Accelerated black holes in an anti-de Sitter universe

The $C$-metric is one of few known exact solutions of Einstein’s field equations which describes the gravitational field of moving sources. For a vanishing or positive cosmological constant, the $C$-metric represents two accelerated black holes in asymptotically flat or de Sitter spacetime. For a negative cosmological constant the structure of the spacetime is more complicated. Depending on the value of the acceleration, it can represent one black hole or a sequence of pairs of accelerated black holes in the spacetime with an anti-de Sitter-like infinity. The global structure of this spacetime is analyzed and compared with an empty anti-de Sitter universe. It is illustrated by 3D conformal-like diagrams.

Figure 1

The diagrams of the allowed range of coordinates $v$ and $\xi$ (the shaded region) in the case of small acceleration $A<1/\ell$. Diagram (a) is applicable in the charged case, (b) is valid for $m\ne 0$, $e=0$. ${\xi }_{\mathrm{b}}$ and ${\xi }_{\mathrm{f}}$ are zeros of the metric function $\mathcal{G}$ closest to $\xi =0$. These values correspond to the axis of $\phi$ symmetry. The diagonal double line represents the infinity, cf. Eq. (3.12). The bottom zigzag line is the singularity at $\upsilon =\infty$. ${\upsilon }_{\mathrm{o}}$ and ${\upsilon }_{\mathrm{i}}$ are zeros of the metric function $\mathcal{F}$. They define the outer and inner black hole horizons. They separate the allowed range of coordinates into regions II, III, and IV. These regions correspond to different domains in spacetime, each of them covered by its own coordinates $\tau$, $\upsilon$, $\xi$, $\phi$. These coordinate systems cannot be smoothly extended over the horizon. Coordinates smooth across the horizon are used in Fig. 2, where sections $\xi =\mathrm{constant}$ are depicted. Such a section is represented in the diagrams above by the vertical thick line.

Figure 2

The conformal diagrams of the sections $\xi ,\phi =\mathrm{constant}$ for (a) charged and (b) uncharged $C$-metric with the acceleration $A<1/\ell$. These diagrams are based on null coordinates $\mathsf{u}$, $\mathsf{v}$ which grow in diagonal directions. Integers ($m$, $n$) in the diagrams, identifying different spacetime domains, are those from definition (2.15). Double lines represent conformal infinity $\mathcal{I}$ (cf. Eq. (3.12)), zigzag lines the singularity at $\upsilon =+\infty$, and thin diagonal lines the outer and inner black hole horizons $\upsilon ={\upsilon }_{\mathrm{o}}$ and $\upsilon ={\upsilon }_{\mathrm{i}}$, respectively. We can recognize familiar structure of the interior of Reissner-Nordström or Schwarzschild black holes, respectively, (domains III and IV). The exterior of the black holes is, however, asymptotically different—it has the asymptotics of anti-de Sitter universe. The whole spacetime consists of more exterior domains II which are connected (not necessary causally) with each other through the black holes. A more detailed diagram of a typical domain outside of the black hole (a darker area indicated above) can be found in Fig. 3b. The thick line corresponds to a section $\tau =\mathrm{constant}$ which is discussed in Fig. 1.

Figure 3

(a) Three-dimensional representation of the exterior of the black hole accelerated in anti-de Sitter universe with acceleration smaller than $1/\ell$. The dark surface represents the outer black hole horizon ${\mathcal{H}}_{\mathrm{o}}$, and the boundary of the diagram corresponds to the conformal infinity $\mathcal{I}$. Embeddings of a typical section $\xi =\mathrm{constant}$ (section $\mathcal{S}$) and of the axis $\xi ={\xi }_{\mathrm{f}},{\xi }_{\mathrm{b}}$ are shown. The nonsymmetric shape of the infinity reflects the fact that the coordinate system used is centered around the black hole which is moving with acceleration with respect to the infinity. (b) Two-dimensional conformal diagram of $\xi =\mathrm{constant}$ section. Only the exterior of the black hole is shown (compare with Fig. 2). This part of the conformal diagram corresponds exactly to the section $\xi =\mathrm{constant}$ indicated in the diagram on the left.

Figure 4

Another three-dimensional representation of the exterior of the accelerated black hole with $A<1/\ell$. The outer black hole horizon of a conical shape from Fig. 3a is here deformed to the surface of a shape of two joined drops. The black hole is thus represented as a localized object. Such a representation is useful for a study of the weak field limit when the black hole changes into the world line of a point particle.

Figure 5

Diagrams analogous to Fig. 1 in the case $A>1/\ell$. The allowed range of coordinates $v$, $\xi$ (shaded area) is again restricted by the infinity (diagonal double line), by the axis (vertical border lines), and by the singularity (zigzag line). Additionally to outer and inner black hole horizons, acceleration and cosmological horizons (at $v=v_{\mathrm{a}}$ and $v=v_{\mathrm{c}}$) are also present. Horizons divide the allowed range into regions O–IV which corresponds to qualitatively different domains of spacetime; cf. Figure 6. Different sections $\xi =\mathrm{constant}$ cross different number of horizons. Typical representatives ${\mathcal{S}}_A$, ${\mathcal{S}}_B$, and ${\mathcal{S}}_C$ of these sections are indicated by thick vertical lines. They correspond to different shapes of the conformal diagrams in Fig. 6.

Figure 6

The conformal diagrams of the sections $\xi ,\phi =\mathrm{constant}$ for $A>1/\ell$. The top diagrams are valid for $m,e\ne 0$, the bottom ones are for the uncharged case. The diagrams are based on coordinates $\mathsf{u}$, $\mathsf{v}$. Integers ($m$, $n$) from definition (2.15) identify different domains of the spacetime. Analogously to Fig. 2, double lines represent the infinity, zigzag lines the singularity, and diagonal lines the horizons. Domains O–II correspond to the exterior of black holes and domains III and IV to interiors of black holes. The interior has a similar causal structure to that of unaccelerated black holes. The spacetime contains more asymptotic domains, one of which is indicated by dark shading. The description below is from a point of view of this domain. Three different shapes of the diagrams correspond to the sections with a different value of the coordinate $\xi$. On the left, section ${\mathcal{S}}_A$ is spanned between two black holes which are moving with respect to each other along a common axis. It is also spanned between different pairs of such black holes through the domains O and I. In the middle, section ${\mathcal{S}}_B$ is spanned only between two black holes. It does not continue to the other pair of black hole because it intersects the conformal infinity in spacelike lines located inside domains I. The section ${\mathcal{S}}_C$, depicted on the right, goes from each black hole directly into infinity—it does not connect different black holes through the exterior domains. These three sections correspond to thick vertical lines in Fig. 5. Thick lines in the diagrams above represent the section $\tau =\mathrm{constant}$, i.e., exactly the section discussed in Fig. 5. The embedding of these two-dimensional diagrams into spacetime is shown in Figs. 8, 9, 10. More detailed two-dimensional diagrams of the exterior of black holes (the dark area above) are also presented there.

Figure 7

Three-dimensional visualizations of the exterior of black holes which are moving with acceleration parameter $A>1/\ell$ in asymptotically anti-de Sitter universe. The diagrams show a compactified picture of the whole universe—borders of the diagrams correspond to the conformal infinity. Diagram (a) is obtained by gluing together two-dimensional diagrams from Fig. 6. Black hole outer horizons ${\mathcal{H}}_{\mathrm{o}}$ are represented by dark surfaces of a conical shape which indicates the null character of these surfaces. In the alternative representation (b), the black hole outer horizons are squeezed into droplike shapes. Such a representation shows the black hole as a localized object and it is useful in the weak field limit when the black hole changes to a pointlike particle—compare with Fig. 17b. The universe represents a sequence of pairs of black holes which repeatedly enter and leave the universe through their timelike infinity—the diagrams should continue periodically in the vertical direction. Black holes of each pair are causally separated by the acceleration horizon ${\mathcal{H}}_{\mathrm{a}}$; consequent pairs of black holes are separated by cosmological horizons ${\mathcal{H}}_{\mathrm{c}}$. These are null surfaces—light cones of the entry points of black holes into the spacetime. Embedding of different types of two-dimensional conformal diagrams into the three-dimensional one is depicted in Figs. 8, 9, 10.

Figure 8

(a) Embedding of section ${\mathcal{S}}_A$ [cf. Figs. 5 and 6a] into a three-dimensional representation of the $C$-metric spacetime. (b) The part of the two-dimensional conformal diagram of ${\mathcal{S}}_A$ representing the exterior of the black holes [corresponds to the dark area in Fig. 6a].

Figure 9

(a) Embedding of section ${\mathcal{S}}_B$ [cf. Figs. 5 and 6b] into a three-dimensional picture of spacetime. (b) The corresponding part of the two-dimensional conformal diagram of ${\mathcal{S}}_B$ [cf. the dark area in Fig. 6b].

Figure 10

(a) Embedding of section ${\mathcal{S}}_C$ [cf. Fig. 5 and 6c] into a three-dimensional diagram. (b) The corresponding part of the two-dimensional conformal diagram of ${\mathcal{S}}_C$ [cf. the dark areas in Fig. 6c].

Figure 11

A shaded region in diagram (a) indicates allowed ranges of coordinates $\upsilon \equiv {\upsilon }_{\mathrm{I}}^{\prime }=\ell /R_{\mathrm{I}}^{\prime }=\cot {\chi }_{\mathrm{I}}^{\prime }$ and $\xi \equiv {\xi }_{\mathrm{I}}^{\prime }=-\cos {\Theta }_{\mathrm{I}}^{\prime }=-\cos {\vartheta }_{\mathrm{I}}^{\prime }$. The diagonal double line corresponds to the infinity, vertical borders to the axis of symmetry, and the bottom line to the origin ${\chi }_{\mathrm{I}}^{\prime }=0$. The diagram (a) is an analogue of Fig. 1. However, this diagram does not respect the angular meaning of the $\xi$ coordinate. A more natural representation (b) of the shaded region is obtained by shrinking the bottom line to a point, forming thus a deformed semicircle.

Figure 12

Conformal diagrams of the section ${\Theta }_{\mathrm{I}}^{\prime },{\Phi }_{\mathrm{I}}^{\prime }=\mathrm{constant}$ (or, equivalently, ${\vartheta }_{\mathrm{I}}^{\prime },{\phi }_{\mathrm{I}}^{\prime }=\mathrm{constant}$). Diagram (a) is based on coordinates ${\overline{\mathsf{u}}}_{\mathrm{I}}^{\prime }$, ${\overline{\mathsf{v}}}_{\mathrm{I}}^{\prime }$. Horizontal and vertical lines are given by coordinate lines ${\tilde{t}}_{\mathrm{I}}^{\prime }=\mathrm{constant}$ and ${\chi }_{\mathrm{I}}^{\prime }=\mathrm{constant}$, since for $m,e=0$, definitions (2.13) and (2.14) give ${\overline{\mathsf{u}}}_{\mathrm{I}}^{\prime }={\chi }_{\mathrm{I}}^{\prime }+{\tilde{t}}_{\mathrm{I}}^{\prime }$ and ${\overline{\mathsf{v}}}_{\mathrm{I}}^{\prime }={\chi }_{\mathrm{I}}^{\prime }-{\tilde{t}}_{\mathrm{I}}^{\prime }$. The coordinate ${\chi }_{\mathrm{I}}^{\prime }$ is a radial coordinate; the left border of the diagram thus corresponds to the world line of an accelerated observer at the origin. The right double line represents conformal infinity $\mathcal{I}$ (formed by limiting end points of spacelike and null geodetics). The diagram should continue infinitely in the vertical direction. (b) Compactified version of the same conformal diagram based on the coordinates ${\mathsf{u}}_{\mathrm{I}}^{\prime }$, ${\mathsf{v}}_{\mathrm{I}}^{\prime }$, related to ${\overline{\mathsf{u}}}_{\mathrm{I}}^{\prime }$, ${\overline{\mathsf{v}}}_{\mathrm{I}}^{\prime }$ by Eq. (2.15). The whole spacetime is here squeezed into a compact region which beside the conformal infinity includes also pointlike future and past infinities (limiting end points of timelike geodesics). This diagram is analogous to those in Fig. 2. An exact position of $\mathcal{I}$ depends on an angular direction of the plane of the diagram (i.e., on a value of coordinate ${\vartheta }_{\mathrm{I}}^{\prime }$) through the relation $\tan {\chi }_{\mathrm{I}}^{\prime }=-\ell \cot {\chi }_{\mathrm{o}}/\cos {\vartheta }_{\mathrm{I}}^{\prime }$ [cf. Eq. (3.12)]. For $A=0$ these diagrams reduce to the standard conformal diagrams based on the cosmological spherical coordinates.

Figure 13

Three-dimensional schematical diagrams of anti-de Sitter universe obtained by rotation (varying angular coordinate ${\vartheta }_{\mathrm{I}}^{\prime }$) of two-dimensional diagrams from Fig. 12. The diagrams are centered on the world line of a static observer (thick line) which is accelerated with acceleration $A<1/\ell$. The horizontal section $\tilde{t}=\mathrm{constant}$ corresponds to two copies of Fig. 11b (one copy for $\phi =0$, another for $\phi =\pi$).

Figure 14

A shaded region in diagram (a) represents the allowed range of coordinates $\upsilon \equiv {\upsilon }_{\mathrm{II}}^{\prime }=\ell /R_{\mathrm{II}}^{\prime }$ and $\xi \equiv {\xi }_{\mathrm{II}}^{\prime }=-\cos {\Theta }_{\mathrm{II}}^{\prime }$. The notation is the same as in Fig. 5, except there are no black hole horizons, and the bottom line does not represent a singularity but poles of the coordinates. Diagram (a) does not respect the bipolar nature of coordinates $\upsilon$ and $\xi$. A more accurate picture of region II is drawn in diagram (b). It depicts section $\tilde{t},\phi =0$ through two spacetime domains of type II. Each of them contains one pole of the coordinate system. Both domains are separated by an acceleration horizon. The coordinate $\upsilon$ decreases from $\upsilon =+\infty$ at poles to $\upsilon ={\upsilon }_{\mathrm{a}}$ at the acceleration horizon, and the coordinate $\xi$ labels different coordinate lines starting from the poles.

Figure 15

Cosmological spherical coordinates $\tilde{t}$, $\chi$ and accelerated spherical coordinates ${\tilde{t}}_{\mathrm{II}}^{\prime }$, ${\chi }_{\mathrm{II}}^{\prime }$ drawn on a two-dimensional section of anti-de Sitter universe. The coordinate systems are related by the ’squeezing transformation’ (5.11). Left: Coordinate lines of both systems drawn in such a way that lines $\tilde{t}=\mathrm{constant}$ and $\chi =\mathrm{constant}$ are horizontal and vertical, respectively. Right: A complementary representation with vertical and horizontal lines given by coordinate system ${\tilde{t}}_{\mathrm{II}}^{\prime }$, ${\chi }_{\mathrm{II}}^{\prime }$. The conformal infinity is given by $\chi =\pi /2$ and is thus deformed in the squeezed diagram on the right.

Figure 16

The conformal diagrams of sections ${\xi }_{\mathrm{II}}^{\prime },{\phi }_{\mathrm{II}}^{\prime }=\mathrm{constant}$ (or, equivalently, ${\rho }_{\mathrm{II}}^{\prime },{\phi }_{\mathrm{II}}^{\prime }=\mathrm{constant}$) for different values of $\xi \equiv {\xi }_{\mathrm{II}}^{\prime }$. The diagrams are based on null coordinates ${\mathsf{u}}_{\mathrm{II}}^{\prime }$, ${\mathsf{v}}_{\mathrm{II}}^{\prime }$. They are spanned between two poles which correspond to observers with uniform acceleration $A>1/\ell$ (straight vertical lines on the border of the diagrams, cf. also Fig. 17 for the three-dimensional localization of the poles). Three different shapes of the diagrams correspond to qualitatively different possibilities of how sections ${\xi }_{\mathrm{II}}^{\prime },{\phi }_{\mathrm{II}}^{\prime }=\mathrm{constant}$ are embedded into the anti-de Sitter universe. They correspond to the three sections $\xi =\mathrm{constant}$ indicated in Fig. 14. Diagonal lines represent acceleration and cosmological horizons. The acceleration horizon causally separates both poles. It is formed by future light cones of points where the poles enter the anti-de Sitter universe. The cosmological horizon is formed by future light cones of points where the poles leave the universe. The thick line is an example of $\upsilon \equiv {\upsilon }_{\mathrm{II}}^{\prime }=\mathrm{constant}$ section—it corresponds to the diagram in Fig. 11. Gluing together diagram (a) for ${\xi }_{\mathrm{II}}^{\prime }=-1$ (the axis ${\rho }_{\mathrm{II}}^{\prime }={\Theta }_{\mathrm{II}}^{\prime }=0$ between poles) with diagram (c) for ${\xi }_{\mathrm{II}}^{\prime }=+1$ (the axis ${\rho }_{\mathrm{II}}^{\prime }={\Theta }_{\mathrm{II}}^{\prime }=\pi$) gives the history of the whole axis of symmetry. It is the same section as that depicted in Fig. 15.

Figure 17

Three-dimensional representations of anti-de Sitter universe based (a) on cosmological coordinates $\tilde{t}$, $\chi$, $\vartheta$, $\phi$ and (b) on ’squeezed’ accelerated coordinates ${\tilde{t}}_{\mathrm{II}}^{\prime }$, ${\chi }_{\mathrm{II}}^{\prime }$, ${\vartheta }_{\mathrm{II}}^{\prime }$, ${\phi }_{\mathrm{II}}^{\prime }$. They can be obtained by a rotation of the corresponding diagrams from Fig. 15. Alternatively, they can be constructed by gluing together two-dimensional diagrams from Fig. 16. These are spanned between world lines of poles moving with the uniform acceleration $A=1/\ell$ along the axis. World lines of the poles are indicated in the diagrams by thick lines. A pair of the poles enter anti-de Sitter universe through the conformal infinity, they approach each other, and then return back to the infinity—all this in a finite cosmological time $\Delta \tilde{t}=\pi$. After a stage without poles, a new pair of poles enters the universe, and so on. The diagram (b) is clearly the limit of Fig. 7b in which the black holes are shrunk to the accelerated particles located at the poles.