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
.