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]

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

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

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

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

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

Reduced field equations for electrovac Weyl solutions

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

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

which respects the source-free covariant Maxwell equations:

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

Eq(5.a) can be simplified to:

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

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

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

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

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

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

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

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

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

We can firstly obtain ${\displaystyle \psi (\rho ,z)}$ by solving Eq(8.b), and then integrate Eq(8.c) and Eq(8.d) for ${\displaystyle \gamma (\rho ,z)}$. Practically, Eq(8.a) arising from ${\displaystyle R=0}$ 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.

Newtonian analogue of metric potential Ψ(ρ,z)

In Weyl's metric Eq(1), ${\displaystyle e^{\pm 2\psi }=\sum _{n=0}^{\infty }{\frac {(\pm 2\psi )^{n}}{n!}}}$; thus in the approximation for weak field limit ${\displaystyle \psi \to 0}$, one has

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

and therefore

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

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]

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

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

${\displaystyle (12)\quad \psi _{SS}={\frac {1}{2}}\ln {\frac {L-M}{L+M}}\,,\quad \gamma _{SS}={\frac {1}{2}}\ln {\frac {L^{2}-M^{2}}{l_{+}l_{-}}}\,,}$

where

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

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

Given ${\displaystyle \psi _{SS}}$ and ${\displaystyle \gamma _{SS}}$, Weyl's metric Eq(\ref{Weyl metric in canonical coordinates}) becomes

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

and after substituting the following mutually consistent relations

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

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

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

The metric Eq(14) cannot be directly transformed into Eq(16) by performing the standard cylindrical-spherical transformation ${\displaystyle (t,\rho ,z,\phi )=(t,r\sin \theta ,r\cos \theta ,\phi )}$, because ${\displaystyle \{t,r,\theta ,\phi \}}$ is complete while ${\displaystyle (t,\rho ,z,\phi )}$ is incomplete. This is why we call ${\displaystyle \{t,\rho ,z,\phi \}}$ in Eq(1) as Weyl's canonical coordinates rather than cylindrical coordinates, although they have a lot in common; for example, the Laplacian ${\displaystyle \nabla ^{2}:=\partial _{\rho \rho }+{\frac {1}{\rho }}\partial _{\rho }+\partial _{zz}}$ 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 (${\displaystyle M>|Q|}$) as solutions to Eqs(7} are given by[1][2][3]

${\displaystyle (17)\quad \psi _{RN}={\frac {1}{2}}\ln {\frac {L^{2}-(M^{2}-Q^{2})}{(L+M)^{2}}}\,,\quad \gamma _{RN}={\frac {1}{2}}\ln {\frac {L^{2}-(M^{2}-Q^{2})}{l_{+}l_{-}}}\,,}$

where

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

Thus, given ${\displaystyle \psi _{RN}}$ and ${\displaystyle \gamma _{RN}}$, Weyl's metric becomes

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

and employing the following transformations

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

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

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

Extremal Reissner–Nordström solution

The potentials generating the extremal Reissner–Nordström solution (${\displaystyle M=|Q|}$) 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.)

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

Thus, the extremal Reissner–Nordström metric reads

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

and by substituting

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

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

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

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

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

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

Weyl vacuum solutions in spherical coordinates

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

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

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

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

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

${\displaystyle (29)\quad \psi (r,\theta )\,=-\sum _{n=0}^{\infty }a_{n}{\frac {P_{n}(\cos \theta )}{r^{n+1}}}\,,}$

where ${\displaystyle P_{n}(\cos \theta )}$ represent Legendre polynomials, and ${\displaystyle a_{n}}$ are multipole coefficients. The other metric potential ${\displaystyle \gamma (r,\theta )}$is given by[1]

${\displaystyle (30)\quad \gamma (r,\theta )\,=-\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }a_{l}a_{m}}$ ${\displaystyle {\frac {(l+1)(m+1)}{l+m+2}}}$ ${\displaystyle {\frac {P_{l}P_{m}-P_{l+1}P_{m+1}}{r^{l+m+2}}}\,.}$