New geometry with the Poincar\'e symmetry
{Chao-Guang Huang}$^{1}$
The new 4d geometry with the Poincar\'e symmetry is presented and its structure is analyzed. The new geometry can be regarded as the Poincar\'e-invariant solution of the vacuum Einstein field equations with a negative cosmological constant and provides a static cosmological space-time with a Lobachesky space. The motion of free particles in the space-time is discussed.
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\begin{document}%\large\bf

%\preprint{}

%\bigskip

\title{New geometry with the Poincar\'e symmetry}%

\author{{Chao-Guang Huang}$^{1}$} \email{huangcg@ihep.ac.cn}
\author{{Yu Tian}$^2$}\email{ytian@gucas.ac.cn}
\author{{Xiao-Ning Wu}$^{3}$}\email{wuxn@amss.ac.cn}
\author{{Zhan Xu}$^{4}$}\email{zx-dmp@tsinghu.edu.cn}
\author{{Bin Zhou}$^5$} \email{zhoub@bnu.edu.cn}


%\affiliation{%
%${}^1$ CCAST (World Laboratory), P.O. Box 8730, Beijing 100080,
%China,}

\affiliation{%
${}^1$Institute of High Energy Physics, and
Theoretical Physics Center for Science Facilities, Chinese Academy of
Sciences, Beijing 100049, China}

\affiliation{%
${}^2$Graduate University of Chinese Academy of
Sciences, Beijing 100049, China}

\affiliation{%
${}^3$Institute of Mathematics, Academy of Mathematics and System Sciences,
Chinese Academy of Sciences, Beijing
100190, China}

\affiliation{%
${}^4$Department of Physics, Tsinghua University, Beijing
100084, China}

\affiliation{%
${}^5$Department of Physics, Beijing Normal University, Beijing
100875, China}

%\date{September  2009}

%\vskip 15mm
%\bigskip

\begin{abstract}
The new 4d geometry with the Poincar\'e symmetry is presented
and its structure is analyzed.  The new geometry can be regarded
as the Poincar\'e-invariant solution of the vacuum Einstein field equations
with a negative cosmological constant and provides
a static cosmological space-time with a Lobachesky space.  The motion
of free particles in the space-time is discussed.
\end{abstract}

\pacs{
%03.30.+p, % Special relativity
04.20.-q, % Classical general relativity
04.90.+e, % Other topics in general relativity and gravitation
04.20.Jb, % Exact solutions to Einstein field equations
02.20.Sv  % Lie algebra of Lie group
%02.90.+p % Other topics in mathematical methods in physics
}



\maketitle


%\tableofcontents
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%%%%%%%%%%%%%%%%%%%%%%%%       Section 1        %%%%%%%%%%%%%%%%%%%%%%%%
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{\it Introduction.} --- The Poincar\'e symmetry is the foundation of Einstein's
special relativity, relativistic field
theories in Minkowski space-time, particle physics, as well as the Poincar\'e gauge
theories of gravity, etc.  It is well known
that conventionally only the Minkowski space-time is invariant under
global Poincar\'e transformations.

Recently, it has been found that as kinematic symmetry
\cite{BLL} there is another Poincar\'e symmetry that preserves the
Minkowski lightcone at origin \cite{GWZ,GHWZ}.

The aim of the letter is to present a new nontrivial 4d geometry with
the Poincar\'e symmetry and to explore its geometric structure.  We show
that the new geometry is the Poincar\'e-invariant solution of the degenerate
extension of the vacuum Einstein
field equations with a negative cosmological constant and provides a
static cosmological space-time with a Lobachesky space. The motion
of free particles in the space-time is also discussed.


{\it The Poincar\'e symmetry.} --- In the algebraic point of
view, an algebra is said to be Poincar\'e one if (1) it is
isomorphic to $\frak{iso}(1,3)$ algebra, (2) the unique Abelean
ideal of the $\frak{iso}(1,3)$ algebra
is regarded as a translation sub-algebra and is divided into the time
translation and space translations as a 1d and a 3d representation,
respectively, of $\mathfrak{so}(3)$ sub-algebra of the
$\frak{so}(1,3)$ sub-algebra and (3) the algebra is invariant under the
suitably defined parity and time-reversal operation \cite{BLL}.

Clearly, in the above algebraic sense, the generator set
$(H, {\bf P}_i, {\bf K}_i, {\bf J}_i)$ with %
\be\label{p}
\begin{array}{l}
H = \r_t ,\ {\bf P}_i = \r_i, \smallskip \\
{\bf K}_i = (t \r_i +\d 1 {c^2} x^i \r_t), \ %
{\bf J}_i = \eps_{ij}^{\phantom{ij} k} x^j\r_k . %&
\end{array}
\ee
spans a Poincar\'e algebra ${\frak p}$
of the ordinary Poincar\'e transformations group%
\be
  {x'}^\mu = L^\mu_{\ \nu}x^\nu+ l a^\mu, \qquad L\in SO(1,3),
\ee%
where $l$ is a constant having dimension of length, and $a^\mu$
are dimensionless parameters.

Remarkably, there exists another generator set
$(H',{\bf P} '_i, {\bf K}_i,{\bf J})$,
\be\label{np2}
\begin{array}{l}
H'  = - c^2 l^{-2} t  x^\ka \r_\ka  \ (=c P_0'), \quad
 {\bf P}'_i  = l^{-2} x^i  x^\ka \r_\ka  \smallskip\\
{\bf K}_i =   (t \r_i +\d 1 {c^2} x^i \r_t), \quad
 {\bf J}_i = \eps_{ij}^{\ \ k} x^j\r_k ,
\end{array}
\ee %
which also spans a Poincar\'e algebra in the above sense \cite{GWZ, GHWZ},
\begin{equation}
\begin{array}{l}
\ [ H', {\bf P}'_i ]=0,  \quad  [ {\bf P}'_i, {\bf P}'_j] = 0, \quad
[H', {\bf K}_i]= {\bf P}'_i, \smallskip \\
\ [ {\bf P}'_i, {\bf K}_j]= \d 1 {c^2} H'\delta_{ij}, \quad
[ {\bf K}_i, {\bf K}_j ] =  \d 1 {c^2}\eps_{ij}^{\ \ k}{\bf J}_k, \smallskip \\
\ [{\bf J}_i, {\bf J}_j]= -\eps_{ij}^{\ \ k}{\bf J}_k ,
\end{array}
\end{equation}
where the indexes are lowered and raised by $\eta_{\mu\nu}={\rm diag}(1,
-1,-1,-1)$.
They are the generators of the Poincar\'e transformation group
\be\label{CoordTrans}
{x'}^\mu = \d {L^\mu_{\ \nu}x^\nu}{1+l^{-1}b_\mu x^\mu}
\ee
with dimensionless parameters $b_\mu$.

It should be noted that the new generator set does not generate the isometry of
the Minkowski space-time.  Instead, it preserves the light cone
at the origin in the Minkowski space-time.
%Minkowski space-time.
Therefore, the Poincar\'e group  with the new generator
set is referred to as the second Poincar\'e group\footnote{From a
different approach, Aldrovandi {\it et al} also mention that
there exists a second Poincar\'e group
\cite{AP,ABCP}. }
and denoted as $P_2$ in the following.  $H'$ and $P'_i$ are called
the pseudo-translations \cite{GWZ, GHWZ} or ${\frak p}_2$-translations.


{\it New geometry with the Poincar\'e symmetry.} --- It is
natural to ask: is there a 4d
geometry which is invariant under the second Poincar\'e
transformations\footnote{The first attempt to find the metric for the second Poincar\'e group
is made by Aldrovandi {\it et al} \cite{AP, ABCP}.  But, they give identically
vanishing metric (i.e. $g_{\mu\nu}\equiv 0$).}?


The following no-go theorem answers the question in conventional way.
{\it There is no tensor field $\vect{g} = g_{\mu\nu} \, dx^\mu \otimes dx^\nu$
with the following three conditions satisfied simultaneously:
(1) $\vect{g}$ is smooth;
(2) $\vect{g}$ is non-degenerate everywhere;
(3) $\vect{g}$ is invariant under the $\mathfrak{p}_2$-translations.}

It implies that the $P_2$-invariant metric on the 4-d underlying manifold must be
degenerate if it exists. For a degenerate geometry, more geometric information
should be assigned.


It can be checked that $(M,\vect{g},\vect{h},\del)$ is invariant under $P_2$
transformations, where $\vect{g}$ is a 4d type-(0,2) degenerate symmetric
tensor field
\be \label{g1}%
\vect{g} &=& g_{\mu\nu}dx^\mu \otimes dx^\nu \nno \\
&=&  \frac {l^2} {(x \cdot x)^2}(\eta_{\mu\nu}\eta_{\rho \tau}
-\eta_{\mu \rho}\eta_{\nu \tau})x^\rho x^\tau dx^\mu dx^\nu,
\ee
%
$\vect{h}$ is a  4d type-(2,0) degenerate symmetric tensor field
\be
\label{gp2inv} \vect{h}=h^{\mu\nu}\r_\mu \otimes \r_\nu =
l^{-4}(x\cdot x)x^\mu x^\nu\r_\mu \r_\nu
\ee%
with $x \cdot x=\eta_{\mu\nu} x^\mu x^\nu >0$,
and $\del$ is a connection compatible to ${\vect g}$ and ${\vect h}$, i.e.
\be
\del_\la \, g_{\mu\nu}= \r_\la g_{\mu\nu}-\Gamma^\ka_{\la\nu}g_{\mu\ka}
-\Gamma^\ka_{\mu\la}g_{\ka\nu} =0
\ee
and
\be
\del_\la \,
{h}^{\mu\nu}= \r_\la {h}^{\mu\nu}+ \Gamma^\nu_{\la\ka}{h}^{\mu\ka}
+\Gamma^\mu_{\la\ka}{h}^{\ka\nu}=0,
\ee
respectively, with connection coefficients,
\be
\label{inv-connectn}%
 \Gamma^\mu_{\ \nu\la} = - \d {(x_\nu
\dl^\mu_\la+\dl^\mu_\nu x_\la) }{x\cdot x}.
\ee
In other words, $\forall \vect\xi \in {\frak p}_2\subset TM$,
\be \label{eq:Lieg}%
&{\cal L}_{\vect \xi} \vect{g} = g_{\mu\nu,\la}\xi^\la+g_{\mu\la}\r_\nu\xi^\la
+g_{\la\nu}\r_\mu\xi^\la=0,&\\
\label{eq:Lieh} %
&{\cal L}_{\vect \xi} {\vect h}={h}^{\mu\nu}_{\ \ ,\la}\xi^\la -
h^{\mu\la}\r_\la\xi^\nu - h^{\la\nu}\r_\la\xi^\mu=0,&\\
\label{eq:LieGamma}%
& [{\cal L}_{\vect \xi}, \del ] =0,&
\ee
are valid simultaneously, or Eqs.(\ref{g1}), (\ref{gp2inv}), and
(\ref{inv-connectn}) are invariant under the coordinate transformation
(\ref{CoordTrans}) and its inverse transformation,
\be \label{InverseTrans}
x=\d {L^{-1} x'}{1-l^{-1}(b\cdot L^{-1}x')}=\d {L^{-1} x'}{1-l^{-1}(b'\cdot x')}.
\ee
The curvature tensor of the space-time is, by definition
\be
R^\si_{\ \mu \nu \rho}&=&\r_\nu\Gamma^\si_{\ \mu \rho}-
\r_\rho \Gamma^\si_{\ \mu  \nu} +
\Gamma^{\si}_{\ \tau \nu} \Gamma^\tau_{\ \mu \rho}-\Gamma^{\si}_{\ \tau \rho}
\Gamma^\tau_{\ \mu \nu}\nno \\
&=&l^{-2}(\dl^\si_\nu g_{\mu\rho}-\dl^\si_\rho g_{\mu\nu}).
\label{curvaturecomponents}
\ee
It is antisymmetric in the latter two indexes and satisfies the Ricci and
Bianchi identities. The Ricci curvature tensor is then
\be \label{riccicurv}
R_{\mu\nu}=R^\si_{\ \mu\nu\si}=  -3l^{-2}g_{\mu\nu}.
\ee
They are obviously invariant under $P_2$ transformations.
Eqs.(\ref{curvaturecomponents}) and
(\ref{riccicurv}) are similar to those of the maximum-symmetric space-times.

In order to see the manifold more transparently,
let us consider the coordinate transformations,
\be \label{Transf2AdS}
\begin{array}{l}
x^0=l^2\eta^{-1} \cosh (r/l) ,  \\
x^1=l^2\eta^{-1} \sinh (r/l) \sin \th \cos\phi ,\\
x^2=l^2\eta^{-1} \sinh (r/l) \sin \th \sin \phi ,\\
x^3=l^2\eta^{-1} \sinh (r/l) \cos \th .
\end{array}
\ee
Under the coordinate transformation, Eqs.(\ref{g1}), (\ref{gp2inv}), and
(\ref{inv-connectn}) become
\be \label{g3}
{\vect g} =-(dr^2 +l^{2}\sinh^2(r/l) d\Om_2^2),
\ee
\be\label{h1}
{\vect h}= (\r_\eta)^2,
\ee
\be \label{connect}\begin{array}{l}
{\bar \Gamma}^\eta_{ij} = -l^{-2}\eta g_{ij} \\
{\bar \Gamma}^r_{\th\th} =-l\sinh(r/l) \cosh(r/l), \quad
{\bar \Gamma}^r_{\phi\phi} ={\bar \Gamma}^r_{\th\th}\sin^2\th   \\
{\bar \Gamma}^\th_{r\th} ={\bar \Gamma}^\th_{\th r} ={\bar \Gamma}^\phi_{r\phi}
={\bar \Gamma}^\phi_{\phi r}
=\d 1 {l\tanh(r/l)} \\
{\bar \Gamma}^\th_{\phi\phi} =-\sin\th\cos\th, \quad
{\bar \Gamma}^\phi_{\th\phi}={\bar \Gamma}^\phi_{\phi\th}=\cot\th  \\
\mbox{others vanish.}
\end{array} %
\ee
respectively, where an over bar represents that the quantity takes the value
in the coordinate system $\{{\bar x}^0, {\bar x}^i\}=\{\eta, r,\th, \phi\}$.
Clearly, Eqs. (\ref{g3}) and (\ref{h1}) define the non-degenerate
3d metric ${\vect g}_3$ and 1d contravariant metric ${\vect h}_1$, respectively.
The above structures show that the manifold is locally
$\mathbb{R} \times \mathbb{H}_3 $, where 3d hyperboloid $\mathbb{H}_3 $
has unit `radius'.  Under the $P_2$-transformations (\ref{CoordTrans}),
points run on $\mathbb{R} \times \mathbb{H}_3 $.  The points satisfying
$1+l^{-1} b\,\cdot x =0$
will be transformed to infinity in the new coordinate system $x'$ under the
coordinate transformation (\ref{CoordTrans}).  In particular, when
$L^\mu_{\ \nu}=\dl^\mu_\nu$, they correspond to the points
being transformed from $(\eta,r,\th,\phi)$ to $(0, r, \th, \phi)$.
Eqs(\ref{g3}), (\ref{h1}), (\ref{connect}) and the above transformation property show the
geometric structure
can be smoothly extended to $\eta=0$ and $\eta<0$.
Therefore, the manifold is $\mathbb{R} \times \mathbb{H}_3 $ globally.


The Ricci curvature (\ref{riccicurv}) reads in the new coordinate system
\be
\bar R_{\mu\nu} = 3l^{-2} {\rm diag}(0, 1, \sinh^2(r/l), \sinh^2(r/l) \sin^2\th).
\ee
Compared
with the reduced vacuum Einstein field equation with a cosmological constant
\be \label{Eeq}
R_{\mu\nu}=\La g_{\mu\nu},
\ee
the new geometry may be regarded as the $P_2$-invariant solution of the
vacuum Einstein field equation with the cosmological constant $-3l^{-2}$.
The solution provides a static cosmological space-time with a Lobachevski space.

Recall that the Ashtekar extension of general relativity admits the degenerate
geometries \cite{Ash,DG}, which satisfy the gauge, vector, and scalar constraints.
For the solution, the non-zero components of the densitized triad and Ashtekar
connection\footnote{In this case, the
extrinsic curvature has no good definition. } $A^i_a=\Ga^i_a$  read
\be\begin{array}{l}
\tilde E^r_1 =l^2 \sinh^2(r/l)\sin\th,  \\
\tilde E^\th_2 = l\sinh(r/l)\sin\th,  \\
\tilde E^\phi_3 =l\sinh(r/l),
\end{array}
\ee
and
\be\begin{array}{l}
\Ga^3_\th=\cosh(r/l), \\
\Ga_\phi^1=\cos\th,  \\
\Ga_\phi^2=-\cosh(r/l)\sin\th,
\end{array}
\ee
respectively.  The constraints are
\be
&&{\cal D}_a\tilde E^a_i=\r_a\tilde E^a_i-\eps_{ij}^{\ \ k}\Ga_a^j\tilde E^a_k=0  \\
&&F_{ab}^i\tilde E^a_i=0,  \\
&&\eps_{i}^{\ jk}F^i_{ab}\tilde E^a_j \tilde E^b_k=l^{-2}\eps^{ijk}\utilde{\eta}_{abc}\tilde E^a_i
\tilde E^b_j \tilde E^c_k,
\ee
where
\be
F^i_{ab}=2\r_{[a} \Ga_{b]}^i-\eps^{i}_{\ jk}\Ga_a^j \Ga_b^k.
\ee
Remember that the indexes are lowered and raised by $\eta_{\mu\nu}
={\rm diag}(1,-1,-1,-1)$ and thus $-\dl_{ij}$.  It again corresponds to the
solution with $\La=-3l^{-2}$ \cite{CC}.


Furthermore, the 4d volume element on the manifold, defined by
\be
\vect{\eps}= l^2\sinh^2(r/l) \sin\th d\eta\wedge dr\wedge d\th \wedge d\phi,
\ee
is invariant under the $P_2$-transformations, so the manifold is orientable.
Moreover, the \omits{invariant }tensor ${\vect h}$ defines an
invariant vector field $\partial_{\eta}$ which is regular on
the whole manifold. Compared with the Newton-Cartan case, it gives an
absolute time direction and, therefore, the space-time is obviously time
orientable.

Finally, one may prove that {\it the obtained covariant
degenerate metric $\vect{g}$, contravariant degenerate metric
$\vect{h}$ and the connection $\nabla$ are unique, nontrivial,
compatible ones which are invariant under the $\mathfrak{p}_2$-transfomrations}.




{\it The motion for free particles.} --- If the motion for
free particles is still determined by the
geodesic equation%
\be
\d {d^2 x^\mu}{d\la^2}+\Gamma^\mu_{\
\nu\la}\d{dx^\nu}{d\la}\d{dx^\la}{d\la}=0,
\ee
as usual, it gives rise to the `uniform rectilinear' motion%
\be
\label{gim}
 x^i= a^i x^0 + lb^i
\ee
with dimensionless constants $a^i$ and $b^i$, in which $x^0$ and
$x^i$ are regarded as the `temporal' and `spatial' coordinates,
respectively.  In the $\mathbb{H}_3$ space,
one may introduce the Beltrami coordinates
$z^i=l {x^i}/{x^0}$.
Then, Eq.(\ref{gim}) reads
\be
z^i= \d {b^i} {\cosh (r/l)} \eta + la^i.
\ee
When $r\ll l$, it reduces to $z^i= b^i \eta + la^i$.  This is a
uniform rectilinear motion, because $\eta$ and $z^i$ are time and
spatial coordinates, respectively, in the conventional sense.


The `uniform rectilinear' motion (\ref{gim}) can also be obtained from the
Lagrangian
\be \label{Lagrangian}
L= \frac {mlc} {x \cdot x}\sqrt{(\eta_{\mu\nu}\eta_{\eta \tau}
-\eta_{\mu \eta}\eta_{\nu \tau}
)x^\eta x^\tau \dot x^\mu \dot x^\nu}.
\ee
The Euler-Lagrangian equation is equivalent to
\be
[(x\cdot x)(\dot x\cdot \dot x)-(x\cdot \dot x)^2 ]
\ddot x_\ka\ +(\dot x\cdot \ddot x)[(x\cdot \dot x) x_\ka -(x\cdot x)
\dot x_\ka] \nno\\
+(x\cdot \ddot x)[(x\cdot \dot x)\dot x_\ka
-(\dot x\cdot \dot x)x_\ka]=0. \qquad \qquad \qquad\nno
\ee
The nonzero determinant of its coefficients for $\ddot x$
requires $\ddot x_\ka= 0.$

It should be noted that even though $c$ is finite and invariant in the
theory, it does not serves as a limit speed.  It only appears in
$x^0$- or $\eta$-expression to ensure their dimension to be $L$.
Similarly, $l$ is not any limit length though it is finite and invariant.




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%%%%%%%%%%%%%%%%%%%%%%%%       Section 4        %%%%%%%%%%%%%%%%%%%%%%%%
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{\it Concluding remarks.} --- In addition to the ordinary Poincar\'e
group preserving the metric of Minkowski space-time,  there is the second
Poincar\'e symmetry preserving the geometric structure $(M,\vect{g},
\vect{h},\del)$. A remarkable feature for the new geometry
$(M,\vect{g}, \vect{h},\del)$ is that it is somewhat like the
Galilei and Carroll space-times in which the metrics of space and
time split, i.e. one may introduce the non-degenerate metrics for
the space and time separately. Therefore, the new kinematics is
non-relativistic-like in this sense.

From the algebraic point of view, ${\frak p}_2$ algebra has the
$SO(1,3)$ isotropy in which ${\bf K}_i$ serve as the boost and $(H', {\bf
P}'_i)$ as the pseudo-translation generators.
The geometrical analysis, however, shows that the generators may have very
different meaning from those appear in the algebra.  In fact, now
the space `translations' on $\mathbb{H}_3$ are generated by
${\vect{\cal P}}_i=\r_{z^i}+ l^{-2}z_iz^j\r_{z^j}=\frac {c}{l}{\bf K}_i$
(where $z_i=-z^i$) but not ${\bf P}'_i$, while the boost generators
${\vect{\cal K}}_i=-l^{-1}z_i\eta\r_\eta= \frac l c {\bf P}'_i$.  Therefore,
the `translations'
in the space-time with ${\frak p}_2$ symmetry are no longer generated by the
Abelean ideal in
the algebra and $SO(1,3)$ subgroup is no longer the
isotropy of the space-time at each point. The isotropic group of the
manifold is now $ISO(3)$ and the space-time is thus
the homogeneous space
\be
M=ISO(1,3)/ISO(3).
\ee
It is a new
geometry satisfying all three assumptions in Ref. \cite{BLL}.

On the new space-time, the motions of free particles can be well
defined.  The mechanics, field theories and even gravity on the
space-time need to be investigated in order to clarify the
application of the new space-time.  In fact, it has been shown that the
$P_2$-invariant electromagnetic field equations can be set up in
this space-time.  We shall explore it elsewhere.

It is straightforward to generalize the 4d degenerate geometries to
higher dimensional degenerate geometries.  On the other hand, in the
higher dimensional theories, there may also be the second Poincar\'e group
as its subgroup of symmetry.  Hence, the geometric structure may appear
in a higher dimension.

Finally, the transformations (\ref{CoordTrans}) are in fact the subset of
the linear
fractional transformations with common denominator.
Thus, it is closely related to the principle of
relativity with two invariant constants \cite{GWZ,GHWZ}.  The relation
of new geometry of the second Poincare group
with these issues and the
physical applications are still need to be explored further.


\begin{acknowledgments}\vskip -4mm
We are grateful to Prof. Guo, who gives us many valuable suggestions.
We would also like to thank Prof. Z.-N. Hu, W. T. Ni and H.-X. Yang and
Dr. H.-t. Wu for their helpful discussion.
This work is supported by NSFC under Grant Nos. 10775140, 10605005,
10705048, 10731080,
 Knowledge
Innovation Funds of CAS (KJCX3-SYW-S03), and the President Fund of
GUCAS.
\end{acknowledgments}



\begin{thebibliography}{99}

\bibitem{BLL} H. Bacry and J.-M. L\'evy-Leblond, Possible kinematics, J. Math. Phys.
{\bf 9} 1605 (1968).

\bibitem{GWZ}  H.-Y. Guo, H.-T. Wu, and B. Zhou, Phys. Lett. B {\bf 670} 437 (2009), arXiv:0809.3562.

\bibitem{GHWZ} H.-Y. Guo, C.-G. Huang, H.-T. Wu and B. Zhou, The principle of relativity,
kinematics and algebraic relations, arXiv: 0812.0871.

\bibitem{AP} R. Aldrovandi and J. G. Pereira, A second Poincar\'e Group,
arXiv:gr-qc/9809061.


\bibitem{ABCP} R. Aldrovandi, A. L. Barbosa, M. Cal\c{c}ada, and J. G. Pereira,
%Kinematics of a spacetime with an infinite cosmological constant,
Found. Phys. {\bf 33} 613 (2003).

\bibitem{Ash}A. Ashtekar, Phys. Rev. Lett. {\bf 57}, 2244 (1986); Phys. Rev.
D {\bf 36}, 1587 (1987).

\bibitem{DG}T. Jacobson and L. Smolin, Nucl. Phys. B {\bf 299}, 295 (1988); I. Bentsson,
Int. J. Mod. Phys. A {\bf 4}, 5527 (1989); Class. Quantum Grav. {\bf 8}, 1847 (1991);
M. Varadarajan, Class. Quantum Grav. {\bf 8}, L235 (1991).

\bibitem{CC} See, for example, L. Smolin, Phys. Lett. A 306,  66 (2002).


\end{thebibliography}




%\begin{widetext}


%\end{widetext}
\end{document}

0909.2773.extracted_bib
\begin{thebibliography}{99}

\bibitem{BLL} H. Bacry and J.-M. L\'evy-Leblond, Possible kinematics, J. Math. Phys.
{\bf 9} 1605 (1968).

\bibitem{GWZ}  H.-Y. Guo, H.-T. Wu, and B. Zhou, Phys. Lett. B {\bf 670} 437 (2009), arXiv:0809.3562.

\bibitem{GHWZ} H.-Y. Guo, C.-G. Huang, H.-T. Wu and B. Zhou, The principle of relativity,
kinematics and algebraic relations, arXiv: 0812.0871.

\bibitem{AP} R. Aldrovandi and J. G. Pereira, A second Poincar\'e Group,
arXiv:gr-qc/9809061.


\bibitem{ABCP} R. Aldrovandi, A. L. Barbosa, M. Cal\c{c}ada, and J. G. Pereira,
%Kinematics of a spacetime with an infinite cosmological constant,
Found. Phys. {\bf 33} 613 (2003).

\bibitem{Ash}A. Ashtekar, Phys. Rev. Lett. {\bf 57}, 2244 (1986); Phys. Rev.
D {\bf 36}, 1587 (1987).

\bibitem{DG}T. Jacobson and L. Smolin, Nucl. Phys. B {\bf 299}, 295 (1988); I. Bentsson,
Int. J. Mod. Phys. A {\bf 4}, 5527 (1989); Class. Quantum Grav. {\bf 8}, 1847 (1991);
M. Varadarajan, Class. Quantum Grav. {\bf 8}, L235 (1991).

\bibitem{CC} See, for example, L. Smolin, Phys. Lett. A 306,  66 (2002).


\end{thebibliography}