In physics , the distorted Schwarzschild metric is the metric of a standard/isolated Schwarzschild spacetime exposed in external fields. In numerical simulation, the Schwarzschild metric can be distorted by almost arbitrary kinds of external energy–momentum distribution . However, in exact analysis, the mature method to distort the standard Schwarzschild metric is restricted to the framework of Weyl metrics .
Standard Schwarzschild as a vacuum Weyl metric
edit
All static axisymmetric solutions of the Einstein–Maxwell equations can be written in the form of Weyl's metric,[ 1]
(
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
,
{\displaystyle (1)\quad ds^{2}=-e^{2\psi (\rho ,z)}dt^{2}+e^{2\gamma (\rho ,z)-2\psi (\rho ,z)}(d\rho ^{2}+dz^{2})+e^{-2\psi (\rho ,z)}\rho ^{2}d\phi ^{2}\,,}
Schwarzschild solution are given by[ 1] [ 2]
(
2
)
ψ
S
S
=
1
2
ln
L
−
M
L
+
M
,
γ
S
S
=
1
2
ln
L
2
−
M
2
l
+
l
−
,
{\displaystyle (2)\quad \psi _{SS}={\frac {1}{2}}\ln {\frac {L-M}{L+M}}\,,\quad \gamma _{SS}={\frac {1}{2}}\ln {\frac {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
,
{\displaystyle (3)\quad L={\frac {1}{2}}{\big (}l_{+}+l_{-}{\big )}\,,\quad l_{+}={\sqrt {\rho ^{2}+(z+M)^{2}}}\,,\quad l_{-}={\sqrt {\rho ^{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
.
{\displaystyle (4)\quad ds^{2}=-{\frac {L-M}{L+M}}dt^{2}+{\frac {(L+M)^{2}}{l_{+}l_{-}}}(d\rho ^{2}+dz^{2})+{\frac {L+M}{L-M}}\,\rho ^{2}d\phi ^{2}\,.}
edit
Vacuum Weyl spacetimes (such as Schwarzschild) respect the following field equations,[ 1] [ 2]
(
5.
a
)
∇
2
ψ
=
0
,
{\displaystyle (5.a)\quad \nabla ^{2}\psi =0\,,}
(
5.
b
)
γ
,
ρ
=
ρ
(
ψ
,
ρ
2
−
ψ
,
z
2
)
,
{\displaystyle (5.b)\quad \gamma _{,\,\rho }=\rho \,{\Big (}\psi _{,\,\rho }^{2}-\psi _{,\,z}^{2}{\Big )}\,,}
(
5.
c
)
γ
,
z
=
2
ρ
ψ
,
ρ
ψ
,
z
,
{\displaystyle (5.c)\quad \gamma _{,\,z}=2\,\rho \,\psi _{,\,\rho }\psi _{,\,z}\,,}
(
5.
d
)
γ
,
ρ
ρ
+
γ
,
z
z
=
−
(
ψ
,
ρ
2
+
ψ
,
z
2
)
,
{\displaystyle (5.d)\quad \gamma _{,\,\rho \rho }+\gamma _{,\,zz}=-{\big (}\psi _{,\,\rho }^{2}+\psi _{,\,z}^{2}{\big )}\,,}
where
∇
2
:=
∂
ρ
ρ
+
1
ρ
∂
ρ
+
∂
z
z
{\displaystyle \nabla ^{2}:=\partial _{\rho \rho }+{\frac {1}{\rho }}\partial _{\rho }+\partial _{zz}}
Laplace operator .
Derivation of vacuum field equations. The vacuum Einstein's equation reads
R
a
b
=
0
{\displaystyle R_{ab}=0}
Moreover, the supplementary relation
R
=
0
{\displaystyle R=0}
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
⟩
}
{\displaystyle \{\psi ^{\langle 1\rangle },\psi ^{\langle 2\rangle }\}}
(
6
)
ψ
~
=
ψ
⟨
1
⟩
+
ψ
⟨
2
⟩
,
{\displaystyle (6)\quad {\tilde {\psi }}\,=\,\psi ^{\langle 1\rangle }+\psi ^{\langle 2\rangle }\,,}
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
}
.
{\displaystyle (7)\quad {\tilde {\gamma }}\,=\,\gamma ^{\langle 1\rangle }+\gamma ^{\langle 2\rangle }+2\int \rho \,{\Big \{}\,{\Big (}\psi _{,\,\rho }^{\langle 1\rangle }\psi _{,\,\rho }^{\langle 2\rangle }-\psi _{,\,z}^{\langle 1\rangle }\psi _{,\,z}^{\langle 2\rangle }{\Big )}\,d\rho +{\Big (}\psi _{,\,\rho }^{\langle 1\rangle }\psi _{,\,z}^{\langle 2\rangle }+\psi _{,\,z}^{\langle 1\rangle }\psi _{,\,\rho }^{\langle 2\rangle }{\Big )}\,dz\,{\Big \}}\,.}
Let
ψ
⟨
1
⟩
=
ψ
S
S
{\displaystyle \psi ^{\langle 1\rangle }=\psi _{SS}}
γ
⟨
1
⟩
=
γ
S
S
{\displaystyle \gamma ^{\langle 1\rangle }=\gamma _{SS}}
ψ
⟨
2
⟩
=
ψ
{\displaystyle \psi ^{\langle 2\rangle }=\psi }
γ
⟨
2
⟩
=
γ
{\displaystyle \gamma ^{\langle 2\rangle }=\gamma }
{
ψ
~
,
γ
~
}
{\displaystyle \{{\tilde {\psi }},{\tilde {\gamma }}\}}
(
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
.
{\displaystyle (8)\quad ds^{2}=-e^{2\psi (\rho ,z)}{\frac {L-M}{L+M}}dt^{2}+e^{2\gamma (\rho ,z)-2\psi (\rho ,z)}{\frac {(L+M)^{2}}{l_{+}l_{-}}}(d\rho ^{2}+dz^{2})+e^{-2\psi (\rho ,z)}{\frac {L+M}{L-M}}\,\rho ^{2}d\phi ^{2}\,.}
With the transformations[ 2]
(
9
)
L
+
M
=
r
,
l
+
+
l
−
=
2
M
cos
θ
,
z
=
(
r
−
M
)
cos
θ
,
{\displaystyle (9)\quad L+M=r\,,\quad l_{+}+l_{-}=2M\cos \theta \,,\quad z=(r-M)\cos \theta \,,}
ρ
=
r
2
−
2
M
r
sin
θ
,
l
+
l
−
=
(
r
−
M
)
2
−
M
2
cos
2
θ
,
{\displaystyle \;\;\quad \rho ={\sqrt {r^{2}-2Mr}}\,\sin \theta \,,\quad l_{+}l_{-}=(r-M)^{2}-M^{2}\cos ^{2}\theta \,,}
one can obtain the superposed Schwarzschild metric in the usual
{
t
,
r
,
θ
,
ϕ
}
{\displaystyle \{t,r,\theta ,\phi \}}
(
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
.
{\displaystyle (10)\quad ds^{2}=-e^{2\psi (r,\theta )}\,{\Big (}1-{\frac {2M}{r}}{\Big )}\,dt^{2}+e^{2\gamma (r,\theta )-2\psi (r,\theta )}{\Big \{}\,{\Big (}1-{\frac {2M}{r}}{\Big )}^{-1}dr^{2}+r^{2}d\theta ^{2}\,{\Big \}}+e^{-2\psi (r,\theta )}r^{2}\sin ^{2}\theta \,d\phi ^{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
}
{\displaystyle \{\psi (\rho ,z)=0,\gamma (\rho ,z)=0\}}
(
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
.
{\displaystyle (11)\quad ds^{2}=-{\Big (}1-{\frac {2M}{r}}{\Big )}\,dt^{2}+{\Big (}1-{\frac {2M}{r}}{\Big )}^{-1}\,dr^{2}+r^{2}\,d\theta ^{2}+r^{2}\sin ^{2}\theta \,d\phi ^{2}\,.}
Weyl-distorted Schwarzschild solution in spherical coordinates
edit
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
)
{\displaystyle \psi (r,\theta )}
multipole expansion [ 3]
(
12
)
ψ
(
r
,
θ
)
=
−
∑
i
=
1
∞
a
i
(
R
n
(
cos
θ
)
M
)
P
i
{\displaystyle (12)\quad \psi (r,\theta )\,=-\sum _{i=1}^{\infty }a_{i}{\Big (}{\frac {R_{n}(\cos \theta )}{M}}{\Big )}P_{i}}
R
:=
[
(
1
−
2
M
r
)
r
2
+
M
2
cos
2
θ
]
1
/
2
{\displaystyle R:={\Big [}{\Big (}1-{\frac {2M}{r}}{\Big )}r^{2}+M^{2}\cos ^{2}\theta {\Big ]}^{1/2}}
where
(
13
)
P
i
:=
p
i
(
(
r
−
m
)
cos
θ
R
)
{\displaystyle (13)\quad P_{i}:=p_{i}{\Big (}{\frac {(r-m)\cos \theta }{R}}{\Big )}}
denotes the Legendre polynomials and
a
i
{\displaystyle a_{i}}
multipole coefficients. The other potential
γ
(
r
,
θ
)
{\displaystyle \gamma (r,\theta )}
(
14
)
γ
(
r
,
θ
)
=
∑
i
=
1
∞
∑
j
=
0
∞
a
i
a
j
{\displaystyle (14)\quad \gamma (r,\theta )\,=\sum _{i=1}^{\infty }\sum _{j=0}^{\infty }a_{i}a_{j}}
(
i
j
i
+
j
)
{\displaystyle {\Big (}{\frac {ij}{i+j}}{\Big )}}
(
R
M
)
i
+
j
{\displaystyle {\Big (}{\frac {R}{M}}{\Big )}^{i+j}}
(
P
i
P
j
−
P
i
−
1
P
j
−
1
)
{\displaystyle (P_{i}P_{j}-P_{i-1}P_{j-1})}
−
1
M
∑
i
=
1
∞
α
i
∑
j
=
0
i
−
1
{\displaystyle -{\frac {1}{M}}\sum _{i=1}^{\infty }\alpha _{i}\sum _{j=0}^{i-1}}
[
(
−
1
)
i
+
j
(
r
−
M
(
1
−
cos
θ
)
)
+
r
−
M
(
1
+
cos
θ
)
]
{\displaystyle {\Big [}(-1)^{i+j}(r-M(1-\cos \theta ))+r-M(1+\cos \theta ){\Big ]}}
(
R
M
)
j
P
j
.
{\displaystyle {\Big (}{\frac {R}{M}}{\Big )}^{j}P_{j}\,.}
^ a b c Jeremy Bransom Griffiths, Jiri Podolsky. Exact Space-Times in Einstein's General Relativity . Cambridge: Cambridge University Press, 2009. Chapter 10.
^ a b c R Gautreau, R B Hoffman, A Armenti. Static multiparticle systems in general relativity . IL NUOVO CIMENTO B, 1972, 7 (1): 71–98.
^ Terry Pilkington, Alexandre Melanson, Joseph Fitzgerald, Ivan Booth. "Trapped and marginally trapped surfaces in Weyl-distorted Schwarzschild solutions". Classical and Quantum Gravity , 2011, 28 (12): 125018. arXiv:1102.0999v2[gr-qc]
![{\displaystyle R:={\Big [}{\Big (}1-{\frac {2M}{r}}{\Big )}r^{2}+M^{2}\cos ^{2}\theta {\Big ]}^{1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83995718d6efbb4af626a13edda00c757738fa14)
![{\displaystyle {\Big [}(-1)^{i+j}(r-M(1-\cos \theta ))+r-M(1+\cos \theta ){\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2dea1772d64930c063e901440786d0a85217bb5)
