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Black Hole Superradiance in Astrophysics

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Superradiance

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

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Abstract

BHs are one of the most striking predictions of Einstein’s GR, or of any relativistic theory of gravity. Since Schmidt’s identification of the first quasar, large consensus in the astronomy community has mounted that nearly all galactic centers harbor a supermassive BH and that compact objects with mass above ∼3M ⊙ should be BHs (we discuss some alternatives to this paradigm in Sect. 5.7.2 below; see Ref. Cardoso and Pani (Living Rev Rel 22(1):4, 2019. arXiv:1904.05363 [gr-qc]) for a recent review). Indeed, strong evidence exists that astrophysical BHs with masses ranging from few solar masses to billions of solar masses are abundant objects.

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Notes

  1. 1.

    In the absence of superradiance the BH would reach extremality in finite time, whereas radiation effects set an upper bound of a∕M ∼ 0.998 [34]. To mimic this upper bound in a simplistic way, a smooth cutoff in the accretion rate for the angular momentum can be introduced [24]. This cutoff merely prevents the BH to reach extremality and does not play any role in the evolution.

  2. 2.

    Note that, through Eq. (5.1), the mass accretion rate only depends on the combination f EddM, so that a BH with mass M = 106M ⊙ and f Edd ∼ 10−3 would have the same accretion rate of a smaller BH with M = 104M ⊙ accreting at rate f Edd ∼ 10−1. Because this is the only relevant scale for a fixed value of μ SM, in our model the evolution of a BH with different mass can be obtained from Fig. 5.2 by rescaling f Edd and μ S.

  3. 3.

    As we already discussed, as m increases, larger values of μ are allowed in the instability region and virtually any value of μ gives some unstable mode in the eikonal (l, m ≫ 1) limit. However, the instability is highly suppressed as l increases so that, in practice, only the first few allowed values of l = m correspond to an effective instability.

  4. 4.

    Some recent observations of ultraluminous X-ray sources suggest that these sources contain intermediate-mass BHs (e.g., [67, 68]), suggesting that the BH mass spectrum might be populated continuously from few solar masses to billions of solar masses.

  5. 5.

    In the spin-2 case there exists a special dipolar mode [18] that does not follow the behavior of Eq. (5.11).

  6. 6.

    This is because the signal is produced by a spinning quadrupolar field and not by a spinning dipolar field [49]. The hexadecapolar nature of the radiation implies that the signal vanishes along the BH spin axis, at variance with the quadrupolar case, for which it is maximum in that direction.

  7. 7.

    The root-mean-square GW strain for a LIGO-like detector is given by \(h=\sqrt {\dot {E}_{\mathrm {GW}}/(5 d^2\pi ^2 f^2)}\) [26], for a source emitting power \(\dot {E}_{\mathrm {GW}}\) at frequency f and at distance d away from the detector.

  8. 8.

    Assuming an observation time T obs, the signal-to-noise (SNR) ratio of a monochromatic signal with frequency f scales as \({\mathrm {SNR}}\propto h\sqrt {T_{\mathrm {obs}}/S_n(f)}\), where S n(f) is the noise PSD. The noise curves we plot in Fig. 5.6 are given by \(\sqrt {S_n(f)/T_{\mathrm {obs}}}\) such that a quick estimate of the SNR can be obtained by taking the ratio between the signal amplitude and the noise curves.

  9. 9.

    Here \(\tilde {n}\) denotes the principal quantum number given by \(\tilde {n}=l+S+n+1\).

  10. 10.

    As previously discussed, observations of LMC X-3 and Cygnus X-1 are consistent with a constant spin on a time scale of the order of 10 years, which gives a very mild bound compared to the one discussed in this section.

  11. 11.

    As was pointed out in Ref. [141], for the (unrealistic) Ernst metric in which radiation cannot escape, the end state is most likely a rotating BH in equilibrium with the outside radiation, similarly to the asymptotically AdS case discussed in Sect. 4.5.1. However, in realistic situations part of the radiation will escape to infinity, reducing the BH spin (see discussion below).

  12. 12.

    The strength of the magnetic field can be measured defining the characteristic magnetic field B M = 1∕M associated to a spacetime curvature of the same order of the horizon curvature. In physical units this is given by \(B_M\sim 2.4\times 10^{19} \left (M_{\odot }/M\right ) {\mathrm {Gauss}}\).

  13. 13.

    Recently, Ref. [159] showed that long-lived modes necessarily exist for matter configurations whose trace of the stress-energy tensor is positive (or zero). For a perfect-fluid star, this requires P > ρ∕3, where P and ρ are the NS pressure and density. This is an extreme configuration which is unlikely to exist in ordinary stars, but it might occur in other models of ultracompact objects, as those discussed in Sect. 5.7.2.

  14. 14.

    Even constant-density NSs have a maximum compactness which is smaller than the BH limit M∕R = 1∕2. Inspection of Eq. (F.2) shows that M∕R ≤ 4∕9 to ensure regularity of the geometry. More realistic equations of state yield a maximum mass and a maximum compactness.

  15. 15.

    The angular momentum of a boson star is quantized [199]; this prevents performing a standard slow-rotation approximation. Furthermore, there are indications that spinning scalar boson stars (at variance with their vector counterpart known as Proca stars [200]) are also subject to another (nonsuperradiant) type of instability, at least in the absence of self-interactions [201].

  16. 16.

    The dependence of \(|\mathcal {R}_{\mathrm {surface}}|{ }^2\) on the combination ω − m Ω should actually be linear, hence the change of sign in the superradiant regime. Heuristically this can be understood as follows. Let us consider a compact object such that the effective potential vanishes at its surface. The field near the surface is \(\Psi \sim e^{-i (\omega -m\Omega ) r_*}\) and the energy flux depends on linear partial derivatives with respect to the tortoise coordinate, \(\partial _{r_*}\Psi \), which brings a linear ω − m Ω dependence.

  17. 17.

    Interestingly the analogy between the BZ process and superradiance might be more than just an analogy. In fact, Ref. [247] recently showed that the BZ process can be interpreted as the long wavelength limit of the superradiant scattering from Alfvénic waves in the plasma.

  18. 18.

    A condition for this to happen is that initially there is a small electric field component parallel to the magnetic field (note that this is a Lorentz invariant condition). In Ref. [252], this was shown to occur for rotating BHs immersed in a magnetic field.

  19. 19.

    The use of a 3 + 1 spacetime decomposition was mainly useful to write the equations in a more familiar form for the astrophysics community. In fact most of the work done in this area in the last decades has been done using this formalism. Recently the GR community has regained interest in the subject and some remarkable effort has been done to develop a fully covariant theory of force-free magnetospheres around rotating BHs [254].

  20. 20.

    This is not to be confused with the jet efficiency, defined by \(\eta _{\mathrm {jet}}=\left <L_{\mathrm {jet}}\right >/\big <\dot {M}\big >\), where \(\left <L_{\mathrm {jet}}\right >\) is the time-average jet luminosity and \(\big <\dot {M}\big >\) is the time-average rate of matter accretion by the BH. Recently, efficiencies up to η jet ∼ 300% have been obtained in GRMHD simulations [245, 249, 250, 266] which is a strong indication that the BZ mechanism is at work.

  21. 21.

    However from the point of view of BH complementarity introduced in Ref. [270], the membrane is real as long as the observers remain outside the horizon, but fictitious for observers who jump inside the BH. Since neither observer can verify a contradiction between each other, the two are complementary in the same sense of the wave–particle duality.

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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). Black Hole Superradiance in Astrophysics. In: Superradiance. Lecture Notes in Physics, vol 971. Springer, Cham. https://doi.org/10.1007/978-3-030-46622-0_5

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