The line element for the conformastatic class of solutions in Weyl's canonical coordinates reads
as a solution to the field equation
Eq(1) has only one metric function to be identified, and for each concrete , Eq(1) would yields a specific conformastatic spacetime.
In consistency with the conformastatic geometry Eq(1), the electrostatic field would arise from an electrostatic potential without spatial symmetry:
which would yield the electromagnetic field tensor by
as well as the corresponding stress–energy tensor by
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 :
where and are respectively the generic Laplace and gradient operators. in Eq(7), run freely over the coordinates .
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):
Hence, a Weyl solution become conformastatic if the metric function vanishes, and the other metric function drops the axial symmetry:
The Weyl electrovac field equations would reduce to the following ones with :
where and 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.
^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.
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^ abcF 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]
^ abcdIvan 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]