Mathematical Principles of Theoretical Physics

194 CHAPTER 4. UNIFIED FIELD THEORY

the 10 unknown functions become
(
−1 0
0 gij

)

, gij=gji for 1≤i,j≤ 3.

This observation implies that the number of independent unknown functions for the Einstein
field equations (4.2.1) is six. Namely,

(4.2.8)

NEQ= 10 ,

NUF= 6 ,

whereNEQis the number of independent equations in (4.2.1), andNUF is the number of
independent unknown functions.
Consequently, the Einstein field equations (4.2.1) have no solutions in the general case.
Some readers may think that the Bianchi identity

(4.2.9) ∇μ

(

Rμ ν−

1

2

gμ νR+

8 πG

c^4

Tμ ν

)

= 0 ,

reduce the numberNEQto six:NEQ=6. But we note that (4.2.9) generates also four new
equations
∇μTμ ν= 0 for 0≤ν≤ 3 ,

because there are unknown functionsgμ νin the covariant derivative operators∇μ; see (3.1.66)
or (2.3.26). Hence, the Einstein field equations (4.2.1) should be in the form

(4.2.10)

Rμ ν−

1

2

gμ νR=−

8 πG

c^4

Tμ ν,

∇μTμ ν= 0.

Thus, the fact (4.2.8) still holds for (4.2.10).
Now we note the gravitational field equations (4.2.7) derived from PID, where there are
four additional new unknown functionsΦν( 0 ≤ν≤ 3 ). In this case, the numbers of inde-
pendent unknown functions and equations are the same.
In the following, we give an example to show the non well-posedness of the classical

2.3.5 Einstein gravitational field equations.

It is known that the metric of central gravitational field takes the form

(4.2.11) ds^2 =c^2 g 00 dt^2 +g 11 dr^2 +r^2 (dθ^2 +sin^2 θdφ^2 ),

where(ct,r,θ,φ)is the spherical coordinate system. The metricgμ νin (4.2.11) can be ex-
pressed in the form

(4.2.12)

g 00 =−eu (u=u(r)),

g 11 =ev (v=v(r)),

g 22 =r^2 ,

g 33 =r^2 sin^2 θ,

gμ ν= 0 forμ 6 =ν.