Conformastatic spacetimes refer to a special class of static solutions to Einstein's equation in general relativity.
Contents
Introduction
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
Reduced electrovac field equations
In consistency with the conformastatic geometry Eq(1), the electrostatic field would arise from an electrostatic potential
which would yield the electromagnetic field tensor
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
Extremal Reissner–Nordström spacetime
The extremal Reissner–Nordström spacetime is a typical conformastatic solution. In this case, the metric function is identified as
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
The Weyl electrovac field equations would reduce to the following ones with
where
It is also noticeable that, Eqs(14) for Weyl are consistent but not identical with the conformastatic Eqs(6)(7) above.