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Invariant description of static and dynamical Brans–Dicke spherically symmetric models

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Abstract

We investigate spherically symmetric static and dynamical Brans–Dicke theory exact solutions using invariants and, in particular, the Newman Penrose formalism utilizing Cartan scalars. In the family of static, spherically symmetric Brans–Dicke solutions, there exists a three-parameter family of solutions, which have a corresponding limit to general relativity. This limit is examined through the use of Cartan invariants via the Cartan–Karlhede algorithm and is additionally supported by analysis of scalar polynomial invariants. It is determined that the appearance of horizons in these spacetimes depends primarily on one of the parameters, n, of the family of solutions. In particular, expansion-free surfaces appear which, for a subset of parameter values, define additional surfaces distinct from the standard surfaces (e.g., apparent horizons) identified in previous work. The “\(r=2M\)” surface in static spherically symmetric Brans–Dicke solutions was previously shown to correspond to the Schwarzschild horizon in general relativity when an appropriate limit exists between the two theories. We show additionally that other geometrically defined horizons exist for these cases, and identify all solutions for which the corresponding general relativity limit is not a Schwarzschild one, yet still contains horizons. The identification of some of these other surfaces was noted in previous work and is characterized invariantly in this work. In the case of the family of dynamical Brans–Dicke solutions, we identify similar invariantly defined surfaces as in the static case and present an invariant characterization of their geometries. Through the analysis of the Cartan invariants, we determine which members of these families of solutions are locally equivalent, through the use of the Cartan–Karlhede algorithm. In addition, we identify black hole surfaces, naked singularities, and wormholes with the Cartan invariants. The aim of this work is to demonstrate the usefulness of Cartan invariants for describing properties of exact solutions like the local equivalence between apparently different solutions, and identifying special surfaces such as black hole horizons.

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Notes

  1. The \(n=2\) solution is not necessarily a Schwarzschild solution, depending on the identification of the constant parameter; rather it is a horizonless ‘negative mass’ Schwarzschild solution if we only consider \(M>0\).

References

  1. Mosani, K., Dey, D., Joshi, P.S.: Strong curvature naked singularities in spherically symmetric perfect fluid collapse. Phys. Rev. D 101(4), 044052 (2020)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  2. Dey, D., Joshi, P.S., Mosani, K., Vertogradov, V.: Causal structure of singularity in non-spherical gravitational collapse. Eur. Phys. J. C 82(5), 431 (2022)

    Article  ADS  CAS  Google Scholar 

  3. Brans, C., Dicke, R.H.: Mach’s principle and a relativistic theory of gravitation. Phys. Rev. 124, 925 (1961)

    Article  ADS  MathSciNet  Google Scholar 

  4. Hochberg, D., Visser, M.: Geometric structure of the generic static traversable wormhole throat. Phys. Rev. D 56, 4745–4755 (1997)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  5. Coley, A., McNutt, D.: Identification of black hole horizons using scalar curvature invariants. Class. Quantum Gravity 35(2), 025013 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  6. Milson, R., Coley, A., Pravda, V., Pravdova, A.: Int. J. Geom. Methods Mod. Phys. 2(01), 41–61 (2005)

    Article  MathSciNet  Google Scholar 

  7. Coley, A.A., McNutt, D.D., Shoom, A.A.: Geometric horizons. Phys. Lett. B 771, 131–135 (2017)

    Article  ADS  CAS  Google Scholar 

  8. Brooks, D., Chavy-Waddy, P.C., Coley, A.A., Forget, A., Gregoris, D., MacCallum, M.A.H., McNutt, D.D.: Cartan invariants and event horizon detection. Gen. Relativ. Gravit. 50, 37 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  9. Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations. Cambridge University Press, Cambridge (2009)

    Google Scholar 

  10. McNutt, D.D., Julius, W., Gorban, M., Mattingly, B., Brown, P., Cleaver, G.: Geometric surfaces: an invariant characterization of spherically symmetric black hole horizons and wormhole throats. Phys. Rev. D 103, 124024 (2021)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  11. Kozak, A., Wojnar, A.: Eur. Phys. J. vC81, 492 (2021)

  12. McNutt, D.D., Page, D.N.: Scalar polynomial curvature invariant vanishing on the event horizon of any black hole metric conformal to a static spherical metric. Phys. Rev. D 95(8), 084044 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  13. Paiva, F.M., Romero, C.: On the limits of Brans–Dicke spacetimes: a coordinate free approach, arXiv:gr-qc/9304031

  14. Letelier, P.S., Wang, A.: Phys. Rev. D 48, 631 (1993)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  15. Brans, C.H.: Phys. Rev. 125, 2194 (1962)

    Article  ADS  MathSciNet  Google Scholar 

  16. Hawking, S.W.: Commun. Math. Phys. 25, 167 (1972)

    Article  ADS  Google Scholar 

  17. Faraoni, V., Hammad, F., Belknap-Keet, S.D.: Phys. Rev. D 94, 104019 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  18. Faraoni, V., Hammad, F., Cardini, A.M., Gobeil, T.: Phys. Rev. D 97, 084033 (2018)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  19. Campanelli, M., Lousto, C.O.: Are black holes in Brans-Dicke theory precisely the same as in general relativity? Int. J. Mod. Phys. D 02, 451 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  20. Bhadra, A., Sarkar, K.: On static spherically symmetric solutions of the vacuum Brans-Dicke theory. Gen. Relativ. Gravit. 37, 2189–2199 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  21. Agnese, A.G., La Camera, M.: Wormholes in the Brans-Dicke theory of gravitation. Phys. Rev. D 51(4), 2011–2013 (1995)

    Article  ADS  CAS  Google Scholar 

  22. Bronnikov, K.A.: Acta Phys. Pol. B 4, 251 (1973)

    Google Scholar 

  23. Morris, M.S., Thorne, K.S.: Wormholes in spacetime and their use for interstellar travel: a tool for teaching general relativity. Am. J. Phys. 56, 395 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  24. Visser, M.: Lorentzian Wormholes: From Einstein to Hawking. Springer-Verlag, New York (1995)

    Google Scholar 

  25. Pimentel, L.O.: Mod. Phys. Lett. A 12, 1865 (1997)

    Article  ADS  Google Scholar 

  26. Husain, V., Martinez, E.A., Núñez, D.: Exact solution for scalar field collapse. Phys. Rev. D.50.3783 (1994) arXiv:gr-qc/9402021

  27. Coley, A., Milson, R., Pravda, V., Pravdova, A.: Class. Quantum Gravity 21, L35 (2004)

    Article  ADS  Google Scholar 

  28. Coley, A., Milson, R., Pravda, V., Pravdova, A.: Class. Quantum Gravity 21, 5519 (2004)

    Article  ADS  Google Scholar 

  29. Ziaie, A.H., Atazadeh, K., Tavakoli, Y.: Class. Quant. Grav. v27, 075016 & 209801 (2010)

  30. Virbhadra, K.S.: Janis Newman Winicour and Wyman Solutions are the Same. Int. J. Modern Phys. A 12, 4831–4835 (1997)

    Article  ADS  MathSciNet  CAS  Google Scholar 

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Acknowledgements

AAC is supported by the Natural Sciences and Engineering Research Council of Canada. DD is supported by an AARMS fellowship. We also thank Valerio Faraoni for helpful comments on Brans Dicke and JNW solutions.

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  1. Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, B3H 3J5, Canada

    Nicholas T. Layden, Alan A. Coley & Dipanjan Dey

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  1. Nicholas T. Layden
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Correspondence to Nicholas T. Layden.

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Layden, N.T., Coley, A.A. & Dey, D. Invariant description of static and dynamical Brans–Dicke spherically symmetric models. Gen Relativ Gravit 56, 10 (2024). https://doi.org/10.1007/s10714-023-03196-3

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