In general relativity, the Weyl metrics (named after the German-American mathematician Hermann Weyl) are a class of static and axisymmetric solutions to Einstein's field equation. Three members in the renowned KerrâÂÂNewman family solutions, namely the Schwarzschild, nonextremal ReissnerâÂÂNordström and extremal ReissnerâÂÂNordström metrics, can be identified as Weyl-type 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 (with two Killing vector fields being and ) 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 stressâÂÂenergy 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 and .
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). As we know, given the electromagnetic four-potential , the anti-symmetric electromagnetic field and the trace-free stressâÂÂenergy tensor will be respectively determined by
which respects the source-free covariant Maxwell equations:
Eq(5.a) can be simplified to:
in the calculations as . Also, since for electrovacuum, Eq(2) reduces to
Now, suppose the Weyl-type axisymmetric electrostatic potential is (the component is actually the electromagnetic scalar potential), and together with the Weyl metric Eq(1), Eqs(3)(4)(5)(6) imply that
where yields Eq(7.a), or yields Eq(7.b), or yields Eq(7.c), yields Eq(7.d), and Eq(5.b) yields Eq(7.e). 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(7.a-e) implies a characteristic relation that
Specifically in the simplest vacuum case with and , Eqs(7.a-7.e) reduce to
We can firstly obtain by solving Eq(8.b), and then integrate Eq(8.c) and Eq(8.d) for . Practically, Eq(8.a) arising from just works as a consistency relation or integrability condition.
Unlike the nonlinear Poisson's equation Eq(7.b), Eq(8.b) is the linear Laplace equation; that is to say, superposition of given vacuum solutions to Eq(8.b) is still a solution. This fact has a widely application, such as to analytically distort a Schwarzschild black hole.
We employed the axisymmetric Laplace and gradient operators to write Eqs(7.a-7.e) and Eqs(8.a-8.d) in a compact way, which is very useful in the derivation of the characteristic relation Eq(7.f). In the literature, Eqs(7.a-7.e) and Eqs(8.a-8.d) are often written in the following forms as well:
and
Considering the interplay between spacetime geometry and energy-matter distributions, it is natural to assume that in Eqs(7.a-7.e) the metric function relates with the electrostatic scalar potential via a function (which means geometry depends on energy), and it follows that
Eq(B.1) immediately turns Eq(7.b) and Eq(7.e) respectively into
which give rise to
Now replace the variable by , and Eq(B.4) is simplified to
Direct quadrature of Eq(B.5) 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(7.a-7.e) that
This relation is important in linearize the Eqs(7.a-7.f) and superpose electrovac Weyl solutions.
In Weyl's metric Eq(1), ; 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(3.a) or Eq(4.a) 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.
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 . (Note: Based on this analogue, important extensions of the Schwarzschild metric have been developed, as discussed in ref.)
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.
The Weyl potentials generating the nonextremal ReissnerâÂÂNordström solution () as solutions to Eqs() are given by
where
Thus, given and , Weyl's metric becomes
and employing the following transformations
one can obtain the common form of non-extremal ReissnerâÂÂNordström metric in the usual coordinates,
The potentials generating the extremal ReissnerâÂÂNordström solution () as solutions to Eqs() are given by (Note: We treat the extremal solution separately because it is much more than the degenerate state of the nonextremal counterpart.)
Thus, the extremal ReissnerâÂÂNordström metric reads
and by substituting
we obtain the extremal ReissnerâÂÂNordström metric in the usual coordinates,
Mathematically, the extremal ReissnerâÂÂNordström 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() with the vanishing potential (like the extremal ReissnerâÂÂNordström metric) constitute a special subclass which have only one metric potential to be identified. 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,
where we use in Eq() as the single metric function in place of in Eq() to emphasize that they are different by axial symmetry (-dependence).
Weyl's metric can also be expressed in spherical coordinates that
which equals Eq() via the coordinate transformation (Note: As shown by Eqs()()(), this transformation is not always applicable.) 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