{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,28]],"date-time":"2026-03-28T08:11:14Z","timestamp":1774685474306,"version":"3.50.1"},"reference-count":68,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2015,10,15]],"date-time":"2015-10-15T00:00:00Z","timestamp":1444867200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100013168","name":"Istituto Nazionale di Fisica Nucleare","doi-asserted-by":"publisher","award":["FLAG"],"award-info":[{"award-number":["FLAG"]}],"id":[{"id":"10.13039\/501100013168","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We review some features of Bose\u2013Einstein condensate (BEC) models of black holes obtained by means of the horizon wave function formalism. We consider the Klein\u2013Gordon equation for a toy graviton field coupled to a static matter current in a spherically-symmetric setup. The classical field reproduces the Newtonian potential generated by the matter source, while the corresponding quantum state is given by a coherent superposition of scalar modes with a continuous occupation number. An attractive self-interaction is needed for bound states to form, the case in which one finds that (approximately) one mode is allowed, and the system of N bosons can be self-confined in a volume of the size of the Schwarzschild radius. The horizon wave function formalism is then used to show that the radius of such a system corresponds to a proper horizon. The uncertainty in the size of the horizon is related to the typical energy of Hawking modes: it decreases with the increasing of the black hole mass (larger number of gravitons), resulting in agreement with the semiclassical calculations and which does not hold for a single very massive particle. The spectrum of these systems has two components: a discrete ground state of energy m (the bosons forming the black hole) and a continuous spectrum with energy \u03c9 &gt; m (representing the Hawking radiation and modeled with a Planckian distribution at the expected Hawking temperature). Assuming the main effect of the internal scatterings is the Hawking radiation, the N-particle state can be collectively described by a single-particle wave-function given by a superposition of a total ground state with energy M = Nm and Entropy 2015, 17 6894 a Planckian distribution for E &gt; M at the same Hawking temperature. This can be used to compute the partition function and to find the usual area law for the entropy, with a logarithmic correction related to the Hawking component. The backreaction of modes with \u03c9 &gt; m is also shown to reduce the Hawking flux. The above corrections suggest that for black holes in this quantum state, the evaporation properly stops for a vanishing mass.<\/jats:p>","DOI":"10.3390\/e17106893","type":"journal-article","created":{"date-parts":[[2015,10,15]],"date-time":"2015-10-15T12:44:06Z","timestamp":1444913046000},"page":"6893-6924","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":33,"title":["Thermal BEC Black Holes"],"prefix":"10.3390","volume":"17","author":[{"given":"Roberto","family":"Casadio","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica e Astronomia, Alma Mater Universit\u00e0 di Bologna, via Irnerio 46, 40126 Bologna, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare (I.N.F.N.), Sezione di Bologna, viale Berti Pichat 6\/2, 40127 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andrea","family":"Giugno","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica e Astronomia, Alma Mater Universit\u00e0 di Bologna, via Irnerio 46, 40126 Bologna, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare (I.N.F.N.), Sezione di Bologna, viale Berti Pichat 6\/2, 40127 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Octavian","family":"Micu","sequence":"additional","affiliation":[{"name":"Institute of Space Science, Atomistilor 409, 077125 Magurele, Ilfov, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alessio","family":"Orlandi","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica e Astronomia, Alma Mater Universit\u00e0 di Bologna, via Irnerio 46, 40126 Bologna, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare (I.N.F.N.), Sezione di Bologna, viale Berti Pichat 6\/2, 40127 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,10,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Oppenheimer, J.R., and Snyder, H. (1939). On Continued Gravitational Contraction. Phys. Rev., 56.","DOI":"10.1103\/PhysRev.56.455"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Oppenheimer, J.R., and Volkoff, G.M. (1939). On Massive neutron cores. Phys. Rev., 55.","DOI":"10.1103\/PhysRev.55.374"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Joshi, P.S. (2007). Gravitational Collapse and Spacetime Singularities, Cambridge University Press.","DOI":"10.1017\/CBO9780511536274"},{"key":"ref_4","unstructured":"Bekenstein, J.D. (2004). Black holes: Physics and astrophysics-stellar-mass, supermassive and primordial black holes, arXiv:astro-ph\/0407560."},{"key":"ref_5","first-page":"231","article-title":"Nonspherical gravitational collapse: A short review","volume":"1","author":"Thorne","year":"1972","journal-title":"Magic Without Magic: John Archibald Wheeler"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"D\u2019Eath, P.D., and Payne, P.N. (1992). Gravitational radiation in black-hole collisions at the speed of light. I. Perturbation treatment of the axisymmetric collision. Phys. Rev. D, 46.","DOI":"10.1103\/PhysRevD.46.658"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"D\u2019Eath, P.D., and Payne, P.N. (1992). Gravitational radiation in black-hole collisions at the speed of light. II. Reduction to two independent variables and calculation of the second-order news function. Phys. Rev. D, 46.","DOI":"10.1103\/PhysRevD.46.675"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"D\u2019Eath, P.D., and Payne, P.N. (1992). Gravitational radiation in black-hole collisions at the speed of light. III. Results and conclusions. Phys. Rev. D, 46.","DOI":"10.1103\/PhysRevD.46.694"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Senovilla, J.M.M. (2008). A Reformulation of the Hoop Conjecture. Europhys. Lett., 81.","DOI":"10.1209\/0295-5075\/81\/20004"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Alberghi, G.L., Casadio, R., Micu, O., and Orlandi, A. (2011). Brane-world black holes and the scale of gravity. J. High Energy Phys., 2011.","DOI":"10.1007\/JHEP09(2011)023"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/S0370-2693(03)00012-1","article-title":"Quantum production of black holes","volume":"555","author":"Hsu","year":"2003","journal-title":"Phys. Lett. B"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1781","DOI":"10.1140\/epjc\/s10052-011-1781-4","article-title":"The flavor of quantum gravity","volume":"71","author":"Calmet","year":"2011","journal-title":"Eur. Phys. J. C"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.physletb.2008.08.011","article-title":"Colorful quantum black holes at the LHC","volume":"668","author":"Calmet","year":"2008","journal-title":"Phys. Lett. B"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1038\/248030a0","article-title":"Black hole explosions?","volume":"248","author":"Hawking","year":"1974","journal-title":"Nature"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1007\/BF02345020","article-title":"Particle Creation by Black Holes","volume":"43","author":"Hawking","year":"1975","journal-title":"Commun. Math. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Dvali, G., and Gomez, C. (2014). Quantum compositeness of gravity: Black holes, AdS and inflation. J. Cosmol. Astropart. Phys., 2014.","DOI":"10.1088\/1475-7516\/2014\/01\/023"},{"key":"ref_17","unstructured":"Dvali, G., and Gomez, C. (2013). Black Hole\u2019s Information Group, arXiv:1307.7630."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Dvali, G., and Gomez, C. (2014). Black Holes as Critical Point of Quantum Phase Transition. Eur. Phys. J. C, 74.","DOI":"10.1140\/epjc\/s10052-014-2752-3"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1016\/j.physletb.2013.01.020","article-title":"Black hole\u2019s 1\/N hair","volume":"719","author":"Dvali","year":"2013","journal-title":"Phys. Lett. B"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1016\/j.physletb.2012.08.019","article-title":"Landau-Ginzburg Limit of Black Hole\u2019s Quantum Portrait: Self Similarity and Critical Exponent","volume":"716","author":"Dvali","year":"2012","journal-title":"Phys. Lett. B"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"742","DOI":"10.1002\/prop.201300001","article-title":"Black hole\u2019s quantum N-portrait","volume":"61","author":"Dvali","year":"2013","journal-title":"Fortsch. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Dvali, G., Gomez, C., and Mukhanov, S. (2011). Black Hole Masses are Quantized, arXiv:1106.5894.","DOI":"10.1007\/JHEP02(2011)012"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Casadio, R. (2013). Localised particles and fuzzy horizons: A tool for probing Quantum Black Holes, arXiv:1305.3195.","DOI":"10.1007\/JHEP08(2013)025"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Casadio, R., and Scardigli, F. (2014). Horizon wave-function for single localized particles: GUP and quantum black hole decay. Eur. Phys. J. C, 74.","DOI":"10.1140\/epjc\/s10052-013-2685-2"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/j.physletb.2014.03.037","article-title":"Quantum hoop conjecture: Black hole formation by particle collisions","volume":"732","author":"Casadio","year":"2014","journal-title":"Phys. Lett. B"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Casadio, R. (2015). Horizons and non-local time evolution of quantum mechanical systems. Eur. Phys. J. C, 75.","DOI":"10.1140\/epjc\/s10052-015-3404-y"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Casadio, R., Micu, O., and Stojkovic, D. (2015). Inner horizon of the quantum Reissner-Nordstr\u00f6m black holes. J. High Energy Phys., 2015.","DOI":"10.1007\/JHEP05(2015)096"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1016\/j.physletb.2015.05.053","article-title":"Horizon Wave-function and the quantum cosmic censorship","volume":"747","author":"Casadio","year":"2015","journal-title":"Phys. Lett. B"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/0370-2693(93)90785-G","article-title":"The algebraic structure of the generalized uncertainty principle","volume":"319","author":"Maggiore","year":"1993","journal-title":"Phys. Lett. B"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Kempf, A., Mangano, G., and Mann, R.B. (1995). Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D, 52.","DOI":"10.1103\/PhysRevD.52.1108"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/S0370-2693(99)00167-7","article-title":"Generalized uncertainty principle in quantum gravity from micro-black hole Gedanken experiment","volume":"452","author":"Scardigli","year":"1999","journal-title":"Phys. Lett. B"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3915","DOI":"10.1088\/0264-9381\/20\/18\/305","article-title":"Generalized uncertainty principle, extra dimensions and holography","volume":"20","author":"Scardigli","year":"2003","journal-title":"Class. Quant. Grav."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1142\/S0218271809014455","article-title":"Is the equivalence principle violated by generalized uncertainty principles and holography in a brane-world?","volume":"18","author":"Scardigli","year":"2009","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1229","DOI":"10.1142\/S0217751X09043353","article-title":"Noncommutative Black Holes, The Final Appeal To Quantum Gravity: A Review","volume":"24","author":"Nicolini","year":"2009","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"084040","DOI":"10.1103\/PhysRevD.90.084040","article-title":"Black holes as self-sustained quantum states, and Hawking radiation","volume":"90","author":"Casadio","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Casadio, R., Giugno, A., and Orlandi, A. (2015). Thermal corpuscular black holes, arXiv:1504.05356.","DOI":"10.1103\/PhysRevD.91.124069"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"M\u00fcck, W. (2013). Counting Photons in Static Electric and Magnetic Fields. Eur. Phys. J. C, 73.","DOI":"10.1140\/epjc\/s10052-013-2679-0"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"M\u00fcck, W., and Pozzo, G. (2014). Quantum portrait of a black hole with P\u00f6schl-Teller potential. J. High Energy Phys., 2014.","DOI":"10.1007\/JHEP05(2014)128"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Foit, V.F., and Wintergerst, N. (2015). Self-similar Evaporation and Collapse in the Quantum Portrait of Black Holes. Phys. Rev. D, 92.","DOI":"10.1103\/PhysRevD.92.064043"},{"key":"ref_40","unstructured":"Hofmann, S., and Rug, T. (2014). A Quantum Bound-State Description of Black Holes, arXiv:1403.3224."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"084024","DOI":"10.1103\/PhysRevD.90.084024","article-title":"Bose-Einstein condensates with derivative and long-range interactions as set-ups for analog black holes","volume":"90","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"K\u00fchnel, F., and Sundborg, B. (2014). Modified Bose-Einstein Condensate Black Holes in d Dimensions, arXiv:1401.6067.","DOI":"10.1007\/JHEP12(2014)016"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"K\u00fchnel, F., and Sundborg, B. (2014). High-Energy Gravitational Scattering and Bose-Einstein Condensates of Gravitons. J. High Energy Phys., 2014.","DOI":"10.1007\/JHEP12(2014)016"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"064025","DOI":"10.1103\/PhysRevD.90.064025","article-title":"Decay of graviton condensates and their generalizations in arbitrary dimensions","volume":"90","author":"Sundborg","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Ruffini, R., and Bonazzola, S. (1969). Systems of selfgravitating particles in general relativity and the concept of an equation of state. Phys. Rev., 187.","DOI":"10.1103\/PhysRev.187.1767"},{"key":"ref_46","unstructured":"Bekenstein, J.D. (1997). Quantum black holes as atoms, arXiv:gr-qc\/9710076."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"K\u00fchnel, F., and Sandstad, M. (2015). Baryon number conservation in Bose-Einstein condensate black holes, arXiv:1506.08823.","DOI":"10.1103\/PhysRevD.92.124028"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Dvali, G., Gomez, C., and Kehagias, A. (2011). Classicalization of Gravitons and Goldstones. J. High Energy Phys., 11.","DOI":"10.1007\/JHEP11(2011)070"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Dvali, G., Giudice, G.F., Gomez, C., and Kehagias, A. (2011). UV-Completion by Classicalization. J. High Energy Phys., 2011.","DOI":"10.1007\/JHEP08(2011)108"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Colpi, M., Shapiro, S.L., and Wasserman, I. (1986). Boson Stars: Gravitational Equilibria of Selfinteracting Scalar Fields. Phys. Rev. Lett., 57.","DOI":"10.1103\/PhysRevLett.57.2485"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"2736","DOI":"10.1103\/PhysRevD.40.2736","article-title":"Newtonian Boson Spheres","volume":"40","author":"Membrado","year":"1989","journal-title":"Phys. Rev. D"},{"key":"ref_52","unstructured":"Balakrishna, J. (1999). A Numerical study of boson stars: Einstein equations with a matter source, arXiv:gr-qc\/9906110."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Nieuwenhuizen, T.M. (2008). Supermassive Black Holes as Giant Bose\u2013Einstein Condensates. Europhys. Lett., 83.","DOI":"10.1209\/0295-5075\/83\/10008"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1016\/j.physe.2009.10.040","article-title":"Bose-Einstein condensed supermassive black holes: A Case of renormalized quantum field theory in curved space-time","volume":"42","author":"Nieuwenhuizen","year":"2010","journal-title":"Physica E"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"064011","DOI":"10.1103\/PhysRevD.86.064011","article-title":"Bose-Einstein Condensate general relativistic stars","volume":"86","author":"Chavanis","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Duff, M.J. (1973). Quantum Tree Graphs and the Schwarzschild Solution. Phys. Rev. D, 7.","DOI":"10.1103\/PhysRevD.7.2317"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1007\/s10714-009-0912-9","article-title":"Gravity from self-interaction redux","volume":"42","author":"Deser","year":"2010","journal-title":"Gen. Rel. Grav."},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Casadio, R., and Orlandi, A. (2013). Quantum Harmonic Black Holes. J. High Energy Phys., 2013.","DOI":"10.1007\/JHEP08(2013)025"},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Harms, B., and Leblanc, Y. (1992). Statistical mechanics of black holes. Phys. Rev. D, 46.","DOI":"10.1103\/PhysRevD.46.2334"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"044014","DOI":"10.1103\/PhysRevD.58.044014","article-title":"Microfield dynamics of black holes","volume":"58","author":"Casadio","year":"1998","journal-title":"Phys. Rev. D"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"502","DOI":"10.3390\/e13020502","article-title":"Microcanonical description of (micro) black holes","volume":"13","author":"Casadio","year":"2011","journal-title":"Entropy"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"084007","DOI":"10.1103\/PhysRevD.87.084007","article-title":"Black holes and quantumness on macroscopic scales","volume":"87","author":"Flassig","year":"2013","journal-title":"Phys. Rev. D"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/j.physletb.2014.07.032","article-title":"Quantum Black Hole Wave Packet: Average Area Entropy and Temperature Dependent Width","volume":"736","author":"Davidson","year":"2014","journal-title":"Phys. Lett. B"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1002\/prop.201300037","article-title":"Origin of the blackhole information paradox","volume":"62","author":"Brustein","year":"2014","journal-title":"Fortsch. Phys."},{"key":"ref_65","doi-asserted-by":"crossref","unstructured":"Brustein, R., and Hadad, M. (2012). Wave function of the quantum black hole. Phys. Lett. B, 718.","DOI":"10.1016\/j.physletb.2012.10.074"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/j.physletb.2014.04.038","article-title":"Singularity free gravitational collapse in an effective dynamical quantum spacetime","volume":"733","author":"Torres","year":"2014","journal-title":"Phys. Lett. B"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.physletb.2014.04.010","article-title":"Singularity-free gravitational collapse and asymptotic safety","volume":"733","author":"Torres","year":"2014","journal-title":"Phys. Lett. B"},{"key":"ref_68","doi-asserted-by":"crossref","unstructured":"Brustein, R., and Medved, A.J.M. (2013). Restoring predictability in semiclassical gravitational collapse. J. High Energy Phys., 2013.","DOI":"10.1007\/JHEP09(2013)015"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/10\/6893\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:50:12Z","timestamp":1760215812000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/10\/6893"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,10,15]]},"references-count":68,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2015,10]]}},"alternative-id":["e17106893"],"URL":"https:\/\/doi.org\/10.3390\/e17106893","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,10,15]]}}}