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.)