Skip to main content
SpringerLink
Log in

Black holes of the Vaidya type with flat and (A)dS asymptotics as point particles

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

A presentation of the Vaidya type Schwarzschild-like black holes with flat, AdS and dS asymptotics in 4-dimensional general relativity in the form of a pointlike mass is given. True singularities are described by making the use of the Dirac \(\delta\)-function in a non-contradictory way. The results essentially generalize previous derivations where the usual Schwarzschild black hole solution is represented in the form of a point particle. The field-theoretical formulation of general relativity, which is equivalent to its standard geometrical formulation, is applied as an alternative mathematical formalism. Then perturbations on a given background are considered as dynamical fields propagating in a given (fixed) spacetime. The energy (mass) distribution of such field configurations is just represented as a point mass. The new description of black holes’ structure can be useful in explaining and understanding their features and can be applied in calculations with black hole models. A possibility of application of the field-theoretical formalism in studying the regular black hole solutions is discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: All calculations have been presented directly in the text.]

Notes

  1. Here, we apply the decomposition (2.5) used in [36, 37] instead of the usual one \(\sqrt{-g}g^{\mu \nu } = \sqrt{-\bar{g}}(\bar{g}^{\mu \nu } + h^{\mu \nu })\), see for example, [32]. Such a different choice does not change the final results, see discussion in [32].

  2. The same background metric in (2.5) can be chosen by different ways that leads to different definitions of the field configuration \(\varkappa _{\alpha \beta }\), where one of them is connected with others by gauge transformations, see, for example, [32, 33].

References

  1. R. Arnowitt, S. Deser, C.W. Misner, Finite self-energy of classical point particles. Phys. Rev. Lett. 4(7), 375 (1960)

    Article  MATH  ADS  Google Scholar 

  2. R. Arnowitt, S. Deser, C.W. Misner, Gravitational-electromagnetic coupling and the classical self-energy problem. Phys. Rev. 120(1), 313 (1960)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  3. F.R. Tangherlini, Nonclassical structure of the energy-momentum tensor of a point mass source for the Schwarzschild field. Phys. Rev. Lett. 6(3), 147 (1961)

    Article  ADS  Google Scholar 

  4. P.E. Parker, Distributional geometry. J. Math. Phys. 20(7), 1423–1426 (1979)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  5. A.H. Taub, Space-times with distribution valued curvature tensors. J. Math. Phys. 21(6), 1423–1431 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  6. C.K. Raju, Junction conditions in general relativity. J. Phys. A: Math. Gen. 15(6), 1785 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  7. J.F. Colombeau, A multiplication of distributions. J. Math. Anal. Appl. 94(1), 96–115 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Geroch, J. Traschen, Strings and other distributional sources in general relativity. Phys. Rev. D 36(4), 1017 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  9. H. Balasin, H. Nachbagauer, Distributional energy-momentum tensor of the Kerr–Newman spacetime family. Class. Quantum Grav. 11(6), 1453 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  10. C.J.S. Clarke, J.A. Vickers, J.P. Wilson, Generalized functions and distributional curvature of cosmic strings. Class. Quantum Grav. 13(9), 2485 (1996)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  11. J.P. Wilson, Distributional curvature of time dependent cosmic strings. Class. Quantum Grav. 14(12), 3337 (1997)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  12. J.A. Vickers, J.P. Wilson, Invariance of the distributional curvature of the cone under smooth diffeomorphisms. Class. Quantum Grav. 16(2), 579 (1999)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  13. N.R. Pantoja, H. Rago, Distributional sources in general relativity: two point-like examples revisited. J. Mod. Phys. D 11(9), 1479–1499 (2002)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  14. M. Melis, S. Mignemi, Two-dimensional static black holes with pointlike sources. Gen. Relat. Grav. 37(July 06), 1313–1322 (2005)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  15. M. Cadoni, S. Mignemi, Two-dimensional description of \(D\)-dimensional static black holes with pointlike source. Mod. Phys. Lett. A 20(38), 2919–2924 (2005)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  16. A.P. Lundgren, B.S. Schmekel, J.W. York Jr., Self-renormalization of the classical quasilocal energy. Phys. Rev. D 75(8), 084026 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  17. I.I. Bulygin, M.V. Sazhin, O.S. Sazhina, Theory of gravitational lensing on a curved cosmic string. Eur. Phys. J. C 83(9), 844 (2023)

    Article  Google Scholar 

  18. D.D. Sokolov, A.A. Starobinsky, The structure of the curvature tensor at conical singularities. Sov. Phys. Doklady 22, 312–314 (1977)

    ADS  Google Scholar 

  19. Ch.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation (W. H. Freeman, San Francisco, 1973)

    Google Scholar 

  20. L. Bel, Schwarzschild singularity. J. Math. Phys. 10(8), 1501–1503 (1969)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  21. T. Kawai, E. Sakane, Distributional energy-momentum densities of Schwarzschild space-time. Prog. Theor. Phys. 98(1), 69–86 (1997)

    Article  MathSciNet  ADS  Google Scholar 

  22. J.M. Heinzle, R. Steinbauer, Remarks on the distributional Schwarzschild geometry. J. Math. Phys. 43(3), 1493–1508 (2002)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  23. M.O. Katanaev, Point massive particle in general relativity. Gen. Relat. Grav. 45(July 24), 1861–1875 (2013)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  24. P.P. Fiziev, The era of gravitational astronomy and gravitational field of non-rotating single point particle in general relativity. Phys. Part. Nuclei 50(6), 944–972 (2019)

    Article  ADS  Google Scholar 

  25. A.N. Petrov, J.V. Narlikar, The energy distribution for a spherically symmetric isolated system in general relativity. Found. Phys. 26(9), 1201–1229 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  26. A.N. Petrov, The Schwarzschild black hole as a point particle. Found. Phys. Lett. 18(10), 477–489 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. A.N. Petrov, A point mass and continuous collapse to a point mass in general relativity. Gen. Relat. Grav. 50(1), 6 (2018)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  28. L.P. Grishchuk, A.N. Petrov, A.D. Popova, Exact theory of the (Einstein) gravitational field in an arbitrary background space-time. Commun. Math. Phys. 94(9), 379–396 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  29. A.D. Popova, A.N. Petrov, The dynamic theories on a fixed background in gravitation. Int. J. Mod. Phys. A 3(11), 2651–2679 (1988)

    Article  ADS  Google Scholar 

  30. L.P. Grishchuk, A.N. Petrov, The Hamiltonian description of the gravitational field and gauge symmertries. Sov. Phys.: JETP 65(1), 5 (1987)

    ADS  Google Scholar 

  31. A.N. Petrov, A note on the Deser–Tekin charges. Class. Quantum Grav. 22(16), L83 (2005)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  32. A.N. Petrov, S.M. Kopeikin, R.R. Lompay, B. Tekin, Metric Theories of Gravity: Perturbations and Conservation Laws, volume 38 of De Gruyter Studies in Mathematical Physics. De Gruyter, 4 (2017)

  33. A.N. Petrov, J.B. Pitts, The field-theoretic approach in general relativity and other metric theories: a review. Space Time Fundam Interact 4, 66–124 (2019)

    Google Scholar 

  34. J.D. Brown, J.W. York, Quasilocal energy and conserved charges derived from the gravitational action. Phys. Rev. D 47(4), 1407 (1993)

    Article  MathSciNet  ADS  Google Scholar 

  35. D. Lovelock, The Einstein tensor and its generalizations. J. Math. Phys. 12(3), 498–501 (1971)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  36. A.N. Petrov, Field-theoretical construction of currents and superpotentials in Lovelock gravity. Class. Quantum Grav. 36(23), 235021 (2019)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  37. A.N. Petrov, Conserved quantities for black hole solutions in pure Lovelock gravity. Class. Quantum Grav. 38(15), 155017 (2021)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  38. N. Dadhich, S.G. Ghosh, S. Jhingan, Gravitational collapse in pure Lovelock gravity in higher dimensions. Phys Rev. D 88(8), 084024 (2013)

    Article  ADS  Google Scholar 

  39. J. Kastikainen, Quasi-local energy and ADM mass in pure Lovelock gravity. Class. Quantum Grav. 37(2), 025001 (2020)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  40. R.-G. Cai, N. Ohta, Black holes in pure lovelock gravities. Phys Rev. D 74(6), 064001 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  41. R.-G. Cai, L.-M. Cao, Y.-P. Hu, S.P. Kim, Generalized Vaidya spacetime in lovelock gravity and thermodynamics on apparent horizon. Phys. Rev. D 78(12), 124012 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  42. T. Damour, P. Jaranowsk, G. Schäfer, Dimensional regularization of the gravitational interaction of point masses. Phys. Lett. B 513(1–2), 147–155 (2001)

    Article  MATH  ADS  Google Scholar 

  43. L. Blanchet, Gravitational radiation from post-Newtonian sources and inspiralling compact binaries. Living Rev. Relat. 17, 2 (2014)

    Article  MATH  ADS  Google Scholar 

  44. J.B. Pitts, W.C. Schieve, Null cones and Einstein’s equations in Minkowski spacetime. Found. Phys. 34(2), 211–238 (2004)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  45. A.Z. Petrov, Einstein Spaces (Pergamon Press, Oxford, 1969)

    Book  MATH  Google Scholar 

  46. A.E. Dominguez, E. Gallo, Radiating black hole solutions in Einstein–Gauss–Bonnet gravity. Phys. Rev. D 73(6), 064018 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  47. S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Space–Time (CUP, Cambridge, 1973)

    Book  MATH  Google Scholar 

  48. H. Nariai, On some static solutions of Einstein’s gravitational field equations in a spherically symmetric case. Gen. Relat. Grav. 31(6), 951–961 (1999)

    Article  MATH  ADS  Google Scholar 

  49. H. Nariai, On a new cosmological solution of Einstein’s field equations of gravitation. Gen. Relat. Grav. 31(6), 963–971 (1999)

    Article  MATH  ADS  Google Scholar 

  50. V. Balasubramanian, J. de Boer, D. Minic, Mass, entropy and holography in asymptotically de Sitter spaces. Phys. Rev. D 65(12), 123508 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  51. R.-G. Cai, Y.S. Myung, Y.-Z. Zhang, Check of the mass bound conjecture in the de Sitter space. Phys. Rev. D 65(8), 084019 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  52. I.M. Gelfand, G.E. Shilov, Generalized Functions. Properties and Operations, vol. 1 (Academic Press, New York, 1964)

    Google Scholar 

  53. G.A. Korn, T.A. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill Book Comp, New York, 1968)

    MATH  Google Scholar 

  54. P.A. Spirin, Some Mathematical Questions of Theoretical Physics. Part 2. Lectures at Physical Faculty (Moscow University Press, Moscow, 2022). (in Russian)

    Google Scholar 

  55. R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio, M. Visser, On the viability of regular black holes. JHEP 2018(07), 023 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  56. L. Sebastiani, S. Zerbini, Some remarks on non-singular spherically symmetric space–times. Astronomy 1(2), 99–125 (2022)

    Article  ADS  Google Scholar 

  57. C. Bambi (ed.), Regular Black Holes: Towards a New Paradigm of Gravitational Collapse (Springer, Singapore, 2023)

    Google Scholar 

  58. D. Malafarina, Semi-classical Dust Collapse and Regular Black Holes. Chapter 12 in book [57] (2023)

Download references

Acknowledgements

The author is very grateful to Brian Pitts and George Alekseev for reading the manuscript and useful comments, he also very thanks Aleksei Staroboinsky, Igor Bulygin and Olga Sazhina for the explanation of their articles. The work has been supported by the Interdisciplinary Scientific and Educational School of Moscow University “Fundamental and Applied Space Research”.

Author information

Authors and Affiliations

  1. Sternberg Astronomical Institute, MV Lomonosov Moscow State University, Universitetskii Pr. 13, Moscow, Russia, 119992

    A. N. Petrov

Authors
  1. A. N. Petrov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. N. Petrov.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Petrov, A.N. Black holes of the Vaidya type with flat and (A)dS asymptotics as point particles. Eur. Phys. J. Plus 138, 879 (2023). https://doi.org/10.1140/epjp/s13360-023-04514-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-023-04514-z

Search

Navigation