% *** ISOMETRIE (ANTI) DE SITTERA ***          *** 5.2.1992 ***
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% final version for pro Czech.J.Phys. (1993)

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\def \citB{s B^2+2x B+1}
\def \citC{-s C^2-2\eta C+1}
\def \citD{s D^2+2y D+1}

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\vsize=8.8truein
{\bf
\centerline{LORENTZ BOOSTS IN DE SITTER AND ANTI-DE SITTER SPACE-TIMES}}
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\centerline {Ji\v r\'\i\   Podolsk\'y}
\centerline {Department of Theoretical Physics, Charles University, Prague}
\centerline {V Hole\v sovi\v ck\'ach 2, 180 00 Praha 8, Czech Republic}
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\midinsert
\narrower\narrower
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.
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\bigskip
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\centerline{{\bf 1. Introduction}}

 \bigskip

 It is well-known that the maximal number of isometries
 of a four-dimensional space-time 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 $\lambda$. The constant curvature $R$ is related to
$\lambda$ by $R=4\lambda$.

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\centerline{{\bf 2. Symmetries of de Sitter space-time}}

\bigskip
De Sitter manifold can be visualized as a hyperboloid
$$-z_0^2+z_1^2+z_2^2+z_3^2+z_4^2=\alpha^2,  \eqno (1)$$
$\alpha=\sqrt{\lambda\over3}$ , embedded in a flat five-dimensional
space-time
$$ds^2=-dz_0^2+dz_1^2+dz_2^2+dz_3^2+dz_4^2\>.   \eqno (2)$$
We use coordinates $(\eta ,x,y,z)$ covering the entire
hyperboloid given by
$$\eqalign {
&z_0={1\over{2\eta}}(\alpha^2 + s)\cr
&z_1=\alpha{x\over\eta}\cr
&z_2=\alpha{y\over\eta}\cr
&z_3=\alpha{z\over\eta}\cr
&z_4={1\over{2\eta}}(\alpha^2 - s)\cr} \eqno (3)$$
where
$s=-\eta^2+x^2+y^2+z^2$  and $\eta,x,y,z\in{(-\infty ,+\infty)}$.
In the global coordinates $(3)$  de Sitter metric can be written as
$$ds^2={\alpha^2\over{\eta^2}}(-d\eta^2 + dx^2 + dy^2 + dz^2)\>.  \eqno (4)$$
This conformally flat metric (see, e.g. [3]) is related to the \lq standard\rq\
form of de Sitter metric
$$ds^2=-dt^2 + \exp {(2{t\over\alpha})}(dx^2+dy^2+dz^2)  \eqno (5)$$
by a simple transformation $\eta=\alpha \exp{(-t/ \alpha)}$. 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
 $\eta=-\alpha \exp{(-t^\ast / \alpha)}$.
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
\lq standard\rq\  coordinates used in $(5)$ and thus have more complicated
forms):
$$\eqalign{
&\xi^\mu_{(1)} =(0,1,0,0)  \cr
&\xi^\mu_{(2)} =(0,0,1,0)  \cr
&\xi^\mu_{(3)} =(0,0,0,1)  \cr
&\xi^\mu_{(4)} =(0,y,-x,0) \cr
&\xi^\mu_{(5)} =(0,z,0,-x) \cr
&\xi^\mu_{(6)} =(0,0,z,-y) \cr
&\xi^\mu_{(7)} =(\eta,x,y,z) \cr
&\xi^\mu_{(8)} =-2x(\eta,x-{s\over{2x}},y,z)  \cr
&\xi^\mu_{(9)} =-2y(\eta,x,y-{s\over{2y}},z)  \cr
&\xi^\mu_{(10)}=-2z(\eta,x,y,z-{s\over{2z}})\>.  \cr}  \eqno (6)$$

By integrating differential equations [2] of the forms
$${{\partial x'^\mu}\over {\partial \varepsilon}}\Bigl|_{\varepsilon=0}
   \  =\  \xi^\mu (x^\alpha)   \eqno (7)$$
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
$$\eqalign {
&\eta \rightarrow \eta '=(A+1)\eta \cr
&x    \rightarrow    x '=(A+1)x    \cr
&y    \rightarrow    y '=(A+1)y    \cr
&z    \rightarrow    z '=(A+1)z    \cr} \eqno (8)$$
where $A=const$. Remaining 3 symmetries are the analogues of
Lorentz boosts. The boost in $x$-direction is given by
$$\eqalign {
&\eta \rightarrow \eta '={\eta \over \citB} \cr
&x    \rightarrow    x '={{sB+x}\over \citB} \cr
&y    \rightarrow    y '={y\over \citB} \cr
&z    \rightarrow    z '={z\over \citB} \cr} \eqno (9)$$
where $B=const$. The boosts in  $y$ , $z$-directions can be obtained from $(9)$
simply by a cyclic permutation $x \rightarrow y \rightarrow z \rightarrow x$.

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\centerline{{\bf 3. Symmetries of anti-de Sitter space-time}}

\bigskip
The anti-de Sitter manifold can be visualized as a hyperboloid
$$-z_0^2-z_1^2+z_2^2+z_3^2+z_4^2=-\beta ^2, \eqno (10)$$
$\beta=\sqrt {-{\lambda \over 3}}$, embedded in a flat
five-dimensional space-time
$$ds^2=-dz_0^2-dz_1^2+dz_2^2+dz_3^2+dz_4^2\>.   \eqno (11)$$
We use coordinates $(\eta,x,y,z)$ covering the entire
hyperboloid given by
$$\eqalign{
&z_0={1\over{2x}}(\beta^2+s) \cr
&z_1=\beta{\eta\over x}      \cr
&z_2=\beta{y\over x}         \cr
&z_3=\beta{z\over x}         \cr
&z_4={1\over{2x}}(\beta^2-s) \cr}  \eqno (12)$$
where $\eta,x,y,z \in (-\infty,+\infty)$. In the global coordintes $(12)$
anti-de Sitter metric can be written as
$$ds^2={\beta^2 \over x^2}(-d\eta^2+dx^2+dy^2+dz^2)\>.  \eqno (13)$$
This conformally flat metric is related to the \lq standard\rq\
form (see, e.g. [1]) of anti-de Sitter metric
$$ds^2=-\cosh^2 r\>dt^2 + \beta^2dr^2+\beta^2\sinh^2 r\> (d\vartheta^2+\sin^2\vartheta\> d\varphi^2)  \eqno (14)$$
by a transformation
$$\eqalign{
&\eta=X\cosh r\>\cos({t\over{\beta}}) \cr
&x=X \cr
&y=X\sinh r\>\cos\vartheta \cr
&z=X\sinh r\>\sin\vartheta\>\cos\varphi \cr}\eqno(15)$$
where $X={\beta/{(\cosh r\>\sin({t\over{\beta}})+\sinh r\>\sin\vartheta\>\sin\varphi)}}.$



In the coordinates (12) Killing vectors have the following form
$$\eqalign {
&\xi^\mu_{(1)} =(1,0,0,0)   \cr
&\xi^\mu_{(4)} =(y,0,\eta,0)   \cr
&\xi^\mu_{(5)} =(z,0,0,\eta)   \cr
&\xi^\mu_{(8)} =2\eta (\eta+{s\over{2\eta}},x,y,z),}  \eqno (16)$$
six remaining vectors having the same form as in (6).
Clearly, there is one time translation (in $\eta$ direction),
two spatial translations (in $y$ , $z$-directions),
two \lq Minkowskian\rq \  boosts
(in $\eta-y$ and $\eta-z$ planes) and one spatial rotation (in $y-z$ plane).
There is also the symmetry given
by $(8)$. Remaining 3 symmetries are given by
$$\eqalign{
&\eta \rightarrow \eta '={{sC+\eta}\over \citC} \cr
&x    \rightarrow     x'={x\over \citC} \cr
&y    \rightarrow     y'={y\over \citC} \cr
&z    \rightarrow     z'={z\over \citC} \cr}  \eqno (17)$$
where $C=const$,
$$\eqalign{
&\eta \rightarrow \eta '={\eta \over \citD}   \cr
&x    \rightarrow     x'={x\over \citD}       \cr
&y    \rightarrow     y'={{sD+y}\over \citD} \cr
&z    \rightarrow     z'={z\over \citD}   \cr} \eqno (18)$$
where $D=const$ and analogously $(18)$ with $y\leftrightarrow z$.
This completes the list of all isometries of de Sitter and
anti-de Sitter space-times.
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\centerline{{\bf 4. Acknowledgments}}
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I would like to thank Prof.Ji\v r\' \i \ Bi\v c\' ak and  Dr.Hans-J\" urgen
Schmidt for useful discussions.

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\centerline{References}

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\noindent
\item{[1]} Hawking S.W. and Ellis G.F.R.: {\it The Large Scale Structure of
    Space-time}, Cambridge University Press,Cambridge,1973.
\item{[2]} Robertson H.P. and Noonan T.W.:{\it Relativity and Cosmology},
    W.B.Saunders ,Phila\-delphia,1969.
\item{[3]} Schmidt H.-J.:{\it Potsdam preprint} PRE-ZIAP 91-04,1991;submitted to
    Fortschr.Phys.
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