Distorted Schwarzschild metric

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

where

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...