Weyl–Lewis–Papapetrou coordinates

General relativity
${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}$
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In general relativity, the Weyl–Lewis–Papapetrou coordinates are used in solutions to the vacuum region surrounding an axisymmetric distribution of mass–energy. They are named for Hermann Weyl, Thomas Lewis, and Achilles Papapetrou.

The square of the line element is of the form:

${\displaystyle ds^{2}=-e^{2\nu }dt^{2}+\rho ^{2}B^{2}e^{-2\nu }(d\phi -\omega dt)^{2}+e^{2(\lambda -\nu )}(d\rho ^{2}+dz^{2})}$

where ${\displaystyle (t,\rho ,\phi ,z)}$are the cylindrical Weyl–Lewis–Papapetrou coordinates in ${\displaystyle 3+1}$-dimensional spacetime, and ${\displaystyle \lambda }$, ${\displaystyle \nu }$, ${\displaystyle \omega }$, and ${\displaystyle B}$, are unknown functions of the spatial non-angular coordinates ${\displaystyle \rho }$and ${\displaystyle z}$only. Different authors define the functions of the coordinates differently.

• Introduction to the mathematics of general relativity
• Stress–energy tensor
• Metric tensor (general relativity)
• Relativistic angular momentum
• Weyl metrics

• Marek, J.; Sloane, A. (May 1979). "A finite rotating body in general relativity". Il Nuovo Cimento B Series 11. 51 (1): 45–52. Bibcode:1979NCimB..51...45M. doi:10.1007/BF02743695. ISSN 1826-9877. S2CID 125042609.
• Richterek, L.; Novotný, J.; Horský, J. (2002). "Einstein-Maxwell fields generated from the gamma-metric and their limits". Czechoslovak Journal of Physics. 52 (9): 1021–1040. arXiv:gr-qc/0209094v1. Bibcode:2002CzJPh..52.1021R. doi:10.1023/A:1020581415399. S2CID 18982611.
• Sharif, M. (December 2007). "Energy-momentum distribution of the Weyl-Lewis-Papapetrou and the Levi-Civita metrics" (PDF). Brazilian Journal of Physics. 37 (4): 1292–1300. arXiv:0711.2721. Bibcode:2007BrJPh..37.1292S. doi:10.1590/S0103-97332007000800017. ISSN 0103-9733. S2CID 15915449.
• Sloane, A (1978). "The Axially Symmetric Stationary Vacuum Field Equations in Einstein's Theory of General Relativity". Australian Journal of Physics. 31 (5): 429. Bibcode:1978AuJPh..31..427S. doi:10.1071/PH780427. ISSN 0004-9506.

• Friedman, John L.; Stergioulas, Nikolaos (2013). Rotating relativistic stars. Cambridge monographs on mathematical physics. Cambridge University Press. p. 151. ISBN 978-0-521-87254-6.
• Macías, A.; Cervantes-Cota, J. L.; Lämmerzahl, C. (2001). Macías, A.; Cervantes Cota, Jorge Luis; Lämmerzahl, C. (eds.). Exact solutions and scalar fields in gravity: recent developments. New York: Kluwer Academic/Plenum Publishers. p. 39. ISBN 978-0-306-46618-2.
• Das, Anadijiban; DeBenedictis, Andrew (2012). The general theory of relativity: a mathematical exposition. New York London: Springer Publishing. p. 317. ISBN 978-1-4614-3658-4.
• Hall, G. S.; Pulham, J. R. (1996). Hall, G. S.; Pulham, J. R. (eds.). General relativity: proceedings of the forty sixth Scottish Universities summer school in physics, Aberdeen, July 1995. SUSSP publications. Vol. 46. Scottish Universities Summer Schools in Physics ; Institute of Physics Publishing. pp. 65, 73, 78. ISBN 978-0-7503-0395-8.