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The Plebański–Demiański Class of Black Hole Space-Times

  • Arne GrenzebachEmail author
Chapter
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Part of the SpringerBriefs in Physics book series (SpringerBriefs in Physics)

Abstract

The Plebański–Demiański class contains stationary, axially symmetric type D solutions of the Einstein–Maxwell equations with a cosmological constant. It covers many well-known black hole space-times like the Schwarzschild, Kerr or the Kottler space-time. The space-times are characterized by seven parameters: mass, spin, electric and magnetic charge, gravitomagnetic NUT charge, a so-called acceleration parameter and the cosmological constant. We review space-time properties like symmetries and isometries as well as the appearance of singularities as ring singularities or axial singularities. Furthermore, we discuss horizons, the ergoregion and a region with causality violation.

Keywords

Plebanski-Demianski Schwarzschild Kerr Kerr-Newman Reissner-Nordstroem NUT C-metric Metric tensor Boyer-Lindquist coordinates Space-time properties Symmetries Isometries Singularities Ring singularity Axial singularity Black hole horizon Ergoregion Causality violation Conformal factor 

References

  1. Aliev AN, Gümrükçüoğlu AE (2005) Charged rotating black holes on a 3-brane. Phys Rev D 71(10):104,027(14). doi: 10.1103/PhysRevD.71.104027
  2. Bonnor WB (1969) A new interpretation of the NUT metric in general relativity. Math Proc Cambridge Philos Soc 66(1):145–151. doi: 10.1017/S0305004100044807 MathSciNetCrossRefzbMATHADSGoogle Scholar
  3. Bonnor WB (1983) The sources of the vacuum \(C\)-Metric. Gen Relativ Gravit 15(6):535–551. doi: 10.1007/BF00759569 MathSciNetCrossRefzbMATHADSGoogle Scholar
  4. Bonnor WB, Davidson W (1992) Interpreting the Levi-Civita vacuum metric. Class Quantum Gravity 9(9):2065–2068. doi: 10.1088/0264-9381/9/9/012 MathSciNetCrossRefADSGoogle Scholar
  5. Carter B (1968) Hamilton-Jacobi and Schrödinger separable solutions of Einstein’s equations. Commun Math Phys 10(4):280–310. http://projecteuclid.org/euclid.cmp/1103841118 Google Scholar
  6. Debever R (1971) On type D expanding solutions of Einstein-Maxwell equations. Bulletin de la Société Mathématique de Belgique 23:360–376MathSciNetzbMATHGoogle Scholar
  7. Ehlers J, Kundt W (1962) Exact solutions of the gravitational field equations. In: Witten, chap 2, pp 49–101Google Scholar
  8. Grenzebach A, Perlick V, Lämmerzahl C (2014) Photon regions and shadows of Kerr–Newman–NUT Black Holes with a cosmological constant. Phys Rev D 89:124,004(12). doi: 10.1103/PhysRevD.89.124004. arXiv:1403.5234
  9. Grenzebach A (2015) Aberrational effects for shadows of black holes. In: Puetzfeld et al Proceedings of the 524th WE-Heraeus-Seminar “Equations of Motion in Relativistic Gravity”, held in Bad Honnef, Germany, 17–23 Feb 2013, pp 823–832. doi: 10.1007/978-3-319-18335-0_25, arXiv:1502.02861 Google Scholar
  10. Grenzebach A, Perlick V, Lämmerzahl C (2015) Photon regions and shadows of accelerated black holes. Int J Mod Phys D 24(9):1542,024(22). doi: 10.1142/S0218271815420249 (“Special Issue Papers” of the “7th Black Holes Workshop”, Aveiro, Portugal, arXiv:1503.03036)
  11. Griffiths JB, Podolský J (2009) Exact space-times in Einstein’s general relativity. Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge. doi: 10.1017/CBO9780511635397
  12. Hong K, Teo E (2003) A new form of the C-metric. Class Quantum Gravity 20(14):3269–3277. doi: 10.1088/0264-9381/20/14/321 MathSciNetCrossRefzbMATHADSGoogle Scholar
  13. Hong K, Teo E (2005) A new form of the rotating C-metric. Class Quantum Gravity 22(1):109–117. doi: 10.1088/0264-9381/22/1/007 MathSciNetCrossRefzbMATHADSGoogle Scholar
  14. Kagramanova V, Kunz J, Hackmann E, Lämmerzahl C (2010) Analytic treatment of complete and incomplete geodesics in Taub–NUT space-times. Phys Rev D 81(12):124,044(17). doi: 10.1103/PhysRevD.81.124044
  15. Kinnersley W, Walker M (1970) Uniformly Accelerating Charged Mass in General Relativity. Phys Rev D 2(8):1359–1370. doi: 10.1103/PhysRevD.2.1359 Google Scholar
  16. Levi-Civita T (1919) \(\rm ds^2\) einsteiniani in campi newtoniani. VIII. Soluzioni binarie de Weyl. Rendiconti della Reale Accademia dei Lincei 28(1):3–13Google Scholar
  17. Manko VS, Ruiz E (2005) Physical interpretation of the NUT family of solutions. Class Quantum Gravity 22(17):3555–3560. doi: 10.1088/0264-9381/22/17/014 MathSciNetCrossRefzbMATHADSGoogle Scholar
  18. Miller JG (1973) Global analysis of the KerrTaubNUT metric. J Math Phys 14(4):486–494. doi: 10.1063/1.1666343 CrossRefADSGoogle Scholar
  19. Misner CW (1963) The flatter regions of Newman, Unti, and Tamburino’s generalized Schwarzschild space. J Math Phys 4(7):924–937. doi: 10.1063/1.1704019 MathSciNetCrossRefADSGoogle Scholar
  20. O’Neill B (1995) The geometry of Kerr black holes. A K Peters, WellesleyGoogle Scholar
  21. Plebański JF, Demiański M (1976) Rotating, charged and uniformly accelerating mass in general relativity. Ann Phys 98(1):98–127. doi: 10.1016/0003-4916(76)90240-2 MathSciNetCrossRefzbMATHADSGoogle Scholar
  22. Stephani H, Kramer D, MacCallum M, Hoenselaers C, Herlt E (2003) Exact solutions of Einstein’s field equations, 2nd edn. Cambridge Monographs on Mathematical Physics, Cambridge University Press, New York. doi: 10.1017/CBO9780511535185
  23. Straumann N (2013) General Relativity, 2nd edn. Graduate texts in physics, Springer, Dordrecht. doi: 10.1007/978-94-007-5410-2 Google Scholar
  24. Weyl H (1919) Raum, Zeit, Materie, 3rd edn. Springer, Berlin. http://www.archive.org/details/raumzeitmateriev00weyl
  25. Weyl H (1917) Zur Gravitationstheorie. Annalen der Physik 359(18):117–145. doi: 10.1002/andp.19173591804 CrossRefADSGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.ZARM—Zentrum für angewandte Raumfahrttechnologie und MikrogravitationUniversität BremenBremenGermany

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