458 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY
whereHis the Hubble constant. We infer then from (7.5.2) and (7.5.6) the ageTof the
Universe as
(7.5.7) T= 1. 4 × 1010 year.
As we approximatively regard the recession velocityvof remote galaxies as the speed of
light, i.e.v≃c, the age (7.5.7) is also considered as the radius of the Universe.
5.Structure of universes.A central topic in modern cosmology is to investigate both the
global topological and geometrical structure of the Universe. A fundamental principle of cos-
mology, called cosmological principle, states that ignoring local irregularities, the Universe
is homogeneous and isotropic.
The cosmological principle is compatible with the Hubble law: Physical Conclusion7.20,
and can be equivalently stated in the following form.
Theorem 7.21(Topological Structure of the Universe).The global topological structure of
the Universe is either S^3 orR^3.
6.Global geometrical structure.Based on the Cosmological Principle7.21, three physi-
cists A. Friedmann (1922), G. Lemaˆıtre (1927), H. P. Robertson (1935), and a mathematician
A. G. Walker (1936), independently derived the globally geometrical structure of the Uni-
verse.
Theorem 7.22(Geometrical Structure of the Universe).The Riemannian metric of 4D space-
time manifold satisfying the Cosmological Principle7.21takes the following form
(7.5.8) ds^2 =−c^2 dt^2 +R(t)
[
dr^2
1 −kr^2+r^2 (dθ^2 +sin^2 θdφ^2 )]
,
where R(t)is the scalar factor representing the radius of the Universe, which depends only
on time t, and k= 1 , 0 ,− 1 stand for the sign of space scalar curvature in the followingsense:
(7.5.9)
k=1 : M=S^3 the 3D sphere with scalar curvature=1
R^2
,
k=0 : M=R^3 the 3D flat Euclidean space,k=−1 : M=L^3 the Lobachevsky space with scalar curvature=−1
R^2
,
where the Lobachevsky space L^3 has the same topological structure asR^3.
Historically, Theorem7.22was rigorously proved by Robertson and Walker, and was
assumed by Friedmann and Lemaˆıtre to derive the field equations satisfied byR(t).
7.The Newton cosmology.The Newton cosmology is based on the Newton Gravitational
Law. By Cosmological Principle (Roos, 2003 ), the universe is spherically symmetric. For
any reference pointp∈M, the motion equation of an object with distancerfrompis
(7.5.10)
d^2 r
dt^2=−
GM(r)
r^2