Matrix splitting

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

$(1)d{s}^2=-e^{2\psi (\rho ,z)}d{t}^2+e^{2\gamma (\rho ,z)-2\psi (\rho ,z)}(d{\rho }^2+d{z}^2)+e^{-2\psi (\rho ,z)}{\rho }^{2}d{\varphi }^2,$

where $\psi (\rho ,z)$ and $\gamma (\rho ,z)$ are two metric potentials dependent on Weyl's canonical coordinates $\{\rho ,z\}$. The coordinate system $\{t,\rho ,z,\varphi \}$ serves best for symmetries of Weyl's spacetime (with two Killing vector fields being ${\xi }^t={\partial }_t$ and ${\xi }^{\varphi }={\partial }_{\varphi }$) and often acts like cylindrical coordinates, 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):

$(2)R_{ab}-\frac{1}{2}R{g}_{ab}=8\pi T_{ab},$

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

$(3)F_{ab}=A_{b;a}-A_{a;b}=A_{b,a}-A_{a,b}$
$(4)T_{ab}=\frac{1}{4\pi }(F_{ac}{F}_b^c-\frac{1}{4}{g}_{ab}{F}_{cd}{F}^{cd}),$

which respects the source-free covariant Maxwell equations:

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

Eq(5.a) can be simplified to:

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

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

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

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

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

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

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

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

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

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.

Box A: Remarks on the electrovac field equation

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:

$(A.1.a){\psi }_{,\rho \rho }+\frac{1}{\rho }{\psi }_{,\rho }+{\psi }_{,zz}=({\psi }_{,\rho }{)}^2+({\psi }_{,z}{)}^2+{\gamma }_{,\rho \rho }+{\gamma }_{,zz}$
$(A.1.b){\psi }_{,\rho \rho }+\frac{1}{\rho }{\psi }_{,\rho }+{\psi }_{,zz}=e^{-2\psi }({\mathrm{\Phi }}_{,\rho }^2+{\mathrm{\Phi }}_{,z}^2)$
$(A.1.c)\frac{1}{\rho }{\gamma }_{,\rho }={\psi }_{,\rho }^2-{\psi }_{,z}^2-e^{-2\psi }({\mathrm{\Phi }}_{,\rho }^2-{\mathrm{\Phi }}_{,z}^2)$
$(A.1.d)\frac{1}{\rho }{\gamma }_{,z}=2{\psi }_{,\rho }{\psi }_{,z}-2e^{-2\psi }{\mathrm{\Phi }}_{,\rho }{\mathrm{\Phi }}_{,z}$
$(A.1.e){\mathrm{\Phi }}_{,\rho \rho }+\frac{1}{\rho }{\mathrm{\Phi }}_{,\rho }+{\mathrm{\Phi }}_{,zz}=2{\psi }_{,\rho }{\mathrm{\Phi }}_{,\rho }+2{\psi }_{,z}{\mathrm{\Phi }}_{,z}$

and

$(A.2.a)({\psi }_{,\rho }{)}^2+({\psi }_{,z}{)}^2=-{\gamma }_{,\rho \rho }-{\gamma }_{,zz}$
$(A.2.b){\psi }_{,\rho \rho }+\frac{1}{\rho }{\psi }_{,\rho }+{\psi }_{,zz}=0$
$(A.2.c){\gamma }_{,\rho }=\rho ({\psi }_{,\rho }^2-{\psi }_{,z}^2)$
$(A.2.d){\gamma }_{,z}=2\rho {\psi }_{,\rho }{\psi }_{,z}.$

Box B: Derivation of the Weyl electrovac $\psi \sim \mathrm{\Phi }$ characteristic relation

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 $\psi (\rho ,z)$ relates with the electrostatic scalar potential $\mathrm{\Phi }(\rho ,z)$ via a function $\psi =\psi (\mathrm{\Phi })$ (which means geometry depends on energy), and it follows that

$(B.1){\psi }_{,i}={\psi }_{,\mathrm{\Phi }}\cdot {\mathrm{\Phi }}_{,i},\mathrm{\nabla }\psi ={\psi }_{,\mathrm{\Phi }}\cdot \mathrm{\nabla }\mathrm{\Phi },{\mathrm{\nabla }}^2\psi ={\psi }_{,\mathrm{\Phi }}\cdot {\mathrm{\nabla }}^2\mathrm{\Phi }+{\psi }_{,\mathrm{\Phi }\mathrm{\Phi }}\cdot (\mathrm{\nabla }\mathrm{\Phi }{)}^2,$

Eq(B.1) immediately turns Eq(7.b) and Eq(7.e) respectively into

$(B.2){\mathrm{\Psi }}_{,\mathrm{\Phi }}\cdot {\mathrm{\nabla }}^2\mathrm{\Phi }=(e^{-2\psi }-{\psi }_{,\mathrm{\Phi }\mathrm{\Phi }})\cdot (\mathrm{\nabla }\mathrm{\Phi }{)}^2,$
$(B.3){\mathrm{\nabla }}^2\mathrm{\Phi }=2{\psi }_{,\mathrm{\Phi }}\cdot (\mathrm{\nabla }\mathrm{\Phi }{)}^2,$

which give rise to

$(B.4){\psi }_{,\mathrm{\Phi }\mathrm{\Phi }}+2({\psi }_{,\mathrm{\Phi }}{)}^2-e^{-2\psi }=0.$

Now replace the variable $\psi$ by $\zeta :=e^{2\psi }$, and Eq(B.4) is simplified to

$(B.5){\zeta }_{,\mathrm{\Phi }\mathrm{\Phi }}-2=0.$

Direct quadrature of Eq(B.5) yields $\zeta =e^{2\psi }={\mathrm{\Phi }}^2+\tilde{C}\mathrm{\Phi }+B$, with $\{B,\tilde{C}\}$ being integral constants. To resume asymptotic flatness at spatial infinity, we need $\underset{\rho ,z\to \mathrm{\infty }}{\lim }\mathrm{\Phi }=0$ and $\underset{\rho ,z\to \mathrm{\infty }}{\lim }{e}^{2\psi }=1$, so there should be $B=1$. Also, rewrite the constant $\tilde{C}$ as $-2C$ for mathematical convenience in subsequent calculations, and one finally obtains the characteristic relation implied by Eqs(7.a-7.e) that

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

This relation is important in linearize the Eqs(7.a-7.f) and superpose electrovac Weyl solutions.

Newtonian analogue of metric potential Ψ(ρ,z)

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

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

and therefore

$(10)d{s}^2\approx -(1+2\psi (\rho ,z))d{t}^2+(1-2\psi (\rho ,z))[e^{2\gamma }(d{\rho }^2+d{z}^2)+{\rho }^{2}d{\varphi }^2].$

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,

$(11)d{s}^2=-(1+2{\mathrm{\Phi }}_N(\rho ,z))d{t}^2+(1-2{\mathrm{\Phi }}_N(\rho ,z))[d{\rho }^2+d{z}^2+{\rho }^{2}d{\varphi }^2].$

where ${\mathrm{\Phi }}_N(\rho ,z)$ is the usual Newtonian potential satisfying Poisson's equation $\underset{L}{\overset{2}{\mathrm{\nabla }}}{\mathrm{\Phi }}_N=4\pi {\varrho }_N$, just like Eq(3.a) or Eq(4.a) for the Weyl metric potential $\psi (\rho ,z)$. The similarities between $\psi (\rho ,z)$ and ${\mathrm{\Phi }}_N(\rho ,z)$ 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.

Schwarzschild solution

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

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

where

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

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.)

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

$(14)d{s}^2=-\frac{L-M}{L+M}d{t}^2+\frac{(L+M{)}^2}{l_{+}{l}_{-}}(d{\rho }^2+d{z}^2)+\frac{L+M}{L-M}{\rho }^{2}d{\varphi }^2,$

and after substituting the following mutually consistent relations

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

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

$(16)d{s}^2=-(1-\frac{2M}{r})d{t}^2+(1-\frac{2M}{r}{)}^{-1}d{r}^2+r^{2}d{\theta }^2+r^2{\sin }^2\theta d{\varphi }^2.$

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

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

where

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

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

$(19)d{s}^2=-\frac{L^2-(M^2-Q^2)}{(L+M{)}^2}d{t}^2+\frac{(L+M{)}^2}{l_{+}{l}_{-}}(d{\rho }^2+d{z}^2)+\frac{(L+M{)}^2}{L^2-(M^2-Q^2)}{\rho }^{2}d{\varphi }^2,$

and employing the following transformations

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

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

$(21)d{s}^2=-(1-\frac{2M}{r}+\frac{Q^2}{r^2})d{t}^2+(1-\frac{2M}{r}+\frac{Q^2}{r^2}{)}^{-1}d{r}^2+r^{2}d{\theta }^2+r^2{\sin }^2\theta d{\varphi }^2.$

Extremal Reissner-Nordström solution

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

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

Thus, the extremal Reissner-Nordström metric reads

$(23)d{s}^2=-\frac{L^2}{(L+M{)}^2}d{t}^2+\frac{(L+M{)}^2}{L^2}(d{\rho }^2+d{z}^2+{\rho }^{2}d{\varphi }^2),$

and by substituting

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

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

$(25)d{s}^2=-(1-\frac{M}{r}{)}^{2}d{t}^2+(1-\frac{M}{r}{)}^{-2}d{r}^2+r^{2}d{\theta }^2+r^2{\sin }^2\theta d{\varphi }^2.$

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,

$(26)d{s}^2=-e^{2\lambda (\rho ,z,\varphi )}d{t}^2+e^{-2\lambda (\rho ,z,\varphi )}(d{\rho }^2+d{z}^2+{\rho }^{2}d{\varphi }^2),$

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 ($\varphi$-dependence).

Weyl vacuum solutions in spherical coordinates

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

$(27)d{s}^2=-e^{2\psi (r,\theta )}d{t}^2+e^{2\gamma (r,\theta )-2\psi (r,\theta )}(d{r}^2+r^{2}d{\theta }^2)+e^{-2\psi (r,\theta )}{\rho }^{2}d{\varphi }^2,$

which equals Eq(1) via the coordinate transformation $(t,\rho ,z,\varphi )\mapsto (t,r\sin \theta ,r\cos \theta ,\varphi )$ (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

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

The asymptotically flat solutions to Eq(28) is

$(29)\psi (r,\theta )=-\sum _{n=0}^{\mathrm{\infty }}{a}_n\frac{P_n(\cos \theta )}{r^{n+1}},$

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

$(30)\gamma (r,\theta )=-\sum _{l=0}^{\mathrm{\infty }}\sum _{m=0}^{\mathrm{\infty }}{a}_l{a}_m$ $\frac{(l+1)(m+1)}{l+m+2}$ $\frac{P_l{P}_m-P_{l+1}{P}_{m+1}}{r^{l+m+2}}.$

See also

• Schwarzschild metric
• Reissner–Nordström metric
• Distorted Schwarzschild metric

References

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