The distorted Schwarzschild metric refers to 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
All static axisymmetric solutions of the Einstein-Maxwell equations can be written in the form of Weyl's metric,
From the Weyl perspective, the metric potentials generating the standard Schwarzschild solution are given by
which yields the Schwarzschild metric in Weyl's canonical coordinates that
Weyl-distortion of Schwarzschild's metric
Vacuum Weyl spacetimes (such as Schwarzschild) respect the following field equations,
where is the Laplace operator.
Derivation of vacuum field equations
The vacuum Einstein's equation reads , which yields Eqs(5.a)-(5.c).
Moreover, the supplementary relation implies Eq(5.d).
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 to Eq(5.a), one can construct a new solution via
and the other metric potential can be obtained by
Let and , while and refer to a second set of Weyl metric potentials. Then, constructed via Eqs(6)(7) leads to the superposed Schwarzschild-Weyl metric
With the transformations
one can obtain the superposed Schwarzschild metric in the usual coordinates,
The superposed metric Eq(10) can be regarded as the standard Schwarzschild metric distorted by external Weyl sources. In the absence of distortion potential , Eq(10) reduces to the standard Schwarzschild metric
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 in Eq(10) is given by the multipole expansion
denotes the Legendre polynomials and are multipole coefficients. The other potential is
- ^ 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]