7.6. THEORY OF DARK MATTER AND DARK ENERGY 479
To this end, we need to make the following transformation to convert the field equations
(7.6.27) into a first order autonomous system:
(7.6.33)
r=es,
w=ev− 1 ,
x 1 (s) =esu′(es),
x 2 (s) =w(es),
x 3 (s) =esφ′(es).Then the equations (7.6.27) can be rewritten as
(7.6.34)
x′ 1 =−x 2 + 2 x 3 −1
2
x^21 −1
2
x 1 x 3 −1
2
x 1 x 2 −1
4
x^21 x 3 ,x′ 2 =−x 2 −1
2
x 1 x 3 −x^22 −1
2
x 1 x 2 x 3 ,x′ 3 =x 1 −x 2 +x 3 −1
2
x 2 x 3 −1
4
x 1 x^23.The system (7.6.34) is supplemented with an initial condition
(7.6.35) (x 1 ,x 2 ,x 3 )(s 0 ) = (α 1 ,α 2 ,α 3 ) withr 0 =es^0.
We now study the problem (7.6.34)-(7.6.35) in a few steps as follows.Step 1. Asymptotic flatness. For a globular matter distribution, its gravitational field
should be asymptotically flat, i.e.
g 00 → − 1 , g 11 →1 if r→∞.It implies thatx=0 is the uniquely physical equilibrium point of (7.6.34) and the following
holds true:
(7.6.36) x(s)→0 if s→∞ (i.e.r→∞).
Step 2. Physical initial values.The physically meaningful initial valuesα= (α 1 ,α 2 ,α 3 )
in (7.6.35) have to satisfy the following two conditions:
(a) The solutionsx(s,α)of (7.6.34)-(7.6.35) are asymptotically flat in the sense of (7.6.36).
Namely, the initial valuesαare in the stable manifoldEsofx=0, defined by(7.6.37) Es={α∈R^3 |x(s,α)→0 fors→∞};(b) The solutionsx(s,α)are near the Schwarzschild solution:(7.6.38) |x 1 −esu′|, |x 2 + 1 −ev|, |x 3 | ≪ 1 ,whereu,vare as in (7.6.30).