Weyl metrics

In general relativity, the Weyl metrics (named after the German-American mathematician Hermann Weyl)[1] are a class of static and axisymmetric solutions to Einstein's field equation.

The Weyl class of solutions has the generic form[2][3] where

are two metric potentials dependent on Weyl's canonical coordinates

serves best for symmetries of Weyl's spacetime (with two Killing vector fields being

) and often acts like cylindrical coordinates,[2] but is incomplete when describing a black hole as

Hence, to determine a static axisymmetric solution corresponding to a specific stress–energy tensor

, we just need to substitute the Weyl metric Eq(1) into Einstein's equation (with c=G=1): and work out the two functions

One of the best investigated and most useful Weyl solutions is the electrovac case, where

comes from the existence of (Weyl-type) electromagnetic field (without matter and current flows).

for electrovacuum, Eq(2) reduces to Now, suppose the Weyl-type axisymmetric electrostatic potential is

in the sense of matter-geometry interplay and assume asymptotic flatness, we will find that Eqs(7.a-e) implies a characteristic relation that Specifically in the simplest vacuum case with

This fact has a widely application, such as to analytically distort a Schwarzschild black hole.

for mathematical convenience in subsequent calculations, and one finally obtains the characteristic relation implied by Eqs(7.a-7.e) that This relation is important in linearize the Eqs(7.a-7.f) and superpose electrovac Weyl solutions.

, one has and therefore This is pretty analogous to the well-known approximate metric for static and weak gravitational fields generated by low-mass celestial bodies like the Sun and Earth,[5] where

is the usual Newtonian potential satisfying Poisson's equation

inspire people to find out the Newtonian analogue of

when studying Weyl class of solutions; that is, to reproduce

[2] The Weyl potentials generating Schwarzschild's metric as solutions to the vacuum equations Eq(8) are given by[2][3][4] where From the perspective of Newtonian analogue,

equals the gravitational potential produced by a rod of mass

(Note: Based on this analogue, important extensions of the Schwarzschild metric have been developed, as discussed in ref.

, Weyl's metric Eq(1) becomes and after substituting the following mutually consistent relations one can obtain the common form of Schwarzschild metric in the usual

in Eq(7) is exactly the two-dimensional geometric Laplacian in cylindrical coordinates.

The Weyl potentials generating the nonextremal Reissner–Nordström solution (

coordinates, The potentials generating the extremal Reissner–Nordström solution (

coordinates, Mathematically, the extremal Reissner–Nordström can be obtained by taking the limit

of the corresponding nonextremal equation, and in the meantime we need to use the L'Hospital rule sometimes.

Remarks: Weyl's metrics Eq(1) with the vanishing potential

Extending this subclass by canceling the restriction of axisymmetry, one obtains another useful class of solutions (still using Weyl's coordinates), namely the conformastatic metrics,[6][7] where we use

in Eq(22) as the single metric function in place of

(Note: As shown by Eqs(15)(21)(24), this transformation is not always applicable.)