Weyl metrics

The Weyl class of solutions has the generic form {{NumBlk|| 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, 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): {{NumBlk|| 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 {{NumBlk|| F_{ab}=A_{b\,;\,a}-A_{a\,;\,b}=A_{b\,,\,a}-A_{a\,,\,b}|}} {{NumBlk|| T_{ab}=\frac{1}{4\pi}\,\left(\, F_{ac}F_b^{\;c} -\frac{1}{4}g_{ab}F_{cd}F^{cd} \right)\,,|}} which respects the source-free covariant Maxwell equations: {{NumBlk|| \big(F^{ab}\big)_{;\,b}=0\,,\quad F_{[ab\,;\,c]}=0\,.|}} Eq(5.a) can be simplified to: {{NumBlk|| \left(\sqrt{-g}\,F^{ab}\right)_{,\,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 {{NumBlk|| 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 {{NumBlk|| \nabla^2 \psi =\,(\nabla\psi)^2 +\gamma_{,\,\rho\rho}+\gamma_{,\,zz}|}} {{NumBlk|| \nabla^2\psi =\,e^{-2\psi} (\nabla\Phi)^2 |}} {{NumBlk|| \frac{1}{\rho}\,\gamma_{,\,\rho} =\,\psi^2_{,\,\rho}-\psi^2_{,\,z}-e^{-2\psi}\big(\Phi^2_{,\,\rho}-\Phi^2_{,\,z}\big) |}} {{NumBlk|| \frac{1}{\rho}\,\gamma_{,\,z} =\,2\psi_{,\,\rho}\psi_{,\,z}- 2e^{-2\psi}\Phi_{,\,\rho}\Phi_{,\,z} |}} 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 Specifically in the simplest vacuum case with \Phi=0 and T_{ab}=0, Eqs(7.a-7.e) reduce to {{NumBlk|| \gamma_{,\,\rho\rho}+\gamma_{,\,zz}=-(\nabla\psi)^2 |}} {{NumBlk|| \gamma_{,\,\rho}=\rho\,\Big(\psi^2_{,\,\rho}-\psi^2_{,\,z} \Big) |}} {{NumBlk|| \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.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: {{NumBlk|| \psi_{,\,\rho\rho}+\frac{1}{\rho}\psi_{,\,\rho}+\psi_{,\,zz}=\,(\psi_{,\,\rho})^2+(\psi_{,\,z})^2 +\gamma_{,\,\rho\rho}+\gamma_{,\,zz}|}} {{NumBlk|| \psi_{,\,\rho\rho}+\frac{1}{\rho} \psi_{,\,\rho} + \psi_{,\,zz}=e^{-2\psi}\big(\Phi^2_{,\,\rho}+\Phi^2_{,\,z}\big)|}} {{NumBlk|| \frac{1}{\rho}\,\gamma_{,\,\rho} =\,\psi^2_{,\,\rho}-\psi^2_{,\,z}-e^{-2\psi}\big(\Phi^2_{,\,\rho}-\Phi^2_{,\,z}\big) |}} {{NumBlk|| \frac{1}{\rho}\,\gamma_{,\,z} =\,2\psi_{,\,\rho}\psi_{,\,z}- 2e^{-2\psi} \Phi_{,\,\rho}\Phi_{,\,z} |}} {{NumBlk|| \Phi_{,\,\rho\rho}+\frac{1}{\rho}\Phi_{,\,\rho}+\Phi_{,\,zz} = \,2\psi_{,\,\rho} \Phi_{,\,\rho} +2\psi_{,\,z}\Phi_{,\,z} |}} and {{NumBlk|| (\psi_{,\,\rho})^2+(\psi_{,\,z})^2=-\gamma_{,\,\rho\rho}-\gamma_{,\,zz} |}} {{NumBlk|| \psi_{,\,\rho\rho}+\frac{1}{\rho}\psi_{,\,\rho}+\psi_{,\,zz} =0 |}} {{NumBlk|| \gamma_{,\,\rho}=\rho\,\Big(\psi^2_{,\,\rho}-\psi^2_{,\,z} \Big) |}} {{NumBlk|| \gamma_{,\,z}=2\,\rho\,\psi_{,\,\rho}\psi_{,\,z} \,.|}} 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 \Phi(\rho,z) via a function \psi=\psi(\Phi) (which means geometry depends on energy), and it follows that {{NumBlk|| \psi_{,\,i}=\psi_{,\,\Phi}\cdot \Phi_{,\,i} \quad,\quad \nabla\psi=\psi_{,\,\Phi}\cdot \nabla \Phi \quad,\quad \nabla^2\psi=\psi_{,\,\Phi}\cdot \nabla^2 \Phi+\psi_{,\,\Phi\Phi}\cdot (\nabla \Phi)^2 ,|}} Eq(B.1) immediately turns Eq(7.b) and Eq(7.e) respectively into {{NumBlk|| \Psi_{,\,\Phi}\cdot \nabla^2\Phi\,=\,\big(e^{-2\psi}-\psi_{,\,\Phi\Phi} \big)\cdot (\nabla\Phi)^2,|}} {{NumBlk|| \nabla^2\Phi\,=\,2\psi_{,\,\Phi}\cdot (\nabla\Phi)^2,|}} which give rise to {{NumBlk|| \psi_{,\,\Phi\Phi}+2 \,\big(\psi_{,\,\Phi}\big)^2-e^{-2\psi}=0.|}} Now replace the variable \psi by \zeta:= e^{2\psi}, and Eq(B.4) is simplified to {{NumBlk|| \zeta_{,\,\Phi\Phi}-2=0.|}} Direct quadrature of Eq(B.5) yields \zeta=e^{2\psi}=\Phi^2+\tilde{C}\Phi+B, with \{B, \tilde{C}\} being integral constants. To resume asymptotic flatness at spatial infinity, we need \lim_{\rho,z\to\infty}\Phi=0 and \lim_{\rho,z\to\infty}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 {{NumBlk|| e^{2\psi}=\Phi^2-2C\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^{\pm2\psi}=\sum_{n=0}^{\infty} \frac{(\pm2\psi)^n}{n!}; thus in the approximation for weak field limit \psi\to 0, one has {{NumBlk|| g_{tt}=-(1+2\psi)-\mathcal {O}(\psi^2)\,,\quad g_{\phi\phi}=1-2\psi+\mathcal {O}(\psi^2)\,,|}} and therefore {{NumBlk|| ds^2\approx-\Big(1+2\psi(\rho,z)\Big)\,dt^2+\Big(1-2\psi(\rho,z)\Big)\left[e^{2\gamma}(d\rho^2+dz^2) + \rho^2 d\phi^2\right]\,.|}} 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, {{NumBlk|| ds^2=-\Big(1+2\Phi_{N}(\rho,z)\Big)\,dt^2+\Big(1-2\Phi_{N}(\rho,z)\Big) \, \left[d\rho^2+dz^2+\rho^2 d\phi^2\right]\,.|}} 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. ==Schwarzschild solution==

The Weyl potentials generating Schwarzschild's metric as solutions to the vacuum equations Eq() are given by {{NumBlk|| \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 {{NumBlk|| 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.) Given \psi_{SS} and \gamma_{SS}, Weyl's metric Eq() becomes {{NumBlk|| 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 {{NumBlk||\begin{align} &L+M=r\,,\quad l_+ - l_- =2M\cos\theta\,,\quad z=(r-M)\cos\theta\,,\\ &\rho=\sqrt{r^2-2Mr}\,\sin\theta\,,\quad l_+ l_-=(r-M)^2-M^2\cos^2\theta\,, \end{align}|}} one can obtain the common form of Schwarzschild metric in the usual \{t,r,\theta,\phi\} coordinates, {{NumBlk|| ds^2 = -\left(1-\frac{2M}{r} \right)\,dt^2+\left(1-\frac{2M}{r} \right)^{-1} dr^2 + r^2 d\theta^2 + r^2\sin^2\theta\, d\phi^2\,.|}} The metric Eq() cannot be directly transformed into Eq() 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() 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() 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() are given by {{NumBlk|| \psi_{RN} = \frac{1}{2}\ln\frac{L^2-\left(M^2-Q^2\right)}{\left(L+M\right)^2} \,, \quad \gamma_{RN} = \frac{1}{2} \ln\frac{L^2-\left(M^2-Q^2\right)}{l_+ l_-}\,,|}} where {{NumBlk|| L = \frac{1}{2}\big(l_+ + l_- \big)\,,\quad l_+ =\sqrt{\rho^2+\left(z+ \sqrt{M^2-Q^2}\right)^2}\,,\quad l_- =\sqrt{\rho^2+\left(z-\sqrt{M^2-Q^2}\right)^2}\,.|}} Thus, given \psi_{RN} and \gamma_{RN}, Weyl's metric becomes {{NumBlk|| ds^2=-\frac{L^2-\left(M^2-Q^2\right)}{\left(L+M\right)^2}dt^2+\frac{\left(L+M\right)^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 {{NumBlk||\begin{align} &L+M=r\,,\quad l_+ - l_- =2\sqrt{M^2-Q^2}\,\cos\theta\,,\quad z=(r-M)\cos\theta\,,\\ &\rho=\sqrt{r^2-2Mr+Q^2}\,\sin\theta\,,\quad l_+ l_-=(r-M)^2-(M^2-Q^2)\cos^2\theta\,, \end{align}|}} one can obtain the common form of non-extremal Reissner–Nordström metric in the usual \{t,r,\theta,\phi\} coordinates, {{NumBlk|| ds^2 = -\left(1-\frac{2M}{r}+\frac{Q^2}{r^2} \right) dt^2+\left(1-\frac{2M}{r}+\frac{Q^2}{r^2} \right)^{-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() are given by {{NumBlk|| 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() as the single metric function in place of \psi in Eq() 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 {{NumBlk|| 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() via the coordinate transformation (t,\rho,z,\phi)\mapsto(t,r\sin\theta,r\cos\theta,\phi) (Note: As shown by Eqs()()(), this transformation is not always applicable.) In the vacuum case, Eq() for \psi(r,\theta) becomes {{NumBlk|| r^2\psi_{,\,rr}+2r\,\psi_{,\,r} + \psi_{,\,\theta\theta} + \cot\theta\cdot\psi_{,\,\theta} \,=\, 0\,.|}} The asymptotically flat solutions to Eq() is {{NumBlk|| \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 {{NumBlk|| \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}}\,.|}} ==See also==