Page 257 - Determinants and Their Applications in Mathematical Physics

242 6. Applications of Determinants in Mathematical Physics

Since det P =1,
1 φ + ψ 2 −ψ
2
−1
P = ,
φ −ψ 1
∂P 1
= −φ ρ 2 φψ ρ − ψφ ρ 2 ,
∂ρ φ 2 φψ ρ − ψφ ρ φ φ ρ +2φψψ ρ − ψ φ ρ
∂P −1 M
P = ,
∂ρ φ 2
∂P −1 N
P = ,
∂z φ 2
where

−(φφ ρ + ψψ ρ )
M = (φ − ψ )ψ ρ − 2φψφ ρ φφ ρ + ψψ ρ
ψ ρ
2
2
and N is the matrix obtained from M by replacing φ ρ by φ z and ψ ρ by ψ z .
The equation above (6.2.6) can now be expressed in the form
M 2
− (φ ρ M + φ z N)+(M ρ + N z ) = 0 (6.2.7)
ρ φ
where
 2 2 
φ(φ + φ )
− ρ z {φ ρ ψ ρ + φ z ψ z }
 +ψ(φ ρ ψ ρ + φ z ψ z ) 
φ ρ M + φ z N =  2 2 2 2  ,
 (φ − ψ )(φ ρ ψ ρ + φ z ψ z ) φ(φ + φ ) 
ρ z
2
2
−2φψ(φ + φ ) +ψ(φ ρ ψ ρ + φ z ψ z )
ρ z
M ρ + N z
φ(φ ρρ + φ zz)+ ψ(ψ ρρ + ψ zz)
/ 0
 
− 2 2 2 2 {ψ ρρ + ψ zz}
=  / 2 2 +φ ρ + φ z + ψ ρ + ψ z 0/ 0 
(φ − ψ )(ψ ρρ + ψ zz) − 2φψ(φ ρρ + φ zz) φ(φ ρρ + φ zz)+ ψ(ψ ρρ + ψ zz)
2
2
2
2
2
2
2
−2ψ(φ ρ + φ z + ψ ρ + ψ z ) +φ ρ + φ z + ψ ρ + ψ z 2
The Einstein equations can now be expressed in the form

f 11 f 12
=0,
f 21 f 22
where
1 1
f 12 = − 2(φ ρ ψ ρ + φ z ψ z ) =0,
ρ
φ φ ψ ρρ + ψ ρ + ψ zz
1 2 2 2 2

− φ − φ + ψ + ψ =0,
ρ ρ z ρ z
f 11 = −ψf 12 − φ φ ρρ + φ ρ + φ zz

1
2 2 − φ − φ + ψ + ψ 2 =0,
2
2
2
ρ ρ z ρ z
f 21 =(φ − ψ )f 12 − 2ψ φ φ ρρ + φ ρ + φ zz
f 22 = −f 11 =0,