Weyl–Lewis–Papapetrou coordinates

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.^[1][2][3]

Details

The square of the line element is of the form:^[4]

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

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.

See also

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

References

1. Weyl, Hermann (1917). "Zur Gravitationstheorie". Annalen der Physik (in German). 359 (18): 117–145. Bibcode:1917AnP...359..117W. doi:10.1002/andp.19173591804. ISSN 0003-3804.
2. Lewis, T. (1932). "Some special solutions of the equations of axially symmetric gravitational fields". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 136 (829): 176–192. Bibcode:1932RSPSA.136..176L. doi:10.1098/rspa.1932.0073. ISSN 0950-1207.
3. Papapetrou, A. (1948). "A static solution of the equations of the gravitatinal field for an arbitrary charge-distribution". Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences. 52: 191–204. JSTOR 20488481.
4. Bičák, Jiří; Semerák, O.; Podolský, Jiří; Žofka, Martin (2002). Bičák, Jiří; Semerák, O.; Podolský, J.; Žofka, M. (eds.). Gravitation, following the Prague inspiration: a volume in celebration of the 60th birthday of Jiří Bičák. River Edge, N.J: World Scientific. p. 122. ISBN 978-981-238-093-7. OCLC 51260088.

Further reading

Selected papers

• 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.

Selected books

• 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.