Conformastatic spacetimes

Conformastatic spacetimes refer to a special class of static solutions to Einstein's equation in general relativity.

Introduction
The line element for the conformastatic class of solutions in Weyl's canonical coordinates reads

$$(1)\qquad ds^2 = - e^{2 \Psi(\rho,\phi,z)} dt^2 + e^{-2 \Psi(\rho,\phi,z) } \Big(d \rho^2 + d z^2 + \rho^2 d \phi^2 \Big)\;,$$

as a solution to the field equation

$$(2)\qquad R_{ab}-\frac{1}{2}Rg_{ab}=8\pi T_{ab}\;.$$

Eq(1) has only one metric function $$\Psi(\rho,\phi,z)$$ to be identified, and for each concrete $$\Psi(\rho,\phi,z)$$, Eq(1) would yields a specific conformastatic spacetime.

Reduced electrovac field equations
In consistency with the conformastatic geometry Eq(1), the electrostatic field would arise from an electrostatic potential $$A_a$$ without spatial symmetry:

$$(3)\qquad A_a = \Phi(\rho,z,\phi) [dt]_a\;,$$

which would yield the electromagnetic field tensor $$F_{ab}$$ by

$$(4)\qquad F_{ab} = A_{b\,;a}-A_{a\,;b}\;,$$

as well as the corresponding stress–energy tensor by

$$(5)\qquad T_{ab}^{(EM)} = \frac{1}{4\pi}\Big(F_{ac}F_b^{\;\;c}-\frac{1}{4}g_{ab}F_{cd}F^{cd} \Big)\;.$$

Plug Eq(1) and Eqs(3)(4)(5) into "trace-free" (R=0) Einstein's field equation, and one could obtain the reduced field equations for the metric function $$\Psi(\rho,\phi,z)$$:

$$(6)\qquad \nabla^2\Psi \,=\,e^{- 2 \Psi} \,\nabla\Phi\, \nabla\Phi$$

$$(7)\qquad \Psi_i \Psi_j = e^{-2 \Psi} \Phi_i \Phi_j $$

where $$\nabla^2 = \partial_{\rho\rho}+\frac{1}{\rho}\,\partial_\rho +\frac{1}{\rho^2}\partial_{\phi\phi}+\partial_{zz}$$ and $$\nabla=\partial_\rho\, \hat{e}_\rho +\frac{1}{\rho}\partial_\phi\, \hat{e}_\phi +\partial_z\, \hat{e}_z $$ are respectively the generic Laplace and gradient operators. in Eq(7), $$i\,,j$$ run freely over the coordinates $$[\rho, z, \phi]$$.

Extremal Reissner–Nordström spacetime
The extremal Reissner–Nordström spacetime is a typical conformastatic solution. In this case, the metric function is identified as

$$(8)\qquad \Psi_{ERN}\,=\,\ln\frac{L}{L+M}\;,\quad L=\sqrt{\rho^2+z^2}\;,$$

which put Eq(1) into the concrete form

$$(9)\qquad ds^2=-\frac{L^2}{(L+M)^2}dt^2+\frac{(L+M)^2}{L^2}\,\big(d\rho^2+dz^2+\rho^2d\varphi^2\big)\;.$$

Applying the transformations

$$(10)\;\;\quad L=r-M\;,\quad z=(r-M)\cos\theta\;,\quad \rho=(r-M)\sin\theta\;,$$

one obtains the usual form of the line element of extremal Reissner–Nordström solution,

$$(11)\;\;\quad ds^2=-\Big(1-\frac{M}{r}\Big)^2 dt^2+\Big(1-\frac{M}{r}\Big)^{-2} dr^2+r^2 \Big(d\theta^2+\sin^2\theta\,d\phi^2\Big)\;.$$

Charged dust disks
Some conformastatic solutions have been adopted to describe charged dust disks.

Comparison with Weyl spacetimes
Many solutions, such as the extremal Reissner–Nordström solution discussed above, can be treated as either a conformastatic metric or Weyl metric, so it would be helpful to make a comparison between them. The Weyl spacetimes refer to the static, axisymmetric class of solutions to Einstein's equation, whose line element takes the following form (still in Weyl's canonical coordinates):

$$(12)\;\;\quad ds^2=-e^{2\psi(\rho,z)}dt^2+e^{2\gamma(\rho,z)-2\psi(\rho,z)}(d\rho^2+dz^2)+e^{-2\psi(\rho,z)}\rho^2 d\phi^2\,. $$

Hence, a Weyl solution become conformastatic if the metric function $$\gamma(\rho,z)$$ vanishes, and the other metric function $$\psi(\rho,z)$$ drops the axial symmetry:

$$(13)\;\;\quad \gamma(\rho,z)\equiv 0\;, \quad \psi(\rho,z)\mapsto \Psi(\rho,\phi,z) \,. $$

The Weyl electrovac field equations would reduce to the following ones with $$\gamma(\rho,z)$$:

$$(14.a)\quad \nabla^2 \psi =\,(\nabla\psi)^2$$

$$(14.b)\quad \nabla^2\psi =\,e^{-2\psi} (\nabla\Phi)^2 $$

$$(14.c)\quad \psi^2_{,\,\rho}-\psi^2_{,\,z}=e^{-2\psi}\big(\Phi^2_{,\,\rho}-\Phi^2_{,\,z}\big) $$

$$(14.d)\quad 2\psi_{,\,\rho}\psi_{,\,z}= 2e^{-2\psi}\Phi_{,\,\rho}\Phi_{,\,z} $$

$$(14.e)\quad \nabla^2\Phi =\,2\nabla\psi \nabla\Phi\,,$$

where $$\nabla^2 = \partial_{\rho\rho}+\frac{1}{\rho}\,\partial_\rho +\partial_{zz}$$ and $$\nabla=\partial_\rho\, \hat{e}_\rho +\partial_z\, \hat{e}_z $$ are respectively the reduced cylindrically symmetric Laplace and gradient operators.

It is also noticeable that, Eqs(14) for Weyl are consistent but not identical with the conformastatic Eqs(6)(7) above.