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Phase diagram of Einstein-Weyl gravity

S. Silveravalle and A. Zuccotti
Phys. Rev. D 107, 064029 – Published 13 March 2023
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  • INTRODUCTION
  • EQUATIONS OF MOTION, ANALYTICAL…
  • SOLUTIONS OF EINSTEIN-WEYL GRAVITY
  • THE PHASE DIAGRAM OF EINSTEIN-WEYL…
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
    • References

    Abstract

    Thanks to their interpretation as the first order correction of general relativity at high energies, quadratic theories of gravity have gained much attention in recent times. Particular attention has been drawn to the Einstein-Weyl theory, where the addition of the squared Weyl tensor to the action opens the possibility of having non-Schwarzschild black holes in the classical spectrum of the theory. Static and spherically symmetric solutions of this theory have been studied and classified in terms of their small scales behavior; however, a classification of these solutions in terms of the asymptotic gravitational field is still lacking. In this paper we address this point and present a phase diagram of the theory, where the different types of solutions are shown in terms of their mass and the strength of a Yukawa-like correction to the gravitational field. In particular we will show that, in the case of compact stars, different equations of state imply different Yukawa corrections to the gravitational potential, with possible phenomenological implications.

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    • Received 1 November 2022
    • Accepted 28 February 2023

    DOI:https://doi.org/10.1103/PhysRevD.107.064029

    © 2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Alternative gravity theoriesAstrophysical studies of gravityClassical black holesGeneral relativity equations & solutionsSingularities In general relativity
    Gravitation, Cosmology & Astrophysics

    Authors & Affiliations

    S. Silveravalle1,2 and A. Zuccotti3

    • 1Università degli Studi di Trento, Via Sommarive, 14, IT-38123, Trento, Italy
    • 2INFN—TIFPA, Via Sommarive, 14, IT-38123, Trento, Italy
    • 3Ghent University, Technologiepark-Zwijnaarde 126, Be-9052 Gent, Belgium

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    Issue

    Vol. 107, Iss. 6 — 15 March 2023

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    Images

    • Figure 1
      Figure 1

      Vacuum solutions of Einstein-Weyl gravity with mass M=0.6. The Schwarzschild BH in dashed black has S2−=0, the non-Schwarzschild BH in orange has S2−=0.101, the type I solution in red has S2−=0.2, the type II solution in green has S2−=0.075, and the type III solution in dotted and solid blue has S2−=−0.2.

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    • Figure 2
      Figure 2

      Conformal diagrams of a naked singularity (on the left), of a no-sy WH (in the center) and of a black hole (on the right); the dotted lines indicate surfaces of constant time and the radius. The conformal diagram of a no-sy WH is taken from [26].

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    • Figure 3
      Figure 3

      Mass-radius relation for polytropic stars in Einstein-Weyl gravity; the equation of state is taken with Γ=2 and k0=6.51185×10−17  cm3/g and the scale is fixed in terms of the length unit l2=1/m2 and the Sun Schwarzschild radius.

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    • Figure 4
      Figure 4

      Phase diagram of vacuum solutions of Einstein-Weyl gravity; the areas populated by type I, II and III solutions are indicated with three different colors, Schwarzschild black holes are indicated with a black dashed line, while non-Schwarzschild black holes are indicated by the blue and red solid lines. The separation between type I and II and III solutions are indicated with a dashed and dotted gray lines.

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    • Figure 5
      Figure 5

      The two triple points of the phase diagram; the first one (Minkowski triple point) is located at the origin of the M−S2− plane, while the second one (massive triple point) is located at M≃0.623, S2−≃0.102, which is the end point of the non-Schwarzschild black hole line.

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    • Figure 6
      Figure 6

      Location of nonvacuum solutions in Einstein-Weyl gravity; the dashed lines indicate solutions with polytropic equations of state with Γ=2, with different colors for different values of k0, while the solid and dotted black lines indicate solutions with an equation of state with Γ=5/3 and Γ=4/3, respectively.

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    • Figure 7
      Figure 7

      Differences in observable masses measured at infinity and at r¯=3M; in the left panel the mass MN is measured by the redshift of a photon, while in the right panel the mass MK is measured using Kepler’s third law. The white region is removed, being populated by wormholes with throat radius rT>3M.

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