Space-time diagrammatics

We introduce a new class of two-dimensional diagrams, the projection diagrams, as a tool to visualize the global structure of space-times. We construct the diagrams for several metrics of interest, including the Kerr-Newman-(anti)de Sitter family, with or without cosmological constant, and the Emparan-Reall black rings.

Figure 1

The conformal diagram for ($1+1$)-dimensional Minkowski space-time.

Figure 2

The radius of the time-machine “left boundary” ${\widehat{r}}_{-}/m$ as a function of $a/m$.

Figure 3

A projection diagram for the Kerr-Newman metrics with two distinct zeros of ${\Delta }_r$ (left diagram) and one double zero (right diagram); see Remark III.1.

Figure 4

The conformal diagram of a maximal analytic extension of the equatorial plane $\theta =\pi /2$ and constant $\tilde{\phi }$ for Kerr metrics with arbitrary $a\in \mathbb{R}$ with $r>0$ (left) and $r<0$ (right).

Figure 5

A projection diagram for overspinning Kerr-Newman space-times.

Figure 6

A projection diagram for the Kerr-Newman-de Sitter metric with four distinct zeros of ${\Delta }_r$; see Remark III.1.

Figure 7

A projection diagram for the Kerr-Newman-de Sitter metrics with three distinct zeros of ${\Delta }_r$, $r_1<0<r_2=r_3<r_4$; see Remark III.1.

Figure 8

A projection diagram for the Kerr-Newman-de Sitter metrics with three distinct zeros of ${\Delta }_r$, $r_1<0<r_2<r_3=r_4$; see Remark III.1. Note that one cannot continue the diagram simultaneously across all boundaries $r=r_3$ on ${\mathbb{R}}^2$, but this can be done on an appropriate Riemann surface.

Figure 9

A projection diagram for the Kerr-Newman-de Sitter metrics with two distinct first-order zeros of ${\Delta }_r$, $r_1<0<r_2$ and $\mu >0$; see Remark III.1. The diagram for a first-order zero at $r_1$ and third-order zero at $r_2=r_3=r_4$ would be identical except for the bifurcation surface of the bifurcate Killing horizon at the intersection of the lines $r=r_2$, which does not exist in the third-order case and has therefore to be removed from the diagram.

Figure 10

The critical mass parameter $m_c\sqrt{|\Lambda /3|}=\Xi {\mu }_c\sqrt{|3/\Lambda |}$ as a function of $|a|\sqrt{|\Lambda /3|}$ when $q=0$.

Figure 11

The critical mass parameter $m_c\sqrt{\frac{|\Lambda |}{3}}$ as a function of $\alpha =a\sqrt{\frac{|\Lambda |}{3}}$ and $q\sqrt{\frac{|\Lambda |}{3}}$.

Figure 12

The projection diagrams for the Kerr-Newman-anti de Sitter metrics with two distinct zeros of ${\Delta }_r$ (left diagram) and one double zero (right diagram); see Remark III.1.

Figure 13

The projection diagram for the Emparan-Reall black rings. The arrows indicate the causal character of the orbits of the isometry group. The boundary $y=-1$ is covered, via the projection map, by the axis of rotation and by spatial infinity $i^0$. Curves approaching the conformal null infinities ${\mathcal{I}}^{\pm }$ asymptote to the missing corners in the diagram.

Figure 14

The projection diagram for the complement of a world-tube $\mathbb{R}\times B(R)$ in five-dimensional Minkowski space-time using spherical coordinates (left figure, where the shaded region has to be removed) or using ring coordinates (right figure). In the right figure the right boundary $y=-1$ is covered, via the projection map, both by the axis of rotation and by spatial infinity, while null infinity projects to the missing points at the top and at the bottom of the diagram.

Figure 15

The sequences $q_i$ and $p_i$. Rotating the figure by integer multiples of 90° shows that the problem of nonunique limits arises on any pair of orbit-adjacent boundaries.

Figure 16

The left diagram is the conformal diagram for an extension of the universal covering space of the sections $\theta =\mathrm{const}$, $\phi ={\mathrm{const}}^{\prime }$, of the Taub space-time. The right diagram represents simultaneously the four possible diagrams for the maximal extensions, within the Taub-NUT class, with compact Cauchy horizons, of the Taub space-time. After invoking the left-right symmetry of the diagram, which lifts to an isometry of the extended space-time, the four diagrams lead to two nonisometric space-times.