Black Hole Waterfall: a unitary phenomenological model for black hole evaporation with Page curve

Abstract We present a unitary phenomenological model for black hole (BH) evaporation based on the analogy of the laboratory process of spontaneous parametric down conversion (Alsing 2015 Class. Quantum Grav. 32 075010; Alsing and Fanto 2016 Class. Quantum Grav. 33 0150005) when the BH (pump) is allowed to deplete to zero mass. The model incorporates an additional new feature that allows for the interior Hawking partner-particles (idlers) behind the horizon to further generate new Hawking particle pairs of lower energy, one of which remains behind the horizon, and the other that adds to the externally emitted Hawking radiation (signals) outside the horizon. This model produces a Page curve for the evolution of the reduced density matrices for the evaporating BH internal degrees of freedom entangled with the generated Hawking radiation pairs entangled across the horizon. The Page curve yields an entropy that rises at early times during the evaporation process as Hawking pairs are generated, reaches a peak midway through the evolution, and then decays to zero upon complete evaporation of the BH. The entire system remains in a pure state at all times undergoing unitary (squeezed state) evolution, with the initial state of the BH modeled as a bosonic Fock state of large, but finite number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:mrow> </mml:math> of particles. For the final state of the system, the BH reaches the vacuum state of zero mass, while the external Hawking radiation carries away the total energy of the initial BH. Inside the horizon there remains <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:mrow> </mml:math> Hawking partner-particles of vanishingly small total energy, reminiscent of the ‘soft-hair’ (zero energy) qubit model of Hotta et al (2018 Phys. Rev. Lett. 120 181301), but now from a Hamiltonian for squeezed state generation perspective. The model presented here can be readily extended to encompass arbitrary initial pure states for the BH, and in falling matter.