Distorted Schwarzschild metric


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

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

Weyl-distortion of Schwarzschild's metric

Vacuum Weyl spacetimes respect the following field equations,
where is the Laplace operator.

Derivation of vacuum field equations


The vacuum Einstein's equation reads, which yields Eqs-.
Moreover, the supplementary relation implies Eq.


Eq is the linear Laplace's equation; that is to say, linear combinations of given solutions are still its solutions. Given two solutions to Eq, 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 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 can be regarded as the standard Schwarzschild metric distorted by external Weyl sources. In the absence of distortion potential, Eq 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. The distortion potential in Eq is given by the multipole expansion
where
denotes the Legendre polynomials and are multipole coefficients. The other potential is