Conformastatic spacetimes

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


The line element for the conformastatic class of solutions in Weyl's canonical coordinates reads[1][2][3][4][5][6]

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[edit]

In consistency with the conformastatic geometry Eq(1), the electrostatic field would arise from an electrostatic potential without spatial symmetry:[3][4][5]

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 :[3][5]

where and are respectively the generic Laplace and gradient operators. in Eq(7), run freely over the coordinates .

Linearization of electrovac field equations[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]

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[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):

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.


  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]