# Weyl metrics

In general relativity, the Weyl metrics (named after the German-American mathematician Hermann Weyl) refer to the 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[1][2]

$(1)\quad ds^2=-e^{2\psi(\rho,z)}dt^2+e^{2\gamma(\rho,z)-2\psi(\rho,z)}(d\rho^2+dz^2)+e^{-2\psi(\rho,z)}\rho^2 d\phi^2\,,$

where $\psi(\rho,z)$ and $\gamma(\rho,z)$ are two metric potentials dependent on Weyl's canonical coordinates $\{\rho\,,z \}$. The coordinate system $\{t,\rho,z,\phi\}$ serves best for symmetries of Weyl's spacetime (with two Killing vector fields being $\xi^t=\partial_t$ and $\xi^\phi=\partial_\phi$) and often acts like cylindrical coordinates,[1] but is incomplete when describing a black hole as $\{\rho\,,z \}$ only cover the horizon and its exteriors.

Hence, to determine a static axisymmetric solution corresponding to a specific stress-energy tensor $T_{ab}$, we just need to substitute the Weyl metric Eq(1) into Einstein's equation (with c=G=1):

$(2)\quad R_{ab}-\frac{1}{2}Rg_{ab}=8\pi T_{ab}\,,$

and work out the two functions $\psi(\rho,z)$ and $\gamma(\rho,z)$.

## Reduced field equations for electrovac Weyl solutions

One of the best investigated and most useful Weyl solutions is the electrovac case, where $T_{ab}$ comes from the existence of (Weyl-type) electromagnetic field (without matter and current flows). As we know, given the electromagnetic four-potential $A_a$, the anti-symmetric electromagnetic field $F_{ab}$ and the trace-free stress-energy tensor $T_{ab}$ $(T=g^{ab}T_{ab}=0)$ will be respectively determined by

$(3)\quad F_{ab}=A_{b\,;\,a}-A_{a\,;\,b}=A_{b\,,\,a}-A_{a\,,\,b}$
$(4)\quad T_{ab}=\frac{1}{4\pi}\,\Big(\, F_{ac}F_b^{\;c} -\frac{1}{4}g_{ab}F_{cd}F^{cd} \Big)\,,$

which respects the source-free covariant Maxwell equations:

$(5.a)\quad \big(F^{ab}\big)_{;\,b}=0\,,\quad F_{[ab\,;\,c]}=0\,.$

Eq(5.a) can be simplified to:

$(5.b)\quad \big(\sqrt{-g}\,F^{ab}\big)_{,\,b}=0\,,\quad F_{[ab\,,\,c]}=0$

in the calculations as $\Gamma^a_{bc}=\Gamma^a_{cb}$. Also, since $R=-8\pi T=0$ for electrovacuum, Eq(2) reduces to

$(6)\quad R_{ab}=8\pi T_{ab}\,.$

Now, suppose the Weyl-type axisymmetric electrostatic potential is $A_a=\Phi(\rho,z)[dt]_a$ (the component $\Phi$ is actually the electromagnetic scalar potential), and together with the Weyl metric Eq(1), Eqs(3)(4)(5)(6) imply that

$(7.a)\quad \nabla^2 \psi =\,(\nabla\psi)^2 +\gamma_{,\,\rho\rho}+\gamma_{,\,zz}$
$(7.b)\quad \nabla^2\psi =\,e^{-2\psi} (\nabla\Phi)^2$
$(7.c)\quad \frac{1}{\rho}\,\gamma_{,\,\rho} =\,\psi^2_{,\,\rho}-\psi^2_{,\,z}-e^{-2\psi}\big(\Phi^2_{,\,\rho}-\Phi^2_{,\,z}\big)$
$(7.d)\quad \frac{1}{\rho}\,\gamma_{,\,z} =\,2\psi_{,\,\rho}\psi_{,\,z}- 2e^{-2\psi}\Phi_{,\,\rho}\Phi_{,\,z}$
$(7.e)\quad \nabla^2\Phi =\,2\nabla\psi \nabla\Phi\,,$

where $R=0$ yields Eq(7.a), $R_{tt}=8\pi T_{tt}$ or $R_{\varphi\varphi}=8\pi T_{\varphi\varphi}$ yields Eq(7.b), $R_{\rho\rho}=8\pi T_{\rho\rho}$ or $R_{zz}=8\pi T_{zz}$ yields Eq(7.c), $R_{\rho z}=8\pi T_{\rho z}$ yields Eq(7.d), and Eq(5.b) yields Eq(7.e). Here $\nabla^2 = \partial_{\rho\rho}+\frac{1}{\rho}\,\partial_\rho +\partial_{zz}$ and $\nabla=\partial_\rho\, \hat{e}_\rho +\partial_z\, \hat{e}_z$ are respectively the Laplace and gradient operators. Moreover, if we suppose $\psi=\psi(\Phi)$ in the sense of matter-geometry interplay and assume asymptotic flatness, we will find that Eqs(7.a-e) implies a characteristic relation that

$(7.f)\quad e^\psi =\,\Phi^2-2C\Phi+1\,.$

Specifically in the simplest vacuum case with $\Phi=0$ and $T_{ab}=0$, Eqs(7.a-7.e) reduce to[3]

$(8.a)\quad \gamma_{,\,\rho\rho}+\gamma_{,\,zz}=-(\nabla\psi)^2$
$(8.b)\quad \nabla^2 \psi =0$
$(8.c)\quad \gamma_{,\,\rho}=\rho\,\Big(\psi^2_{,\,\rho}-\psi^2_{,\,z} \Big)$
$(8.d)\quad \gamma_{,\,z}=2\,\rho\,\psi_{,\,\rho}\psi_{,\,z} \,.$

We can firstly obtain $\psi(\rho,z)$ by solving Eq(8.b), and then integrate Eq(8.c) and Eq(8.d) for $\gamma(\rho,z)$. Practically, Eq(8.a) arising from $R=0$ just works as a consistency relation or integrability condition.

Unlike the nonlinear Poisson's equation Eq(7.b), Eq(8.a) is the linear Laplace equation; that is to say, superposition of given vacuum solutions to Eq(8.a) is still a solution. This fact has a widely application, such as to analytically distort a Schwarzschild black hole.

## Newtonian analogue of metric potential Ψ(ρ,z)

In Weyl's metric Eq(1), $e^{\pm2\psi}=\sum_{n=0}^{\infty} \frac{(\pm2\psi)^n}{n!}$; thus in the approximation for weak field limit $\psi\to 0$, one has

$(9)\quad g_{tt}=-(1+2\psi)-\mathcal {O}(\psi^2)\,,\quad g_{\phi\phi}=1-2\psi+\mathcal {O}(\psi^2)\,,$

and therefore

$(10)\quad ds^2\approx-\Big(1+2\psi(\rho,z)\Big)\,dt^2+\Big(1-2\psi(\rho,z)\Big)\Big[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\phi^2\Big]\,.$

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,[4]

$(11)\quad ds^2=-\Big(1+2\Phi_{N}(\rho,z)\Big)\,dt^2+\Big(1-2\Phi_{N}(\rho,z)\Big)\,\Big[d\rho^2+dz^2+\rho^2d\phi^2\Big]\,.$

where $\Phi_{N}(\rho,z)$ is the usual Newtonian potential satisfying Poisson's equation $\nabla^2_{L}\Phi_{N}=4\pi\varrho_{N}$, just like Eq(3.a) or Eq(4.a) for the Weyl metric potential $\psi(\rho,z)$. The similarities between $\psi(\rho,z)$ and $\Phi_{N}(\rho,z)$ inspire people to find out the Newtonian analogue of $\psi(\rho,z)$ when studying Weyl class of solutions; that is, to reproduce $\psi(\rho,z)$ nonrelativistically by certain type of Newtonian sources. The Newtonian analogue of $\psi(\rho,z)$ proves quite helpful in specifying particular Weyl-type solutions and extending existing Weyl-type solutions.[1]

## Schwarzschild solution

The Weyl potentials generating Schwarzschild's metric as solutions to the vacuum equations Eq(8) are given by[1][2][3]

$(12)\quad \psi_{SS}=\frac{1}{2}\ln\frac{L-M}{L+M}\,,\quad \gamma_{SS}=\frac{1}{2}\ln\frac{L^2-M^2}{l_+ l_-}\,,$

where

$(13)\quad L=\frac{1}{2}\big(l_+ + l_- \big)\,,\quad l_+ =\sqrt{\rho^2+(z+M)^2}\,,\quad l_- =\sqrt{\rho^2+(z-M)^2}\,.$

From the perspective of Newtonian analogue, $\psi_{SS}$ equals the gravitational potential produced by a rod of mass $M$ and length $2M$ placed symmetrically on the $z$-axis; that is, by a line mass of uniform density $\sigma=1/2$ embedded the interval $z\in[-M,M]$. (Note: Based on this analogue, important extensions of the Schwarzschild metric have been developed, as discussed in ref.[1])

Given $\psi_{SS}$ and $\gamma_{SS}$, Weyl's metric Eq(\ref{Weyl metric in canonical coordinates}) becomes

$(14)\quad ds^2=-\frac{L-M}{L+M}dt^2+\frac{(L+M)^2}{l_+ l_-}(d\rho^2+dz^2)+\frac{L+M}{L-M}\,\rho^2 d\phi^2\,,$

and after substituting the following mutually consistent relations

$(15)\quad L+M=r\,,\quad l_+ + l_- =2M\cos\theta\,,\quad z=(r-M)\cos\theta\,,$
$\;\;\quad \rho=\sqrt{r^2-2Mr}\,\sin\theta\,,\quad l_+ l_-=(r-M)^2-M^2\cos^2\theta\,,$

one can obtain the common form of Schwarzschild metric in the usual $\{t,r,\theta,\phi\}$ coordinates,

$(16)\quad ds^2=-\Big(1-\frac{2M}{r} \Big)\,dt^2+\Big(1-\frac{2M}{r} \Big)^{-1}dr^2+r^2d\theta^2+r^2\sin^2\theta\, d\phi^2\,.$

The metric Eq(14) cannot be directly transformed into Eq(16) by performing the standard cylindrical-spherical transformation $(t,\rho,z,\phi)=(t,r\sin\theta,r\cos\theta,\phi)$, because $\{t,r,\theta,\phi\}$ is complete while $(t,\rho,z,\phi)$ is incomplete. This is why we call $\{t,\rho,z,\phi\}$ in Eq(1) as Weyl's canonical coordinates rather than cylindrical coordinates, although they have a lot in common; for example, the Laplacian $\nabla^2:= \partial_{\rho\rho}+\frac{1}{\rho}\partial_\rho +\partial_{zz}$ in Eq(7) is exactly the two-dimensional geometric Laplacian in cylindrical coordinates.

## Nonextremal Reissner-Nordström solution

The Weyl potentials generating the nonextremal Reissner-Nordström solution ($M>|Q|$) as solutions to Eqs(7} are given by[1][2][3]

$(17)\quad \psi_{RN}=\frac{1}{2}\ln\frac{L^2-(M^2-Q^2)}{(L+M)^2} \,, \quad \gamma_{RN}=\frac{1}{2}\ln\frac{L^2-(M^2-Q^2)}{l_+ l_-}\,,$

where

$(18)\quad L=\frac{1}{2}\big(l_+ + l_- \big)\,,\quad l_+ =\sqrt{\rho^2+(z+ \sqrt{M^2-Q^2})^2}\,,\quad l_- =\sqrt{\rho^2+(z-\sqrt{M^2-Q^2})^2}\,.$

Thus, given $\psi_{RN}$ and $\gamma_{RN}$, Weyl's metric becomes

$(19)\quad ds^2=-\frac{L^2-(M^2-Q^2)}{(L+M)^2}dt^2+\frac{(L+M)^2}{l_+ l_-}(d\rho^2+dz^2)+\frac{(L+M)^2}{L^2-(M^2-Q^2)}\rho^2 d\phi^2\,,$

and employing the following transformations

$(20)\quad L+M=r\,,\quad l_+ + l_- =2\sqrt{M^2-Q^2}\,\cos\theta\,,\quad z=(r-M)\cos\theta\,,$
$\;\;\quad \rho=\sqrt{r^2-2Mr+Q^2}\,\sin\theta\,,\quad l_+ l_-=(r-M)^2-(M^2-Q^2)\cos^2\theta\,,$

one can obtain the common form of non-extremal Reissner-Nordström metric in the usual $\{t,r,\theta,\phi\}$ coordinates,

$(21)\quad ds^2=-\Big(1-\frac{2M}{r}+\frac{Q^2}{r^2} \Big)\,dt^2+\Big(1-\frac{2M}{r}+\frac{Q^2}{r^2} \Big)^{-1}dr^2+r^2d\theta^2+r^2\sin^2\theta\, d\phi^2\,.$

## Extremal Reissner-Nordström solution

The potentials generating the extremal Reissner-Nordström solution ($M=|Q|$) as solutions to Eqs(7} are given by[3] (Note: We treat the extremal solution separately because it is much more than the degenerate state of the nonextremal counterpart.)

$(22)\quad \psi_{ERN}=\frac{1}{2}\ln\frac{L^2}{(L+M)^2}\,,\quad \gamma_{ERN}=0\,,\quad\text{with}\quad L=\sqrt{\rho^2+z^2}\,.$

Thus, the extremal Reissner-Nordström metric reads

$(23)\quad ds^2=-\frac{L^2}{(L+M)^2}dt^2+\frac{(L+M)^2}{L^2}(d\rho^2+dz^2+\rho^2d\phi^2)\,,$

and by substituting

$(24)\quad L+M=r\,,\quad z=L\cos\theta\,,\quad \rho=L\sin\theta\,,$

we obtain the extremal Reissner-Nordström metric in the usual $\{t,r,\theta,\phi\}$ coordinates,

$(25)\quad ds^2=-\Big(1-\frac{M}{r} \Big)^2 dt^2+\Big(1-\frac{M}{r} \Big)^{-2}dr^2+r^2d\theta^2+r^2\sin^2\theta\, d\phi^2\,.$

Mathematically, the extremal Reissner-Nordström can be obtained by taking the limit $Q\to M$ 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 $\gamma(\rho,z)$ (like the extremal Reissner-Nordström metric) constitute a special subclass which have only one metric potential $\psi(\rho,z)$ 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,[5][6]

$(26)\quad ds^2\,=-e^{2\lambda(\rho,z,\phi)}dt^2+e^{-2\lambda(\rho,z,\phi)}\Big(d\rho^2+dz^2+\rho^2 d\phi^2 \Big)\,,$

where we use $\lambda$ in Eq(22) as the single metric function in place of $\psi$ in Eq(1) to emphasize that they are different by axial symmetry ($\phi$-dependence).

## Weyl vacuum solutions in spherical coordinates

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

$(27)\quad ds^2\,=-e^{2\psi(r,\theta)}dt^2+e^{2\gamma(r,\theta)-2\psi(r,\theta)}(dr^2+r^2d\theta^2)+e^{-2\psi(r,\theta)}\rho^2 d\phi^2\,,$

which equals Eq(1) via the coordinate transformation $(t,\rho,z,\phi)\mapsto(t,r\sin\theta,r\cos\theta,\phi)$ (Note: As shown by Eqs(15)(21)(24), this transformation is not always applicable.) In the vacuum case, Eq(8.b) for $\psi(r,\theta)$ becomes

$(28)\quad r^2\psi_{,\,rr}+2r\,\psi_{,\,r}+\psi_{,\,\theta\theta}+\cot\theta\cdot\psi_{,\,\theta}\,=\,0\,.$

The asymptotically flat solutions to Eq(28) is[1]

$(29)\quad \psi(r,\theta)\,=-\sum_{n=0}^\infty a_n \frac{P_n(\cos\theta)}{r^{n+1}}\,,$

where $P_n(\cos\theta)$ represent Legendre polynomials, and $a_n$ are multipole coefficients. The other metric potential $\gamma(r,\theta)$is given by[1]

$(30)\quad \gamma(r,\theta)\,=-\sum_{l=0}^\infty \sum_{m=0}^\infty a_l a_m$ $\frac{(l+1)(m+1)}{l+m+2}$ $\frac{P_l P_m-P_{l+1}P_{m+1}}{r^{l+m+2}}\,.$