# 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 $(1)\qquad ds^{2}=-e^{2\Psi (\rho ,\phi ,z)}dt^{2}+e^{-2\Psi (\rho ,\phi ,z)}{\Bigd\rho ^{2}+dz^{2}+\rho ^{2}d\phi ^{2}{\Big )}\;,$ 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
$(5)\qquad T_{ab}^{(EM)}={\frac {1}{4\pi }}{\BigF_{ac}F_{b}^{\;\;c}-{\frac {1}{4}}g_{ab}F_{cd}F^{cd}{\Big )}\;.$
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 $[\rho ,z,\phi ]$.

## Examples
### 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
$(9)\qquad ds^{2}=-{\frac {L^{2}}{(L+M)^{2}}}dt^{2}+{\frac {(L+M)^{2}}{L^{2}}}\,{\bigd\rho ^{2}+dz^{2}+\rho ^{2}d\varphi ^{2}{\big )}\;.$
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,
$(11)\;\;\quad ds^{2}=-{\Big1-{\frac {M}{r}}{\Big )}^{2}dt^{2}+{\Big1-{\frac {M}{r}}{\Big )}^{-2}dr^{2}+r^{2}{\Bigd\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}{\Big )}\;.$

### 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 $\gamma (\rho ,z)$:
$(14.a)\quad \nabla ^{2}\psi =\,(\nabla \psi )^{2}$ $(14.b)\quad \nabla ^{2}\psi =\,e^{-2\psi }(\nabla \Phi )^{2}$ $(14.c)\quad \psi _{,\,\rho }^{2}-\psi _{,\,z}^{2}=e^{-2\psi }{\big\Phi _{,\,\rho }^{2}-\Phi _{,\,z}^{2}{\big )}$ $(14.d)\quad 2\psi _{,\,\rho }\psi _{,\,z}=2e^{-2\psi }\Phi _{,\,\rho }\Phi _{,\,z}$ $(14.e)\quad \nabla ^{2}\Phi =\,2\nabla \psi \nabla \Phi \,,$
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.