Mathematical Principles of Theoretical Physics

7.5. THE UNIVERSE 461

then the metric (7.5.8) satisfies the following equations

R ̈=−^4 πG

3

(

ρ+

3 p

c^2

)

R+

Λc^2

3

(7.5.24) R,

( ̇

R

R

) 2

=

8 πG

3

ρ+

Λc^2

3

−

kc^2

R^2

(7.5.25) ,

ρ ̇+ 3

( ̇

R

R

)(

ρ+

p

c^2

)

(7.5.26) = 0.

The equations (7.5.24)-(7.5.26) are known as the Lemaˆıtre cosmological model, or theΛ-
cosmological model, which leads to the following conclusions ofΛ-cosmology:

Conclusions ofΛ-Cosmology 7.24.

1) The Universe is temporally open: R→∞as t→∞;

2) There is a critical radius Rc, such that

the universe is decelerating forR<Rc,

the universe is accelerating forR>Rc,

where Rc≃

(

4 πGρ 0

Λc^2

) 1 / 3

;

3) As t→∞andρ→ 0 , then we deduce from (7.5.25) that the cosmological radius R has

the asymptotic relation

(7.5.27) R∼e

√

Λc^2 / 3 t ast→∞.

Namely

R(t)/e

√

Λc^2 / 3 t=const. ast→∞;

4) The total kinetic energy E is given by

(7.5.28) E=











3

5

GM^2

R

+

Λ

10

Mc^2 R^2 fork= 0 ,

2

3 π

GM^2

R

+

Λ

6

Mc^2 R^2 −

1

2

Mc^2 fork= 1.

Remark 7.25.The field equations (7.5.23) with a cosmological constantΛlead to a special
conclusion that in the expansion process, there are a large quantities of energy to be created,
and the added energy in (7.5.28) is generated byΛis as

1

6

Mc^2 ΛR^2 (k= 1 ) and

Λ

10

Mc^2 R^2 (k= 0 ).

It implies that the total energy is not conserved in theΛ-model.