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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.
Reduced electrovac field equations
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
Linearization of electrovac field equations
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Extremal Reissner–Nordström spacetime
which put Eq(1) into the concrete form
Applying the transformations
one obtains the usual form of the line element of extremal Reissner–Nordström solution,
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):
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.
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