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The '''distorted Schwarzschild metric''' refers to the metric of a standard/isolated [[Schwarzschild metric|Schwarzschild spacetime]] exposed in external fields. In numerical simulation, the Schwarzschild metric can be distorted by almost arbitrary kinds of external [[Stress-energy tensor|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==

All static axisymmetric solutions of the [[Einstein-Maxwell equations]] can be written in the form of Weyl's metric,<ref name=Weyl1>Jeremy Bransom Griffiths, Jiri Podolsky. ''Exact Space-Times in Einstein's General Relativity''. Cambridge: Cambridge University Press, 2009. Chapter 10.</ref>

<br />
<math>(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\,,</math>

<br />
From the Weyl perspective, the metric potentials generating the standard [[Schwarzschild metric|Schwarzschild solution]] are given by<ref name="Weyl1" /><ref name=Weyl4>R Gautreau, R B Hoffman, A Armenti. ''Static multiparticle systems in general relativity''. IL NUOVO CIMENTO B, 1972, '''7'''(1): 71-98.</ref>

<br />
<math>(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_-}\,,</math>

where

<br />
<math>(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}\,,</math>

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

<br />
<math>(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\,.</math>

==Weyl-distortion of Schwarzschild's metric==

Vacuum Weyl spacetimes (such as Schwarzschild) respect the following field equations,<ref name="Weyl1" /><ref name="Weyl4" />

<br />
<math>(5.a)\quad \nabla^2 \psi =0\,,</math><br />
<math>(5.b)\quad \gamma_{,\,\rho}=\rho\,\Big(\psi^2_{,\,\rho}-\psi^2_{,\,z} \Big)\,,</math><br />
<math>(5.c)\quad \gamma_{,\,z}=2\,\rho\,\psi_{,\,\rho}\psi_{,\,z}\,,</math><br />
<math>(5.d)\quad \gamma_{,\,\rho\rho}+\gamma_{,\,zz}=-\big(\psi^2_{,\,\rho}+\psi^2_{,\,z} \big)\,,</math>

<br />
where <math>\nabla^2:= \partial_{\rho\rho}+\frac{1}{\rho}\partial_\rho +\partial_{zz}</math> is the [[Laplace operator]].

<div style="clear:both;width:65%;" class="NavFrame collapsed">
<div class="NavHead" style="background-color:#FFFFFF; text-align:left; font-size:larger;">Derivation of vacuum field equations</div>
<div class="NavContent" style="text-align:left;">

The vacuum Einstein's equation reads <math>R_{ab}=0</math>, which yields Eqs(5.a)-(5.c).

Moreover, the supplementary relation <math>R=0</math> implies Eq(5.d).

</div>
</div>

Eq(5.a) is the ''linear'' [[Laplace equation|Laplace's equation]]; that is to say, linear combinations of given solutions are still its solutions. Given two solutions <math>\{\psi^{\langle1\rangle}, \psi^{\langle2\rangle}\}</math> to Eq(5.a), one can construct a new solution via

<math>(6)\quad
\tilde\psi\,=\,\psi^{\langle1\rangle}+\psi^{\langle2\rangle}\,,
</math>

and the other metric potential can be obtained by

<math>(7)\quad
\tilde\gamma\,=\,\gamma^{\langle1\rangle}+\gamma^{\langle2\rangle}+2\int\rho\,\Big\{\,\Big( \psi^{\langle1\rangle}_{,\,\rho}\psi^{\langle2\rangle}_{,\,\rho}-\psi^{\langle1\rangle}_{,\,z}\psi^{\langle2\rangle}_{,\,z} \Big)\,d\rho +\Big( \psi^{\langle1\rangle}_{,\,\rho}\psi^{\langle2\rangle}_{,\,z}+\psi^{\langle1\rangle}_{,\,z}\psi^{\langle2\rangle}_{,\,\rho} \Big)\,dz \, \Big\}\,.
</math>

Let <math>\psi^{\langle1\rangle}=\psi_{SS}</math> and <math>\gamma^{\langle1\rangle}=\gamma_{SS}</math>, while <math>\psi^{\langle2\rangle}=\psi</math> and <math>\gamma^{\langle2\rangle}=\gamma</math> refer to a second set of Weyl metric potentials. Then, <math>\{\tilde\psi, \tilde\gamma \}</math> constructed via
Eqs(6)(7) leads to the superposed Schwarzschild-Weyl metric

<math>(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\,.
</math>

With the transformations<ref name="Weyl4" />

<br />
<math>(9)\quad L+M=r\,,\quad l_+ + l_- =2M\cos\theta\,,\quad z=(r-M)\cos\theta\,,</math><br />
<math>\;\;\quad \rho=\sqrt{r^2-2Mr}\,\sin\theta\,,\quad l_+ l_-=(r-M)^2-M^2\cos^2\theta\,,</math>

one can obtain the superposed Schwarzschild metric in the usual <math>\{t,r,\theta,\phi\}</math> coordinates,

<math>(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^2d\theta^2\,\Big\}+e^{-2\psi(r,\theta)}r^2\sin^2\theta\, d\phi^2\,.
</math>

The superposed metric Eq(10) can be regarded as the standard Schwarzschild metric distorted by external Weyl sources. In the absence of distortion potential <math>\{\psi(\rho,z)=0, \gamma(\rho,z)=0\}</math>, Eq(10) reduces to the standard Schwarzschild metric

<br />
<math>(11)\quad ds^2=-\Big(1-\frac{2M}{r} \Big)\,dt^2+\Big(1-\frac{2M}{r} \Big)^{-1}dr^2+r^2d\theta^2+r^2\sin^2\theta\, d\phi^2\,.</math>

==Weyl-distorted Schwarzschild solution in spherical coordinates==

Similar to the [[Weyl_metrics#Weyl_vacuum_solutions_in_spherical_coordinates|exact vacuum solutions]] to Weyl's metric in [[spherical coordinates]], we also have [[Series solution of differential equations|series solutions]] to Eq(10). The distortion potential <math>\psi(r,\theta)</math> in Eq(10) is given by the [[multipole expansion]]<ref>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. [http://arxiv.org/abs/1102.0999 arXiv:1102.0999v2[gr-qc]]</ref>

<math>(12)\quad \psi(r,\theta)\,=-\sum_{i=1}^\infty a_i \Big(\frac{R_n(\cos\theta)}{M}\Big) P_i</math> with <math>R:=\Big[\Big(1-\frac{2M}{r} \Big) r^2 +M^2\cos^2\theta \Big]^{1/2}</math>

where

<math>(13)\quad P_i:=p_i\Big(\frac{(r-m)\cos\theta}{R} \Big)</math>

denotes the [[Legendre polynomials]] and <math>a_i</math> are [[Multipole moment|multipole]] coefficients. The other potential <math>\gamma(r,\theta)</math> is

<math>(14)\quad \gamma(r,\theta)\,=\sum_{i=1}^\infty \sum_{j=0}^\infty a_i a_j</math> <math>\Big(\frac{ij}{i+j}\Big)</math> <math>\Big(\frac{R}{M} \Big)^{i+j}</math><math>(P_i P_j-P_{i-1}P_{j-1})</math><math>-\frac{1}{M}\sum_{i=1}^\infty \alpha_i \sum_{j=0}^{i-1}</math> <math>\Big[(-1)^{i+j}(r-M(1-\cos\theta))+r-M(1+\cos\theta) \Big]</math><math>\Big(\frac{R}{M} \Big)^j P_j\,.</math>

==See also==

*[[Weyl metrics]]
* [[Schwarzschild metric]]

==References==
{{reflist}}

[[Category:Black holes]]
[[Category:General relativity]]
[[Category:Exact solutions in general relativity]]

History of experiments - Revision history

en>KConWiki at 15:29, 23 November 2012