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Black Holes in Asymptotically Safe Gravity and Beyond

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Regular Black Holes

Part of the book series: Springer Series in Astrophysics and Cosmology ((SSAC))

Abstract

Asymptotically safe quantum gravity is an approach to quantum gravity that achieves formulating a standard quantum field theory for the metric. Therefore, even the deep quantum gravity regime, that is expected to determine the true structure of the core of black holes, is described by a spacetime metric. The essence of asymptotic safety lies in a new symmetry of the theory – quantum scale symmetry – which characterizes the short-distance regime of quantum gravity. It implies the absence of physical scales. Therefore, the Newton coupling, which corresponds to a scale, namely the Planck length, must vanish asymptotically in the short-distance regime. This implies a weakening of the gravitational interaction, from which a resolution of classical spacetime singularities can be expected. In practise, properties of black holes in asymptotically safe quantum gravity cannot yet be derived from first principles, but are constructed using a heuristic procedure known as Renormalization Group improvement. The resulting asymptotic-safety inspired black holes have been constructed both for vanishing and for nonvanishing spin parameter. They are characterized by (i) the absence of curvature singularities, (ii) a more compact event horizon and photon sphere, (iii) a second (inner) horizon even at vanishing spin and (iv) a cold remnant as a possible final product of the Hawking evaporation. Observations can start to constrain the quantum-gravity scale that can be treated as a free parameter in asymptotic-safety inspired black holes. For slowly-spinning black holes, constraints from the Event Horizon Telescope and X-ray observations can only constrain quantum-gravity scales far above the Planck length. In the limit of near-critical spin, asymptotic-safety inspired black holes may “light up” in a way the next-generation Event Horizon Telescope may be sensitive to, even for a quantum-gravity scale equalling the Planck length. Finally, a connection to gravitational-wave observations of the ringdown phase can currently only be established under very strong theoretical assumptions, due to a lack of a dynamical equation to which asymptotic-safety inspired black holes constitute a solution.

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Notes

  1. 1.

    Attempts at formulating asymptotically safe gravity in terms of, e.g., the vielbein [64], the vielbein and the connection [34], a unimodular metric [44] or a generalized connection [62] also exist; however, metric gravity is by far the most explored of these options, see [97, 106] for textbooks and [8, 46, 49, 92, 98, 102] for reviews. The other options all start from theories which are classically equivalent to GR.

  2. 2.

    In technical language, we state here that a theory with quantum scale symmetry but without classical scale symmetry has at least one non-vanishing canonically relevant or irrelevant coupling and not just canonically marginal ones. Due to its canonical dimension, the coupling implicitly defines a mass scale.

  3. 3.

    There are exceptions to this argument, namely when a theory has vanishing mass parameters and the diverging mass scale is related to a relevant interaction. In that case quantum scale symmetry corresponds to a strongly-coupled regime.

  4. 4.

    Incidentally, this also implies that a “folk theorem” about the violation of global symmetries from quantum gravity may not be applicable to asymptotically safe quantum gravity.

  5. 5.

    In the above, one-loop examples, the scale-dependence of couplings on k agrees with that on \(\mu \), the RG scale that appears in perturbative renormalization.

  6. 6.

    We put “classical” in quotation marks, because it is not the \(\hslash \rightarrow 0\) regime – in nature, \(\hslash \rightarrow 0\) is at best an approximately observable limit – instead, the IR is the setting in which all quantum fluctuations in the path integral are present, and we are simply probing the effective action of the theory in its low-curvature regime.

  7. 7.

    Note that such constraints on \(g_{N,\, *}\) should not be misinterpreted as actual constraints on the fixed-point value. The latter is a non-universal quantity that is not directly accessible to measurements. It makes its way into observable quantities in the RG improvement procedure, because the procedure is an approximate one. Observations constrain a physical scale, namely the scale at which fixed-point scaling sets in. This scale depends on various couplings of the theory, but in our simple approximation it is only set by the fixed-point value of the Newton coupling.

  8. 8.

    More complete studies with higher-order gravitational couplings would have to start from black-hole solutions in corresponding higher-order theories, which are typically not known at all or only for small or vanishing spin.

  9. 9.

    We use the term ‘curvature invariants’ to refer to Riemann invariants, i.e., scalars built from contractions of any number of Riemann tensors and the metric. This does not include derivative invariants, i.e., those which involve additional covariant derivatives.

  10. 10.

    Heuristically, one can think of the non-zero energy-momentum tensor as a contribution on the left-hand-side of the generalized Einstein equations, i.e., a contribution from higher-order curvature terms which arise in asymptotically safe gravity.

  11. 11.

    The definition of the coordinates \((r,\,\theta )\) is equivalent in many of the standard coordinate systems for Kerr spacetime and, in particular, agrees with the \((r,\,\theta )\) as defined in ingoing Kerr and Boyer-Lindquist coordinates.

  12. 12.

    It has not been proven that the displayed invariants are polynomially independent, hence there could, in principle, be further polynomial relations among them.

  13. 13.

    Curvature invariants are finite, but not single-valued at \(r \rightarrow 0, \chi \rightarrow 0\). For \(r>0\), the spacetime is not geodesically complete and needs an extension to \(r<0\). There, the dependence on \(r^2\) (instead of r) in Eq. (5.58) ensures the absence of curvature singularities, see [110].

  14. 14.

    In the spherically-symmetric case, closed photon orbits can only arise at a 2-dimensional surface at fixed radius, cf. Sect. 5.4.3 – hence the name photon sphere. For stationary axisymmetric spacetimes and, in particular, for Kerr spacetime [2, 108], several classes of closed photon orbits can occur which cover an extended 3-dimensional region – hence, the name photon shell.

  15. 15.

    Consequences for black hole thermodynamics have not been explored yet.

  16. 16.

    Because Killing symmetries can be made manifest in the spacetime metric, RG improvement can always be made to respect Killing symmetries. However, any symmetry that is not a Killing symmetry, but instead only expressible as a condition on the Riemann tensor, need not be respected by the RG improvement.

  17. 17.

    See, however, [120] for a proof that circularity holds for black holes which are perturbatively connected to the Kerr solution.

  18. 18.

    The word ‘physical’ refers to physically realistic matter models [96, 119], see also the discussion below Eq. (5.65). The word ‘generic’ is important since non-generic initial conditions result in counter-examples to weak cosmic censorship [29], even in restricted sectors (spherically symmetric GR with a real massless scalar field) for which the mathematical conjecture has been proven [30].

  19. 19.

    Note that there are spacetimes which satisfy weak cosmic censorship in GR, but their RG-improved counterparts do not satisfy “quantum gravity censorship”: For instance, a Kerr-black hole, with sub-, but near-critical spin is an example, since its RG-improved counterpart can be horizonless, cf. [53] and Sect. 5.5.4.

  20. 20.

    Our simple RG-improvement procedure can of course not account for this possibility, because it starts from a unique starting point, namely a Kerr black hole.

  21. 21.

    An inclusion of such higher-order couplings for black holes would require the construction of spinning black-hole solutions in higher-curvature gravity, followed by an RG improvement. Due to the technical complexity of this task, it has not been attempted.

  22. 22.

    This reasoning does not invalidate effective field theory, because not all terms in the effective action remain low. Because perturbations become blueshifted, the kinetic terms describing perturbations of scalars/fermions/vectors/metric fluctuations become large, signalling the breakdown of effective field theory and the need for a quantum theory of gravity.

  23. 23.

    Whether or not a regular black hole can be overspun is not settled; see the discussion and references in [53].

  24. 24.

    For smaller mass, there is no longer an event horizon.

  25. 25.

    For which we are not aware of a reason why it should work beyond GR in the sense of upgrading a spherically symmetric solution of the theory to an axisymmetric solution; instead, beyond GR, it is just one particular way of reducing spherical symmetry to axisymmetry.

  26. 26.

    It is typically not made explicit.

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Acknowledgements

AE is supported by a research grant (29405) from VILLUM FONDEN. The work leading to this publication was supported by the PRIME programme of the German Academic Exchange Service (DAAD) with funds from the German Federal Ministry of Education and Research (BMBF). A. Held acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) under Grant no 406116891 within the Research Training Group RTG 2522/1.

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Authors and Affiliations

  1. CP3-Origins, University of Southern Denmark, Campusvej 55, 5230, Odense M, Denmark

    Astrid Eichhorn

  2. Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743, Jena, Germany

    Aaron Held

  3. The Princeton Gravity Initiative, Jadwin Hall, Princeton University, Princeton, NJ, 08544, United States

    Aaron Held

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  1. Astrid Eichhorn
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Correspondence to Astrid Eichhorn .

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  1. Department of Physics, Fudan University, Shanghai, China

    Cosimo Bambi

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Eichhorn, A., Held, A. (2023). Black Holes in Asymptotically Safe Gravity and Beyond. In: Bambi, C. (eds) Regular Black Holes. Springer Series in Astrophysics and Cosmology. Springer, Singapore. https://doi.org/10.1007/978-981-99-1596-5_5

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