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Fix CS1 deprecated date parameter errors

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'''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<ref name=CS1>John Lighton Synge. ''Relativity: The General Theory'', Chapter VIII. Amsterdam: North-Holland Publishing Company (Interscience), 1960.</ref><ref name=CS2>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.</ref><ref name=CS3>Guillermo A Gonzalez, Antonio C Gutierrez-Pineres, Paolo A Ospina. ''Finite axisymmetric charged dust disks in conformastatic spacetimes''. Physical Review D '''78''' (2008): 064058. [http://arxiv.org/abs/0806.4285 arXiv:0806.4285[gr-qc]]</ref><ref name=CS4>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. [http://arxiv.org/abs/1009.1005 arXiv:1009.1005[gr-qc]]</ref><ref name=CS5>Ivan Booth, David Wenjie Tian. ''Some spacetimes containing non-rotating extremal isolated horizons''. Accepted by Classical and Quantum Gravity. [http://arxiv.org/abs/1210.6889 arXiv:1210.6889[gr-qc]]</ref><ref name=CS6>Antonio C Gutierrez-Pineres, Guillermo A Gonzalez, Hernando Quevedo. ''Conformastatic disk-haloes in Einstein-Maxwell gravity''. Physical Review D '''87''' (2013): 044010. [http://arxiv.org/abs/1211.4941 arXiv:1211.4941[gr-qc]]</ref> 
<math>(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)\;,</math> 
as a solution to the field equation 
<math>(2)\qquad R_{ab}-\frac{1}{2}Rg_{ab}=8\pi T_{ab}\;.</math> 
Eq(1) has only one metric function <math>\Psi(\rho,\phi,z)</math> to be identified, and for each concrete <math>\Psi(\rho,\phi,z)</math>, 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 <math>A_a</math> without spatial symmetry:<ref name=CS3 /><ref name=CS4 /><ref name=CS5 /> 
<math>(3)\qquad A_a = \Phi(\rho,z,\phi) [dt]_a\;,</math> 
which would yield the electromagnetic field tensor <math>F_{ab}</math> by 
<math>(4)\qquad F_{ab} = A_{b\,;a}-A_{a\,;b}\;,</math> 
as well as the corresponding [[stress-energy tensor]] by 
<math>(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)\;.</math>

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 <math>\Psi(\rho,\phi,z)</math>:<ref name=CS3 /><ref name=CS5 />

<math>(6)\qquad \nabla^2\Psi \,=\,e^{- 2 \Psi} \,\nabla\Phi\, \nabla\Phi</math> 
<math>(7)\qquad \Psi_i \Psi_j = e^{-2 \Psi} \Phi_i \Phi_j </math>

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

== Linearization of electrovac field equations ==

{{Empty section|date=June 2013}}

== Examples ==

=== Extremal Reissner-Nordström spacetime ===

The extremal Reissner-Nordström spacetime is a typical conformastatic solution. In this case, the metric function is identified as<ref name=CS4 /><ref name=CS5 />

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

which put Eq(1) into the concrete form 

<math>(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)\;.</math>

Applying the transformations

<math>(10)\;\;\quad L=r-M\;,\quad z=(r-M)\cos\theta\;,\quad \rho=(r-M)\sin\theta\;,</math>

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

<math>(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)\;.</math>

=== Charged dust disks ===

Some conformastatic solutions have been adopted to describe charged dust disks.<ref name=CS3 />

== 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 metrics|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):
 
<math>(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\,.
</math> 
Hence, a Weyl solution become conformastatic if the metric function <math>\gamma(\rho,z)</math> vanishes, and the other metric function <math>\psi(\rho,z)</math> drops the axial symmetry: 
<math>(13)\;\;\quad \gamma(\rho,z)\equiv 0\;, \quad \psi(\rho,z)\mapsto \Psi(\rho,\phi,z) \,.
</math> 
The [[Weyl_metrics#Reduced_field_equations_for_electrovac_Weyl_solutions|Weyl electrovac field equations]] would reduce to the following ones with <math>\gamma(\rho,z)</math>:

<math>(14.a)\quad \nabla^2 \psi =\,(\nabla\psi)^2</math> 
<math>(14.b)\quad \nabla^2\psi =\,e^{-2\psi} (\nabla\Phi)^2 </math> 
<math>(14.c)\quad \psi^2_{,\,\rho}-\psi^2_{,\,z}=e^{-2\psi}\big(\Phi^2_{,\,\rho}-\Phi^2_{,\,z}\big) </math> 
<math>(14.d)\quad 2\psi_{,\,\rho}\psi_{,\,z}= 2e^{-2\psi}\Phi_{,\,\rho}\Phi_{,\,z} </math> 
<math>(14.e)\quad \nabla^2\Phi =\,2\nabla\psi \nabla\Phi\,,</math> 

where <math>\nabla^2 = \partial_{\rho\rho}+\frac{1}{\rho}\,\partial_\rho +\partial_{zz}</math> and <math>\nabla=\partial_\rho\, \hat{e}_\rho +\partial_z\, \hat{e}_z </math> 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 ==

{{reflist}}

== See also ==

*[[Weyl metrics]]
*[[Reissner–Nordström metric]]



[[Category:General relativity]]

Diophantine quintuple - Revision history

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