Determinants and Their Applications in Mathematical Physics

242 6. Applications of Determinants in Mathematical Physics

Since detP=1,

P

− 1

=

1

φ

[

φ

2

+ψ

2

−ψ

−ψ 1

]

,

∂P

∂ρ

=

1

φ

2

[

−φρ φψρ−ψφρ

φψρ−ψφρ φ

2

φρ+2φψψρ−ψ

2

φρ

]

,

∂P

∂ρ

P

− 1

=

M

φ

2

,

∂P

∂z

P

− 1

=

N

φ

2

,

where

M=

[

−(φφρ+ψψρ) ψρ

(φ

2

−ψ

2

)ψρ− 2 φψφρ φφρ+ψψρ

]

andNis the matrix obtained fromMby replacingφρbyφzandψρbyψz.

The equation above (6.2.6) can now be expressed in the form

M

ρ

−

2

φ

(φρM+φzN)+(Mρ+Nz) = 0 (6.2.7)

where

φρM+φzN=









−

{

φ(φ

2

ρ

+φ

2

z

)

+ψ(φρψρ+φzψz)

}

{φρψρ+φzψz}

{

(φ

2

−ψ

2

)(φρψρ+φzψz)

− 2 φψ(φ

2

ρ

+φ

2

z

)

}{

φ(φ

2

ρ

+φ

2

z

)

+ψ(φρψρ+φzψz)

}









,

Mρ+Nz

=





−

{

φ(φρρ+φzz)+ψ(ψρρ+ψzz)

+φ

2

ρ+φ

2

z+ψ

2

ρ+ψ

2

z

}

{ψρρ+ψzz}

{

(φ

2

−ψ

2

)(ψρρ+ψzz)− 2 φψ(φρρ+φzz)

− 2 ψ(φ

2

ρ+φ

2

z+ψ

2

ρ+ψ

2

z)

}{

φ(φρρ+φzz)+ψ(ψρρ+ψzz)

+φ

2

ρ+φ

2

z+ψ

2

ρ+ψ

2

z

}





The Einstein equations can now be expressed in the form

[

f 11 f 12

f 21 f 22

]

=0,

where

f 12 =

1

φ

[

φ

(

ψρρ+

1

ρ

ψρ+ψzz

)

−2(φρψρ+φzψz)

]

=0,

f 11 =−ψf 12 −

[

φ

(

φρρ+

1

ρ

φρ+φzz

)

−φ

2

ρ

−φ

2

z

+ψ

2

ρ

+ψ

2

z

]

=0,

f 21 =(φ

2

−ψ

2

)f 12 − 2 ψ

[

φ

(

φρρ+

1

ρ

φρ+φzz

)

−φ

2

ρ−φ

2

z+ψ

2

ρ+ψ

2

z

]

=0,

f 22 =−f 11 =0,