LORENTZ BOOSTS IN DE SITTER AND ANTI-DE SITTER SPACE-TIMES

Jiří Podolský Department of Theoretical Physics, Charles University, Prague V Holešovičkách 2, 180 00 Praha 8, Czech Republic

Symmetries of de Sitter and anti-de Sitter space-times are discussed. In particular, Lorentz boosts, which appear to be given previously only in the infinitesimal form, are presented here in closed finite forms.

1. Introduction

It is well-known that the maximal number of isometries of a four-dimensional spacetime is 10 corresponding to the same number of independent Killing vectors (infinitesimal generators of the symmetry transformations). Such maximally symmetric space-times have a constant curvature R (see, e.g.[1]). According to the sign of R, there are three possibilities: one gets flat Minkowski space-time (R = 0), de Sitter space-time (R > 0) and anti-de Sitter space-time (R < 0). These space-times are the only conformally flat vacuum solutions of Einstein’s field equations with cosmological constant λ. The constant curvature R is related to λ by R = 4λ.

2. Symmetries of de Sitter space-time

De Sitter manifold can be visualized as a hyperboloid

−z_{0}^{2} + z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + z_{4}^{2} = α^{2},  (1)

α = \sqrt{\frac{λ}{3}} , embedded in a flat five-dimensional space-time

ds^{2} = −dz_{0}^{2} + dz_{1}^{2} + dz_{2}^{2} + dz_{3}^{2} + dz_{4}^{2} .  (2)

We use coordinates (η, x, y, z) covering the entire hyperboloid given by

\begin{matrix}z_{0} = \frac{1}{2η} (α^{2} + s) \\ z_{1} = α\frac{x}{η} \\ z_{2} = α\frac{y}{η} \\ z_{3} = α\frac{z}{η} \\ z_{4} = \frac{1}{2η} (α^{2} − s) \end{matrix}  (3)

where s = −η^{2} + x^{2} + y^{2} + z^{2} and η, x, y, z ∈ (−∞, +∞). In the global coordinates (3) de Sitter metric can be written as

ds^{2} = \frac{α^{2}}{η^{2}} (−dη^{2} + dx^{2} + dy^{2} + dz^{2}) .  (4)

This conformally flat metric (see, e.g. [3]) is related to the ‘standard’ form of de Sitter metric

ds^{2} = −dt^{2} + exp (2\frac{t}{α})(dx^{2} + dy^{2} + dz^{2})  (5)

by a simple transformation η = α exp (−t/α). However, the metric (5) covers only one half (z_{0} + z_{4} > 0) of the manifold (1). Thus, one needs another coordinate chart of the form (5) to cover also z_{0} + z_{4} < 0 ; such a chart is given by η = −α exp (−t^{∗}/α). We prefer coordinates (3) since they do not only cover the entire manifold but the symmetries can be seen most easily just in these coordinates.

Killing vectors are (see [2],[3], where, however, the expressions are given in ‘standard’ coordinates used in (5) and thus have more complicated forms):

\begin{matrix}ξ_{(1)}^{μ} = (0, 1, 0, 0) \\ ξ_{(2)}^{μ} = (0, 0, 1, 0) \\ ξ_{(3)}^{μ} = (0, 0, 0, 1) \\ ξ_{(4)}^{μ} = (0, y, −x, 0) \\ ξ_{(5)}^{μ} = (0, z, 0, −x) \\ ξ_{(6)}^{μ} = (0, 0, z, −y) \\ ξ_{(7)}^{μ} = (η, x, y, z) \\ ξ_{(8)}^{μ} = −2x(η, x − \frac{s}{2x}, y, z) \\ ξ_{(9)}^{μ} = −2y(η, x, y − \frac{s}{2y} , z) \\ ξ_{(10)}^{μ} = −2z(η, x, y, z − \frac{s}{2z}) . \end{matrix}  (6)

By integrating differential equations [2] of the forms

\frac{∂x^{′μ}}{∂ε} |_{ε=0} = ξ^{μ}(x^{α})  (7)

we find finite symmetry transformations. Clearly, there are 3 spatial translations and 3 spatial rotations of the form exactly the same as in the flat Minkowski space-time. There is one time translation accompanied by a spatial dilatation given by

\begin{matrix}η → η′ = (A + 1)η \\ x → x′ = (A + 1)x \\ y → y′ = (A + 1)y \\ z → z′ = (A + 1)z \end{matrix}  (8)

where A = const. Remaining 3 symmetries are the analogues of Lorentz boosts. The boost in x-direction is given by

\begin{matrix}η → η′ = \frac{η}{sB^{2} + 2xB + 1} \\ x → x′ = \frac{sB + x}{sB^{2} + 2xB + 1} \\ y → y′ = \frac{y}{sB^{2} + 2xB + 1} \\ z → z′ = \frac{z}{sB^{2} + 2xB + 1} \end{matrix}  (9)

where B = const. The boosts in y , z-directions can be obtained from (9) simply by a cyclic permutation x → y → z → x.

3. Symmetries of anti-de Sitter space-time

The anti-de Sitter manifold can be visualized as a hyperboloid

−z_{0}^{2} − z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + z_{4}^{2} = −β^{2},  (10)

β = \sqrt{−\frac{λ}{3}}, embedded in a flat five-dimensional space-time

ds^{2} = −dz_{0}^{2} − dz_{1}^{2} + dz_{2}^{2} + dz_{3}^{2} + dz_{4}^{2} .  (11)

We use coordinates (η, x, y, z) covering the entire hyperboloid given by

\begin{matrix}z_{0} = \frac{1}{2x}(β^{2} + s) \\ z_{1} = β \frac{η}{x} \\ z_{2} = β \frac{y}{x} \\ z_{3} = β \frac{z}{x} \\ z_{4} = \frac{1}{2x}(β^{2} − s) \end{matrix}  (12)

where η, x, y, z ∈ (−∞, +∞). In the global coordintes (12) anti-de Sitter metric can be written as

ds^{2} = \frac{β^{2}}{x^{2}} (−dη^{2} + dx^{2} + dy^{2} + dz^{2}) .  (13)

This conformally flat metric is related to the ‘standard’ form (see, e.g. [1]) of anti-de Sitter metric

ds^{2} = − cosh^{2} r dt^{2} + β^{2}dr^{2} + β^{2} sinh^{2} r (dϑ^{2} + sin^{2} ϑ dφ^{2})  (14)

by a transformation

η = X cosh r cos(\frac{t}{β})

x = X

y = X sinh r cos ϑ z = X sinh r sin ϑ cos φ

(15)

where X = β/(cosh r sin(\frac{t}{β}) + sinh r sin ϑ sin φ).

In the coordinates (12) Killing vectors have the following form

\begin{matrix}ξ_{(1)}^{μ} = (1, 0, 0, 0) \\ ξ_{(4)}^{μ} = (y, 0, η, 0) \\ ξ_{(5)}^{μ} = (z, 0, 0, η) \\ ξ_{(8)}^{μ} = 2η(η + \frac{s}{2η} , x, y, z), \end{matrix}  (16)

six remaining vectors having the same form as in (6). Clearly, there is one time translation (in η direction), two spatial translations (in y , z-directions), two ‘Minkowskian’ boosts (in η − y and η − z planes) and one spatial rotation (in y − z plane). There is also the symmetry given by (8). Remaining 3 symmetries are given by

\begin{matrix}η → η′ = \frac{sC + η}{−sC^{2} − 2ηC + 1} \\ x → x′ = \frac{x}{−sC^{2} − 2ηC + 1} \\ y → y′ = \frac{y}{−sC^{2} − 2ηC + 1} \\ z → z′ = \frac{z}{−sC^{2} − 2ηC + 1} \end{matrix}  (17)

where C = const,

\begin{matrix}η → η′ = \frac{η}{sD^{2} + 2yD + 1} \\ x → x′ = \frac{x}{sD^{2} + 2yD + 1} \\ y → y′ = \frac{sD + y}{sD^{2} + 2yD + 1} \\ z → z′ = \frac{z}{sD^{2} + 2yD + 1} \end{matrix}  (18)

where D = const and analogously (18) with y ↔ z. This completes the list of all isometries of de Sitter and anti-de Sitter space-times.

4. Acknowledgments

I would like to thank Prof.Jiří Bičák and Dr.Hans-Jürgen Schmidt for useful discussions.

References

[1] Hawking S.W. and Ellis G.F.R.: The Large Scale Structure of Space-time, Cambridge University Press,Cambridge,1973.

[2] Robertson H.P. and Noonan T.W.:Relativity and Cosmology, W.B.Saunders ,Philadelphia,1969.

[3] Schmidt H.-J.:Potsdam preprint PRE-ZIAP 91-04,1991;submitted to Fortschr.Phys.