Weyl metrics


In general relativity, the Weyl metrics are a class of static and axisymmetric solutions to Einstein's field equation. Three members in the renowned Kerr–Newman family solutions, namely the Schwarzschild, nonextremal Reissner–Nordström and extremal Reissner–Nordström metrics, can be identified as Weyl-type metrics.

Standard Weyl metrics

The Weyl class of solutions has the generic form


where and are two metric potentials dependent on Weyl's canonical coordinates. The coordinate system serves best for symmetries of Weyl's spacetime and often acts like cylindrical coordinates, but is incomplete when describing a black hole as only cover the horizon and its exteriors.
Hence, to determine a static axisymmetric solution corresponding to a specific stress–energy tensor, we just need to substitute the Weyl metric Eq into Einstein's equation :


and work out the two functions and.

Reduced field equations for electrovac Weyl solutions

One of the best investigated and most useful Weyl solutions is the electrovac case, where comes from the existence of electromagnetic field. As we know, given the electromagnetic four-potential, the anti-symmetric electromagnetic field and the trace-free stress–energy tensor will be respectively determined by


which respects the source-free covariant Maxwell equations:
Eq can be simplified to:
in the calculations as. Also, since for electrovacuum, Eq reduces to


Now, suppose the Weyl-type axisymmetric electrostatic potential is , and together with the Weyl metric Eq, Eqs imply that








where yields Eq, or yields Eq, or yields Eq, yields Eq, and Eq yields Eq. Here and are respectively the Laplace and gradient operators. Moreover, if we suppose in the sense of matter-geometry interplay and assume asymptotic flatness, we will find that Eqs implies a characteristic relation that
Specifically in the simplest vacuum case with and, Eqs reduce to








We can firstly obtain by solving Eq, and then integrate Eq and Eq for. Practically, Eq arising from just works as a consistency relation or integrability condition.
Unlike the nonlinear Poisson's equation Eq, Eq is the linear Laplace equation; that is to say, superposition of given vacuum solutions to Eq is still a solution. This fact has a widely application, such as to analytically distort a Schwarzschild black hole.

Box A: Remarks on the electrovac field equation


We employed the axisymmetric Laplace and gradient operators to write Eqs and Eqs in a compact way, which is very useful in the derivation of the characteristic relation Eq. In the literature, Eqs and Eqs are often written in the following forms as well:








and











Box B: Derivation of the Weyl electrovac characteristic relation


Considering the interplay between spacetime geometry and energy-matter distributions, it is natural to assume that in Eqs the metric function relates with the electrostatic scalar potential via a function , and it follows that
Eq immediately turns Eq and Eq respectively into


which give rise to
Now replace the variable by, and Eq is simplified to
Direct quadrature of Eq yields, with being integral constants. To resume asymptotic flatness at spatial infinity, we need and, so there should be. Also, rewrite the constant as for mathematical convenience in subsequent calculations, and one finally obtains the characteristic relation implied by Eqs that
This relation is important in linearize the Eqs and superpose electrovac Weyl solutions.

Newtonian analogue of metric potential Ψ(ρ,z)

In Weyl's metric Eq, ; thus in the approximation for weak field limit, 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,


where is the usual Newtonian potential satisfying Poisson's equation , just like Eq or Eq for the Weyl metric potential. The similarities between and inspire people to find out the Newtonian analogue of when studying Weyl class of solutions; that is, to reproduce nonrelativistically by certain type of Newtonian sources. The Newtonian analogue of proves quite helpful in specifying particular Weyl-type solutions and extending existing Weyl-type solutions.

Schwarzschild solution

The Weyl potentials generating Schwarzschild's metric as solutions to the vacuum equations Eq are given by


where


From the perspective of Newtonian analogue, equals the gravitational potential produced by a rod of mass and length placed symmetrically on the -axis; that is, by a line mass of uniform density embedded the interval.
Given and, Weyl's metric Eq becomes


and after substituting the following mutually consistent relations




one can obtain the common form of Schwarzschild metric in the usual coordinates,


The metric Eq cannot be directly transformed into Eq by performing the standard cylindrical-spherical transformation, because is complete while is incomplete. This is why we call in Eq as Weyl's canonical coordinates rather than cylindrical coordinates, although they have a lot in common; for example, the Laplacian in Eq is exactly the two-dimensional geometric Laplacian in cylindrical coordinates.

Nonextremal Reissner–Nordström solution

The Weyl potentials generating the nonextremal Reissner–Nordström solution as solutions to Eqs(7

Extremal Reissner–Nordström solution

The potentials generating the extremal Reissner–Nordström solution as solutions to Eqs(7

Weyl vacuum solutions in spherical coordinates

Weyl's metric can also be expressed in spherical coordinates that


which equals Eq via the coordinate transformation In the vacuum case, Eq for becomes


The asymptotically flat solutions to Eq is


where represent Legendre polynomials, and are multipole coefficients. The other metric potential is given by