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]^[2]^[3]^[4]^[5]^[6]
$(1)\qquad ds^{2}=-e^{2\Psi (\rho ,\phi ,z)}dt^{2}+e^{-2\Psi (\rho ,\phi ,z)}{\Big (}d\rho ^{2}+dz^{2}+\rho ^{2}d\phi ^{2}{\Big )}\;,$
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]^[4]^[5]
$(3)\qquad A_{a}=\Phi (\rho ,z,\phi )[dt]_{a}\;,$
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 $\Psi (\rho ,\phi ,z)$ :^[3]^[5]

$(6)\qquad \nabla ^{2}\Psi \,=\,e^{-2\Psi }\,\nabla \Phi \,\nabla \Phi$
$(7)\qquad \Psi _{i}\Psi _{j}=e^{-2\Psi }\Phi _{i}\Phi _{j}$

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^[4]^[5]

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

which put Eq(1) into the concrete form

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

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,

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

Charged dust disks

Some conformastatic solutions have been adopted to describe charged dust disks.^[3]

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.

References

^ John Lighton Synge. Relativity: The General Theory, Chapter VIII. Amsterdam: North-Holland Publishing Company (Interscience), 1960.
^ Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, Eduard Herlt . Exact Solutions of Einstein's Field Equations (2nd Edition), Chapter 18. Cambridge: Cambridge University Press, 2003.
^ ^a ^b ^c ^d Guillermo A Gonzalez, Antonio C Gutierrez-Pineres, Paolo A Ospina. Finite axisymmetric charged dust disks in conformastatic spacetimes. Physical Review D 78 (2008): 064058. arXiv:0806.4285[gr-qc]
^ ^a ^b ^c F D Lora-Clavijo, P A Ospina-Henao, J F Pedraza. Charged annular disks and Reissner–Nordström type black holes from extremal dust. Physical Review D 82 (2010): 084005. arXiv:1009.1005[gr-qc]
^ ^a ^b ^c ^d Ivan Booth, David Wenjie Tian. Some spacetimes containing non-rotating extremal isolated horizons. Accepted by Classical and Quantum Gravity. arXiv:1210.6889[gr-qc]
^ Antonio C Gutierrez-Pineres, Guillermo A Gonzalez, Hernando Quevedo. Conformastatic disk-haloes in Einstein-Maxwell gravity. Physical Review D 87 (2013): 044010. arXiv:1211.4941[gr-qc]