The most symmetric *vacuum* solution to Einstein's equation, of
course, is the flat spacetime. If we now add
the cosmological constant as the only source of curvature in Einstein's
equation, the resulting spacetime
is also highly symmetric and has an interesting geometrical
structure. In the case of a positive cosmological constant, this is the
*de Sitter manifold* and in the case of negative cosmological
constant, it is known as *anti-de Sitter manifold*. We shall now
discuss some features of the former, corresponding to the positive
cosmological constant.

To understand the geometrical structure of the de Sitter spacetime,
let us begin by noting that a spacetime with the source
*T*_{b}^{a} =
_{}
^{a}_{b}
must have 3-dimensional section which are homogeneous and
isotropic. This will lead us to the Einstein's equations for a
FRW universe with cosmological constant as source

(130) |

This equation can be solved with any of the following three forms of
(*k*, *a*(*t*)) pair. The first
pair is the spatially flat universe with
(*k* = 0, *a* = *e*^{H t}). The second corresponds to
spatially open universe with
(*k* = - 1, *a* = *H*^{-1}sinh *H t*) and the
third will be
(*k* = + 1, *a* = *H*^{-1}cosh *H t*). Of
these, the last pair gives a coordinate system
which covers the full de Sitter manifold. In fact, this is the metric on
a 4-dimensional hyperboloid, embedded in a 5 dimensional Minkowski space
with the metric

(131) |

The equation of the hyperboloid in 5-D space is

(132) |

We can introduce a parametric representation of the hyperbola with the four variables (, , , ) where

(133) |

This set, of course, satisfies (132). Using (131), we can compute the metric induced on the hyperboloid which - when expressed in terms of the four coordinates (, , , ) - is given by

(134) |

This is precisely the de Sitter manifold with closed spatial sections.

All the three forms of FRW universes with *k* = 0, ± 1 arise
by taking different cuts in this 4-dimensional hyperboloid embedded in
the 5-dimensional spacetime. Since two of these dimensions
(corresponding to the polar angles
and
) merely
go for a ride, it is more convenient (for visualization) to work with a
3-dimensional spacetime having the metric

(135) |

instead of the 5-dimensional metric (131). Every point in this 3-dimensional space corresponds to a 2-sphere whose coordinates and are suppressed for simplicity. The (1 + 1) de Sitter spacetime is the 2-dimensional hyperboloid [instead of the four dimensional hyperboloid of (132)] with the equation

(136) |

embedded in the 3-dimensional space with metric (135). The three different coordinate systems which are natural on this hyperboloid are the following:

*Closed Spatial Sections*: This is obtained by introducing the coordinates*t*=*H*^{-1}sinh(*H*);*x*=*H*^{-1}cosh(*H*) sin;*v*=*H*^{-1}cosh(*H*) cos on the hyperboloid, in terms of which the induced metric on the hyperboloid has the form(137) This is the two dimensional de Sitter space which is analogous to the 4-dimensional case described by (134).

*Open Spatial Sections*: These are obtained by using the coordinates*t*=*H*^{-1}sinh(*H*) cosh;*x*=*H*^{-1}sinh(*H*) sinh;*v*=*H*^{-1}cosh(*H*) on the hyperboloid in terms of which the induced metric on the hyperboloid has the form(138) *Flat Spatial Sections*: This corresponds to the choice*t*=*H*^{-1}sinh(*H*) + (*H*^{-1}/2)^{2}exp(*H*);*x*=*H*^{-1}cosh(*H*) - (*H*^{-1}/2)^{2}exp(*H*);*v*= exp(*H*) leading to the metric(139) This covers one half of the de Sitter hyperboloid bounded by the null rays

*t*+*x*= 0.

All these metrics have an apparent time dependence. But, in
the absence of any source other than cosmological constant, there is no
preferred notion of
time and the spacetime manifold cannot have any intrinsic time dependence.
This is indeed true, in spite of the expansion factor *a*(*t*)
ostensibly depending on time. The translation along the time direction
merely slides the point on the surface of the hyperboloid.
[This is obvious in the coordinates
(*k* = 0, *a*
*e*^{H t}) in which the time translation
*t*
*t* + merely
rescales the coordinates by
(exp *H* ).]

The time independence of the metric can be made explicit in another set of coordinates called `static coordinates'. To motivate these coordinates, let us note that a spacetime with only cosmological constant as the source is certainly static and possesses spherical symmetry. Hence we can also express the metric in the form

(140) |

where and
are functions of
*r*. The Einstein's equations for this metric has the solution
*e*^{} =
*e*^{-}
= (1 - *H*^{2} *r*^{2}) leading to

(141) |

This form of the metric makes the static nature apparent. This metric also describes a hyperboloid embedded in a higher dimensional flat space. For example, in the (1 + 1) case (with , suppressed) this metric can be obtained by the following parameterization of the hyperboloid in equation (136):

(142) |

The key feature of the manifold, revealed by
equation (141) is the existence of a horizon at *r* =
*H*^{-1}. It also shows that
*t* is a time-like coordinate only in the region *r* <
*H*^{-1}.

The structure of the metric is very similar to the Schwarzschild metric:

(143) |

Both the metrics (143) and (141) are spherically symmetric with
*g*_{00} = - (1/*g*_{11}).
Just as the Schwarzschild metric
has a horizon at *r* = 2*M* (indicated by
*g*_{00}
0,
*g*_{11}
),
the de Sitter metric also has a horizon at *r* =
*H*^{-1}. From the slope of the light cones
(*dt* / *dr*) = ± (1 - *H*^{2}
*r*^{2})^{-1} [corresponding to
*ds* = 0 = *d*
= *d*
in (142)] it is clear that
signals sent from the region *r* < *H*^{-1} cannot
go beyond the surface *r* = *H*^{-1}.

This feature, of course, is independent of the coordinate system used.
To see how the horizon in de Sitter universe arises in the
FRW coordinates, let us recall the equation governing the propagation of
light signals between the events (*t*_{1},
*r*_{1}) and (*t*, *r*):

(144) |

Consider a photon emitted by an observer at the origin at the present
epoch (*r*_{1} = 0, *t*_{1} =
*t*_{0}). The maximum *coordinate* distance
*x*_{H} reached by this photon as
*t*
is determined
by the equation

(145) |

If the integral on the right hand side diverges as
*t*
, then, in the same
limit, *x*_{H}
and
an observer can send signals to any event provided (s)he waits
for a sufficiently long time. But if the integral on the right hand side
converges to a finite value as
*t*
, then
there is a finite horizon radius beyond which
the observer's signals will not reach even if (s)he waits
for infinite time. In the de Sitter universe with *k* = 0 and
*a*(*t*) = *e*^{H t},
*x*_{H} = *H*^{-1}
*e*^{-H t0}; the corresponding maximum proper
distance up to which the signals can reach is
*r*_{H} = *a*(*t*_{0})
*x*_{H} = *H*^{-1}.
Thus we get the same result in any other coordinate system.

Since the result depends essentially on the behaviour of
*a*(*t*) as
*t*
, it will persist even
in the case of a universe containing
*both* non relativistic matter and cosmological constant. For example,
in our universe, we can ask what is the highest redshift
source from which we can ever receive a light signal, if the signal
was sent today. To compute this explicitly, consider a model with
_{NR} +
_{} =
1. Let us assume that light from an event at
(*r*_{H}, *z*_{H}) reaches *r* = 0 at
*z* = 0 giving

(146) |

If we take *r*_{H} to be the size of the horizon, then it
also follows that the light emitted today from this event will just
reach us at *t* = .
This gives

(147) |

Equating the two expressions, we get an implicit expression for
*z*_{H}. If
_{NR} = 0.3,
the limiting redshift is quite small:
*z*_{H}
1.8. This implies that sources with *z* > *z*_{H}
can never be influenced by light signals from us
in a model with cosmological constant
[286,
287].