Conformastatic spacetimes

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

Introduction[edit]

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 + 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[edit]

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].

Linearization of electrovac field equations[edit]

Examples[edit]

Extremal Reissner–Nordström spacetime[edit]

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^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[edit]

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

Comparison with Weyl spacetimes[edit]

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.

References[edit]

  1. ^ John Lighton Synge. Relativity: The General Theory, Chapter VIII. Amsterdam: North-Holland Publishing Company (Interscience), 1960.
  2. ^ 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.
  3. ^ 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]
  4. ^ 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]
  5. ^ 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]
  6. ^ 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]

See also[edit]