We measure the correlation spectrum of the Hawking radiation emitted by an analogue black hole and find it to be thermal at the Hawking temperature implied by the analogue surface gravity. The Hawking radiation is in the regime of linear dispersion, in analogy with a real black hole. Furthermore, the radiation inside of the black hole is seen to be composed of negative-energy partners only. This work confirms the prediction of Hawking's theory regarding the value of the Hawking temperature, as well as the thermality of the spectrum. The thermality of Hawking radiation is the root of the information paradox. The correlations between the Hawking and partner particles imply that the analogue black hole has no analogue firewall.It was a profound realization that the entropy of a black hole [1] and Hawking radiation [2,3] should have the same temperature, within a numerical factor on the order of unity. It was further asserted that Hawking radiation should have a thermal spectrum, which creates an information paradox [4,5]. Furthermore, it was proposed that the physics of Hawking radiation could be verified in an analogue system [6]. This proposal was carefully studied and developed theoretically [7][8][9][10][11][12][13][14][15][16][17][18][19]. Classical white and black-hole analogues were also studied experimentally [20][21][22][23], as well as a variety of other analogue gravitational systems [24][25][26][27][28][29][30]. The theoretical works, combined with our long-term study of this subject [14,[31][32][33][34], allowed for the observation of spontaneous Hawking radiation in an analogue black hole [35]. Several theoretical works studied our observation [35], and made predictions about the thermality and Hawking temperature [36][37][38][39][40]. During the years since our observation [35], we have made many improvements to the experimental apparatus. This allows us to study the thermality of the Hawking spectrum, and compare its temperature with the prediction given by the analogue of the surface gravity. In this work, we find that the spectrum of Hawking radiation agrees well with a thermal spectrum, and its temperature is given by Hawking's prediction.The analogue black hole consists of a flowing Bose-Einstein condensate. The flow velocity out in the region < 0 is less than the speed of sound out , as indicated in Fig. 1a. This region corresponds to the outside of the black hole. For > 0, the flow is supersonic ( in > in ),

We observe the time dependence of the Hawking radiation in an analogue black hole. Soon after the formation of the horizon, there is little or no Hawking radiation. The Hawking radiation then ramps up during approximately one period of oscillation, until it reaches the quantity expected for spontaneous emission. This is similar to a black hole created from gravitational collapse. The spectrum remains approximately constant at the spontaneous level for some time, similar to a stationary black hole. An inner horizon then forms, in analogy with a charged black hole. The inner horizon causes stimulated Hawking radiation. Both types of stimulation predicted by Ted Jacobson and coworkers likely contribute, but the monochromatic stimulation probably contributes more than does the black-hole lasing.

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The evolution of two grains, which lie on a substrate and are in contact with each other, can be roughly described by a simple model in which the exterior surfaces of the grains evolve by surface diffusion and the grain boundary, namely the contact surface between the adjacent grains, evolves by motion by mean curvature. For simplicity we consider an axi-symmetric two grain system, which is contained within an inert bounding semi-infinite cylinder with unit radius and which is bounded below by a planar substrate. The two grain system is assumed to have a hole along the axis of symmetry, where the substrate is exposed. Boundary conditions are imposed reflecting the considerations of W.W. Mullins, 1958. The resultant dynamic problem conserves the total volume of the two grain system and dissipates the total energy, where the total energy is defined as sum of the areas of the various participating surfaces, weighted according to their respective surface free energies.We focus here on the steady states of this system. We demonstrate that at steady state, the exterior surfaces have constant and equal mean curvatures, and the grain boundary has zero mean curvature. Taking into account the geometry and the boundary conditions, it then follows that the exterior surfaces are nodoids and the grain boundary surface is a catenoid. The physical parameters in the model can be expressed via the angles β and θc, which depend on the surface free energies, where, in the meridian cross-section, β ∈ (π/2, π) is the angle between the grain boundary and each of the exterior surfaces, and θc ∈ (0, π], the contact angle, is the angle between the inner grain and the substrate. Typically if a steady state solution exists for given values of (β, θc), then there exists a continuum of such solutions with varying volumes and energies. In particular, we prove that there exists a continuum of solutions with θc = π for any β ∈ (π/2, π). While many open questions remain, with regard to both the steady states and the full dynamic problem, our study already provides insight into the possible steady states and their structure. In particular, the relative volumes and heights of the inner and outer grain can be seen to be roughly in accordance with experimental predictions, for realistic values of the physical parameters.