Standard Weyl metrics
The Weyl class of solutions has the generic form
( 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 ,
where ψ ( ρ , z ) and γ ( ρ , z ) are two metric potentials dependent on Weyl's canonical coordinates { ρ , z } . The coordinate system { t , ρ , z , ϕ } serves best for symmetries of Weyl's spacetime (with two Killing vector fields being ξ t = ∂ t and ξ ϕ = ∂ ϕ ) and often acts like cylindrical coordinates, but is incomplete when describing a black hole as { ρ , z } only cover the horizon and its exteriors.
Hence, to determine a static axisymmetric solution corresponding to a specific stress–energy tensor T a b , we just need to substitute the Weyl metric Eq(1) into Einstein's equation (with c=G=1):
( 2 ) R a b − 1 2 R g a b = 8 π T a b ,
and work out the two functions ψ ( ρ , z ) and γ ( ρ , z ) .
One of the best investigated and most useful Weyl solutions is the electrovac case, where T a b comes from the existence of (Weyl-type) electromagnetic field (without matter and current flows). As we know, given the electromagnetic four-potential A a , the anti-symmetric electromagnetic field F a b and the trace-free stress–energy tensor T a b ( T = g a b T a b = 0 ) will be respectively determined by
( 3 ) F a b = A b ; a − A a ; b = A b , a − A a , b
( 4 ) T a b = 1 4 π ( F a c F b c − 1 4 g a b F c d F c d ) ,
which respects the source-free covariant Maxwell equations:
( 5. a ) ( F a b ) ; b = 0 , F [ a b ; c ] = 0 .
Eq(5.a) can be simplified to:
( 5. b ) ( − g F a b ) , b = 0 , F [ a b , c ] = 0
in the calculations as Γ b c a = Γ c b a . Also, since R = − 8 π T = 0 for electrovacuum, Eq(2) reduces to
( 6 ) R a b = 8 π T a b .
Now, suppose the Weyl-type axisymmetric electrostatic potential is A a = Φ ( ρ , z ) [ d t ] a (the component Φ is actually the electromagnetic scalar potential), and together with the Weyl metric Eq(1), Eqs(3)(4)(5)(6) imply that
( 7. a ) ∇ 2 ψ = ( ∇ ψ ) 2 + γ , ρ ρ + γ , z z
( 7. b ) ∇ 2 ψ = e − 2 ψ ( ∇ Φ ) 2
( 7. c ) 1 ρ γ , ρ = ψ , ρ 2 − ψ , z 2 − e − 2 ψ ( Φ , ρ 2 − Φ , z 2 )
( 7. d ) 1 ρ γ , z = 2 ψ , ρ ψ , z − 2 e − 2 ψ Φ , ρ Φ , z
( 7. e ) ∇ 2 Φ = 2 ∇ ψ ∇ Φ ,
where R = 0 yields Eq(7.a), R t t = 8 π T t t or R φ φ = 8 π T φ φ yields Eq(7.b), R ρ ρ = 8 π T ρ ρ or R z z = 8 π T z z yields Eq(7.c), R ρ z = 8 π T ρ z yields Eq(7.d), and Eq(5.b) yields Eq(7.e). Here ∇ 2 = ∂ ρ ρ + 1 ρ ∂ ρ + ∂ z z and ∇ = ∂ ρ e ^ ρ + ∂ z e ^ z are respectively the Laplace and gradient operators. Moreover, if we suppose ψ = ψ ( Φ ) in the sense of matter-geometry interplay and assume asymptotic flatness, we will find that Eqs(7.a-e) implies a characteristic relation that
( 7. f ) e ψ = Φ 2 − 2 C Φ + 1 .
Specifically in the simplest vacuum case with Φ = 0 and T a b = 0 , Eqs(7.a-7.e) reduce to
( 8. a ) γ , ρ ρ + γ , z z = − ( ∇ ψ ) 2
( 8. b ) ∇ 2 ψ = 0
( 8. c ) γ , ρ = ρ ( ψ , ρ 2 − ψ , z 2 )
( 8. d ) γ , z = 2 ρ ψ , ρ ψ , z .
We can firstly obtain ψ ( ρ , z ) by solving Eq(8.b), and then integrate Eq(8.c) and Eq(8.d) for γ ( ρ , z ) . Practically, Eq(8.a) arising from R = 0 just works as a consistency relation or integrability condition.
Unlike the nonlinear Poisson's equation Eq(7.b), Eq(8.b) is the linear Laplace equation; that is to say, superposition of given vacuum solutions to Eq(8.b) is still a solution. This fact has a widely application, such as to analytically distort a Schwarzschild black hole.
In Weyl's metric Eq(1), e ± 2 ψ = ∑ n = 0 ∞ ( ± 2 ψ ) n n ! ; thus in the approximation for weak field limit ψ → 0 , one has
( 9 ) g t t = − ( 1 + 2 ψ ) − O ( ψ 2 ) , g ϕ ϕ = 1 − 2 ψ + O ( ψ 2 ) ,
and therefore
( 10 ) d s 2 ≈ − ( 1 + 2 ψ ( ρ , z ) ) d t 2 + ( 1 − 2 ψ ( ρ , z ) ) [ e 2 γ ( d ρ 2 + d z 2 ) + ρ 2 d ϕ 2 ] .
This is pretty analogous to the well-known approximate metric for static and weak gravitational fields generated by low-mass celestial bodies like the Sun and Earth,
( 11 ) d s 2 = − ( 1 + 2 Φ N ( ρ , z ) ) d t 2 + ( 1 − 2 Φ N ( ρ , z ) ) [ d ρ 2 + d z 2 + ρ 2 d ϕ 2 ] .
where Φ N ( ρ , z ) is the usual Newtonian potential satisfying Poisson's equation ∇ L 2 Φ N = 4 π ϱ N , just like Eq(3.a) or Eq(4.a) for the Weyl metric potential ψ ( ρ , z ) . The similarities between ψ ( ρ , z ) and Φ N ( ρ , z ) inspire people to find out the Newtonian analogue of ψ ( ρ , z ) when studying Weyl class of solutions; that is, to reproduce ψ ( ρ , z ) nonrelativistically by certain type of Newtonian sources. The Newtonian analogue of ψ ( ρ , z ) proves quite helpful in specifying particular Weyl-type solutions and extending existing Weyl-type solutions.
The Weyl potentials generating Schwarzschild's metric as solutions to the vacuum equations Eq(8) are given by
( 12 ) ψ S S = 1 2 ln L − M L + M , γ S S = 1 2 ln L 2 − M 2 l + l − ,
where
( 13 ) L = 1 2 ( l + + l − ) , l + = ρ 2 + ( z + M ) 2 , l − = ρ 2 + ( z − M ) 2 .
From the perspective of Newtonian analogue, ψ S S equals the gravitational potential produced by a rod of mass M and length 2 M placed symmetrically on the z -axis; that is, by a line mass of uniform density σ = 1 / 2 embedded the interval z ∈ [ − M , M ] . (Note: Based on this analogue, important extensions of the Schwarzschild metric have been developed, as discussed in ref.)
Given ψ S S and γ S S , Weyl's metric Eq(ef{Weyl metric in canonical coordinates}) becomes
( 14 ) 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 ,
and after substituting the following mutually consistent relations
( 15 ) 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 common form of Schwarzschild metric in the usual { t , r , θ , ϕ } coordinates,
( 16 ) 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 .
The metric Eq(14) cannot be directly transformed into Eq(16) by performing the standard cylindrical-spherical transformation ( t , ρ , z , ϕ ) = ( t , r sin θ , r cos θ , ϕ ) , because { t , r , θ , ϕ } is complete while ( t , ρ , z , ϕ ) is incomplete. This is why we call { t , ρ , z , ϕ } in Eq(1) as Weyl's canonical coordinates rather than cylindrical coordinates, although they have a lot in common; for example, the Laplacian ∇ 2 := ∂ ρ ρ + 1 ρ ∂ ρ + ∂ z z in Eq(7) is exactly the two-dimensional geometric Laplacian in cylindrical coordinates.
The Weyl potentials generating the nonextremal Reissner–Nordström solution ( M > | Q | ) as solutions to Eqs(7} are given by
( 17 ) ψ R N = 1 2 ln L 2 − ( M 2 − Q 2 ) ( L + M ) 2 , γ R N = 1 2 ln L 2 − ( M 2 − Q 2 ) l + l − ,
where
( 18 ) L = 1 2 ( l + + l − ) , l + = ρ 2 + ( z + M 2 − Q 2 ) 2 , l − = ρ 2 + ( z − M 2 − Q 2 ) 2 .
Thus, given ψ R N and γ R N , Weyl's metric becomes
( 19 ) d s 2 = − L 2 − ( M 2 − Q 2 ) ( L + M ) 2 d t 2 + ( L + M ) 2 l + l − ( d ρ 2 + d z 2 ) + ( L + M ) 2 L 2 − ( M 2 − Q 2 ) ρ 2 d ϕ 2 ,
and employing the following transformations
( 20 ) L + M = r , l + + l − = 2 M 2 − Q 2 cos θ , z = ( r − M ) cos θ ,
ρ = r 2 − 2 M r + Q 2 sin θ , l + l − = ( r − M ) 2 − ( M 2 − Q 2 ) cos 2 θ ,
one can obtain the common form of non-extremal Reissner–Nordström metric in the usual { t , r , θ , ϕ } coordinates,
( 21 ) d s 2 = − ( 1 − 2 M r + Q 2 r 2 ) d t 2 + ( 1 − 2 M r + Q 2 r 2 ) − 1 d r 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 .
The potentials generating the extremal Reissner–Nordström solution ( M = | Q | ) as solutions to Eqs(7} are given by (Note: We treat the extremal solution separately because it is much more than the degenerate state of the nonextremal counterpart.)
( 22 ) ψ E R N = 1 2 ln L 2 ( L + M ) 2 , γ E R N = 0 , with L = ρ 2 + z 2 .
Thus, the extremal Reissner–Nordström metric reads
( 23 ) d s 2 = − L 2 ( L + M ) 2 d t 2 + ( L + M ) 2 L 2 ( d ρ 2 + d z 2 + ρ 2 d ϕ 2 ) ,
and by substituting
( 24 ) L + M = r , z = L cos θ , ρ = L sin θ ,
we obtain the extremal Reissner–Nordström metric in the usual { t , r , θ , ϕ } coordinates,
( 25 ) d s 2 = − ( 1 − M r ) 2 d t 2 + ( 1 − M r ) − 2 d r 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 .
Mathematically, the extremal Reissner–Nordström can be obtained by taking the limit Q → M of the corresponding nonextremal equation, and in the meantime we need to use the L'Hospital rule sometimes.
Remarks: Weyl's metrics Eq(1) with the vanishing potential γ ( ρ , z ) (like the extremal Reissner–Nordström metric) constitute a special subclass which have only one metric potential ψ ( ρ , z ) to be identified. Extending this subclass by canceling the restriction of axisymmetry, one obtains another useful class of solutions (still using Weyl's coordinates), namely the conformastatic metrics,
( 26 ) d s 2 = − e 2 λ ( ρ , z , ϕ ) d t 2 + e − 2 λ ( ρ , z , ϕ ) ( d ρ 2 + d z 2 + ρ 2 d ϕ 2 ) ,
where we use λ in Eq(22) as the single metric function in place of ψ in Eq(1) to emphasize that they are different by axial symmetry ( ϕ -dependence).
Weyl's metric can also be expressed in spherical coordinates that
( 27 ) d s 2 = − e 2 ψ ( r , θ ) d t 2 + e 2 γ ( r , θ ) − 2 ψ ( r , θ ) ( d r 2 + r 2 d θ 2 ) + e − 2 ψ ( r , θ ) ρ 2 d ϕ 2 ,
which equals Eq(1) via the coordinate transformation ( t , ρ , z , ϕ ) ↦ ( t , r sin θ , r cos θ , ϕ ) (Note: As shown by Eqs(15)(21)(24), this transformation is not always applicable.) In the vacuum case, Eq(8.b) for ψ ( r , θ ) becomes
( 28 ) r 2 ψ , r r + 2 r ψ , r + ψ , θ θ + cot θ ⋅ ψ , θ = 0 .
The asymptotically flat solutions to Eq(28) is
( 29 ) ψ ( r , θ ) = − ∑ n = 0 ∞ a n P n ( cos θ ) r n + 1 ,
where P n ( cos θ ) represent Legendre polynomials, and a n are multipole coefficients. The other metric potential γ ( r , θ ) is given by
( 30 ) γ ( r , θ ) = − ∑ l = 0 ∞ ∑ m = 0 ∞ a l a m ( l + 1 ) ( m + 1 ) l + m + 2 P l P m − P l + 1 P m + 1 r l + m + 2 .
