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Superradiance in Black-Hole Physics

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Superradiance

Part of the book series: Lecture Notes in Physics ((LNP,volume 971))

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Abstract

As discussed in the previous section, superradiance requires dissipation. The latter can emerge in various forms, e.g. viscosity, friction, turbulence, radiative cooling, etc. All these forms of dissipation are associated with some medium or some matter field that provides the arena for superradiance. It is thus truly remarkable that—when spacetime is curved—superradiance can also occur in vacuum, even at the classical level. In this section we discuss in detail BH superradiance, which is the main topic of this work.

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Notes

  1. 1.

    We also require the spacetime to be invariant under the “circularity condition,” t →−t and φ →−φ, which implies g t𝜗 = g tφ = g r𝜗 = g rφ = 0 [6]. While the circularity condition follows from Einstein and Maxwell equations in electrovacuum, it might not hold true in modified gravities or for exotic matter fields.

  2. 2.

    This possibility was at some stage considered of potential interest for the physics of jets emitted by quasars.

  3. 3.

    Interestingly, in the case of a naked singularity large-curvature regions become accessible to outside observers and g tt can be arbitrarily large. This suggests that the Penrose effects around spinning naked singularities can be very efficient. It is also possible that rotating wormholes are prone to efficient Penrose-like processes, although to the best of our knowledge a detailed investigation has not been performed.

  4. 4.

    A special feature of vacuum stationary GR solutions is their axisymmetry [43]. This simplifies considerably the treatment of superradiant instabilities in GR, as it excludes mixing between modes with different azimuthal number m.

  5. 5.

    As we shall discuss, this condition holds for scalar perturbations of spinning and charged BHs, but it does not hold in other cases of interest, for example, for EM and gravitational perturbations satisfying the Teukolsky equation for a Kerr BH. In the latter cases, it is convenient to make a change of variables by introducing the Detweiler’s function, which can be chosen such that the effective potential is real [44, 45].

  6. 6.

    Note that a planar tensor wave along γ = 0 in Cartesian coordinates will have a sin 2φ modulation when transformed to spherical coordinates, in which the multipolar decomposition is performed. This explains why Eq. (3.114) depends only on |m| = 2 and on a sum over all multipolar indices l ≤ 2. Likewise, an EM wave along γ = 0 would be modulated by \(\sin \varphi \) and its cross-section would only depend on |m| = 1, whereas the cross-section (3.111) for a scalar wave along γ = 0 only depends on m = 0.

  7. 7.

    There are no gravitational degrees of freedom in less than 4 dimensions, and a BH solution only exists for a negative cosmological constant, the so-called BTZ solution [127]. This solution has some similarities with the Kerr-AdS metric and, as we shall discuss in Sect. 3.12, superradiance does not occur when reflective boundary conditions at infinity are imposed [128].

  8. 8.

    It would be interesting to understand the large amplification of the superradiance energy in terms of violation of some energy condition due to the effective coupling that appears in scalar–tensor theories.

  9. 9.

    The geometry used in the original Kerr/CFT duality is the so-called near-horizon extreme Kerr “NHEK” geometry found by Bardeen and Horowitz [181] which is not asymptotically flat but resembles AdS3. That this geometry could have a dual CFT description was first pointed out in Ref. [181].

  10. 10.

    This example was suggested to us by Luis Lehner and Frans Pretorius.

  11. 11.

    We follow the terminology of Dias, Emparan and Maccarrone who, in a completely different context, arrived at conclusions very similar to ours, see Section 2.4 in Ref. [177].

  12. 12.

    This simple proof was suggested to us by Roberto Emparan.

  13. 13.

    The only exception to this rule concerns BHs surrounded by matter coupled to scalar fields, where the amplification factors can become unbounded (see Sect. 3.13). Because the laws of BH mechanics will be different, these fall outside the scope of this discussion.

  14. 14.

    We thank Shahar Hod for drawing our attention to this point.

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  1. Department of Physics, Sapienza University of Rome, Rome, Italy

    Richard Brito & Paolo Pani

  2. Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal

    Vitor Cardoso

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Brito, R., Cardoso, V., Pani, P. (2020). Superradiance in Black-Hole Physics. In: Superradiance. Lecture Notes in Physics, vol 971. Springer, Cham. https://doi.org/10.1007/978-3-030-46622-0_3

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