Mathematical Principles of Theoretical Physics

482 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

It is easy to show that the orbits inΩ 1 withx 1 >0 will eventually cross thex 2 −axis. Thus,
using the three invariant sets in (7.6.45), we obtain the phase diagram of (7.6.43) onx 2 > 0
as shown in Figure7.16. In this diagram, we see that, the orbits inΩ 2 andΩ 3 will not cross
thex 2 -axis, but these inΩ 1 withx 1 >0 will do.

x 1

Ω 3

x 2 =x 1

Ω 2

x 2 = 2x 1

x 2

Ω 1

x 2 =−x 1

Figure 7.16: Only the orbits onΩ 1 withx 1 >0 will eventually cross thex 2 -axis, leading to
the sign change ofx 1 , and to a repelling gravitational force corresponding to dark energy.

Step 6. Asymptotic repulsion theorem of gravity.We now derive an asymptotic repulsion
theorem of gravity, based on the phase diagram in Figure7.16. In fact, by (7.6.28) and
(7.6.29), the gravitational forceFreads as

(7.6.46) F=−

mc^2

2

euu′.

It is known that

F< 0 represents attraction,

F> 0 represents repelling.

Hence, byx 1 =ru′(r)and (7.6.46), the phase diagram shows that an orbit inΩ 1 , starting with
x 1 >0, will cross thex 2 -axis, and the sign ofx 1 changes from positive to negative, leading
consequently to a repulsive gravitational forceF. Namely, we have obtained the following
theorem.

Theorem 7.30(Asymptotic Repulsion of Gravitation).For a central gravitational field, the
following assertions hold true:

1) The gravitational force F is given by (7.6.29), and is asymptotic zero:

(7.6.47) F→0 if r→∞.

2) If the initial valueαin (7.6.35) is near the Schwarzschild solution (7.6.31) with 0 <

α 1 <α 2 / 2 , then there exists a sufficiently large r 1 such that the gravitational force F

is repulsive for r>r 1 :

(7.6.48) F>0 for r>r 1.