Yet another family of diagonal metrics for de Sitter and anti-de Sitter spacetimes

TL;DR

This paper introduces a new class of diagonal, static, and axially symmetric coordinate metrics for de Sitter and anti-de Sitter spacetimes, based on a 2+2 foliation, expanding the understanding of their geometric and physical properties.

Contribution

It presents a novel family of coordinate representations for (A)dS spacetimes using a 2+2 foliation, including explicit parametrizations, visualizations, and transformations to standard forms.

Findings

New diagonal, static, axially symmetric metrics for (A)dS spacetimes.

Explicit parametrizations and visualizations of the metrics.

Physical interpretation of B-metrics as tachyon gravitational fields.

Abstract

In this work we present and analyze a new class of coordinate representations of de Sitter and anti-de Sitter spacetimes for which the metrics are diagonal and (typically) static and axially symmetric. Contrary to the well-known forms of these fundamental geometries, that usually correspond to a 1+3 foliation with the 3-space of a constant spatial curvature, the new metrics are adapted to a 2+2 foliation, and are warped products of two 2-spaces of constant curvature. This new class of (anti-)de Sitter metrics depends on the value of cosmological constant Lambda and two discrete parameters +1,0,-1 related to the curvature of the 2-spaces. The class admits 3 distinct subcases for Lambda>0 and 8 subcases for Lambda<0. We systematically study all these possibilities. In particular, we explicitly present the corresponding parametrizations of the (anti-)de Sitter hyperboloid, visualize the coordinate lines and surfaces within the global conformal cylinder, investigate their mutual relations, present some closely related forms of the metrics, and give transformations to standard de Sitter and anti-de Sitter metrics. Using these results, we also provide a physical interpretation of B-metrics as exact gravitational fields of a tachyon.

Tables

Table 1. Table 1: Summary of all possible subcases for Λ > 0 Λ 0 \Lambda>0 and different values of ϵ 2 subscript italic-ϵ 2 \epsilon_{2} , ϵ 0 subscript italic-ϵ 0 \epsilon_{0} . The symbol ✓  indicates admitted solutions (their geometry is p 2 ​ d ​ S 2 × S 2 superscript 𝑝 2 𝑑 subscript 𝑆 2 superscript 𝑆 2 {\,p^{2}{dS}_{2}\times{S}^{2}} ), while those with × \times are forbidden.

${ϵ}_2$	${ϵ}_0$	$\exists$	$P$	range of $p$	$Q$	range of $q$
1	1	✓	$1-p^2/a^2$	$(-a,a)$	$1-q^2$	$ℝ\setminus \left\{\pm 1\right\}$
1	0	✓	$1-p^2/a^2$	$[0,a)$	$-q^2$	$ℝ\setminus \left\{0\right\}$
1	$-1$	✓	$1-p^2/a^2$	$[0,a)$	$-1-q^2$	$ℝ$
0	0,$\pm 1$	$\times$
$-1$	0,$\pm 1$	$\times$

Table 2. Table 2: Summary of all possible subcases for Λ < 0 Λ 0 \Lambda<0 and different values of ϵ 2 subscript italic-ϵ 2 \epsilon_{2} , ϵ 0 subscript italic-ϵ 0 \epsilon_{0} . Their geometry is p 2 ​ d ​ S 2 × H 2 superscript 𝑝 2 𝑑 subscript 𝑆 2 superscript 𝐻 2 {\,p^{2}{dS}_{2}\times{H}^{2}} for ϵ 2 = 1 subscript italic-ϵ 2 1 {\epsilon_{2}=1} , p 2 ​ M 2 × H 2 superscript 𝑝 2 subscript 𝑀 2 superscript 𝐻 2 {\,p^{2}{M}_{2}\times{H}^{2}} for ϵ 2 = 0 subscript italic-ϵ 2 0 {\epsilon_{2}=0} , and p 2 ​ A ​ d ​ S 2 × H 2 superscript 𝑝 2 𝐴 𝑑 subscript 𝑆 2 superscript 𝐻 2 {\,p^{2}{AdS}_{2}\times{H}^{2}} for ϵ 2 = − 1 subscript italic-ϵ 2 1 {\epsilon_{2}=-1} .

${ϵ}_2$	${ϵ}_0$	$\exists$	$P$	range of $p$	$Q$	range of $q$
1	1	✓	$1+p^2/a^2$	$ℝ$	$1-q^2$	$ℝ\setminus \left\{\pm 1\right\}$
1	0	✓	$1+p^2/a^2$	$[0,\infty )$	$-q^2$	$ℝ\setminus \left\{0\right\}$
1	$-1$	✓	$1+p^2/a^2$	$[0,\infty )$	$-1-q^2$	$ℝ$
0	$1$	✓	$p^2/a^2$	$ℝ$	$1$	$ℝ$
0	0	$\times$
0	$-1$	✓	$p^2/a^2$	$ℝ$	$-1$	$ℝ$
$-1$	1	✓	$-1+p^2/a^2$	$[a,\infty )$	$1+q^2$	$ℝ$
$-1$	0	✓	$-1+p^2/a^2$	$[a,\infty )$	$q^2$	$ℝ\setminus \left\{0\right\}$
$-1$	$-1$	✓	$-1+p^2/a^2$	$ℝ\setminus (-a,a)$	$-1+q^2$	$ℝ\setminus \left\{\pm 1\right\}$

Table 3. Table 3: Specific functions used in the unified parametrization ( 8 ) for Λ > 0 Λ 0 {\Lambda>0} .

${ϵ}_2$	${ϵ}_0$	$T_1$	$T_2$	$S$	$Y_1$	$Y_2$	$a$	$b$	$c$	$d$	$e$	Eq.
1	1	$\sinh t$	$\cosh t$	0	$\cos \varphi$	$\sin \varphi$	0	1	2	3	4	(20)
1	1	$\cosh t$	$\sinh t$	0	$\cos \varphi$	$\sin \varphi$	0	1	2	3	4	(21)
1	0	$\frac{1}{2}signq$	$\frac{1}{2}signq$	$t^2-q^{-2}$	$\cos \varphi$	$\sin \varphi$	0	1	2	3	4	(28)
1	$-1$	$\sin t$	$\cos t$	0	$\cos \varphi$	$\sin \varphi$	2	1	0	3	4	(31)

Table 4. Table 4: Specific functions used in the unified parametrizations ( 8 ) for Λ < 0 Λ 0 {\Lambda<0} . Here s = a 2 ​ ( − t 2 + q 2 + ζ 2 + a 2 / p 2 ) 𝑠 superscript 𝑎 2 superscript 𝑡 2 superscript 𝑞 2 superscript 𝜁 2 superscript 𝑎 2 superscript 𝑝 2 {s=a^{2}\left(-t^{2}+q^{2}+\zeta^{2}+a^{2}/p^{2}\right)} . For ϵ 2 = 1 subscript italic-ϵ 2 1 {\epsilon_{2}=1} we also consider the second chart with − Y 1 subscript 𝑌 1 -Y_{1} and − Y 2 subscript 𝑌 2 -Y_{2} in order to obtain Z e < 0 subscript 𝑍 𝑒 0 {Z_{e}<0} . For ϵ 2 = 0 subscript italic-ϵ 2 0 {\epsilon_{2}=0} we also consider − Z 2 subscript 𝑍 2 -Z_{2} , − Z 3 subscript 𝑍 3 -Z_{3} and − Z 4 subscript 𝑍 4 -Z_{4} .

${ϵ}_2$	${ϵ}_0$	$T_1$	$T_2$	$S$	$Y_1$	$Y_2$	$a$	$b$	$c$	$d$	$e$	Eq.
1	1	$\sinh t$	$\cosh t$	0	$\sinh \zeta$	$\cosh \zeta$	0	1	2	3	4	(34)
1	1	$\cosh t$	$\sinh t$	0	$\sinh \zeta$	$\cosh \zeta$	0	1	2	3	4	(35)
1	0	$\frac{1}{2}signq$	$\frac{1}{2}signq$	$t^2-q^{-2}$	$\sinh \zeta$	$\cosh \zeta$	0	1	2	3	4	(46)
1	$-1$	$\sin t$	$\cos t$	0	$\sinh \zeta$	$\cosh \zeta$	2	1	0	3	4	(54)
0	1	$\frac{1}{2}$	$\frac{1}{2}$	$s/a^2$	$\zeta$	$t$	0	1	2	3	4	(64)
$-1$	1	$\cos t$	$\sin t$	0	$\sin \varphi$	$\cos \varphi$	0	4	1	3	2	(68)
$-1$	0	$\frac{1}{2}signq$	$\frac{1}{2}signq$	$q^{-2}-t^2$	$\sin \varphi$	$\cos \varphi$	0	1	4	3	2	(71)
$-1$	$-1$	$\sinh t$	$\cosh t$	0	$\sin \varphi$	$\cos \varphi$	0	1	4	3	2	(74)
$-1$	$-1$	$\cosh t$	$\sinh t$	0	$\sin \varphi$	$\cos \varphi$	0	1	4	3	2	(75)

Equations

$${\mathrm{d}}s^{2}=p^{2}\Big{(}\!-Q\,{\mathrm{d}}t^{2}+\frac{{\mathrm{d}}q^{2}}{Q}\Big{)}+\frac{{\mathrm{d}}p^{2}}{P}+P\,a^{2}{\mathrm{d}}\phi^{2}\,,$$

$$Q(q)$$

$$P(p)$$

$$Q\equiv \tilde{\mathcal{Q}}={ϵ}_0-{ϵ}_2{q}^2,P\equiv \frac{\mathcal{P}}{p^2}=\frac{2n}{p}+{ϵ}_2-\frac{\Lambda }{3}{p}^2.$$

$$-Z_0^2+Z_1^2+Z_2^2+Z_3^2+Z_4^2=a^2,wherea=\sqrt{3/\Lambda },$$

$$\mathrm{d}{s}^2=-\mathrm{d}{Z}_0^2+\mathrm{d}{Z}_1^2+\mathrm{d}{Z}_2^2+\mathrm{d}{Z}_3^2+\mathrm{d}{Z}_4^2.$$

$$Z_0=a\sinh \frac{t}{a},$$

$$Z_1=a\cosh \frac{t}{a}\cos \chi ,$$

$$Z_2=a\cosh \frac{t}{a}\sin \chi \cos \theta ,$$

$$Z_3=a\cosh \frac{t}{a}\sin \chi \sin \theta \cos \varphi ,$$

$$Z_4=a\cosh \frac{t}{a}\sin \chi \sin \theta \sin \varphi ,$$

$${\mathrm{d}}s^{2}=-{\mathrm{d}}t^{2}+a^{2}\cosh^{2}\frac{t}{a}\,\Big{(}{\mathrm{d}}\chi^{2}+\sin^{2}\chi\big{(}{\mathrm{d}}\theta^{2}+\sin^{2}\theta\>{\mathrm{d}}\phi^{2}\big{)}\Big{)}\,.$$

$$\Omega =\sin \eta =\mathrm{sech}\frac{t}{a}.$$

$${\mathrm{d}}s^{2}=\frac{a^{2}}{\Omega^{2}}\Big{(}-{\mathrm{d}}\eta^{2}+{\mathrm{d}}\chi^{2}+\sin^{2}\chi\big{(}{\mathrm{d}}\theta^{2}+\sin^{2}\theta\>{\mathrm{d}}\phi^{2}\big{)}\Big{)}\,,$$

$$Z_0=a\cot \eta ,$$

$$Z_1=\frac{a}{\sin \eta }\cos \chi ,$$

$$Z_2=\frac{a}{\sin \eta }\sin \chi \cos \theta ,$$

$$Z_3=\frac{a}{\sin \eta }\sin \chi \sin \theta \cos \varphi ,$$

$$Z_4=\frac{a}{\sin \eta }\sin \chi \sin \theta \sin \varphi .$$

$$-Z_0^2+Z_1^2+Z_2^2+Z_3^2-Z_4^2=-a^2,wherea=\sqrt{-3/\Lambda },$$

$$\mathrm{d}{s}^2=-\mathrm{d}{Z}_0^2+\mathrm{d}{Z}_1^2+\mathrm{d}{Z}_2^2+\mathrm{d}{Z}_3^2-\mathrm{d}{Z}_4^2,$$

$$Z_0=a\cosh r\cos \frac{T}{a},$$

$$Z_1=a\sinh r\cos \theta ,$$

$$Z_2=a\sinh r\sin \theta \cos \varphi ,$$

$$Z_3=a\sinh r\sin \theta \sin \varphi ,$$

$$Z_4=a\cosh r\sin \frac{T}{a},$$

$${\mathrm{d}}s^{2}=-\cosh^{2}r\>{\mathrm{d}}T^{2}+a^{2}\Big{(}{\mathrm{d}}r^{2}+\sinh^{2}r\>\big{(}{\mathrm{d}}\theta^{2}+\sin^{2}\theta\>{\mathrm{d}}\phi^{2}\big{)}\Big{)}\,.$$

$$\tan \chi =\sinh r,\Omega =\cos \chi .$$

$${\mathrm{d}}s^{2}=\frac{a^{2}}{\Omega^{2}}\Big{(}-{\mathrm{d}}\eta^{2}+{\mathrm{d}}\chi^{2}+\sin^{2}\chi\big{(}{\mathrm{d}}\theta^{2}+\sin^{2}\theta\>{\mathrm{d}}\phi^{2}\big{)}\Big{)}\,,$$

$$Z_0=a\frac{\cos \eta }{\cos \chi },$$

$$Z_1=a\tan \chi \cos \theta ,$$

$$Z_2=a\tan \chi \sin \theta \cos \varphi ,$$

$$Z_3=a\tan \chi \sin \theta \sin \varphi ,$$

$$Z_4=a\frac{\sin \eta }{\cos \chi }.$$

$${\mathrm{d}}s^{2}=p^{2}\Big{(}\!-(1-q^{2})\,{\mathrm{d}}t^{2}+\frac{{\mathrm{d}}q^{2}}{1-q^{2}}\Big{)}+\frac{a^{2}\,{\mathrm{d}}p^{2}}{a^{2}-p^{2}}+(a^{2}-p^{2})\,{\mathrm{d}}\phi^{2}\,,$$

$$\left.\begin{array}[]{l}Z_{0}={\displaystyle p\,\sqrt{1-q^{2}}\,\sinh t}\,,\\[8.0pt] Z_{1}={\displaystyle p\,\sqrt{1-q^{2}}\,\cosh t}\,,\\[8.0pt] Z_{2}={\displaystyle|p|\,q}\,,\\[8.0pt] Z_{3}={\displaystyle\sqrt{a^{2}-p^{2}}\,\cos\phi}\,,\\[8.0pt] Z_{4}={\displaystyle\sqrt{a^{2}-p^{2}}\,\sin\phi}\,,\end{array}\!\right\}\ \Leftrightarrow\ \left\{\!\begin{array}[]{l}\tanh t={\displaystyle\frac{Z_{0}}{Z_{1}}}\,,\\[8.0pt] \tan\phi={\displaystyle\frac{Z_{4}}{Z_{3}}}\,,\\[6.0pt] p={\displaystyle\mathop{\rm sign}\nolimits(Z_{1})\,\sqrt{-Z_{0}^{2}+Z_{1}^{2}+Z_{2}^{2}}}\\[2.0pt] \quad={\displaystyle\mathop{\rm sign}\nolimits(Z_{1})\,\sqrt{a^{2}-Z_{3}^{2}-Z_{4}^{2}}}\,,\\[6.0pt] q={\displaystyle\frac{Z_{2}}{\sqrt{-Z_{0}^{2}+Z_{1}^{2}+Z_{2}^{2}}}}\,.\end{array}\right.$$

$$\left.\begin{array}[]{l}Z_{0}={\displaystyle p\,\sqrt{q^{2}-1}\,\cosh t}\,,\\[10.0pt] Z_{1}={\displaystyle p\,\sqrt{q^{2}-1}\,\sinh t}\,,\end{array}\!\right\}\ \Leftrightarrow\ \left\{\!\begin{array}[]{l}\tanh t={\displaystyle\frac{Z_{1}}{Z_{0}}}\,,\\[6.0pt] p={\displaystyle\mathop{\rm sign}\nolimits(Z_{0})\,\sqrt{-Z_{0}^{2}+Z_{1}^{2}+Z_{2}^{2}}}\,.\end{array}\right.$$

$$\displaystyle\begin{array}[]{rll}a\,\cot\eta&\!\!\!=p\,\sqrt{1-q^{2}}\,\sinh t&\ \textrm{for}\ |q|<1\,,\qquad a\,\cot\eta=p\,\sqrt{q^{2}-1}\,\cosh t\hskip 19.91692pt\textrm{for}\ |q|>1\,,\\ \cot\chi&\!\!\!=\frac{\displaystyle p\,\sqrt{1-q^{2}}\,\cosh t}{\displaystyle\sqrt{a^{2}-p^{2}(1-q^{2})}}&\ \textrm{for}\ |q|<1\,,\hskip 31.29802pt\cot\chi=\frac{\displaystyle p\,\sqrt{q^{2}-1}\,\sinh t}{\displaystyle\sqrt{a^{2}-p^{2}(1-q^{2})}}\quad\textrm{for}\ |q|>1\,,\\ \cot\theta&\!\!\!=\frac{\displaystyle|p|\,q}{\displaystyle\sqrt{a^{2}-p^{2}}}\,.&\end{array}$$

$$\tanh t$$

Full text

Yet another family of diagonal metrics

for de Sitter and anti-de Sitter spacetimes

Jiří  Podolský, Ondřej Hruška

Institute of Theoretical Physics, Faculty of Mathematics and Physics,

Charles University, Prague, V Holešovičkách 2, 180 00 Praha 8, Czech Republic

podolsky@mbox.troja.mff.cuni.cz and HruskaOndrej@seznam.cz

Abstract

In this work we present and analyze a new class of coordinate representations of de Sitter and anti-de Sitter spacetimes for which the metrics are diagonal and (typically) static and axially symmetric. Contrary to the well-known forms of these fundamental geometries, that usually correspond to a ${1+3}$ foliation with the 3-space of a constant spatial curvature, the new metrics are adapted to a ${2+2}$ foliation, and are warped products of two 2-spaces of constant curvature. This new class of (anti-)de Sitter metrics depends on the value of cosmological constant $\Lambda$ and two discrete parameters ${+1,0,-1}$ related to the curvature of the 2-spaces. The class admits 3 distinct subcases for ${\Lambda>0}$ and 8 subcases for ${\Lambda<0}$. We systematically study all these possibilities. In particular, we explicitly present the corresponding parametrizations of the (anti-)de Sitter hyperboloid, visualize the coordinate lines and surfaces within the global conformal cylinder, investigate their mutual relations, present some closely related forms of the metrics, and give transformations to standard de Sitter and anti-de Sitter metrics. Using these results, we also provide a physical interpretation of $B$-metrics as exact gravitational fields of a tachyon.

1 Introduction

Almost exactly 100 years ago in 1917 Willem de Sitter published two seminal papers [1, 2] in which he presented his, now famous, exact solution of Einstein’s gravitational field equations without matter but with a positive cosmological constant $\Lambda$. Together with the pioneering work from the same year by Einstein himself [3], they mark the very origin of relativistic cosmology.

Since then, the vacuum de Sitter universe with ${\Lambda>0}$ and its formal antipode with ${\Lambda<0}$, later nicknamed the anti-de Sitter universe, have become truly fundamental spacetimes. They admit a cornucopia of applications, ranging from purely mathematical studies, theoretical investigations in classical general relativity, quantum field theory and string theory in higher dimensions, to contemporary “inflationary” and “dark matter” cosmologies, modeling naturally the observed accelerated expansion of the universe, both in its distant past and future.

This remarkable number of applications in various research branches clearly stems from the fact that the (anti-)de Sitter spacetime is highly symmetric and simple, yet it is non-trivial by being curved everywhere. In fact, it is maximally symmetric and has a constant curvature, just as the flat Minkowski space. Of course, this greatly simplifies all analyses and studies.

One would be inclined to expect that, after a century of thorough investigation of various aspects of these important spacetimes, there is not much left to add or discover. Quite surprisingly, still there are interesting new properties which deserve some attention. To celebrate the centenary of de Sitter space, we take the liberty of presenting here yet another family of coordinate representations of de Sitter and anti-de Sitter 4-dimensional solutions, in which the metric is diagonal and typically static and axially symmetric. It naturally arises in the context of $B$-metrics [4, 5, 6] with a cosmological constant $\Lambda$, which belong to a wider class of Plebański–Demiański non-expanding spacetimes [7, 8]. As far as we know, this family has not yet been considered and systematically described. We hope that it could find useful applications in general relativity and high energy physics theories.

In particular, in our work we will present and analyze the family of metrics for de Sitter and anti–de Sitter spacetimes which can be written in a unified form

[TABLE]

with

[TABLE]

where ${\,a^{2}=3/|\Lambda|\,}$ and ${\,\epsilon_{0},\epsilon_{2}=1,0,-1\,}$ are two independent discrete parameters.

The metric has the geometry of a warped product of two 2-spaces of constant curvature, namely ${{dS}_{2},{M}_{2},{AdS}_{2}}$ (according to the sign of $\epsilon_{2}$) spanned by ${t,q}$ , and ${{S}^{2},{E}^{2},{H}^{2}}$ (according to the sign of $\Lambda$) spanned by ${p,\phi}$. The warp factor is $p^{2}$. Also, there are two obvious symmetries corresponding to the Killing vectors $\partial_{t}$ and $\partial_{\phi}$. The metric is clearly static when ${Q>0}$. It is also axially symmetric when the axis given by ${P=0}$ is admitted (for ${\Lambda>0,\epsilon_{2}=1}$ and for ${\Lambda<0,\epsilon_{2}=-1}$), in which case this axis is regular for ${\phi\in[0,2\pi)}$.

The metric (1)–(3) is a special subcase of the Plebański–Demiański non-expanding (and thus Kundt) type D solutions, see metric (16.27) of [6]. It is obtained by setting ${\gamma=0}$ and ${e=0=g}$ (so that ${m=0=k}$) with the identification

[TABLE]

This gives vacuum $B$-metrics, as classified by Ehlers and Kundt [4], generalized here to any value of the cosmological constant $\Lambda$. When ${n=0}$, the curvature singularity at ${p=0}$ disappears and the metric reduces to the conformally flat (anti-)de Sitter space with (3). Therefore, the family of de Sitter and anti–de Sitter metrics (1)–(3) can be viewed as a natural background for the B-metrics with $\Lambda$, and will thus play a key role in understanding their geometrical and physical properties. This is our main motivation.

The structure of the present work is simple. In the next Section 2 we recall basic properties of (anti-)de Sitter spacetimes, introducing the notation. In Section 3 we identify all distinct subclasses of the general metric (1)–(3). These are studied in detail in subsequent Section 4 for the de Sitter case ${\Lambda>0}$, and for the anti-de Sitter case ${\Lambda<0}$ in Section 5. An interesting physical application is given in Section 6. There are also four appendices in which we present a unified form of the new parametrizations, their mutual relations, further metric forms, and transformations to standard (anti-)de Sitter metrics.

2 The de Sitter and anti-de Sitter spacetimes

Minkowski, de Sitter and anti-de Sitter spacetimes are the most fundamental and simplest exact solutions of Einstein’s field equations. They are maximally symmetric, have constant curvature, and are the only vacuum solutions with a vanishing Weyl tensor — they are conformally flat and satisfy the field equations ${R_{\alpha\beta}=\Lambda g_{\alpha\beta}}$, where $\Lambda$ is a cosmological constant, so that in four spacetime dimensions the curvature is ${R=4\Lambda}$. Minkowski, de Sitter and anti-de Sitter spacetimes are thus Einstein spaces with vanishing, positive, and negative cosmological constant $\Lambda$, respectively.

Specific properties of these classic spacetimes have been investigated and employed in various contexts for a century. They were described in many reviews, such as by Schrödinger [9], Penrose [10], Weinberg [11], Møller [12], Hawking and Ellis [13], Birell and Davies [14], Bičák [15], Bičák and Krtouš [16], Grøn and Hervik [17], or Griffiths and Podolský [6]. Let us recall here only those expressions and properties that will be important for our analysis.

2.1 The de Sitter spacetime

Since the original works by de Sitter [1, 2] and Lanczos [18] it is known that the de Sitter manifold can be conveniently visualized as the hyperboloid

[TABLE]

embedded in a flat five-dimensional Minkowski space

[TABLE]

This geometrical representation of de Sitter spacetime is related to its symmetry structure characterised by the ten-parameter group SO(1,4). The entire hyperboloid (5) is covered by coordinates ${t\in(-\infty,+\infty)}$, ${\chi\in[0,\pi]}$, ${\theta\in[0,\pi]}$, ${\phi\in[0,2\pi)}$ such that

[TABLE]

in which the de Sitter metric takes the FLRW form

[TABLE]

The spatial sections at a fixed synchronous time $t$ are 3-spheres $S^{3}$ of constant positive curvature which have radius ${\>a\cosh\frac{t}{a}\>}$. These contract to a minimum size $a$ at ${t=0}$, and then re-expand. The de Sitter spacetime thus has a natural topology $R^{1}\times S^{3}$. This most natural parametrisation of the hyperboloid is illustrated in the left part of Figure 1.

Global causal structure of the de Sitter spacetime can be analyzed by introducing a conformal time $\eta$ and the conformal factor $\Omega$ by

[TABLE]

The metric (8) then becomes

[TABLE]

corresponding to the parametrization

[TABLE]

The de Sitter spacetime is thus conformal to a part of the Einstein static universe. Its boundary given by ${\Omega=0}$ is located at ${\eta=0}$ and ${\eta=\pi}$, which correspond to past and future conformal infinities ${\mathcal{I}}^{-}$ and ${\mathcal{I}}^{+}$, respectively, as illustrated in the right part of Figure 1. These infinities have a spacelike character. For more details see, e.g., [6, 17, 13].

2.2 The anti-de Sitter spacetime

Analogously, the anti-de Sitter manifold can be viewed as the hyperboloid

[TABLE]

embedded in a flat five-dimensional space

[TABLE]

which has two temporal dimensions $Z_{0}$ and $Z_{4}$. This representation of anti-de Sitter space reflects its symmetry structure characterised by the ten-parameter group of isometries SO(2,3).

The most natural static coordinates ${(T,r,\theta,\phi)}$ covering the entire hyperboloid are111Notice the swap ${Z_{0}\leftrightarrow Z_{4}}$ with respect to the parametrization (5.3) employed in [6].

[TABLE]

see the left part of Figure 2, in which the anti-de Sitter metric reads

[TABLE]

Any spatial section ${T=}$ const. is a 3-space of constant negative curvature spanned by ${r\in[0,\infty)}$, ${\theta\in[0,\pi]}$, ${\phi\in[0,2\pi)}$. The singularities at ${r=0}$ and ${\theta=0,\pi}$ are only coordinate singularities.

The two temporal dimensions $Z_{0}$ and $Z_{4}$ in the 5D flat space are parametrised using (14) by a single time coordinate $T$ that is periodic: values of $T$ which differ by a multiple of ${2\pi a}$ represent the same points on the hyperboloid. Thus, the anti-de Sitter spacetime defined in this way has the topology $S^{1}\times R^{3}$, and contains closed timelike worldlines. This periodicity of $T$ is not evident in the four-dimensional metric (15), and it is possible to take ${T\in(-\infty,+\infty)}$. Such a range of coordinates corresponds to an infinite number of turns around the hyperboloid. It is usual to unwrap the circle $S^{1}$ and extend it to the whole $R^{1}$ instead, without reference to the parametrisation (14). One thus obtains a universal covering space of the anti-de Sitter universe with topology $R^{4}$ without closed timelike curves.

To represent global causal structure of the anti-de Sitter spacetime, a conformal spatial coordinate $\chi$ and the conformal factor $\Omega$ may be introduced as

[TABLE]

Writing ${\>\eta\equiv T/a\>}$, the metric (15) then takes the form

[TABLE]

corresponding to the parametrization

[TABLE]

The whole (universal) anti-de Sitter spacetime is thus conformal to the region ${\chi\in[0,\frac{\pi}{2})}$ of the Einstein static universe, see the right part of Figure 2. The anti-de Sitter conformal infinity ${\mathcal{I}}$, given by ${\Omega=0}$, is located at the boundary ${\chi=\frac{\pi}{2}}$ (corresponding to ${r=\infty}$). In contrast to the de Sitter space, the conformal infinity ${\mathcal{I}}$ in the anti-de Sitter spacetime forms a timelike surface ${\chi=\frac{\pi}{2}}$.

3 Subcases of the new (anti-)de Sitter metric for distinct values of $\Lambda$, $\epsilon_{2}$, $\epsilon_{0}$

First, we are going to summarize different possible forms of the metric (1) with (2), (3) for the (anti-)de Sitter space, depending on the cosmological constant ${\Lambda>0}$ or ${\Lambda<0}$, and the two discrete parameters ${\epsilon_{2}=+1,0,-1\,}$ and ${\epsilon_{0}=+1,0,-1\,}$.

To keep the correct signature of the metric, the metric function $P(p)$ must be positive, ${P>0}$ (otherwise there would be two additional temporal coordinates $p$ and $\phi$). This puts a constraint on the parameter $\epsilon_{2}$ and the range of $p$. On the other hand, the function $Q(q)$ can be both positive and negative, depending on $\epsilon_{0}$, $\epsilon_{2}$, and the range of $q$. For ${Q>0}$, the coordinate $t$ is temporal and $q$ is spatial, while for ${Q<0}$, $q$ is temporal and $t$ is spatial. The boundary ${Q(q)=0}$ localizes the Killing horizon related to the Killing vector field $\partial_{t}$. In the case of de Sitter universe, this coincides with the cosmological horizon separating the static and dynamic regions.

The coordinate singularity at ${P(p)=0}$, where the norm of the Killing vector field $\partial_{\phi}$ vanishes, localizes the axis of symmetry. This occurs either when ${\Lambda>0,\epsilon_{2}=1}$ or ${\Lambda<0,\epsilon_{2}=-1}$. In both these cases such an axis is given by ${p=\pm a}$, and it is regular for the range ${\phi\in[0,2\pi)}$.

The list of all possible subcases of the general metric (1)–(3), with the admitted ranges of $p$ and $q$, are summarized in Table 1 for ${\Lambda>0}$, and in Table 2 for ${\Lambda<0}$.

We conclude that for ${\Lambda>0}$ there are 3 distinct subcases (namely ${\epsilon_{2}=1}$ with any $\epsilon_{0}$) while for ${\Lambda<0}$ we must discuss 8 subcases (namely all combinations of ${\epsilon_{2},\epsilon_{0}}$, except ${\epsilon_{2}=0=\epsilon_{0}}$). These will now be analyzed in the following sections 4 and 5, respectively.

4 New parametrizations of the de Sitter spacetime

4.1 Subcase ${\Lambda>0}$, ${\epsilon_{2}=1}$, ${\epsilon_{0}=1}$

In this case, the metric (1)–(3) of de Sitter universe has the form

[TABLE]

where ${|p|<a\equiv\sqrt{3/\Lambda}}$, ${q\in\mathbb{R}\setminus\{\pm 1\}}$, ${t\in\mathbb{R}}$, ${\phi\in[0,2\pi)}$, with ${|p|=a}$ representing the axis of symmetry. Here ${Q=0}$ is the horizon: for ${Q>0\Leftrightarrow|q|<1}$ the coordinate $q$ is spatial and $t$ is temporal, and vice versa for ${Q<0\Leftrightarrow|q|>1}$. These two cases must be discussed separately:

$\bullet$ For ${|q|<1}$, the coordinates of the metric (19) parametrize the de Sitter hyperboloid (5) as

[TABLE]

Actually, the parametrization (20) represents two maps covering the de Sitter manifold: the first one for ${p>0}$ covers the part ${Z_{1}>0}$, while the other for ${p<0}$ covers ${Z_{1}<0}$. Moreover, ${q>0}$ corresponds to ${Z_{2}>0}$, while ${q<0}$ corresponds to ${Z_{2}<0}$.

$\bullet$ For ${|q|>1}$, the parametrization is the same as (20), except that now

[TABLE]

Again, these are two maps: ${p>0}$ covers the part ${Z_{0}>0}$, while ${p<0}$ covers ${Z_{0}<0}$.

Specific character of these coordinates covering the de Sitter hyperboloid (5) is illustrated for two different sections in Figure 3.

Global conformal representation: To understand the global character of the coordinates ${(t,q,p,\phi)}$ of (19), it is best to plot them in the conformal representation of de Sitter spacetime, see the cylinder shown in the right part of Figure 1. This is achieved by comparing the 5D-parametrization (20), (21) of the de Sitter hyperboloid with the standard conformal parametrization (11) corresponding to the metric (10). The explicit relations are

[TABLE]

or, inversely,

[TABLE]

Using (25) we can now visualize the main surfaces ${p=\mathop{\rm const.}\nolimits}$ and ${q=\mathop{\rm const.}\nolimits}$ in the conformal de Sitter cylinder, see Figure 4–Figure 7. It is usual that the coordinate $\theta$ is suppressed and the full cylinder is drawn with the angular coordinate $\phi\in[0,2\pi)$, see Figure 1. However, in (4.1) the new coordinates $p,q$ depend on $\theta$, while they are independent of $\phi$. Therefore, in the subsequent figures we will suppress $\phi$ instead of $\theta$, so we will plot half-cylinders with the angular coordinate ${\theta\in[0,\pi]}$. Indeed, the global conformal metric (10) is the same both for ${\theta=\frac{\pi}{2}}$ and for ${\phi=\mathop{\rm const.}\nolimits}$ if we relabel $\phi\leftrightarrow\theta$, only the domain of these coordinates differ.

4.2 Subcase ${\Lambda>0}$, ${\epsilon_{2}=1}$, ${\epsilon_{0}=0}$

In this case, the de Sitter metric (1)–(3) has the form

[TABLE]

where ${p\in[0,a)}$, ${q\in\mathbb{R}\setminus\{0\}}$, ${t\in\mathbb{R}}$, ${\phi\in[0,2\pi)}$, with ${p=a}$ representing the axis of symmetry. Clearly, $t$ is now a spatial coordinate, whereas $q$ is temporal. These coordinates cover the de Sitter hyperboloid (5) by

[TABLE]

They are illustrated for specific sections in Figure 8.

Global conformal representation is obtained using the transformation

[TABLE]

relating (27) to the global de Sitter metric (10) — compare (28) to (11). This enables us to understand the global character of the coordinates of the metric (27). Because in both cases ${\epsilon_{0}=1}$ and ${\epsilon_{0}=0}$ the surfaces ${p=\mathop{\rm const.}\nolimits>0}$ are given by the same relations (compare (4.1) and (29)), their form is the same as in Figure 4. However, the surfaces ${q=\mathop{\rm const.}\nolimits}$ are now different — they are shown in Figure 9.

4.3 Subcase ${\Lambda>0}$, ${\epsilon_{2}=1}$, ${\epsilon_{0}=-1}$

The metric (1)–(3) for this third (and last) de Sitter subcase reads

[TABLE]

where ${p\in[0,a)}$, ${q\in\mathbb{R}}$, ${t\in[0,2\pi)}$, ${\phi\in[0,2\pi)}$. Since ${-Q=1+q^{2}\not=0}$, there is no Killing horizon related to the vector field $\partial_{t}$ in this metric. In fact, $t$ is everywhere a spatial angular coordinate, while $q$ is temporal. These coordinates cover the (part of) de Sitter hyperboloid (5) as

[TABLE]

This parametrization is visualized in different sections of the de Sitter hyperboloid in Figure 10.

Global conformal representation is obtained by combining (31) with (11):

[TABLE]

Specific character of the coordinates of metric (30) can thus be visualized in terms of the global de Sitter metric (10). Again, the surfaces ${p=\mathop{\rm const.}\nolimits>0}$ are the same as in Figure 4 because the relations (4.1) and (32) are identical. The different surfaces ${q=\mathop{\rm const.}\nolimits}$ are shown in Figure 11.

5 New parametrizations of the anti-de Sitter spacetime

5.1 Subcase ${\Lambda<0}$, ${\epsilon_{2}=1}$, ${\epsilon_{0}=1}$

The metric (1)–(3) of the anti-de Sitter universe for this choice of ${\epsilon_{2},\epsilon_{0}}$ takes the form

[TABLE]

where ${a=\sqrt{-3/\Lambda}\,}$ and ${\,p,t,\zeta\in\mathbb{R}}$, ${q\in\mathbb{R}\setminus\{\pm 1\}}$. Since $\phi$ employed in (1) now does not play the role of an angular coordinate, we have relabeled it as $\zeta$. The condition ${Q=1-q^{2}=0}$ identifies the Killing horizon associated with the vector field $\partial_{t}$: for ${|q|<1}$ the coordinate $q$ is spatial and $t$ is temporal, while for ${|q|>1}$ the coordinate $q$ is temporal and $t$ is spatial. However, contrary to the analogous de Sitter case discussed in Section 4.1, there is no cosmological horizon in the anti-de Sitter universe since there exists another Killing vector field which is everywhere timelike (see, e.g.,[6]). The distinct cases ${|q|<1}$ and ${|q|>1}$ are:

$\bullet$ For ${|q|<1}$, the coordinates of (33) parametrize the anti-de Sitter hyperboloid (12) as

[TABLE]

This parametrization gives two maps covering the anti-de Sitter manifold, namely the coordinate map ${Z_{4}\geq a}$ for the “$+$” sign, and ${Z_{4}\leq a}$ for the “$-$” sign (and again two maps ${p>0}$ and ${p<0}$). Moreover, ${q>0}$ corresponds to ${Z_{2}>0}$, while ${q<0}$ corresponds to ${Z_{2}<0}$.

$\bullet$ For ${|q|>1}$, the parametrization is the same as (34), except that now

[TABLE]

Again, these are two maps: ${p>0}$ covers the part ${Z_{0}>0}$, while ${p<0}$ covers ${Z_{0}<0}$.

We immediately observe that the coordinates ${t,p,q}$ depend on the coordinates ${Z_{0},Z_{1},Z_{2}}$ in exactly the same way as in the analogous case of the de Sitter spacetime, cf. (34) with (20), and (35) with (21). Therefore, sections through the anti-de Sitter spacetime in the subspace ${Z_{0},Z_{1},Z_{2}}$ are the same as the corresponding sections through the de Sitter spacetime, except that now the hyperbolas ${p=\mathop{\rm const.}\nolimits}$ are not bounded by ${p=a}$. Character of these coordinates covering the anti-de Sitter hyperboloid (12) is illustrated for two such sections in Figure 12.

Global conformal representation: To visualize the global character of these coordinates ${(t,q,p,\zeta)}$ of (33), we will plot them in the standard conformal representation of anti-de Sitter spacetime, see the right part of Figure 2. This is achieved by comparing the 5D-parametrization (34), (35) with the standard conformal parametrization (18) corresponding to the metric (17). We thus obtain the following explicit relations

[TABLE]

where the signs “$\pm$” again correspond to two coordinate maps ${Z_{4}>0}$ and ${Z_{4}<0}$, respectively.

The inverse relations to (43) are

[TABLE]

We use the relations (43) to draw the surfaces ${p=\mathop{\rm const.}\nolimits}$ (Figure 13), ${q=\mathop{\rm const.}\nolimits}$ (Figure 14), ${t=\mathop{\rm const.}\nolimits}$ (left part of Figure 15), and $\zeta=\mathop{\rm const.}\nolimits$ (right part of Figure 15), respectively. In order to draw these global conformal pictures, it was necessary to suppress one coordinate. Because $p,q$ and $\zeta$ now explicitly depend on $\phi$, we cannot simply suppress it (unlike in the previous cases of de Sitter universe). It can be seen from (5.1) that for larger $\phi$ there is a greater constraint on the ranges of $p$ and $q$. Therefore, the coordinates $p$ and $q$ cover a smaller portion of the section. As an important illustration we draw the section ${\phi=0}$ (and also ${\phi=\frac{\pi}{4}}$ and ${\frac{\pi}{2}}$ in Figure 13). For $\phi$ shifted by $\pi$ we would obtain the same pictures (possibly with $\pm$).

5.2 Subcase ${\Lambda<0}$, ${\epsilon_{2}=1}$, ${\epsilon_{0}=0}$

In this case, the anti-de Sitter metric (1)–(3) with ${\phi=\zeta}$ reads

[TABLE]

where ${p\in[0,\infty)}$, ${q\in\mathbb{R}\setminus\{0\}}$, ${t,\zeta\in\mathbb{R}}$. Clearly, $t$ is a spatial coordinate, whereas $q$ is temporal. Such coordinates cover the anti-de Sitter hyperboloid (12) as

[TABLE]

where the “$+$” sign corresponds to the coordinate chart ${Z_{4}>0}$, while the “$-$” sign corresponds to ${Z_{4}<0}$. These coordinates are visualized in Figure 16.

Global conformal representation of the anti-de Sitter metric (45) is given by the transformation (obtained by comparing (46) with (18))

[TABLE]

where the signs “$\pm$” correspond to the coordinate charts ${Z_{4}>0}$ and ${Z_{4}<0}$, respectively. An inverse transformation then reads

[TABLE]

The surfaces ${p=\mathop{\rm const.}\nolimits}$ and ${\zeta=\mathop{\rm const.}\nolimits}$ are given by the same relations in both cases ${\epsilon_{0}=1}$ and ${\epsilon_{0}=0}$ (compare (5.1) and (5.2)) so that their form is the same as in Figures 13 and 15. The surfaces ${q=\mathop{\rm const.}\nolimits}$ are now different — they are shown in Figure 17 for ${\phi=0}$ and ${\phi=\frac{\pi}{2}}$.

5.3 Subcase ${\Lambda<0}$, ${\epsilon_{2}=1}$, ${\epsilon_{0}=-1}$

The anti-de Sitter metric (1)–(3) with ${\phi=\zeta}$ now takes the form

[TABLE]

where ${p\in[0,\infty)}$, ${t\in[0,2\pi)}$, ${q,\zeta\in\mathbb{R}}$. Again, $t$ is a spatial angular coordinate, whereas $q$ is temporal. These coordinates cover the anti-de Sitter hyperboloid (12) as

[TABLE]

This is visualized in different sections through the anti-de Sitter hyperboloid in Figure 18.

Global conformal representation of the anti-de Sitter metric (53) is obtained by comparing (54) with (18):

[TABLE]

where the signs “$\pm$” correspond to the coordinate charts ${Z_{4}>0}$ and ${Z_{4}<0}$, respectively. An inverse transformation is

[TABLE]

The surfaces ${p=\mathop{\rm const.}\nolimits}$ and ${\zeta=\mathop{\rm const.}\nolimits}$ are given by the same relations in all the three cases ${\epsilon_{0}=1,0,-1}$, so that their form is the same as in Figure 13 and Figure 15. However, the surfaces ${q=\mathop{\rm const.}\nolimits}$ are now different, as shown in Figure 19 for ${\phi=0}$ and ${\phi=\frac{\pi}{2}}$.

5.4 Subcase ${\Lambda<0}$, ${\epsilon_{2}=0}$, ${\epsilon_{0}=1}$

In this case ${Q=1}$, and the corresponding anti-de Sitter metric (1)–(3) with ${\phi=\zeta}$ is

[TABLE]

in which ${p,q,t,\zeta\in\mathbb{R}}$. With a simple transformation

[TABLE]

where ${\eta_{{}_{0}},x,y,z\in\mathbb{R}}$, we immediately obtain the metric

[TABLE]

This is exactly the well-known conformally flat Poincaré form, see e.g. the metric (5.14) in [6]. These coordinates on the anti-de Sitter hyperboloid are visualized on Fig. 5.5 therein, as well as the explicit parametrization Eq. (5.13). Combining this with relations (62) we obtain

[TABLE]

where

[TABLE]

The metric (63) of anti-de Sitter spacetime has been thoroughly described and employed in literature, for example in the works on AdS/CFT correspondence. Of course, via the simple relations (62), all this can be equivalently re-expresed in terms of the alternative metric form (61) presented here.

The coordinate surfaces of constant $p$ and $t$ within the conformal cylinder representing the global structure of anti-de Sitter spacetime (shown in the right part of Figure 2) are plotted in Figure 20. Recall that the conformal infinity ${\cal I}$ is the outer boundary ${\chi=\frac{\pi}{2}}$. It is well known that this corresponds to ${x=0}$ (see e.g. Fig. 5.6 in [6]), which is here equivalent to ${p=\infty}$ in the new coordinates of (61). In fact, it seems more natural to represent the conformal infinity ${\cal I}$ by the infinite value of the coordinate $p$ rather than the zero value of the usual coordinate $x$.

The corresponding two-dimensional Penrose conformal diagram of anti-de Sitter spacetime (the vertical shaded plane in the right part of Figure 2), with the coordinate lines of $p$ and $t$, is plotted in the bootom right part of Figure 20.

5.5 Subcase ${\Lambda<0}$, ${\epsilon_{2}=0}$, ${\epsilon_{0}=-1}$

In such a case ${Q=-1}$, and the metric is thus

[TABLE]

This is clearly the same form as the previous metric (61) if we swap the coordinates ${t\leftrightarrow q}$.

5.6 Subcase ${\Lambda<0}$, ${\epsilon_{2}=0}$, ${\epsilon_{0}=0}$

For this choice ${Q=0}$, so that the metric (1)–(3) degenerates. This case is thus forbidden.

5.7 Subcase ${\Lambda<0}$, ${\epsilon_{2}=-1}$, ${\epsilon_{0}=1}$

Finally, we will discuss three subcases of the anti-de Sitter metric (1)–(3) for ${\Lambda<0}$ with ${\epsilon_{2}=-1}$ and ${\epsilon_{0}=1,0,-1}$ (which are not possible for ${\Lambda\geq 0}$). For ${\epsilon_{0}=1}$ the metric reads

[TABLE]

where ${a=\sqrt{-3/\Lambda}\,}$ and ${p\in[a,\infty)}$, ${q\in\mathbb{R}}$, ${t,\phi\in[0,2\pi)}$, with ${p=a}$ representing the axis of symmetry. This arises as the parametrization

[TABLE]

of the anti-de Sitter hyperboloid (12) in the flat space (13). Let us recall that $Z_{0}$ a $Z_{4}$ are two temporal coordinates expressed here by the natural single temporal coordinate ${t\in[0,2\pi)}$. The covering space is obtained by allowing ${t\in\mathbb{R}}$ in (67).

It is interesting to notice that after the formal relabeling $Z_{0}\leftrightarrow Z_{1}$ and ${Z_{2}\rightarrow Z_{3}\rightarrow Z_{4}\rightarrow Z_{2}}$ we obtain basically the same expressions as (31) for the de Sitter subcase ${\Lambda>0}$, ${\epsilon_{2}=1}$, ${\epsilon_{0}=-1}$. Therefore, the sections through the anti-de Sitter hyperboloid closely resemble those shown in Figure 10, after the relabeling of the axes $Z_{a}$ and reconsidering different ranges of the coordinates. In particular, in Figure 21 we plot the sections ${Z_{4}=0}$ and ${Z_{1}=\mathop{\rm const.}\nolimits>0}$, respectively. It can be seen that (unlike in the cases ${\epsilon_{2}=1}$) for ${\epsilon_{2}=-1}$ the coordinates cover the whole anti-de Sitter universe, which is also true for the related subcases ${\epsilon_{0}=0}$ and ${\epsilon_{0}=-1}$.

Global conformal representation is obtained by combining (68) with (18):

[TABLE]

Specific character of the coordinates in metric (67) can thus be visualized within the anti-de Sitter global conformal cylinder. In Figure 22 we plot both the surfaces ${p=\mathop{\rm const.}\nolimits}$ and the surfaces ${q=\mathop{\rm const.}\nolimits}$ It can be seen that they foliate the entire anti-de Sitter spacetime in a very natural way. Moreover, the conformal infinity ${\cal I}$, given by ${\chi=\frac{\pi}{2}}$, is now completely represented by the boundary ${p=\infty}$.

5.8 Subcase ${\Lambda<0}$, ${\epsilon_{2}=-1}$, ${\epsilon_{0}=0}$

In this case, the anti-de Sitter metric takes the form

[TABLE]

where ${p\in[a,\infty)}$, ${q\in\mathbb{R}\setminus\{0\}}$, $t\in\mathbb{R}$, ${\phi\in[0,2\pi)}$. Again, ${p=a}$ represents the axis of symmetry, $t$ is a temporal coordinate, whereas $q$ is spatial. These coordinates are given by

[TABLE]

Global conformal representation is now

[TABLE]

In view of (69), the surfaces ${p=\mathop{\rm const.}\nolimits}$ are the same as in Figure 22, but the surfaces ${q=\mathop{\rm const.}\nolimits}$ have a different shape shown in Figure 23.

5.9 Subcase ${\Lambda<0}$, ${\epsilon_{2}=-1}$, ${\epsilon_{0}=-1}$

The anti-de Sitter metric (1)–(3) in this final case reads

[TABLE]

where ${p\in(-\infty,-a]\cup\,[a,\infty)}$, ${q\in\mathbb{R}\setminus\{\pm 1\}}$, $t\in\mathbb{R}$, ${\phi\in[0,2\pi)}$, with ${p=\pm a}$ representing the axis of symmetry. For $|q|>1$ the coordinate $t$ is temporal and $q$ is spatial, whereas for $|q|<1$ the coordinate $t$ is spatial and $q$ is temporal. These two cases thus need to be discussed separately:

$\bullet$ For ${|q|>1}$, the coordinates of the metric (73) parametrize the anti-de Sitter hyperboloid (12)

[TABLE]

The parametrization (74) represents two maps covering the anti-de Sitter manifold: the first one for ${p>0}$ covers the part ${Z_{1}>0}$, while the other for ${p<0}$ covers ${Z_{1}<0}$. Moreover, ${q>0}$ corresponds to ${Z_{4}>0}$, while ${q<0}$ corresponds to ${Z_{4}<0}$.

$\bullet$ For ${|q|<1}$, the parametrization is the same as (74), except that now

[TABLE]

Again, these are two maps: ${p>0}$ covers the part ${Z_{0}>0}$, while ${p<0}$ covers ${Z_{0}<0}$.

Global conformal representation is obtained by comparing (74), (75) with (18):

[TABLE]

or, inversely,

[TABLE]

This can be used for plotting the coordinate surfaces of the metric (73) in terms of the standard global conformal representation of the entire anti-de Sitter spacetime. The surfaces ${p=\mathop{\rm const.}\nolimits}$ are the same as in the case ${\epsilon_{0}=1}$ shown in Figure 22, except that now there are two complementary regions ${p>0}$ and ${p<0}$, see the conformal infinity ${\mathcal{I}}$ shown in the left part of Figure 24. Specific character of the surfaces ${q=\mathop{\rm const.}\nolimits}$ is shown for two views in the right part of Figure 24.

6 Physical application: exact gravitational field of a tachyon moving in the (anti-)de Sitter spacetime

To illustrate one of possible applications of the new coordinate representation (1)–(3) of the (anti-)de Sitter spacetime, analyzed in this contribution, let us finally outline a physical interpretation of the family of vacuum $B$-metrics with any value of the cosmological constant $\Lambda$. Although this family has been known for a long time [4, 5, 6, 7, 8], it has not yet been systematically studied from the physical point of view.

The $B$-metrics with $\Lambda$ can be written in the form (1) with the functions $Q(q)$ and $P(p)$ given by (4), that is

[TABLE]

Notice that $Q(q)$ is exactly the same as in (2), and $P(p)$ immediately reduces to (3) when ${n=0}$. Therefore, the specific form (1)–(3) of the (anti-)de Sitter metric, which we have studied in this work, can be understood as the natural background for the $B$-metrics with $\Lambda$: It is obtained in the weak-field limit as ${n\to 0}$. In fact, the $B$-metrics are of algebraic type D, with a curvature singularity located at ${p=0}$. Setting ${n=0}$, this singularity disappears and (84) becomes conformally flat vacuum solution with $\Lambda$, that is the maximally symmetric (anti-)de Sitter spacetime with $P(p)$ given by (3). Various subcases of the (anti-)de Sitter metrics (1)–(3) — in particular, the character of the coordinates employed — are thus fundamental for an understanding of geometrical and physical properties of the whole $B$-metrics family (84).

The $B$-metrics can be directly compared to the family of $A$-metrics

[TABLE]

It can be seen that (85) is obtained from (84) by formal substitutions of the coordinates ${p\rightarrow r}$, ${t\rightarrow{\rm i}\,\phi}$, ${a\,\phi\rightarrow{\rm i}\,t}$. In particular, the complex unit “i” formally interchanges the temporal and spatial character of the coordinates $t$ and $\phi$. For three distinct signs of the Gaussian curvature parameter ${\epsilon_{2}}$ of the spatial 2-surfaces on which $r$ and $t$ are constant, there are three subclasses of the $A$-metrics, namely $AI$ for ${\epsilon_{2}=1}$, $AII$ for ${\epsilon_{2}=-1}$, and $AIII$ for ${\epsilon_{2}=0}$. These include various (topological) black holes which were studied in a number of works, see Chapter 9 in [6] for a review and number of references. In particular, for the $AI$ subcase ${\epsilon_{2}=1=\epsilon_{0}}$ we can introduce an angular coordinate $\theta$ by ${q=\cos\theta}$ and relabel the parameter $n$ as $-m$, obtaining thus the standard form of the famous Schwarzschild–(anti-)de Sitter metric

[TABLE]

It is also of type D, with a curvature singularity at ${r=0}$. This unique spherically symmetric solution was discovered by Kottler, Weyl, and Trefftz [19, 21, 20]. Of course, for ${\Lambda=0}$ it is just the first exact solution to Einstien’s field equations, found by Karl Schwarzschild [22].

Analogously to $A$-metrics, there exist three distinct subclasses of the $B$-metrics (84), denoted as $BI$ for ${\epsilon_{2}=1}$, $BII$ for ${\epsilon_{2}=-1}$, and $BIII$ for ${\epsilon_{2}=0}$. However, ${\epsilon_{2}}$ now determines the Gaussian curvature of the Lorentzian 2-surfaces on which $p$ and $\phi$ are constant.

Applying the results of the present paper, we are now suggesting a physical interpretation of the $BI$-metric defined by ${\epsilon_{2}=1}$, which is the direct counterpart of the Schwarzschild–(anti-)de Sitter metric (86) representing a static (black hole) matter source. For the convenient choice ${\epsilon_{0}=-1}$, a simple transformation ${q=\sinh\tau}$, $t=\chi$ puts such metric (84) into the form

[TABLE]

In fact, we are extending the idea presented in 1974 in an interesting work [23] by J. R. Gott. In the case ${\Lambda=0}$, he argued that the $BI$-metric represents an exact gravitational field of a tachyonic source moving with a superluminal velocity ${v>c\equiv 1}$ along the spatial axis of symmetry. We now claim that the metric (87) generalizes this physical situation to a tachyonic-type singular source moving along the spatial axis of $\partial_{\phi}$ at ${p=0}$ in the de Sitter universe with ${\Lambda>0}$ (and similarly in the anti-de Sitter universe when ${\Lambda<0}$). This is the counterpart to the static source moving along the temporal axis of $\partial_{t}$ at ${r=0}$ in the Schwarzschild–(anti-)de Sitter $AI$-metric (86). Indeed, in the weak-field limit ${n\to 0}$ we immediately observe that the $BI$-metric (87) for ${p=0}$ reduces to ${{\mathrm{d}}s^{2}=a^{2}{\mathrm{d}}\phi^{2}}$, yielding a positive norm of $\partial_{\phi}$ (contrary to the analogous limit ${m\to 0}$ of the $AI$-metric (86) which for ${r=0}$ yields ${{\mathrm{d}}s^{2}=-{\mathrm{d}}t^{2}}$, and the norm of $\partial_{t}$ is thus negative).

Moreover, in the limit ${n\to 0}$ we observe that the metric (87), which is equivalent to (84) with ${\epsilon_{2}=1}$ and ${\epsilon_{0}=-1}$, reduces to the metric form (30) of the de Sitter spacetime studied in Subsection 4.3. In that part of this work we have analyzed and visualized the character of the coordinates, and the corresponding foliation of the background spacetime, see Figure 10 and Figures 4 and 11.

In particular, we immediately obtain from the specific parametrization (31) of the de Sitter hyperboloid (5) in (6) by these coordinates that the source at ${p=0}$ is, in fact, located at

[TABLE]

In other words, the trajectory of the source is given by

[TABLE]

with ${Z_{0}=Z_{1}=Z_{2}=0}$. This fully confirms that the trajectory of the source of the $BI$-metric is indeed spacelike, and thus “tachyonic”. In fact, it is exactly the closed spatial loop around the “neck” of the de Sitter hyperboloid at ${Z_{0}}$, see the left part of Figure 25. Geometrically, it is one of the main circles of the spatial 3-sphere. Notice also that in the global conformal representation (32) this geodesic corresponds to a special curve given by ${\eta=\frac{\pi}{2}}$, ${\chi=\frac{\pi}{2}}$, ${\theta=\frac{\pi}{2}}$ with the angular coordinate taking the range ${\phi\in[0,2\pi)}$, i.e., it is a closed circle around “the middle” of the conformal 4D representation of the whole universe.

Since this source of the field (87) moves with a superluminal velocity, it generates gravitational Cherenkov shock cone. Its location is obtained by substituting (89) into (5), i.e.,

[TABLE]

In fact, there are two Cherenkov cones: the first given by ${\sqrt{Z_{1}^{2}+Z_{2}^{2}}=Z_{0}}$ (rear one) is expanding, while the second given by ${\sqrt{Z_{1}^{2}+Z_{2}^{2}}=-Z_{0}}$ (front one) is contracting. The tachyonic source is located at their common vertex. The background de Sitter spacetime is divided by these Cherenkov cones into three distinct regions, namely region 1 given by ${Z_{0}>\sqrt{Z_{1}^{2}+Z_{2}^{2}}}$, region 2 given by ${Z_{0}<-\sqrt{Z_{1}^{2}+Z_{2}^{2}}}$, and region 3 defined by ${Z_{0}^{2}<Z_{1}^{2}+Z_{2}^{2}}$. As in the case ${\Lambda=0}$ studied by Gott in [23], detailed analysis shows that for ${n\not=0}$ the curved region 3 outside the Cherenkov cones is covered by the $BI$-metric (87), while each of the regions 1 and 2 is covered by the $AII$-metric (85).

The source of the gravitational field (87) moves along (88) with an infinite velocity ${v=\infty}$. Its Cherenkov cone (90) at ${Z_{0}=0}$ thus degenerates to the single circular axis along which the tachyon moves. To obtain a nondegenerate Cherenkov cone, it is necessary to “slow down” the tachyon to a finite velocity $v$ such that ${v>1}$. This is achieved by performing an appropriate Lorentz boost in the representation of the de Sitter spacetime as the hyperboloid (5) embedded in the five-dimensional Minkowski space (6), for example in the $Z_{3}$-direction,

[TABLE]

With respect to the new frame, the tachyonic source (88) localized along ${Z_{0}=0}$ now moves along ${Z_{3}^{\prime}/Z_{0}^{\prime}=v}$. This is illustrated in the right part of Figure 25. The corresponding Cherenkov cone (90) then takes the form

[TABLE]

with its source (vertex) given by ${Z_{3}^{\prime}=vZ_{0}^{\prime}}$, i.e., moving with the velocity $v$ along the axis $Z_{3}^{\prime}$. Both parts of this Cherenkov shock (the expanding part of the cone and the contracting one) are visualized in Figure 26 for four typical values of the fixed global time ${Z_{0}^{\prime}=}$const., together with the spatial representation of the de Sitter spacetime as a sphere. As explained in Subsection 2.1, the spatial geometry of the de Sitter spacetime is a 3-sphere $S^{3}$ which contracts to a minimal size $a$ at ${Z_{0}^{\prime}=0}$ and then re-expands. At a given time $Z_{0}^{\prime}$, the position of both the Cherenkov shocks are two intersections of the cone with this sphere. In the full de Sitter space $S^{3}$, the Cherenkov shocks are two 2-spheres (one of which is expanding while the other is contracting) but since in Figure 26 we have suppressed one angular coordinate, the de Sitter space is represented just by a 2-sphere while the Cherenkov shocks are circles.

7 Summary and conclusions

In our contribution we have presented 3 interesting forms (1)–(3) of the de Sitter spacetime and 8 forms of the anti-de Sitter spacetime in four dimensions, according to the values of two discrete parameters ${\epsilon_{0},\epsilon_{2}=1,0,-1}$, see Tables 1 and 2. As far as we know, these have not yet been explicitly identified, described and analyzed in (otherwise vast) literature on these fundamental solutions to Einstein’s field equations with a cosmological constant $\Lambda$.

Such new forms of maximally symmetric vacuum spacetimes with $\Lambda$ naturally arise in the context of $B$-metrics with a cosmological constant, when they are expressed in a unified Plebański–Demiański form of the Kundt type D solutions. Quite surprisingly, all the metric forms (1)–(3) of the (anti-)de Sitter spacetime are very simple: they are diagonal, represent a warped product of two 2-dimensional geometries of constant curvature, and many of them explicitly exhibit stationarity and axial symmetry. The coordinates of (1) are naturally adapted to a ${2+2}$ foliation of the spacetime, in contrast to the coordinates of standard well-known forms, which correspond to a ${1+3}$ foliation with the 3-space of a constant spatial curvature.

In each subcase, determined by ${\epsilon_{0},\epsilon_{2}=1,0,-1}$, we found the corresponding parametrizations of the de Sitter or anti-de Sitter hyperboloid, and then we plotted the coordinate lines and surfaces in the global conformal cylinder. This clearly demonstrated specific character of the new coordinates. In the appendices we also present mutual relations between various subcases, some other related forms of the metrics, and also transformations to standard coordinates of the de Sitter and anti-de Sitter spacetimes.

Our initial motivation was to investigate geometrical and physical properties of the $B$-metrics with a cosmological constant, which can be understood as type D extensions of the conformally flat (anti-)de Sitter metric (1). In the final Section 6 we outlined such a study. Using the results given in the main part of this paper, we demonstrated that the family of $B$-metrics can be physically interpreted as representing exact gravitational field of a tachyon moving in the de Sitter or anti-de Sitter spacetime. A detailed analysis will be performed in our future work, as we think that the new coordinate representations of the de Sitter and anti-de Sitter spacetime presented in the current contribution can be interesting by itself. We hope that they could also be employed for various other investigations of these fundamental spacetimes, discovered 100 years ago.

Acknowledgements

We are grateful to Robert Švarc for reading the manuscript and some useful suggestions. This work has been supported by the grant GAČR 17-01625S. O.H. also acknowledges the support by the Charles University Grant GAUK 196516.

8 Appendix A: Unified form of all parametrizations

All new parametrizations of the de Sitter or anti-de Sitter hyperboloid, presented and studied in previous Sections 4 and 5, can be written in a unified way as:

[TABLE]

where ${Q=\epsilon_{0}-\epsilon_{2}\,q^{2}}$, ${P=\epsilon_{2}-\frac{\Lambda}{3}\,p^{2}}$, as in (2), (3). The indices $a,b,c,d,e$ and also the functions $S,T_{1},T_{2},Y_{1},Y_{2}$ for various $\epsilon_{2}$ and $\epsilon_{0}$ are listed in Tables 3 and 4 for ${\Lambda>0}$ or ${\Lambda<0}$, respectively. The last column refers to the corresponding equation in the main text.

The coordinates $(p,q,t,\phi)$ only cover the part ${Z_{3},Z_{4}\in(-a,a)}$ of de Sitter spacetime, while they cover the whole anti-de Sitter spacetime (except in the cases ${\epsilon_{2}=1}$ for which ${Z_{4}\in\mathbb{R}\setminus(-a,a)}$).

9 Appendix B: Mutual relations

Case ${\Lambda>0}$: The de Sitter metric (19) for ${\epsilon_{2}=1}$, ${\epsilon_{0}=1}$ (in which the coordinates ${p,q,t}$ are relabeled to ${\tilde{p},\tilde{q},\tilde{t}}$) can be explicitly transformed to the metric (27) for ${\epsilon_{2}=1}$, ${\epsilon_{0}=0}$ by applying

[TABLE]

Similarly, the metric (19) for ${|q|<1}$ can be explicitly transformed to the metric (30) for ${\epsilon_{2}=1}$, ${\epsilon_{0}=-1}$ by applying

[TABLE]

These relations are easily obtained by comparing (20) to (28), and (20) to (31), respectively.

Case ${\Lambda<0}$: Interestingly, the same transformations (94) and (95) relate the anti-de Sitter metric (33) for ${\epsilon_{2}=1}$, ${\epsilon_{0}=1}$ (with the coordinates relabeled as ${\tilde{p},\tilde{q},\tilde{t}}$) to (45) for ${\epsilon_{2}=1}$, ${\epsilon_{0}=0}$ and (53) for ${\epsilon_{2}=1}$, ${\epsilon_{0}=-1}$, respectively.

Comparing the parameterizations (46) and (64) we find a transformation between the anti-de Sitter metric (45) for ${\epsilon_{2}=1}$, ${\epsilon_{0}=0}$ and the form (61) for ${\epsilon_{2}=0}$, ${\epsilon_{0}=1}$ (relabeled to ${\tilde{p},\tilde{q},\tilde{t},\tilde{\zeta}}$):

[TABLE]

Also, the anti-de Sitter metric (67) for ${\epsilon_{2}=-1}$, ${\epsilon_{0}=1}$ (relabeled to ${\tilde{p},\tilde{q},\tilde{t}}$) is transformed to the metric (70) for ${\epsilon_{2}=-1}$, ${\epsilon_{0}=0}$ by

[TABLE]

while (67) for ${\epsilon_{2}=-1}$, ${\epsilon_{0}=1}$ is transformed to the metric (73) for ${\epsilon_{2}=-1}$, ${\epsilon_{0}=-1}$ by

[TABLE]

10 Appendix C: Further related metric forms of the de Sitter and anti-de Sitter spacetimes

The new metrics analyzed in this work can also be expressed in closely related forms. In these, both (warped) parts representing the constant-curvature 2-spaces are more clearly seen because they are all put into the “canonical” forms using triginometric and/or hyperbolic functions. In this appendix we describe all such related metrics. For each subcase, we present the transformation, the resulting metric, and the corresponding paramaterization of the 5D hyperboloid:

10.1 The de Sitter spacetime (${\Lambda>0}$)

Subcase ${\epsilon_{2}=1}$, ${\epsilon_{0}=1}$:

[TABLE]

transforms (19) to

[TABLE]

respectively, corresponding to the parametrizations

[TABLE]

Subcase ${\epsilon_{2}=1}$, ${\epsilon_{0}=0}$:

[TABLE]

transforms (27) to

[TABLE]

corresponding to

[TABLE]

Subcase ${\epsilon_{2}=1}$, ${\epsilon_{0}=-1}$:

[TABLE]

transforms (30) to

[TABLE]

corresponding to

[TABLE]

10.2 The anti-de Sitter spacetime (${\Lambda<0}$)

Subcase ${\epsilon_{2}=1}$, ${\epsilon_{0}=1}$:

[TABLE]

transforms (33) to

[TABLE]

respectively, corresponding to the parametrizations

[TABLE]

Subcase ${\epsilon_{2}=1}$, ${\epsilon_{0}=0}$:

[TABLE]

transforms (45) to

[TABLE]

corresponding to

[TABLE]

Subcase ${\epsilon_{2}=1}$, ${\epsilon_{0}=-1}$:

[TABLE]

transforms (53) to

[TABLE]

corresponding to

[TABLE]

Subcase ${\epsilon_{2}=0}$, ${\epsilon_{0}=\pm 1}$:

The transformation (62) puts (61) into (63), and similarly for (66).

Subcase ${\epsilon_{2}=-1}$, ${\epsilon_{0}=1}$:

[TABLE]

transforms (67) to

[TABLE]

corresponding to

[TABLE]

Subcase ${\epsilon_{2}=-1}$, ${\epsilon_{0}=0}$:

[TABLE]

transforms (70) to

[TABLE]

corresponding to

[TABLE]

Subcase ${\epsilon_{2}=-1}$, ${\epsilon_{0}=-1}$:

[TABLE]

transforms (73) to

[TABLE]

respectively, corresponding to the parametrizations

[TABLE]

11 Appendix D: Transformations to standard metric forms of (anti-)de Sitter spacetime

11.1 The de Sitter spacetime (${\Lambda>0}$)

Subcase ${\epsilon_{2}=1}$, ${\epsilon_{0}=1}$: By applying transformation

[TABLE]

the metric (19) is put into standard spherical form of the de Sitter spacetime (see (4.9) in [6])

[TABLE]

Subcase ${\epsilon_{2}=1}$, ${\epsilon_{0}=0}$: With

[TABLE]

the metric (27) is put into the well-known exponentially expanding flat FLRW form of the de Sitter spacetime (see (4.14) in [6]) with ${y=\rho\cos\phi}$, ${z=\rho\sin\phi}$:

[TABLE]

Subcase ${\epsilon_{2}=1}$, ${\epsilon_{0}=-1}$: With

[TABLE]

the de Sitter metric (30) transforms into the Bianchi III form (see (4.21) in [6]):

[TABLE]

Interestingly, inverse transformations to (126), (128), (130) have very similar structure, namely

[TABLE]

11.2 The anti-de Sitter spacetime (${\Lambda<0}$)

Subcase ${\epsilon_{2}=1}$, ${\epsilon_{0}=1}$: Using

[TABLE]

the metric (33) for ${|q|<1}$ is transformed into

[TABLE]

which is simply related via ${\eta_{{}_{0}}=\tau\cosh\zeta}$, ${z=\tau\sinh\zeta}$ to standard form of the anti-de Sitter spacetime (see (5.16) in [6])

[TABLE]

Subcase ${\epsilon_{2}=1}$, ${\epsilon_{0}=0}$: Similarly, with

[TABLE]

the metric (45) is transformed into the metric (134).

Subcase ${\epsilon_{2}=1}$, ${\epsilon_{0}=-1}$: By

[TABLE]

the metric (53) is also put into (134).

Subcase ${\epsilon_{2}=-1}$, ${\epsilon_{0}=1}$: With

[TABLE]

the metric (67) transforms into the anti-de Sitter metric (15) in the global static coordinates (see (5.4) in [6])

[TABLE]

Subcase ${\epsilon_{2}=-1}$, ${\epsilon_{0}=0}$: By applying

[TABLE]

the metric (70) is put into the axially symmetric form

[TABLE]

which is related via ${y=\rho\cos\phi}$, ${z=\rho\sin\phi}$ to the standard form (135).

Subcase ${\epsilon_{2}=-1}$, ${\epsilon_{0}=-1}$: With

[TABLE]

for ${|q|>1}$, (73) is transformed into the anti-de Sitter metric (15) in the global static coordinates (for ${|q|<1}$, the function $\tanh t$ is replaced by $\coth t$).