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Hairy black holes in Einstein-Weyl gravity

Joseph Sultana
Phys. Rev. D 101, 084027 – Published 10 April 2020
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  • INTRODUCTION
  • EINSTEIN-WEYL-SCALAR GRAVITY
  • NUMERICAL SOLUTIONS
  • ASYMPTOTIC SOLUTIONS AT INFINITY
  • DISCUSSION AND CONCLUSION
    • References

    Abstract

    Higher derivative extensions of Einstein’s general relativity are commonly viewed as alternative and effective theories of gravity, not just due to the fact that these arise naturally in the string theory approach to gravity, but mainly for their use in cosmology to generate geometric dark energy models. In a recent paper [Phys. Rev. Lett. 114, 171601 (2015)] Lü et al. used numerical methods to obtain static, spherically symmetric and asymptotically flat black hole solutions in vacuum Einstein-Weyl gravity and showed that these are different than the Schwarzschild black hole. Inspired by the absence of any no-hair theorem for Einstein-Weyl gravity, in this paper we derive numerical examples of black hole solutions with massive scalar hair, and study the effect of the scalar field on the black hole structure. We limit ourselves to static and spherically symmetric solutions which are asymptotically flat such that the scalar field is regular on the horizon and vanishes at infinity.

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    • Received 25 October 2019
    • Accepted 30 March 2020

    DOI:https://doi.org/10.1103/PhysRevD.101.084027

    © 2020 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Alternative gravity theoriesClassical black holes
    Gravitation, Cosmology & Astrophysics

    Authors & Affiliations

    Joseph Sultana*

    • Department of Mathematics, Faculty of Science, University of Malta, Msida MSD2080, Malta

    • *joseph.sultana@um.edu.mt

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    Issue

    Vol. 101, Iss. 8 — 15 April 2020

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    Images

    • Figure 1
      Figure 1

      The metric functions for the numerical hairy black hole solutions to (6)–(8) with r0=1, for (a) different values of the scalar field on the horizon ϕ1=0.1 (solid curve) and ϕ1=0.3 (dashed curve), and (b) different scalar field masses squared m2=−1 (solid curve) and m2=−1/4 (dashed curve). In each plot the upper curve represents f(r) and the lower curve is h(r). For clarity we have rescaled h(r) such that it approaches 0.75 instead of unity, to avoid overlapping of the two curves in the asymptotic region. In Fig. 1 we fix m2=−1, while in Fig. 1 we let ϕ1=0.1.

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    • Figure 2
      Figure 2

      The scalar field for (a) different values of the initial value of the field on the horizon ϕ1=0.1 (solid curve) and ϕ1=0.3 (dashed curve), and (b) different values of the scalar field mass m2=−1 (solid curve) and m2=−1/4 (dashed curve). In (a) we fix m2=−1, while in (b) we take ϕ1=0.1.

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    • Figure 3
      Figure 3

      The difference in the Kretschmann scalar κdiff=κ(ϕ=0)−κ(ϕ≠0) between the non-Schwarzschild black hole obtained in Ref. [30] and the hairy black holes obtained in this paper, for (a) ϕ1=0.1 (solid curve), ϕ1=0.3 (dashed curve), and (b) m2=−1 (solid curve), m2=−1/4 (dashed curve). Again in (a) we take m2=−1 and in (b) we assume ϕ1=0.1.

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    • Figure 4
      Figure 4

      The radial component of the energy momentum tensor Trr for (a) different values of the initial value of the field on the horizon ϕ1=0.1 (solid curve) and ϕ1=0.3 (dashed curve), and (b) different values of the scalar field mass m2=−1 (solid curve) and m2=−1/4 (dashed curve). In (a) we fix m2=−1, while in (b) we take ϕ1=0.1.

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    • Figure 5
      Figure 5

      The variation of the asymptotic mass M of the black hole on the initial value of the scalar field ϕ1 for the case m2=−1.

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    • Figure 6
      Figure 6

      The variation of the Hawking temperature on the horizon TH on the initial value of the scalar field ϕ1 for the case m2=−1.

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    • Figure 7
      Figure 7

      The numerical solutions and the corresponding analytical approximations in (15) for (a) the metric functions f(r) and (rescaled) h(r) and (b) the scalar field ϕ(r). In both sub-figures the dashed curves represent the analytical approximations. We consider the case m2=−1 and ϕ1=0.1.

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