Mathematical Principles of Theoretical Physics

480 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

In fact, by (7.6.31) and (7.6.33) we can see that all Schwarzschild solutions lie on the line

(7.6.39) L={(x 1 ,x 2 , 0 )|x 1 =x 2 ,x 1 ,x 2 > 0 }.

In particular, the lineLis on the stable manifoldEsof (7.6.37).

Step 3. Stable manifold Es.The equations (7.6.34) can be written as

x ̇=Ax+O(|x|^2 ),

where

(7.6.40) A=





0 −1 2

0 −1 0

1 −1 1



.

The dimension of the stable manifoldEsis the number of negative eigenvalues of the matrix
A. It is easy to see that the eigenvalues ofAare given by

λ 1 =− 1 , λ 2 =− 1 , λ 3 = 2.

Hence, the dimension ofEsis two:

dimEs= 2.

Consequently, the initial valueαof an asymptotically flat solution has only two independent
components due toα∈Es, which is of two dimensional. Namely, we arrive at the following
conclusion.

Physical Conclusion 7.29.In the gravitation formula (7.6.29) there are two free parameters
to be determined by experiments (or by astronomical measurements).

In fact, the two free parameters will be determined by the Rubin rotational curve and the
repulsive property of gravity at large distance.

Step 4. Local expression of Es.In order to derive the asymptotic property of the grav-
itational forceFof (7.6.29), we need to derive the local expression of the stable manifold
Esnearx=0. Since the tangent space ofEsatx=0 is spanned by the two eigenvectors
( 1 , 1 , 0 )tand( 1 ,− 1 ,− 1 )tcorresponding to the two negative eigenvaluesλ 1 =λ 2 =−1, the
coordinate vector( 0 , 0 , 1 )ofx 3 is not contained inEs. This implies that the stable manifold
can be expressed nearx=0 in the form

(7.6.41) x 3 =h(x 1 ,x 2 ).

Inserting the Taylor expansion for (7.6.41) into (7.6.34), and comparing the coefficients,
we derive the following local expression of (7.6.41) of the stable manifold function:

(7.6.42) h(x 1 ,x 2 ) =−

1

2

x 1 +

1

2

x 2 +

1

16

x^21 −

1

16

x^22 +O(|x|^3 ).