ISSN 0253-2778

CN 34-1054/N

Open AccessOpen Access JUSTC Physics/Astronomy 16 May 2022

Unruh effect of multiparticle states and black hole radiation

Cite this:
https://doi.org/10.52396/JUSTC-2022-0039
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  • Author Bio:

    Jianyu Wang is a postgraduate student of Department of Astronomy, University of Science and Technology of China. His research interests include black holes and Hawking radiation

  • Corresponding author: E-mail: emc@mail.ustc.edu.cn
  • Received Date: 23 February 2022
  • Accepted Date: 01 April 2022
  • Available Online: 16 May 2022
  • In this study, we investigated the field under the Unruh effect. The energy and entanglement properties of the single-mode $q$-particle states were discussed. We found that in the non-inertial reference frame $|q, 0\rangle_\alpha$ states exhibit a similar energy spectrum to vacuum $|0, 0\rangle_\alpha$, but with different entanglement properties. With respect to an application, we proposed a black hole radiation model, assuming that states near the horizon are constructed via $q$-particle states. We calculated the evolution of the entanglement entropy of radiation and proved that our model can reproduce the Page curve. Hence, this can be considered as an indication solution of the black hole information paradox.
    The state of the black hole horizon will shift after particle radiated in our toy model. Such a black hole will have the memory effect and the entropy of radiation can reproduce the Page curve. At the same time this horizon shift cannot be measured locally.
    In this study, we investigated the field under the Unruh effect. The energy and entanglement properties of the single-mode $q$-particle states were discussed. We found that in the non-inertial reference frame $|q, 0\rangle_\alpha$ states exhibit a similar energy spectrum to vacuum $|0, 0\rangle_\alpha$, but with different entanglement properties. With respect to an application, we proposed a black hole radiation model, assuming that states near the horizon are constructed via $q$-particle states. We calculated the evolution of the entanglement entropy of radiation and proved that our model can reproduce the Page curve. Hence, this can be considered as an indication solution of the black hole information paradox.
    • Unruh effect of multiparticle is studied in both left and right wedge of Rindler spacetime.
    • A toy model of quantum black hole is constructed. The state of black hole horizon will evolve when particles fall in or emit out.
    • This black hole radiation model reproduces the Page curve.

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  • [1]
    Fulling S A. Nonuniqueness of canonical field quantization in Riemannian space-time. Physical Review D, 1973, 7 (10): 2850–2862. doi: 10.1103/PhysRevD.7.2850
    [2]
    Unruh W G, Wald R M. Acceleration radiation and the generalized second law of thermodynamics. Physical Review D, 1982, 25 (4): 942–958. doi: 10.1103/PhysRevD.25.942
    [3]
    Unruh W G, Wald R M. What happens when an accelerating observer detects a Rindler particle. Physical Review D, 1984, 29 (6): 1047–1056. doi: 10.1103/PhysRevD.29.1047
    [4]
    Unruh W G. Thermal bath and decoherence of Rindler spacetimes. Physical Review D, 1992, 46 (8): 3271–3277. doi: 10.1103/physrevd.46.3271
    [5]
    Unruh W G. Acceleration radiation for orbiting electrons. Physics Reports, 1998, 307: 163–171.
    [6]
    Crispino L C, Higuchi A, Matsas G E. The Unruh effect and its applications. Reviews of Modern Physics, 2008, 80 (3): 787–838. doi: 10.1103/RevModPhys.80.787
    [7]
    Bekenstein J D. Generalized second law of thermodynamics in black-hole physics. Physical Review D, 1974, 9 (12): 3292–3300. doi: 10.1103/PhysRevD.9.3292
    [8]
    Unruh W G. Second quantization in the Kerr metric. Physical Review D, 1974, 10 (10): 3194–3205. doi: 10.1103/PhysRevD.10.3194
    [9]
    Hawking S W. Particle creation by black holes. Communications in Mathematical Physics, 1975, 43 (3): 199–220. doi: 10.1007/BF02345020
    [10]
    Unruh W G. Notes on black-hole evaporation. Physical Review D, 1976, 14 (4): 870–892. doi: 10.1103/PhysRevD.14.870
    [11]
    Dabholkar A, Nampuri S. Quantum black holes. In: Baumgartl M, Brunner I, Haack M, editors. Strings and Fundamental Physics. Berlin: Springer, 2012: 165–232.
    [12]
    Lambert P H. Introduction to black hole evaporation. Proceedings of Science, 2014: PoS(Modave 2013)001.
    [13]
    Socolovsky M. Rindler space, Unruh effect and Hawking temperature. Annales de la Fondation Louis de Broglie, 2014, 39: 1–49.
    [14]
    Alsing P M, Fuentes-Schuller I, Mann R B, et al. Entanglement of Dirac fields in noninertial frames. Physical Review A, 2006, 74 (3): 032326. doi: 10.1103/PhysRevA.74.032326
    [15]
    Martín-Martínez E, León J. Quantum correlations through event horizons: Fermionic versus bosonic entanglement. Physical Review A, 2010, 81 (3): 032320. doi: 10.1103/PhysRevA.81.032320
    [16]
    Martín-Martínez E, Fuentes I. Redistribution of particle and antiparticle entanglement in noninertial frames. Physical Review A, 2011, 83 (5): 052306. doi: 10.1103/PhysRevA.83.052306
    [17]
    Wang J, Jing J. Multipartite entanglement of fermionic systems in noninertial frames. Physical Review A, 2011, 83 (2): 022314. doi: 10.1103/PhysRevA.83.022314
    [18]
    Wipf A. Quantum fields near black holes. In: Hehl F W, Kiefer C, Metzler R J K, editors. Black Holes: Theory and Observation. Berlin: Springer, 2003: 385–415.
    [19]
    Susskind L, Lindesay J. An Introduction to Black Holes, Information and the String Theory Revolution: The Holographic Universe. Singapore: World Scientific Publishing Co. Pte. Ltd., 2004.
    [20]
    Fuentes-Schuller I, Mann R B. Alice falls into a black hole: Entanglement in noninertial frames. Physical Review Letters, 2005, 95 (12): 120404. doi: 10.1103/PhysRevLett.95.120404
    [21]
    Jacobson T. Introduction to quantum fields in curved spacetime and the Hawking effect. In: Gomberoff A, Marolf D, editors. Lectures on Quantum Gravity. Boston, MA: Springer, 2005: 39–89.
    [22]
    Semay C. Penrose-Carter diagram for a uniformly accelerated observer. European Journal of Physics, 2007, 28 (5): 877–887. doi: 10.1088/0143-0807/28/5/011
    [23]
    Alsing P M, Fuentes I. Observer-dependent entanglement. Classical and Quantum Gravity, 2012, 29 (22): 224001. doi: 10.1088/0264-9381/29/22/224001
    [24]
    Higuchi A, Iso S, Ueda K, et al. Entanglement of the vacuum between left, right, future, and past: The origin of entanglement-induced quantum radiation. Physical Review D, 2017, 96 (8): 083531. doi: 10.1103/PhysRevD.96.083531
    [25]
    Bruschi D E, Louko J, Martín-Martínez E, et al. Unruh effect in quantum information beyond the single-mode approximation. Physical Review A, 2010, 82 (4): 042332. doi: 10.1103/PhysRevA.82.042332
    [26]
    Adesso G, Fuentes-Schuller I, Ericsson M. Continuous-variable entanglement sharing in noninertial frames. Physical Review A, 2007, 76 (6): 062112. doi: 10.1103/PhysRevA.76.062112
    [27]
    Alsing P M, Milburn G J. Teleportation with a uniformly accelerated partner. Physical Review Letters, 2003, 91 (18): 180404. doi: 10.1103/PhysRevLett.91.180404
    [28]
    Dai Y, Shen Z, Shi Y. Killing quantum entanglement by acceleration or a black hole. Journal of High Energy Physics, 2015, 2015: 71. doi: 10.1007/JHEP09(2015)071
    [29]
    Datta A. Quantum discord between relatively accelerated observers. Physical Review A, 2009, 80 (5): 052304. doi: 10.1103/PhysRevA.80.052304
    [30]
    Santana A E, Malbouisson J M C, Malbouisson A P C, et al. Thermal field theory: Algebraic aspects and applications to confined systems. In: Khanna F, Matrasulov D, editors. Non-Linear Dynamics and Fundamental Interactions. Dordrecht, Netherlands: Springer, 2006: 187–213.
    [31]
    Mukohyama S. Hartle-Hawking state is a maximum of entanglement entropy. Phys. Rev. D, 2000, 61: 064015. doi: 10.1103/PhysRevD.61.064015
    [32]
    Jacobson T. Black holes and Hawking radiation in spacetime and its analogues. In: Faccio D, Francesco B, Cacciatori S, et al, editors. Analogue Gravity Phenomenology. Cham, Switzerland: Springer, 2013: 1–29.
    [33]
    Martín-Martínez E, Garay L J, León J. Unveiling quantum entanglement degradation near a Schwarzschild black hole. Physical Review D, 2010, 82 (6): 064006. doi: 10.1103/PhysRevD.82.064006
    [34]
    Venkataratnam K K. Analytical study of two-mode thermal squeezed states and black holes. International Journal of Theoretical Physics, 2017, 56 (2): 377–385. doi: 10.1007/s10773-016-3178-5
    [35]
    Dhayal R, Rathore M, Venkataratnam K K. Single-mode squeezed thermal states and black holes. International Journal of Theoretical Physics, 2019, 58 (12): 4311–4322. doi: 10.1007/s10773-019-04303-4
    [36]
    Hawking S W. Breakdown of predictability in gravitational collapse. Physical Review D, 1976, 14 (10): 2460–2473. doi: 10.1103/PhysRevD.14.2460
    [37]
    Mathur S D. The information paradox: A pedagogical introduction. Classical and Quantum Gravity, 2009, 26 (22): 224001. doi: 10.1088/0264-9381/26/22/224001
    [38]
    Unruh W G, Wald R M. Information loss. Reports on Progress in Physics, 2017, 80 (9): 092002. doi: 10.1088/1361-6633/aa778e
    [39]
    Page D N. Average entropy of a subsystem. Physical Review Letters, 1993, 71 (9): 1291–1294. doi: 10.1103/PhysRevLett.71.1291
    [40]
    Page D N. Information in black hole radiation. Physical Review Letters, 1993, 71 (23): 3743–3746. doi: 10.1103/PhysRevLett.71.3743
  • 加载中

Catalog

    Figure  1.  Rindler space. Spacetime is divided into four wedges, left “L”, right “R”, future “F”, and past “P”. The black full line represents the trajectory with constant acceleration. The blue long dashed line is an example of the Cauchy surface. Both the left and right wedges are required to cover the entire space.

    Figure  2.  Coefficient function $f_n(q, \tanh r)$ for different $q$. The horizontal axis represents $n$, and the vertical axis represents the value of $f_n(q, \tanh r)$. The figures show that as $q$ increases, the peak of $f_n(q, \tanh r)$ moves to the right, and the distribution becomes more “even”.

    Figure  3.  Coefficient function $f_n(q, \tanh r)$ for different $\tanh r$ (different acceleration). The horizontal axis represents $n$, and the vertical axis represents the value of $f_n(q, \tanh r)$. The figures show that as $\tanh r$ (larger acceleration) increases, the peak of $f_n(q, \tanh r)$ moves to the right, and the distribution becomes more “even”.

    Figure  4.  Right wedge von Neumann entropy for different $|q, 0\rangle_\alpha$ states along with $\tanh r$. Four curves from bottom to top are $q=0, 1, 10, 50.$ Entropy increases as acceleration increases. Simultaneously, a larger $q$ leads to larger entropy.

    Figure  5.  “Normalized entropy” along with $\tanh r$ to compare right wedge entropy of different $q$ when fixing the energy. Four curves from bottom to top are $q=50 $, $10,\; 1,\; 0$. This shows that vacuum $q=0$ leads to the largest entropy, and a larger $q$ leads to less “normalized entropy”.

    Figure  6.  Entropy of the radiation after $m$ steps when considering the simplification situation that assumes fixed surface gravity and $m\to \infty$. The vertical axis corresponds to the von Neumann entropy of radiation, and the horizontal axis is $\tanh {r_{m-1}}$. This quantifies the surface gravity of the last radiating process. The results show that the radiation entropy increases from zero at zero surface gravity and decreases to zero at infinitely large surface gravity.

    Figure  7.  Entropy of the radiation after $m$ steps when considering surface gravity changes during the radiation process. In this case, we manually select certain values of $r_i$. The vertical axis is the von Neumann entropy of radiation, and the horizontal axis is $\tanh {r_{m-1}}$, which quantifies the surface gravity of the last radiating process. The results show that the radiation entropy increases from zero at zero surface gravity and decreases to zero at infinitely large surface gravity.

    [1]
    Fulling S A. Nonuniqueness of canonical field quantization in Riemannian space-time. Physical Review D, 1973, 7 (10): 2850–2862. doi: 10.1103/PhysRevD.7.2850
    [2]
    Unruh W G, Wald R M. Acceleration radiation and the generalized second law of thermodynamics. Physical Review D, 1982, 25 (4): 942–958. doi: 10.1103/PhysRevD.25.942
    [3]
    Unruh W G, Wald R M. What happens when an accelerating observer detects a Rindler particle. Physical Review D, 1984, 29 (6): 1047–1056. doi: 10.1103/PhysRevD.29.1047
    [4]
    Unruh W G. Thermal bath and decoherence of Rindler spacetimes. Physical Review D, 1992, 46 (8): 3271–3277. doi: 10.1103/physrevd.46.3271
    [5]
    Unruh W G. Acceleration radiation for orbiting electrons. Physics Reports, 1998, 307: 163–171.
    [6]
    Crispino L C, Higuchi A, Matsas G E. The Unruh effect and its applications. Reviews of Modern Physics, 2008, 80 (3): 787–838. doi: 10.1103/RevModPhys.80.787
    [7]
    Bekenstein J D. Generalized second law of thermodynamics in black-hole physics. Physical Review D, 1974, 9 (12): 3292–3300. doi: 10.1103/PhysRevD.9.3292
    [8]
    Unruh W G. Second quantization in the Kerr metric. Physical Review D, 1974, 10 (10): 3194–3205. doi: 10.1103/PhysRevD.10.3194
    [9]
    Hawking S W. Particle creation by black holes. Communications in Mathematical Physics, 1975, 43 (3): 199–220. doi: 10.1007/BF02345020
    [10]
    Unruh W G. Notes on black-hole evaporation. Physical Review D, 1976, 14 (4): 870–892. doi: 10.1103/PhysRevD.14.870
    [11]
    Dabholkar A, Nampuri S. Quantum black holes. In: Baumgartl M, Brunner I, Haack M, editors. Strings and Fundamental Physics. Berlin: Springer, 2012: 165–232.
    [12]
    Lambert P H. Introduction to black hole evaporation. Proceedings of Science, 2014: PoS(Modave 2013)001.
    [13]
    Socolovsky M. Rindler space, Unruh effect and Hawking temperature. Annales de la Fondation Louis de Broglie, 2014, 39: 1–49.
    [14]
    Alsing P M, Fuentes-Schuller I, Mann R B, et al. Entanglement of Dirac fields in noninertial frames. Physical Review A, 2006, 74 (3): 032326. doi: 10.1103/PhysRevA.74.032326
    [15]
    Martín-Martínez E, León J. Quantum correlations through event horizons: Fermionic versus bosonic entanglement. Physical Review A, 2010, 81 (3): 032320. doi: 10.1103/PhysRevA.81.032320
    [16]
    Martín-Martínez E, Fuentes I. Redistribution of particle and antiparticle entanglement in noninertial frames. Physical Review A, 2011, 83 (5): 052306. doi: 10.1103/PhysRevA.83.052306
    [17]
    Wang J, Jing J. Multipartite entanglement of fermionic systems in noninertial frames. Physical Review A, 2011, 83 (2): 022314. doi: 10.1103/PhysRevA.83.022314
    [18]
    Wipf A. Quantum fields near black holes. In: Hehl F W, Kiefer C, Metzler R J K, editors. Black Holes: Theory and Observation. Berlin: Springer, 2003: 385–415.
    [19]
    Susskind L, Lindesay J. An Introduction to Black Holes, Information and the String Theory Revolution: The Holographic Universe. Singapore: World Scientific Publishing Co. Pte. Ltd., 2004.
    [20]
    Fuentes-Schuller I, Mann R B. Alice falls into a black hole: Entanglement in noninertial frames. Physical Review Letters, 2005, 95 (12): 120404. doi: 10.1103/PhysRevLett.95.120404
    [21]
    Jacobson T. Introduction to quantum fields in curved spacetime and the Hawking effect. In: Gomberoff A, Marolf D, editors. Lectures on Quantum Gravity. Boston, MA: Springer, 2005: 39–89.
    [22]
    Semay C. Penrose-Carter diagram for a uniformly accelerated observer. European Journal of Physics, 2007, 28 (5): 877–887. doi: 10.1088/0143-0807/28/5/011
    [23]
    Alsing P M, Fuentes I. Observer-dependent entanglement. Classical and Quantum Gravity, 2012, 29 (22): 224001. doi: 10.1088/0264-9381/29/22/224001
    [24]
    Higuchi A, Iso S, Ueda K, et al. Entanglement of the vacuum between left, right, future, and past: The origin of entanglement-induced quantum radiation. Physical Review D, 2017, 96 (8): 083531. doi: 10.1103/PhysRevD.96.083531
    [25]
    Bruschi D E, Louko J, Martín-Martínez E, et al. Unruh effect in quantum information beyond the single-mode approximation. Physical Review A, 2010, 82 (4): 042332. doi: 10.1103/PhysRevA.82.042332
    [26]
    Adesso G, Fuentes-Schuller I, Ericsson M. Continuous-variable entanglement sharing in noninertial frames. Physical Review A, 2007, 76 (6): 062112. doi: 10.1103/PhysRevA.76.062112
    [27]
    Alsing P M, Milburn G J. Teleportation with a uniformly accelerated partner. Physical Review Letters, 2003, 91 (18): 180404. doi: 10.1103/PhysRevLett.91.180404
    [28]
    Dai Y, Shen Z, Shi Y. Killing quantum entanglement by acceleration or a black hole. Journal of High Energy Physics, 2015, 2015: 71. doi: 10.1007/JHEP09(2015)071
    [29]
    Datta A. Quantum discord between relatively accelerated observers. Physical Review A, 2009, 80 (5): 052304. doi: 10.1103/PhysRevA.80.052304
    [30]
    Santana A E, Malbouisson J M C, Malbouisson A P C, et al. Thermal field theory: Algebraic aspects and applications to confined systems. In: Khanna F, Matrasulov D, editors. Non-Linear Dynamics and Fundamental Interactions. Dordrecht, Netherlands: Springer, 2006: 187–213.
    [31]
    Mukohyama S. Hartle-Hawking state is a maximum of entanglement entropy. Phys. Rev. D, 2000, 61: 064015. doi: 10.1103/PhysRevD.61.064015
    [32]
    Jacobson T. Black holes and Hawking radiation in spacetime and its analogues. In: Faccio D, Francesco B, Cacciatori S, et al, editors. Analogue Gravity Phenomenology. Cham, Switzerland: Springer, 2013: 1–29.
    [33]
    Martín-Martínez E, Garay L J, León J. Unveiling quantum entanglement degradation near a Schwarzschild black hole. Physical Review D, 2010, 82 (6): 064006. doi: 10.1103/PhysRevD.82.064006
    [34]
    Venkataratnam K K. Analytical study of two-mode thermal squeezed states and black holes. International Journal of Theoretical Physics, 2017, 56 (2): 377–385. doi: 10.1007/s10773-016-3178-5
    [35]
    Dhayal R, Rathore M, Venkataratnam K K. Single-mode squeezed thermal states and black holes. International Journal of Theoretical Physics, 2019, 58 (12): 4311–4322. doi: 10.1007/s10773-019-04303-4
    [36]
    Hawking S W. Breakdown of predictability in gravitational collapse. Physical Review D, 1976, 14 (10): 2460–2473. doi: 10.1103/PhysRevD.14.2460
    [37]
    Mathur S D. The information paradox: A pedagogical introduction. Classical and Quantum Gravity, 2009, 26 (22): 224001. doi: 10.1088/0264-9381/26/22/224001
    [38]
    Unruh W G, Wald R M. Information loss. Reports on Progress in Physics, 2017, 80 (9): 092002. doi: 10.1088/1361-6633/aa778e
    [39]
    Page D N. Average entropy of a subsystem. Physical Review Letters, 1993, 71 (9): 1291–1294. doi: 10.1103/PhysRevLett.71.1291
    [40]
    Page D N. Information in black hole radiation. Physical Review Letters, 1993, 71 (23): 3743–3746. doi: 10.1103/PhysRevLett.71.3743

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