Symmetry axes of Kerr-NUT-(A)dS spacetimes

We study fixed points of isometries of the higher-dimensional Kerr-NUT-(A)dS spacetimes that form generalizations of symmetry axes. It turns out that, in the presence of nonzero NUT charges, some parts of the symmetry axes are necessarily singular and their intersections are surrounded by regions with closed timelike curves. Motivated by similarities with the spacetime of a spinning cosmic string, we introduce geometric quantities that characterize various types of singularities on symmetry axes. Expanding the Kerr-NUT-(A)dS spacetimes around candidates for possible fixed points, we find the Killing vectors associated with generalized symmetry axes. By means of these Killing vectors, we calculate the introduced geometric quantities describing axial singularities and show their relation to the parameters of the Kerr-NUT-(A)dS spacetimes. In addition, we identify the Killing coordinates that may be regarded as generalization of the Boyer-Lindquist coordinates.

Figure 1

Construction of the cosmic string spacetime from the Minkowski spacetime.

Figure 2

Construction of the spinning cosmic string spacetime from the Minkowski spacetime.

Figure 3

Orbits of the axial Killing vector ${\mathbf{\partial }}_{\varphi }$ that give rise to the symmetry axis at $\rho =0$. Top: the closed orbit of ${\mathbf{\partial }}_{\varphi }$ in the nonspinning spacetime—${\mathbf{\partial }}_{\varphi }$ is aligned with the cyclic Killing vector ${\mathbf{\partial }}_{\phi }$. Bottom: the open orbit of ${\mathbf{\partial }}_{\varphi }$ in the spacetime of the spinning cosmic string—${\mathbf{\partial }}_{\varphi }$ differs from the cyclic Killing vector ${\mathbf{\partial }}_{\phi }$. Pictures are drawn after the smooth periodic gluing (indicated in Figs. 1 and 2); i.e., $\rho$, $\phi$, and $\tau$ are represented as cylindrical coordinates.

Figure 4

Future null cones of the spinning cosmic string spacetime on the surface of constant $\tau$. The circle denotes the position of the ergosurface, $\rho =|\alpha |/\gamma$. Outside the region, the cones lie completely above the surface, while inside the region, the cones intersect the surface.

Figure 5

Generalized definition of the conicity in the presence of other axial singularities.

Figure 6

Graph of the polynomial $\lambda \mathcal{J}(x^2)$. Intersections of the polynomial and the lines $2b_{\overline{\mu }}x$ passing through the origin determine the ranges of coordinates $x_{\overline{\mu }}$; see (3.10) and (3.18).

Figure 7

Graph of the polynomial $-\lambda \mathcal{J}(-r^2)$. Intersections of the polynomial and the line $2mr$ passing through the origin represent the horizons ${}^{\pm }r$, ${{}^{\pm }r}^{(\lambda )}$.

Figure 8

Coordinates $x_1=x$, $x_2=ir$ with respect to ${\rho }_0$, ${\rho }_1$ for $D=4$, $\lambda =0$. Coordinate values $x=\pm a$ correspond to the regular axis at ${\rho }_1=0$.

Figure 9

Coordinates $x_1$, $x_2$, $x_3=ir$ (top, center, and bottom) with respect to ${\rho }_0$, ${\rho }_1$, ${\rho }_2$ for $D=6$, $\lambda =0$. Coordinate values $x_1=\pm a_1$, $x_2=a_1$ correspond to the regular axis at ${\rho }_1=0$, and values $x_2=a_2$ correspond to the regular axis at ${\rho }_2=0$.

Figure 10

Ergosurfaces of the cyclic Killing vector ${\mathbf{\partial }}_{{\phi }_1}$ for $D=4$, $\lambda =0$. The graphs show spacetimes with the regular part of the axis $(\overline{\mu },p)=(1,-1)$ (left) and $(\overline{\mu },p)=(1,0)$ (right), i.e., ${\stackrel{\circ }{x}}_1^2={{}^{-}x}_1^2$ and ${\stackrel{\circ }{x}}_1^2={{}^{+}x}_1^2$, respectively. The lighter gray area denote regions where ${\mathbf{\partial }}_{{\phi }_1}$ is spacelike, ${\mathbf{\partial }}_{{\phi }_1}^2>0$, while the darker gray area stands for regions where ${\mathbf{\partial }}_{{\phi }_1}$ is timelike, ${\mathbf{\partial }}_{{\phi }_1}^2<0$. Dashed lines represent ergosurfaces or fixed points of ${\mathbf{\partial }}_{{\phi }_1}$, ${\mathbf{\partial }}_{{\phi }_1}^2=0$. The highlighted coordinates $x_1=x$, $x_2=ir$ are drawn with respect to ${\tilde{\rho }}_0$, ${\tilde{\rho }}_1$ [cf., Eqs. (4.11) and (4.12)]. Outer and inner horizons are represented by solid lines.

Figure 11

Ergosurfaces of the cyclic Killing vector ${\mathbf{\partial }}_{{\phi }_1}$ for $D=6$, $\lambda =0$. The graphs show spacetimes with the regular part of the axis $(\overline{\mu },p)=(1,-1)$ (left), $(\overline{\mu },p)=(1,0)$ (center), and $(\overline{\mu },p)=(1,+1)$ (right), i.e., ${\stackrel{\circ }{x}}_1^2={{}^{-}x}_1^2$, ${\stackrel{\circ }{x}}_1^2={{}^{+}x}_1^2$, and ${\stackrel{\circ }{x}}_1^2={{}^{-}x}_2^2$, respectively. Top: the sections corresponding to the axis ${\tilde{\rho }}_1=0$. Bottom: the sections corresponding to the axis ${\tilde{\rho }}_2=0$. The lighter gray area denote regions where ${\mathbf{\partial }}_{{\phi }_1}$ is spacelike, ${\mathbf{\partial }}_{{\phi }_1}^2>0$, while the darker gray area stands for regions where ${\mathbf{\partial }}_{{\phi }_1}$ is timelike, ${\mathbf{\partial }}_{{\phi }_1}^2<0$. Dashed lines and white areas represent either ergosurfaces or fixed points of ${\mathbf{\partial }}_{{\phi }_1}$, ${\mathbf{\partial }}_{{\phi }_1}^2=0$. The highlighted coordinates $x_1$, $x_2$, $x_3=ir$ are drawn with respect to ${\tilde{\rho }}_0$, ${\tilde{\rho }}_1$, ${\tilde{\rho }}_2$ [cf., Eqs. (4.11) and (4.12)]. Outer and inner horizons are represented by solid lines.

Figure 12

Ergosurfaces of the cyclic Killing vector ${\mathbf{\partial }}_{{\phi }_2}$ for $D=6$, $\lambda =0$. Graphs show a spacetime with the regular part of the axis $(\overline{\mu },p)=(2,0)$, i.e., ${\stackrel{\circ }{x}}_2^2={{}^{+}x}_2^2$. Left: the section corresponding to the axis ${\tilde{\rho }}_1=0$. Right: the section corresponding to the axis ${\tilde{\rho }}_2=0$. The lighter gray area denotes regions where ${\mathbf{\partial }}_{{\phi }_2}$ is spacelike, ${\mathbf{\partial }}_{{\phi }_2}^2>0$, while the darker gray area stands for regions where ${\mathbf{\partial }}_{{\phi }_2}$ is spacelike, ${\mathbf{\partial }}_{{\phi }_2}^2<0$. Dashed lines and white areas represent either ergosurfaces or fixed points of ${\mathbf{\partial }}_{{\phi }_2}$, ${\mathbf{\partial }}_{{\phi }_2}^2=0$. The highlighted coordinates $x_1$, $x_2$, $x_3=ir$ are drawn with respect to ${\tilde{\rho }}_0$, ${\tilde{\rho }}_1$, ${\tilde{\rho }}_2$ [cf., Eqs. (4.11) and (4.12)]. Outer and inner horizons are represented by solid lines.

Figure 13

Examples of closed orbits on ${\mathbb{T}}^2$ corresponding to different winding numbers $n_a^b$. Top: orbits with winding numbers $n_1^b=(1,0)$ (solid line) and $n_2^b=(0,1)$ (dashed line). Bottom: orbits with winding numbers $n_1^b=(5,1)$ (solid line) and $n_2^b=(1,2)$ (dashed line).