Solitonic generation of vacuum solutions in five-dimensional general relativity

We describe a solitonic solution-generating technique in the five-dimensional general relativity. Reducing the five-dimensional problem to the four-dimensional one, we can systematically obtain single-rotational axially symmetric vacuum solutions. Applying the technique for a simple seed solution, we have previously obtained the series of stationary solutions which includes the $S^2$-rotating black ring. We analyze the qualitative features of these solutions, e.g., conical singularities, closed timelike curves, and spacetime curvatures. We investigate the rod structures of seed and solitonic solutions. We examine the relation between the expressions of the metric in the prolate-spheroidal coordinates and in the $C$-metric coordinates.

Figure 1

Schematic diagram of a local ringlike object which resides in the spacetime. Generally some singular behavior appears near the horizon.

Figure 2

Schematic pictures of rod structures. The left panel shows the rod structure of Minkowski spacetime which is a seed of the $S^2$-rotating black ring. The right panel shows the rod structure of the $S^2$-rotating black ring. The segment [$-\sigma$, $\sigma$] of the semi-infinite rod of the seed becomes a finite timelike rod and changes its direction by the solution-generating transformation. Here we put the finite timelike rod between $x^0$ and $\varphi$ lines because the corresponding eigenvector has nonzero $x^0$ and $\varphi$ components. Note that the segment [$-\lambda \sigma$, $-\sigma$] also changes its direction by the transformation in general.

Figure 3

The behavior of the 0-0 component of the metric (34) in the case of $(\alpha ,\beta ,\lambda )=(0.5,-0.195752,2)$. The region where the component function is above the level zero corresponds to the ergo region.

Figure 4

Ergo regions for the cases of $\alpha =0.5$, 0.7, 0.9 with $\lambda =2$. The values of $\beta$ are determined by Eq. (36). In the shaded regions the values of $g_{00}$ are positive.

Figure 5

Ergo regions for singular cases where $\alpha =0.5$ and $\lambda =2$. In the shaded regions the values of $g_{00}$ become positive. The values of $\beta$ do not satisfy Eq. (36). When $\beta =\frac{\alpha (1+\lambda )}{2}$, the ergo region disappears.

Figure 6

The behavior of the $\varphi \mathrm{\text{−}}\varphi$ component of the metric in the cases of $(\alpha ,\beta ,\lambda )=(0.5,0.75,2)$ and (0.5, $-0.195752$, 2) which satisfies the Eq. (36). The corresponding component in the right panel is always non-negative, while for the general case the component becomes negative near the horizon, which means the existence of CTC regions.

Figure 7

CTC regions. In the shaded regions the values of $g_{\varphi \varphi }$ are negative. Note that the $y$-range of the upper-right panel and the $x$-range of the lower-left panel are different from the others. The lower-left panel corresponds to $\beta ={\beta }_{+}$.

Figure 8

Plots of the $\psi \mathrm{\text{−}}\psi$ components of the $S^2$-rotating black ring for $a_0$ with $m^2=3$ and $h=0.5$. The upper, middle, and lower lines correspond with the outer edge ($y=1$), middle ($y=0$), and inner edge ($y=-1$) of the ring, respectively.

Figure 9

Plots of Kretchman invariants for the cases of $(\lambda ,\alpha ,\beta ,)=(2,0.5,{\beta }_{-})$, (2, 0.5, $\beta +$), (1, 0.5, $\beta +$), and (1, 0.5, $\beta -$). In the upper-right panel we plot the value of $\ln |K|$. We can see that $K$ grows disastrously around the point where $B=0$.

Figure 10

The contour lines of $x$, $y$, $u$, and $v$ in $\chi -r$ plane. In the left panel, the solid lines and the dotted lines are $x=\mathrm{const}.$ and $y=\mathrm{const}.$ lines, respectively. In the right panel, the solid lines and the dotted lines are $u=\mathrm{const}.$ and $v=\mathrm{const}.$ lines, respectively.