16 Kinematics and dynamics of an expanding universe
the metric in (1.30) becomes
dl^2 =dr′^2
1 −(r′^2 /a^2 )+r′^2 dφ^2. (1.32)The limita^2 →∞corresponds to a (flat) plane. We can also formally takea^2
to be negative and then metric (1.32) describes a homogeneous, isotropic two-
dimensional space with constant negative curvature, known as Lobachevski space.
Unlike the flat plane or the two-dimensional sphere, Lobachevski space cannot be
embedded in Euclidean three-dimensional space because the radius of the “sphere”
ais imaginary (this is why this space is called a pseudo-sphere or hyperbolic
space). Of course, this does not mean that this space cannot exist. Any curved
space can be described entirely in terms of its internal geometry without referring
to its embedding.
Problem 1.8Lobachevski space can be visualized as a hyperboloid in Lorentzian
three-dimensional space (Figure 1.5). Verify that the embedding of the surface
x^2 +y^2 −z^2 =−a^2 , wherea^2 is positive, in the space with metricdl^2 =dx^2 +
dy^2 −dz^2 gives a Lobachevski space.
Introducing the rescaled coordinater=r′/√
|a^2 |, we can recast metric (1.32)
as
dl^2 =|a^2 |(
dr^2
1 −kr^2+r^2 dφ^2)
, (1.33)
wherek=+1 for the sphere (a^2 >0),k=−1 for the pseudo-sphere (a^2 <0) and
k=0 for the plane (two-dimensional flat space). In curved space,|a^2 |characterizes
the radius of curvature. In flat space, however, the normalization of|a^2 |does not
have any physical meaning and this factor can be absorbed by redefinition of the
coordinates. The generalization of the above consideration to three dimensions is
straightforward.
Fig. 1.5.