Interpreting $A$ and $B$ metrics with $\mathrm{\Lambda }$ as gravitational field of a tachyon in an (anti–)de Sitter universe

We investigate the main properties and mutual relations of the so-called $A$ and $B$ metrics with any value of the cosmological constant. In particular, we explicitly show that both the $AII$ and $BI$ metrics are, in fact, the famous Schwarzschild–(anti–)de Sitter spacetime (that is the $AI$ metric) boosted to superluminal speed. Together, they can be combined to form a complete gravitational field of a tachyon in an asymptotically Minkowski or (anti–)de Sitter background. The boundary separating the $AII$ and $BI$ regions is the Mach-Cherenkov shockwave on which the curvature is unbounded. We analyze various geometric features of such spacetimes, provide their natural physical interpretation, and visualize them using convenient background coordinates and embeddings.

Figure 1

Complete Minkowski spacetime divided into Regions 1 and 2 covered by two “background” $AII$ metrics (Region 1 for $T>0$ and Region 2 for $T<0$) and Region 3 covered by the “background” $BI$ metric. The left part is the $T-X$ section with $Y=0$ and $Z$ arbitrary, while the right part shows the $X$, $Y$, $Z$ subspace with $T=\mathrm{const}>0$. For $T>0$ the boundary is an expanding cylinder whose interior is Region 1, while for $T<0$ it is a contracting cylinder whose interior is Region 2. Outside this cylinder lies Region 3 covered by the $BI$ metric. The tachyon moves along the $Z$ axis with infinite speed.

Figure 2

The $T$, $X$, $Z$ subspace of Minkowski spacetime for $Y=0$ (left) and for a constant $Y\ne 0$ (right). It is divided into the two disconnected Regions 1 and 2 with the $AII$ metrics, and Region 3 with the $BI$ metric.

Figure 3

Left: Global conformal diagram for the $AII$ metric (32) with coordinates $\vartheta ,\phi$ suppressed (in particular, $\vartheta =0$). Here $i^{\pm }$ denote past/future timelike infinities, while ${\mathcal{I}}^{\pm }$ are null conformal infinities. The horizons $\sigma =2M$ are null, while the singularities $\sigma =0$ are timelike. Right: Embedding diagram of the $BI$ metric (40) for the section $z=\text{const}$ and $\cosh \tau \equiv C=\text{const}$. For larger $C$, the neck diameter grows, while the surface becomes more restricted.

Figure 4

Minkowski spacetime separated into Regions 1, 2, and 3 by the Mach-Cherenkov shock cones generated by the tachyon slowed down by a boost to finite superluminal speed. The spacetime structure is visualized in the sections $Y=0$ (left) and $T^{\prime }=\mathrm{const}>0$ (right). The tachyon moves in the $Z^{\prime }$ direction, and generates an expanding rear cone and contracting forward cone.

Figure 5

The spacetime structure (left), consisting of the $AII$ and $BI$ metrics separated by the Mach-Cherenkov shock waves (right) for various superluminal speeds $v$ of the tachyonic source.

Figure 6

The visualizations of de Sitter spacetime (left) and anti–de Sitter spacetime (right) as hyperboloids (45) embedded in a flat five-dimensional spacetime (46). The remaining coordinates ($Z_3$, $Z_4$ for $\mathrm{\Lambda }>0$, and $Z_2$, $Z_3$ for $\mathrm{\Lambda }<0$) are suppressed. For more details see Ref. [4].

Figure 7

Allowed ranges of the coordinate $p$ for $\mathrm{\Lambda }>0$ (left) and $\mathrm{\Lambda }<0$ (right) in the $BI$ metric (57). The horizontal lines correspond to different values of $-2n$. The parts of these lines that lie under the curve $-\frac{\mathrm{\Lambda }}{3}{p}^3+p$ determine the admitted range of $p$. Their parts above the curve are dashed: there is no solution for these values because the metric would no longer have the correct signature. The intersections mark the roots $p_0$, $p_1$ of Eq. (58), with $0<p_0<p_1$.

Figure 8

Allowed ranges of $p$ for $\mathrm{\Lambda }>0$ (left) and $\mathrm{\Lambda }<0$ (right) in the $BII$ metric (61). They are determined by those parts of the horizonal lines $-2n$ that lie under the curve $-\frac{\mathrm{\Lambda }}{3}{p}^3-p$.

Figure 9

Allowed ranges of $p$ for $\mathrm{\Lambda }>0$ (left) and $\mathrm{\Lambda }<0$ (right) in the $BIII$ metric (65). They are determined by the root $p_0=\sqrt[3]{6n/\mathrm{\Lambda }}$ of Eq. (66).

Figure 10

Motion of the infinitely fast test tachyon in (anti–)de Sitter background space. For $\mathrm{\Lambda }>0$, the tachyon runs with infinite speed in a closed circle around the neck of the de Sitter hyperboloid (left). For $\mathrm{\Lambda }<0$, there are two tachyons running along two hyperbolic lines on opposite sides, corresponding to the signs $\pm$ in Eq. (75), of the anti–de Sitter hyperboloid (right).

Figure 11

Separation of the background de Sitter space into Regions 1, 2 and 3, covered by the $AII$ and $BI$ metrics, for $Z_2=0$ and $|Z_4|<a$ (upper left part), $|Z_4|>a$ (upper right part) and $|Z_4|=a$ (lower part). For $|Z_4|<a$ the separation surface $Z_1^2+Z_2^2=Z_0^2$ cuts the de Sitter hyperboloid, for $|Z_4|=a$ it is tangent to it, and for $|Z_4|>a$ this surface does not intersect the hyperboloid. Because the coordinate $Z_4$ is suppressed here, the tachyon motion is not visualized very well: in this section, it corresponds just to two points at the intersection of the $Z_3$ axis with the hyperboloid when $|Z_4|<a$, and one point when $|Z_4|=a$. The tachyon motion is better seen in Fig. 10 where both nontrivial coordinates $Z_3$ and $Z_4$ are visible.

Figure 12

Separation of the background anti–de Sitter space into Regions 1, 2 and 3 for the section $Z_2=0$, $Z_3=\mathrm{const}$. Unlike for de Sitter space, visualized in Fig. 11, this picture looks qualitatively the same for any value of $Z_3$. The motion of the tachyon is better seen in Fig. 10.

Figure 13

The position of the tachyon and the Mach-Cherenkov shock wave separating the $AII$ and $BI$ metrics in the global conformal representation of de Sitter spacetime for arbitrary $\varphi =\mathrm{const}$. Since the angular coordinate $\varphi \in [0,2\pi )$ is suppressed, the tachyon is visible here only as a single point at $\eta =\frac{\pi }{2},\chi =\frac{\pi }{2},\theta =\frac{\pi }{2}$. The bottom spacelike surface $\eta =0$ is the past conformal infinity ${\mathcal{I}}^{-}$ of the de Sitter universe, while the top surface $\eta =\pi$ is its future conformal infinity ${\mathcal{I}}^{+}$.

Figure 14

The position of the tachyon and the Mach-Cherenkov shock wave separating the $AII$ and $BI$ metrics in the global conformal representation of anti–de Sitter spacetime for $\varphi =0$ (left), $\varphi =\frac{\pi }{4}$ (middle) and $\varphi =\frac{\pi }{2}$ (right). Unlike the case $\mathrm{\Lambda }>0$ shown in Fig. 13, these conformal diagrams depend on $\varphi$. In the unfolded covering space, the tachyon periodically reappears in time, namely at $\eta =\frac{\pi }{2}+k\pi$, $k\in \mathbb{Z}$. Its full trajectory is only visible in the section $\varphi =\frac{\pi }{2}$. The outer cylindrical boundary $\chi =\frac{\pi }{2}$ is the null conformal infinity $\mathcal{I}$ of the anti–de Sitter universe.

Figure 15

Motion of a slowed tachyon in de Sitter (left) and anti–de Sitter (right) background space in boosted coordinates. Its trajectory is given by $Z_3^{\prime }/Z_0^{\prime }=v>1$ and $Z_1=0=Z_2$.

Figure 16

Separation of the de Sitter universe into Regions 1, 2 and 3 in the boosted coordinates for $|Z_4|<a$ (upper left part), $|Z_4|>a$ (upper right part) and $|Z_4|=a$ (lower part), with $Z_2=0$. The intersections of these separation boundaries with the hyperboloid in the upper left part give the position of the Mach-Cherenkov shock cones in the de Sitter universe. The tachyon is always located at their joint vertex at $Z_1=0=Z_2$.

Figure 17

Separation of the anti–de Sitter universe into Regions 1, 2 and 3 in the boosted coordinates. Notice that Regions 1 and 2, covered by the $AII$ metrics, are larger, whereas Region 3 covered by the $BI$ metric is smaller, compared to Fig. 12 (i.e., without the boost).

Figure 18

A time sequence visualizing the de Sitter universe—in this section $Z_1$, $Z_2$, $Z_3^{\prime }$ represented as a sphere—at five different times $Z_0^{\prime }=\mathrm{const}$ (and fixed $Z_4<a$). The universe contracts, reaches its minimal size at $Z_0^{\prime }=0$, and then reexpands. The intersections of the sphere with the cones give the actual position of the two Mach-Cherenkov shocks in the universe. For $Z_0^{\prime }<-Z_{0\max }^{\prime }$ both shocks are contracting on the sphere towards its poles, while for $Z_0^{\prime }>Z_{0\max }^{\prime }$ they are both expanding from the poles. The tachyon is always located at the joint vertex of the cones. Analogous pictures apply to the anti–de Sitter universe expressed in global static coordinates because the 2-spaces of constant $T$ and $r$ are also spheres; see the metric (5.4) in Ref. [4].