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Let an electromagnetic field be scattered by a Kerr black-hole; then its amplitude will be amplified, but its frequency will remain same (super-radiant amplification). Hence though its intensity will be increased, its energy should not be changed (as its frequency remains same).

Still why the popular books like,

(i) Padmanabhan, T. (2010) Gravitation Foundation and Frontiers. Cambridge University Press, Cambridge.

(ii) Chandrasekhar, S. (1983). The Mathematical Theory of Black Holes.

(iii) Lecture Notes by Prof. Emanuele Berti at the summer school at ICTS (2016). Black Hole Perturbation Theory.

have written that the energy of the field has been increased in this case (and hence the mass and rotation of the black-hole has been decreased)?

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  • $\begingroup$ When you studied classical electromagnetism, did you learn the formula for the energy density of a classical electromagnetic field? $\endgroup$
    – Ghoster
    Commented Jan 8, 2023 at 5:47
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    $\begingroup$ I did. But that was a classical formulation! I've understood that here 'energy is increased' is used to mean that the amplitude of the wave is increasing. But if I consider the quantum picture (or if I treat the EM field as photon), then the change in frequency is required to change its energy (as $E = \hbar\omega$). So... $\endgroup$
    – SCh
    Commented Jan 8, 2023 at 6:22
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    $\begingroup$ All of the calculations you mention are of the scattering of a classical EM wave by a black hole; they’re not doing quantum gravity. In any case, if you want to imagine the classical EM wave as consisting of photons, its energy doesn’t depend only on the frequency; it depends also on how many photons there are (which is related to the wave amplitude). $\endgroup$
    – Ghoster
    Commented Jan 8, 2023 at 6:37
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    $\begingroup$ Okay, I got it. $\endgroup$
    – SCh
    Commented Jan 8, 2023 at 6:38
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    $\begingroup$ The quantum picture would be that more photons come out than you send in, all of the same frequency. $\endgroup$
    – Ghoster
    Commented Jan 8, 2023 at 6:40

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