Black holes as self-sustained quantum states and Hawking radiation

Abstract

We employ the recently proposed formalism of the "horizon wave-function" to investigate the emergence of a horizon in models of black holes as Bose-Einstein condensates of gravitons. We start from the Klein-Gordon equation for a massless scalar (toy graviton) field coupled to a static matter current. The (spherically symmetric) classical field reproduces the Newtonian potential generated by the matter source, and the corresponding quantum state is given by a coherent superposition of scalar modes with continuous occupation number. Assuming an attractive self-interaction that allows for bound states, one finds that (approximately) only one mode is allowed, and the system can be confined in a region of the size of the Schwarzschild radius. This radius is then shown to correspond to a proper horizon, by means of the horizon wave-function of the quantum system, with an uncertainty in size naturally related to the expected typical energy of Hawking modes. In particular, this uncertainty decreases for larger black hole mass (with larger number of light scalar quanta), in agreement with semiclassical expectations, a result which does not hold for a single very massive particle. We finally speculate that a phase transition should occur during the gravitational collapse of a star, ideally represented by a static matter current and Newtonian potential, that leads to a black hole, again ideally represented by the condensate of toy gravitons, and suggest an effective order parameter that could be used to investigate this transition. *

High-energy gravitational scattering and Bose-Einstein condensates of gravitons

“…3 These estimates are precisely the same as in the quantum N -portrait [1][2][3][4][5][6][7] of black holes, i.e. we have in particular 15) JHEP12 (2014)016 where we used that r S ∼ s 1/(2D−6) and s ∼ N (2D−6)/(D−2) (cf. eq.…”

“…A number of essential black-hole features have been matched by estimates in this language such as Bekenstein entropy and Hawking temperature (cf. also [8][9][10][11][12][13][14][15]). Such a picture can potentially improve our understanding conceptually, but also quantitatively, if 1/N corrections to the leading behaviour can be evaluated.…”

Quantum state of the black hole interior

“…Let us now verify that |S is a highly quantum state as prescribed. This can be accomplished by calculating expectation values for field operators such as 20) where the f 's are complex numbers with order-one magnitude. Expectation values of odd powers of Φ, including Φ S ≡ S| Φ|S , are all trivially vanishing and, in a way, this is already enough because then the ratio of the variance of Φ to the square of its expectation value is formally divergent.…”

“…This perspective leads one to a third choice: uniformly distributed matter -a stance that has JHEP08(2015)082 been advocated, in particular, by Dvali and Gomez in a recent series of articles [13][14][15][16][17][18][19]. (See [20,21] for related work.) Those authors have reasoned that the BH interior is composed, for the most part, by N ∼ S BH weakly coupled, horizon-sized gravitons whose state is that of a "leaky" Bose-Einstein condensate.…”

Corpuscular consideration of eternal inflation

“…In fact, some phenomena, like Hawking radiation, the Bekenstein entropy, or, the information paradox, can only be fully understood in this quantum picture [9][10][11][12][13][14][15] (cf. also [16][17][18][19][20][21][22][23][24][25][26][27] for recent progress).…”