Black holes as self-sustained quantum states and Hawking radiation

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 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 a 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.

Figure 1

Scalar field mode of momentum number $k_{\mathrm{c}}$: the ideal approximation in Eq. (3.9) (thick solid line) is compared to the exact $j_0(k_{\mathrm{c}}r)$ (dashed line). The thin solid line represents a Gaussian distribution of the kind considered in Ref. [5].

Figure 2

Modes of momentum number $k=k_{\mathrm{c}}$ (thick solid line), $k=(5/4)k_{\mathrm{c}}$ (dashed line), $k=(3/2)k_{\mathrm{c}}$ (dotted line), $k=(7/4)k_{\mathrm{c}}$ (dash-dotted line), and $k=2k_{\mathrm{c}}$ (thin solid line). The relative weight is determined according to Eq. (3.18).

Figure 3

Probability density of finding the horizon with radius $r_{\mathrm{H}}$ for $N=1$ ($R_{\mathrm{H}}=2{\ell }_{\mathrm{p}}$; thin solid line), $N=4$ ($R_{\mathrm{H}}=4{\ell }_{\mathrm{p}}$; dotted line), $N=9$ ($R_{\mathrm{H}}=6{\ell }_{\mathrm{p}}$; dashed line), $N=16$ ($R_{\mathrm{H}}=8{\ell }_{\mathrm{p}}$; solid line), and $N=25$ ($R_{\mathrm{H}}=10{\ell }_{\mathrm{p}}$; thick solid line). The curves clearly become narrower as $N$ becomes larger.

Figure 4

Monte Carlo estimate of the spectral coefficient in Eq. (b3) (dots) compared to its analytical approximation (3.27) (solid line), both normalized according to Eq. (c1), for $N=10$ (left panel) and $N=50$ (right panel).

Figure 5

Monte Carlo estimate of the spectral coefficient in Eq. (b3) (dots) compared to its analytical approximation (3.27) (solid line), both normalized according to Eq. (c1), for $N=100$ (left panel) and $N=200$ (right panel).

Figure 6

Monte Carlo estimate of the spectral coefficient in Eq. (b3) (dots) compared to its analytical approximation (3.27) (solid line), both normalized according to Eq. (c1), for $N=300$ (left panel) and $N=500$ (right panel).