lower left) on the angular momentum $J$. The values of $M_{+}, M_{-}, Q$ and $J$ are normalized by the area of the throat $A_{\text{th}}$ (with $G=1$), with color variations corresponding to the value of $a_{-\infty}$ for fixed $r_0$. The red curves correspond to the Myers--Perry solution, and these endpoints, the red points at $\abs{J}/A_{\text{th}} \sim 0.04$, indicate the extremal solutions. The blue-dashed curves represent the extrapolation toward the red points for $a_{-\infty}=-8$ (upper left and lower left) and $a_{-\infty}=3$ (upper right), converging to the limits $a_{-\infty}\to -\infty$ and $a_{-\infty}\to \infty$, respectively. The gray-shaded region shows the gap between the red curve and the blue-dashed curve.
• Figure 3: The dependence of the cross-sectional area $A$ on the proper length $l_{p}$ for symmetric case ($a_{-\infty}~=~0$). The solid curves are the result of the numerical integration for the cases of the wormhole, and the dotted horizontal line corresponds to the case of the extremal Myers--Perry black hole.
• Figure 4: The solution of the scalar field $\phi$ (upper) and the null energy condition $\Xi$ (lower) as a function of area $A$ (with $G=1$). The left panels show the results for fixed $a_{-\infty}=0$ and $c_{\omega}=0, 0.1, 0.5, 1, 3$, while the right panels show those for fixed $c_{\omega}=0.5$ and $a_{-\infty}=-2, -1, 0, 1, 2$. For graphical convenience, the values of $\phi$ are shifted so that they become zero at the throat. The solid and dashed curves describe the profiles in the region $l>l_{\rm th}$ and $l<l_{\rm th}$, respectively.
• Figure 5: The contours of $\Xi_{\text{th}} A_{\text{th}}^{2/3}$, the quantity evaluated at the throat, are overlaid on the phase diagrams shown in Fig. \ref{['fig_JvsQorM']}, plotted on the $|J|-M_+$ plane (upper left), the $|J|-M_-$ plane (upper right), and the $|J|-Q$ plane (lower left). The red curve, blue-dashed curve, and gray-shaded region are identical to those in Fig. \ref{['fig_JvsQorM']}, representing the sequence of the Myers--Perry solutions, the limit curve of the wormhole solutions, and the gap region between the two curves, respectively. The color scale of the contours indicates the value of $\Xi_{\text{th}} A_{\text{th}}^{2/3}$: brighter colors correspond to weaker NEC violation.