Distorted Schwarzschild metric

Standard Schwarzschild as a vacuum Weyl metric

All static axisymmetric solutions of the Einstein-Maxwell equations can be written in the form of Weyl's metric,

( 1 ) d s 2 = − e 2 ψ ( ρ , z ) d t 2 + e 2 γ ( ρ , z ) − 2 ψ ( ρ , z ) ( d ρ 2 + d z 2 ) + e − 2 ψ ( ρ , z ) ρ 2 d ϕ 2 ,

From the Weyl perspective, the metric potentials generating the standard Schwarzschild solution are given by

( 2 ) ψ S S = 1 2 ln ⁡ L − M L + M , γ S S = 1 2 ln ⁡ L 2 − M 2 l + l − ,

where

( 3 ) L = 1 2 ( l + + l − ) , l + = ρ 2 + ( z + M ) 2 , l − = ρ 2 + ( z − M ) 2 ,

which yields the Schwarzschild metric in Weyl's canonical coordinates that

( 4 ) d s 2 = − L − M L + M d t 2 + ( L + M ) 2 l + l − ( d ρ 2 + d z 2 ) + L + M L − M ρ 2 d ϕ 2 .

Weyl-distortion of Schwarzschild's metric

Vacuum Weyl spacetimes (such as Schwarzschild) respect the following field equations,

( 5. a ) ∇ 2 ψ = 0 ,
( 5. b ) γ , ρ = ρ ( ψ , ρ 2 − ψ , z 2 ) ,
( 5. c ) γ , z = 2 ρ ψ , ρ ψ , z ,
( 5. d ) γ , ρ ρ + γ , z z = − ( ψ , ρ 2 + ψ , z 2 ) ,

where ∇ 2 := ∂ ρ ρ + 1 ρ ∂ ρ + ∂ z z is the Laplace operator.

Eq(5.a) is the linear Laplace's equation; that is to say, linear combinations of given solutions are still its solutions. Given two solutions { ψ ⟨ 1 ⟩ , ψ ⟨ 2 ⟩ } to Eq(5.a), one can construct a new solution via

( 6 ) ψ ~ = ψ ⟨ 1 ⟩ + ψ ⟨ 2 ⟩ ,

and the other metric potential can be obtained by

( 7 ) γ ~ = γ ⟨ 1 ⟩ + γ ⟨ 2 ⟩ + 2 ∫ ρ { ( ψ , ρ ⟨ 1 ⟩ ψ , ρ ⟨ 2 ⟩ − ψ , z ⟨ 1 ⟩ ψ , z ⟨ 2 ⟩ ) d ρ + ( ψ , ρ ⟨ 1 ⟩ ψ , z ⟨ 2 ⟩ + ψ , z ⟨ 1 ⟩ ψ , ρ ⟨ 2 ⟩ ) d z } .

Let ψ ⟨ 1 ⟩ = ψ S S and γ ⟨ 1 ⟩ = γ S S , while ψ ⟨ 2 ⟩ = ψ and γ ⟨ 2 ⟩ = γ refer to a second set of Weyl metric potentials. Then, { ψ ~ , γ ~ } constructed via Eqs(6)(7) leads to the superposed Schwarzschild-Weyl metric

( 8 ) d s 2 = − e 2 ψ ( ρ , z ) L − M L + M d t 2 + e 2 γ ( ρ , z ) − 2 ψ ( ρ , z ) ( L + M ) 2 l + l − ( d ρ 2 + d z 2 ) + e − 2 ψ ( ρ , z ) L + M L − M ρ 2 d ϕ 2 .

With the transformations

( 9 ) L + M = r , l + + l − = 2 M cos ⁡ θ , z = ( r − M ) cos ⁡ θ ,
ρ = r 2 − 2 M r sin ⁡ θ , l + l − = ( r − M ) 2 − M 2 cos 2 ⁡ θ ,

one can obtain the superposed Schwarzschild metric in the usual { t , r , θ , ϕ } coordinates,

( 10 ) d s 2 = − e 2 ψ ( r , θ ) ( 1 − 2 M r ) d t 2 + e 2 γ ( r , θ ) − 2 ψ ( r , θ ) { ( 1 − 2 M r ) − 1 d r 2 + r 2 d θ 2 } + e − 2 ψ ( r , θ ) r 2 sin 2 ⁡ θ d ϕ 2 .

The superposed metric Eq(10) can be regarded as the standard Schwarzschild metric distorted by external Weyl sources. In the absence of distortion potential { ψ ( ρ , z ) = 0 , γ ( ρ , z ) = 0 } , Eq(10) reduces to the standard Schwarzschild metric

( 11 ) d s 2 = − ( 1 − 2 M r ) d t 2 + ( 1 − 2 M r ) − 1 d r 2 + r 2 d θ 2 + r 2 sin 2 ⁡ θ d ϕ 2 .

Weyl-distorted Schwarzschild solution in spherical coordinates

Similar to the exact vacuum solutions to Weyl's metric in spherical coordinates, we also have series solutions to Eq(10). The distortion potential ψ ( r , θ ) in Eq(10) is given by the multipole expansion

( 12 ) ψ ( r , θ ) = − ∑ i = 1 ∞ a i ( R n ( cos ⁡ θ ) M ) P i with R := [ ( 1 − 2 M r ) r 2 + M 2 cos 2 ⁡ θ ] 1 / 2

where

( 13 ) P i := p i ( ( r − m ) cos ⁡ θ R )

denotes the Legendre polynomials and a i are multipole coefficients. The other potential γ ( r , θ ) is

( 14 ) γ ( r , θ ) = ∑ i = 1 ∞ ∑ j = 0 ∞ a i a j ( i j i + j ) ( R M ) i + j ( P i P j − P i − 1 P j − 1 ) − 1 M ∑ i = 1 ∞ α i ∑ j = 0 i − 1 [ ( − 1 ) i + j ( r − M ( 1 − cos ⁡ θ ) ) + r − M ( 1 + cos ⁡ θ ) ] ( R M ) j P j .