Mathematical Principles of Theoretical Physics

7.5. THE UNIVERSE 457

Assume that ̇ris proportional tor, i.e.

(7.5.5) r ̇=Hr, (His a constant).

Then the Hubble law (7.5.1) follows from (7.5.4) and (7.5.5).

θ

p 1

R 1

p 2

R 2

Figure 7.12

The Hubble Law (7.5.1) also manifests that the recession velocity is independentof the
direction. Hence, it implies that the Universe is isotropic. This very property gives rise to a
strong restriction on the global structure of the Universe:there are only two types of geometry
for our Universe: either the 3D sphere (7.5.3) or the 3D Euclidian spaceR^3. Namely, we have
reached the following physical conclusion.

Physical Conclusion 7.20.The Hubble law allows only two types of topological structure,
either S^3 or R^3 , as our Universe. Further, the Hubble Law requires that the Universe is
expanding.

3.Big-Bang theory.In 1927, the Belgium cosmologist G. Lemaˆıtre first proposedthat
the Universe begins with a big explosion, known as the Big-Bang theory. After the Hubble
law was discovered, in 1931, G. Lemaˆıtre regarded the recessions of remote galaxies as the
expanding of the Universe, and he thought that as the time inverses, the early universe must
be in a high temperature and dense state, and the stage is considered as the beginning of the
Big-Bang.
The Big-Bang theory appears to be consistent with the three most important astronom-
ical facts: the Hubble law, the cosmic microwave background, discovered in 1965 by two
physicists A. Penzias and R. Wilson, and the abundance of helium.

4.Age of the Universe.Based on the Big-Bang theory, the age of the Universe is finite.
LetTbe the age, then the present-day distance between any two points is

R=vT.

It follows from (7.5.1) that

(7.5.6) T=

1

H

,